The World of Computer Software/The World of Computer Software.iso/spectr20.zip/SPECTRUM.DOC

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[WS 1.5 in]

                         SPECTRAL  ANALYSIS  ON  A  PC

[WS 3.5 in]

                                 David E. Hess
                                Fluid Flow Group
                         Process Measurements Division
                   Chemical Science and Technology Laboratory
                 National Institute of Standards and Technology
                            Email: [e-mail]
                             Phone: [phone]

[WS 0.5 in]

                                  [December 1991]


                                    Abstract

     These notes are intended to serve as a brief instruction manual for
engineers engaged in the spectral analysis of experimentally-derived random
data.  This manual is intended to accompany a software routine (Spectrum
Calculation Routine, V 1.0, David E. Hess) designed to compute amplitude
density, autospectral (power) density and cross-spectral density functions for
discrete, stationary random time series.  This program will run on any
PC-compatible desktop computer and accepts input data files obtained from
experiments in which analog data has been digitized by an analog-to-digital
converter.
     A collection of techniques are described which are necessary for the
proper software implementation of spectral analysis procedures: data
transformation, mean value and linear trend removal, digital filtering, time
history tapering and data overlapping.  The various spectral density functions
are then defined and explained along with a means for the calculation of the
error associated with a particular estimate.  Several detailed examples are
included which serve to illustrate the methods described.  The manual also
contains a brief description of the considerations necessary when sampling the
data to be analyzed.  References to more complete sources of information are
provided as required.


                                Notice to Users

     This program was developed for use with the ongoing Compliant Surface
Program in the Fluid Flow Group; this group is a part of the Process
Measurements Division at the National Institute of Standards and Technology.
With the evolution of the SPECTRUM routine to its present form, the author felt
that the program may perhaps be of some utility to others.  Therefore, SPECTRUM
is being released as freeware.  This program, the source code and accompanying
utilities may be freely copied and distributed, but may not be sold.  There is
no warranty of SPECTRUM's suitability for any purpose, nor any acceptance of
liability, express or implied.
     Originally, the SPECTRUM routine was designed to run on any IBM-PC or
compatible computer using the Intel 80x86 series of microprocessors.  However,
the code was compiled using the Microsoft Fortran Optimizing Compiler, V 5.00,
and this compiler allows the selection of a variety of options at compile-time
which enhance the speed of operation of the program but limit its suitability
for use with certain computers.  The version contained on the diskette requires
a computer equipped with an 80286 or better microprocessor and an accompanying
80287 or better coprocessor.  A version of the routine which will run on
computers with an 8086 or 8088 processor, or a version which does not require a
numeric coprocessor can be obtained by contacting the author.  The program uses
about 450K bytes of available RAM, and requires that a hard disk or RAM disk be
available for storage of temporary files.  The amount of space to be reserved
depends upon the size of the data file to be processed, but can easily be as
much as several megabytes.
     Microsoft has announced the release of V 5.1 of their Fortran compiler.
To this version the capability of porting programs to the Windows environment
has been added.  This is significant only in that it allows the use of the DOS
extender which is a part of the Windows environment.  Fortran programs written
for Windows are not limited by the DOS memory restriction of 640K, but instead
can access substantially larger amounts of available RAM at run-time.  This can
greatly increase the utility and speed of the SPECTRUM routine, and a version
incorporating these improvements may be available at a later date.
     This program has been subjected to extensive testing and appears to be
free of any bugs; however, it is possible that not all situations have been
anticipated.  The author encourages any modifications to or customization of
the source code as required by the user, and welcomes any suggested
improvements or bug-fixes.  Please direct all correspondence to the address
given below.


                       National Institute of Standards
                               and Technology
                       Building 230   Room 105
                       Gaithersburg, MD 20899
                       Attn   : David E. Hess

                       Phone  : [phone]
                       Fax    : [phone]
                       Email  : [e-mail]

[PB Odd Page]

                               Table of Contents

1 Overview and Parameters [LN String:"."]                                1

2 Input Data Files [LN String:"."]                                       3
   2.1 Naming Convention [LN String:"."]                                 3
   2.2 File Structure [LN String:"."]                                    3
      2.2.1 File Header [LN String:"."]                                  3
      2.2.2 Remainder of Data File [LN String:"."]                       4
      2.2.3 Writing and reading the Data File [LN String:"."]            5
   2.3 Multiple File Processing [LN String:"."]                          6
      2.3.1 Consecutively numbered files [LN String:"."]                 6
      2.3.2 Arbitrarily numbered files [LN String:"."]                   7
      2.3.3 Files entered on command line [LN String:"."]                7
   2.4 Coefficient Data Files [LN String:"."]                            8
      2.4.1 Naming Convention [LN String:"."]                            8
      2.4.2 Unformatted File Structure [LN String:"."]                   8
      2.4.3 Formatted Data Files [LN String:"."]                         10
3 Data Preparation [LN String:"."]                                       11
   3.1 Transformation [LN String:"."]                                    11
   3.2 Mean Removal [LN String:"."]                                      13
   3.3 Removal of Linear Trend [LN String:"."]                           14
   3.4 Filtering [LN String:"."]                                         14
   3.5 Tapering [LN String:"."]                                          18
4 Spectrum Computation [LN String:"."]                                   25
   4.1 Fast Fourier Transform [LN String:"."]                            25
   4.2 Autospectral (Power) Density [LN String:"."]                      27
   4.3 Cross-spectral Density [LN String:"."]                            29
      4.3.1 Calculation Procedure [LN String:"."]                        29
      4.3.2 Adjustment of Phase Angle [LN String:"."]                    30
      4.3.3 Coherence Function [LN String:"."]                           35
      4.3.4 Error Estimates [LN String:"."]                              35
   4.4 Amplitude Spectral Density [LN String:"."]                        37
5 Output Data Files [LN String:"."]                                      41
   5.1 Naming Convention [LN String:"."]                                 41
   5.2 File Structure [LN String:"."]                                    41
      5.2.1 Power Spectrum Output Files [LN String:"."]                  41
      5.2.2 Cross Spectrum Output Files [LN String:"."]                  42
      5.2.3 Amplitude Spectrum Output Files [LN String:"."]              43
      5.2.4 Error Estimate Output Files [LN String:"."]                  44
6 Run-time Considerations [LN String:"."]                                45
   6.1 Configuration File [LN String:"."]                                45
   6.2 Temporary Files [LN String:"."]                                   46
7 Examples [LN String:"."]                                               49
   7.1 Colored Noise [LN String:"."]                                     49
   7.2 Sinusoidal Input [LN String:"."]                                  56
   7.3 Experimental Noise Input [LN String:"."]                          61
   7.4 Sampling Considerations [LN String:"."]                           64
8 Utility Routines [LN String:"."]                                       67
   8.1 MAKDAT routine [LN String:"."]                                    67
   8.2 MAKCOEF routine [LN String:"."]                                   69
9 References [LN String:"."]                                             73


                 ^S▒1 Overview and Parameters

     This program is designed to compute amplitude density, autospectral
(power) density and cross-spectral density functions for discrete, stationary
random time series in the form


[EQUATION "center x sub n #3#3 = #3#3 x #3 (t sub 0 + n Delta t) #4#4 #R[for]
#3



where  [EQUATION "N", Position:In Text]  is the number of data points per
record.  Typically,  [EQUATION "M", Position:In Text]  such records are stored
in a data file, and each record is a contiguous segment of the stationary
process.  Most often, this data is in the form of 2-byte integers after having
been digitized by an analog to digital converter.  This routine reads this data
file and then transforms the integer data back into voltages or optionally into
some other desired quantity.  This transformation into a dimensional quantity
is by means of a polynomial of up to fourth degree.  The coefficients necessary
to complete the transformation are placed in coefficient data files which are
read by the routine.  Alternatively, the routine can read 4-byte floating-point
data which has already been transformed by some other program.  The input data
may consist of one or two separate time series which are to be analyzed.

     After transformation, various cosmetic operations may be performed on the
data prior to the spectrum computation.  The mean value, which is calculated
during the transformation process, may be subtracted from the data; or instead,
a linear trend in the data may be removed using a least squares fit technique.
Recursive digital filters including lowpass, highpass, bandpass and bandstop
have been incorporated and may be applied prior to the calculation of the
spectra.  Finally, to reduce the leakage of power from other frequencies due to
the fact that the input time series are of finite length and are then
truncated, a tapering operation has been included.  A choice of four tapering
functions is supplied, and the appropriate scale factors are computed
automatically.  The use of tapering may introduce some additional variability
in the resulting spectral estimate; to counteract this tendency, a means for
overlapping the input data records is provided.  This option employs 50%
overlap to reduce overall error at the expense of computation time (which is
doubled); however, the use of overlapping requires that the data be contiguous
in time from record to record.

     After computation of the selected spectral density function, the data are
stored to an ASCII output file which may then be imported into a plotting
program.  The routine is designed for multiple file processing; many input data
files may be processed sequentially if certain naming conventions are
followed.  The maximum number of files is currently set to IFMAX = 100 in order
to allocate storage for data.  Entering data file names on the command line
upon invocation of the program is also permitted and the maximum is set to
JFMAX = 5.  Each record of the input data file may contain no more than NMAX =
16384 points; note that in the case of two channels of data per record, each
channel may contain no more than NMAX / 2 points.  As described below, the
order of the recursive digital filters is set by the number of cascaded stages
desired; the maximum number of stages is currently placed at NSMAX = 20.  For
convenience, the base 10 logarithms of certain data in the output are computed;
to avoid taking the logarithm of zero, any number in the output data less than
[EQUATION "epsilon #3 = #3 1.0 #3 x #3 10 super [-20]", Position:In Text]  is
set equal to  [EQUATION "epsilon", Position:In Text] .  Any of these maximum
values may be altered by making a change in the appropriate PARAMETER statement
near the beginning of the code and then recompiling.

     The speed of the routine is limited by the large number of I/O operations
that are required; reduce the number of points per record and/or the number of
records per file for increased speed.  Alternatively, specify in the
configuration file that the routine use a RAM disk for storage of temporary
files.  The size of the executable file is largely governed by the setting of
NMAX; since this parameter must be a power of two, the largest possible value
is 16384 for a DOS-based computer which results in an executable file size of
430K bytes.  The CHKDSK command must report that at least 450K bytes of memory
are present for the routine to execute for this value of NMAX.  When compiling
the source code using Microsoft Fortran V 5.00, at least 603K of RAM must be
available for full optimization of the code.

     When the data to be processed by the spectrum routine are samples of a
continuous function of time at a single spatial location, then the calculated
spectral density functions are a function of frequency.  The user should keep
in mind that if the input data are samples of a continuous function of one
spatial direction at an instant of time (such as intensity data for horizontal
or vertical rows of pixels on a photograph from a CCD camera), then this
routine can be used to determine one-dimensional estimates of wavenumber
spectra.

                    ^S▒2 Input Data Files

     For any routine in which the processing of large amounts of data is
required, a rather rigid structure is necessarily imposed on the manner in
which the data is transferred into the program for manipulation.  The spectrum
routine is certainly no exception to this rule.  Furthermore, since the
spectrum routine has the ability to process data files in batches, additional
rules are required, such as naming conventions, to allow this feature to be
implemented.  Although the proliferation of rules can be a bit wearying, later
sections discuss some situations in which the rules can be bent or discarded.

2.1 Naming Convention

     A convention for naming the input data files was chosen to facilitate
multiple file processing.  The selected method resulted from the ease with
which the file names could be generated by a few lines of code.  Each file name
consists of eight characters: the first character must be alphabetic (A-Z), the
next three characters are digits (0-9) and the remaining four characters are a
period followed by the letters DAT.  Note that the routine is not
case-sensitive, lower-case as well as upper-case characters may be entered;
however, all lower-case inputs are converted to upper-case internally.  The
routine automatically appends the suffix [.DAT] and therefore it should never
be entered upon input.  The data file must reside in the same directory as the
executable file unless specified otherwise in the configuration file (discussed
later).  Examples of acceptable data file names are: A001.DAT, z087.dat and
M418.DAT.  The latter filename would require that IFMAX
[EQUATION ">=", Position:In Text] 418.  Every rule has an exception; data file
names entered on the command line when calling the routine need not be so
restrictive as outlined above.  They must still consist of eight characters and
the first character must still be alphabetic (A-Z); however, the next three
characters can be any of the alphanumeric characters.  The blank character is
used to delimit the names on the command line and is therefore not a valid
character for a file name.  The remaining four characters must be a period
followed by the letters DAT.  Examples of acceptable filenames for files
entered on the command line are: TEST.DAT, x_in.dat and S1$$.DAT.  When
actually entering the names on the command line, only the first four characters
are entered and the program appends [.DAT].

2.2 File Structure

     All input data files must be FORTRAN data files opened as sequential,
unformatted files.  The minimal amount of internal formatting allows the files
to be more compact which is a consideration when dealing with large quantities
of data.

     2.2.1 File Header

     The first record of the data file consists of a header containing four
quantities which characterize the data which follows. The first quantity is a
2-byte integer which gives the number of channels of data per record (1 or 2).
The next number is the record size in bytes and may be either a 2-byte or a
4-byte integer.  The record size is twice the number of points per record if
the data consists of 2-byte integer values, or the record size is four times
the number of points per record if the data consists of 4-byte real data.  The
third quantity is a 2-byte integer which gives the number of records of data
which follow, and the last quantity is the time interval in microseconds
between data points of the same channel expressed as a 2-byte integer.  Recall
that the largest positive signed number that may be stored as a 2-byte integer
is 32,767 and, for a 4-byte integer, the value is 2,147,483,647.

     For example, suppose the data file consists of 100 records of one channel
of data saved as 2-byte integers and sampled at 10000 samples per second with
2048 data points per record.  The file header would contain the four values:


[EQUATION "center 1 #4#4 4096 #4#4 100 #4#4 100 #4#4"]


Now suppose that two channels of data were sampled with a time interval of
0.016383 seconds between each successive data point resulting in a time
interval of 0.032766 seconds between each successive data point of the same
channel.  Furthermore, 40 records of data were sampled with 8192 data points
per record such that each channel of data consisted of 4096 data points.
Finally, the data were transformed into desired quantities and saved as 4-byte
floating-point numbers.  The header for this data file should consist of the
following four numbers:


[EQUATION "center 2 #4#4 32768 #4#4 40 #4#4 32766"]


     2.2.2 Remainder of Data File

     Following the first record containing the file header is the actual data.
When the data comes directly from an analog to digital converter (12-bit, say),
then the values range from -2048 to 2047 (or 0 to 4095) and are appropriately
stored as 2-byte integers.  Alternatively, if some preprocessing is typically
performed on the data, the data may be stored as 4-byte floating-point
numbers.  As a result of the Fast Fourier Transform (FFT) algorithm used in
this routine, the number of data points per channel per record must be a power
of 2.  One or two separate time series may be stored per record.  If only one
channel of data is sampled then an even number of records following the file
header should be stored in the data file.  This is necessary because in the
case of one channel of data, two records are submitted to the FFT algorithm to
be transformed simultaneously as a time-saving measure.  If an odd number of
records is detected, then the program automatically appends an additional
record of data containing a copy of the data values of the previous record.
When two channels of data are sampled, the number of records may be even or
odd.  The method of storing the data values in each record follows the standard
procedure used by analog to digital converters:


[EQUATION "center #R[one] #4 #R[channel] #^ #R[:] #4#4 x sub 0 #3 x sub 1 #3 x
s



where the notation is as defined above.  This convention must be followed even
if the data are processed and then stored as floating-point numbers.  When two
channels of data are sampled and then stored to a data file, the values for
each channel of data should be of the same data type: either 2-byte integers or
4-byte floating-point numbers.  If the user is unfamiliar with the task of
creating unformatted FORTRAN data files, then the user may instead create these
files in an ASCII file format.  A utility MAKDAT which is included on the
diskette can then be used to write the file header and transform the ASCII data
files into the unformatted FORTRAN data files required by the spectrum
routine.  See the section on utility files.

     2.2.3 Writing and reading the Data File

     This section gives a few lines of code which are representative of the
manner in which these data files are written and read.  When writing, the file
is first opened, the file header is written, the data is written and then the
file is closed.


                   INTEGER*2 ICHANS, IRSIZE, NUMREC, IDELTMS
                   INTEGER*2 NDATA (NMAX)

                   OPEN (2, FILE = 'A001.DAT', STATUS = 'UNKNOWN',
                         ACCESS = 'SEQUENTIAL', FORM = 'UNFORMATTED')

                   WRITE (2) ICHANS, IRSIZE, NUMREC, IDELTMS

                   DO J = 1, NUMREC
                      ... put data record into NDATA array ...
                     WRITE (2) (NDATA (I), I = 1, N)
                      ... get next record of data ...
                   ENDDO

                   CLOSE (2, STATUS = 'KEEP')

     The same procedure is followed when the data file is opened and read by
the spectrum routine.  The file is opened, the file header is read, the data is
read and then the routine is closed.


                   INTEGER*2 ICHANS, IRSIZE, NUMREC, IDELTMS
                   INTEGER*2 NDATA (NMAX)

                   OPEN (2, FILE = 'A001.DAT', STATUS = 'OLD',
                         ACCESS = 'SEQUENTIAL', FORM = 'UNFORMATTED')

                   READ (2) ICHANS, IRSIZE, NUMREC, IDELTMS

                   N = IRSIZE/2
                   DELT = FLOAT (IDELTMS) / 1000000.0

                   DO J = 1, NUMREC
                     READ (2) (NDATA (I), I = 1, N)
                      ... process this record of data ...
                   ENDDO

                   CLOSE (2, STATUS = 'KEEP')

2.3 Multiple File Processing

     This routine is capable of sequentially processing many files in a batch.
The program will handle up to IFMAX files which are consecutively numbered or
which are not consecutively numbered but have an arbitrary numbering
arrangement.  All data files in the batch must start with the same first
letter.  The letter and number combination in the file name are used to
associate each data file to a corresponding set of coefficients (if needed) in
an accompanying coefficient data file; this is the reason for the rather rigid
naming convention.  Alternatively, up to JFMAX files may be entered on the
command line when calling the program, and these will be processed
sequentially.  These files need not have the same first letter; however, these
data files also cannot use coefficient data files.  This is further clarified
below.  All data files must conform to the naming convention described above.
The selection of options to perform various operations on the data will then be
applied to all files in the batch.

     2.3.1 Consecutively numbered files

     If it is desired to process several files in a consecutive order, then
this option is selected. The routine will ask for the beginning and ending file
numbers and then the first letter in the file names. As an example, suppose it
was desired to process files C031.DAT through C094.DAT, then the user would
respond as follows:

spectrum

Process files (C)onsecutively or in an (A)rbitrary order : c

Enter starting & ending data file # : 31 94  (or 31,94)

Enter first letter of data file : c

     2.3.2 Arbitrarily numbered files

     Perhaps it is not desirable to process all of the files indicated above,
but only selected ones from the set.  If it were necessary to process C031.DAT,
C042.DAT, C067.DAT and C085.DAT, then the user would respond as follows.

SPECTRUM

Process files (C)onsecutively or in an (A)rbitrary order : A

Enter number of files to process : 4

Enter last digits of file   1 : 31
Enter last digits of file   2 : 42
Enter last digits of file   3 : 67
Enter last digits of file   4 : 85

Files are :
           31 42 67 85        (routine echoes file numbers selected)

Enter first letter of data file : C

     2.3.3 Files entered on command line

     This option was added as a means of processing a few files without having
to answer all of the prompts above and to allow a bit more freedom in the
selection of file names.  Suppose it was desired to process TSTX.DAT, X_IN.DAT
and WOW!.DAT, then the user would enter the following command line.

SPECTRUM TSTX X_IN WOW!

By relaxing the naming convention, it is no longer possible to easily associate
a coefficient data file with the correct input data file.  Therefore, if the
user desires to transform the integer data in each of the input files to
voltages only, or if the input data in each file already consists of
transformed floating-point numbers (as a result of some other program), then
coefficient data files are not necessary and this method for quickly processing
a small number of files may be used.  If the integer data in each input file
must be transformed into a desired floating-point quantity using a polynomial
transformation prior to the spectral calculation, then the program must be
initiated using the methods described in sections 2.3.1 or 2.3.2.  Further
discussion on the manner in which coefficient data files are associated with
the respective input data files appears below.

2.4 Coefficient Data Files

     These data files may be sequential unformatted or sequential formatted
FORTRAN data files.  They contain the constants which are used as the
coefficients of the polynomial that transforms the input data from voltages to
some other desired quantity.

     2.4.1 Naming Convention

     The coefficient data file names consist of eight characters.  The first
letter must be alphabetic (A-Z) and must be the same letter as the input data
file (or series of input data files) with which it is associated.  The spectrum
routine assigns the remaining seven characters of the name as CON.DAT (if the
file is unformatted) or CON.PRN (if the file is formatted).  If the input data
file D061.DAT were to be transformed from voltages into velocities, then the
coefficients of the polynomial would be found in the coefficient data file
named DCON.DAT (if unformatted) or DCON.PRN (if formatted), and this file must
reside in the same directory as that of the executable routine and the input
data file unless specified otherwise in the configuration file (discussed
later).  If D061.DAT contained two channels of data, then the coefficients
needed to transform the data for the second channel would be found in the
coefficient data file named DCON2.DAT (if unformatted) or DCON2.PRN (if
formatted).  Again, the routine requests that the user enter the first letter
only and the remaining eight characters are automatically assigned by the
routine.

     2.4.2 Unformatted File Structure

     These coefficient data files are accessed only if the input data are
2-byte integers and the user desires to convert the integers first into
voltages and then into another quantity.  The voltage transformation factor is
not included in these files; instead, it is entered into the routine at the
beginning and therefore applies to all files processed.  If a series of files
are to be processed, the coefficients for the transformation from voltages into
the other quantity may not be the same for all files in the batch, and this
situation is addressed as follows.  The conversion from a voltage into a
velocity, say, for each data point is carried out using


[EQUATION "center u sub n #3#3 = #3#3 b sub 0 #3 + #3 b sub 1 v sub n #3 + #3 b



where  [EQUATION "v sub n", Position:In Text]  is the nth data point expressed
as a voltage,
[EQUATION "b sub 0 #1 , #3 b sub 1 #1 , ... , b sub 4", Position:In Text]  are
the coefficients of the velocity transformation and
[EQUATION "u sub n", Position:In Text]  is the nth data point which has been
converted to a velocity.  The five coefficients of the transformation remain
the same for each data point in each record of the file, but are permitted to
vary from file to file.  These constants are stored in a two-dimensional array
of 4-byte floating-point numbers.  The first index of the array is the number
of the file to which the coefficients are to apply (1 to IFMAX), and the second
index is the number of the coefficient (1=
[EQUATION "b sub 0", Position:In Text] to 5=
[EQUATION "b sub 4", Position:In Text] ).  The first record of the unformatted
coefficient data file is a file header containing two 2-byte integer
quantities: the first is the number of sets of coefficients to read, and the
second is the starting file number to which these coefficients should be
associated.  The remainder of the data file consists of one large record
containing all of the coefficients.  If the user wishes to first create these
coefficient data files in an ASCII file format (without the header), the
utility MAKCOEF has been included on the diskette which will convert these
ASCII files into the unformatted FORTRAN files needed by the spectrum routine.
See the section on utility routines.  An example of the code used to write and
read a sequential unformatted coefficient data file follows.


         INTEGER*2 NUMSETS, NUMCON, NSTART
         REAL*4 CONST (IFMAX, 5)

         NUMCON = 5

         OPEN (2, FILE = 'DCON.DAT', STATUS = 'UNKNOWN',
               ACCESS = 'SEQUENTIAL', FORM = 'UNFORMATTED')

         WRITE (2) NUMSETS, NSTART
         WRITE (2) ((CONST (I,J), J = 1, NUMCON), I = 1, NUMSETS)

         CLOSE (2, STATUS = 'KEEP')


The spectrum routine then reads the coefficient data file using code of the
following form.


         INTEGER*2 NUMSETS, NUMCON, NSTART
         REAL*4 CONST (IFMAX, 5)

         OPEN (2, FILE = 'DCON.DAT', STATUS = 'OLD',
               ACCESS = 'SEQUENTIAL', FORM = 'UNFORMATTED')

         READ (2) NUMSETS, NSTART
         READ (2) ((CONST (NSTART + I - 1, J), J = 1, NUMCON), I = 1, NUMSETS)

         CLOSE (2, STATUS = 'KEEP')


The coefficients
[EQUATION "b sub 0, #1 b sub 1, #1 ... #1 , #1 b sub 4", Position:In Text]  for
the data file D061.DAT are then given by CONST (61,1), CONST (61,2), ... ,
CONST (61,5).

     2.4.3 Formatted Data Files

     It is not always convenient to store coefficients in sequential
unformatted data files.  This option was included to allow a quick way of
storing coefficients.  A sequential formatted file is exactly that created by
any simple text editor.  To create a sequential formatted coefficient data file
for the input data file D061.DAT, enter a text editor or word processor
(capable of creating ASCII files) and give the file the name DCON.PRN.  Next,
enter the five coefficients,
[EQUATION "b sub 0, #1 b sub 1, #1 ... #1 , #1 b sub 4", Position:In Text], one
per line and then save the file.  If you display the file to the screen, it
should look something like the following.  Note that the numbers should all be
floating-point numbers.


                   3.14159265359            (
[EQUATION "b sub 0", Position:In Text])
                   0.0                      (
[EQUATION "b sub 1", Position:In Text])
                   1.012345E-03             (
[EQUATION "b sub 2", Position:In Text])
                   0.1                      (
[EQUATION "b sub 3", Position:In Text])
                   2.3E-05                  (
[EQUATION "b sub 4", Position:In Text])


An example of the code that the spectrum routine will use to read in this data
file follows.


                   REAL*4 B (5), CONST (IFMAX, 5)

                   OPEN (2, FILE = 'DCON.PRN', STATUS = 'OLD')
                     READ (2, *) (B (I), I = 1, 5)
                   CLOSE (2, STATUS = 'KEEP')

                   DO I = 1, IFMAX
                     DO J = 1, 5
                       CONST (I, J) = B (J)
                     ENDDO
                   ENDDO


As is seen from the above code, the routine assigns the five constants to apply
to all of the IFMAX different input data files that might be read.  If a
different set of constants is required for each data file, then the files must
be processed one at a time, or sequential unformatted data files as described
above must be used.  If an input data file (D061.DAT, say) contains two
channels of information, then a second sequential formatted data file with the
name DCON2.PRN should be prepared as described above.

                    ^S^^Data Preparation

     This section deals with a sequence of cosmetic operations which may be
applied to the data prior to performing spectral calculations.  The first
section discusses the transformation of the input data from integer values into
voltages and then optionally into some other desired dimensional quantity (for
example, velocities).  If the input data consists of floating-point values,
then the transformation operation is skipped.  If desired, the mean value may
be removed from each record of the input data, or a linear trend resulting,
perhaps, from instrumentation drift may be removed from each record.  Unwanted
frequencies in the input data can be removed using the recursive digital
filtering routines discussed below.  Finally, a tapering operation which may be
coupled with a 50% overlap of the input data records is included to eliminate
the discontinuities found at the beginning and end of these records resulting
from the fact that only a finite amount of data can be sampled.  Currently, the
user has a choice of four different window functions.  If the input consists of
two channels of data, then the first three operations: transformation, mean or
trend removal and filtering may be different for each of the two input
channels.  The choice of whether or not to employ tapering as well as the
choice of window function, however, applies to both input channels.

3.1 Transformation

     Experimental data sampled by means of an analog-to-digital (A/D) converter
typically results in data files containing an array of two-byte integer
values.  This is due to the fact that a two-byte integer is capable of
representing numbers in a range from -32768 to 32767.  This range is sufficient
for 12, 14 and 16-bit converters and is used to approximate the magnitude of an
analog voltage at a given instant of time.  This voltage is constrained to lie
within a specified input voltage range which may or may not be programmable and
which is particular to the A/D converter used.  Before proceeding to the
spectral calculation, the integer input data must be transformed back into
voltages using the scale factor appropriate for the input voltage range of the
A/D converter used.

     During development of this routine, a 12-bit A/D converter was used which
was capable of digitizing three different bipolar voltage ranges: -1V to 1V,
-5V to 5V and -10V to 10V.  The use of 12 bits allows integer values ranging
from -2048 to 2047 for a total of 4096 digital levels to be used to quantize
each of the above input voltage ranges.  The appropriate scale factors for
conversion of the integer data back into voltages are given by


[EQUATION "center [2 #3 V] over [4096 #3 units] #4 , #4 [10 #3 V] over [4096 #3



respectively.  The user may choose from one of these scale factors or may enter
an alternative scale factor from the keyboard.  The routine first prompts the
user with:

Transform into (V)oltages or into (O)ther quantity
or read (R)eal data which is already transformed :

If the input data consists of floating-point values, then the letter R should
be entered, and the routine skips the rest of the transformation operation.
For integer input data, the user responds with V for transformation into
voltages, or O for voltage transformation followed by an additional
transformation into a second dimensional quantity; the following prompt is then
displayed:

Enter VOFST 1) 10/4096, 2) 20/4096, 3) 2/4096 or 4) other :

To this prompt, the user responds with a digit (1-4) indicating the desired
scale factor for voltage transformation.  Selection of item (4) leads to the
prompt:

Enter VOFST :

and, at this point, the user enters a floating-point number representing the
desired scale factor.

     The input data will now be converted back into voltages using the scale
factor specified above.  Since the analog voltage sampled by the A/D converter
often comes from a transducer, which measures some dimensional quantity and
then converts it to a voltage signal, provision for transforming the digitized
input voltages back into the original transduced quantity is included.
Knowledge of calibration data for the particular transducer is required for
this transformation, and the calibration curve may be linear, quadratic,
exponential, etc.  It was impossible during development of the code to
anticipate all forms that the calibration curve might take; therefore, a
polynomial fit was decided upon in hopes that it would satisfy the calibration
requirements of most users.  The polynomial may consist of terms up to fourth
degree; however, lower degree polynomial representations may be used by setting
the coefficient for the appropriate term to zero.  The conversion from a
voltage into a velocity, say, for each data point is carried out using


[EQUATION "center u sub n #3#3 = #3#3 b sub 0 #3 + #3 b sub 1 v sub n #3 + #3 b



where  [EQUATION "b sub 0", Position:In Text]  through
[EQUATION "b sub 4", Position:In Text]  are the five calibration coefficients,
[EQUATION "v sub n", Position:In Text]  is the
[EQUATION "nth", Position:In Text]  data point of the current record which has
just been transformed into a velocity as described above and
[EQUATION "u sub n", Position:In Text]  is the resulting
[EQUATION "nth", Position:In Text]  velocity value.  The coefficients necessary
to complete the transformation must be included in a suitable format in
coefficient data files which are then read by the routine; instructions for the
creation of these coefficient data files are contained in section 2.4.

     An alternative to the transformation operations described herein is also
available.  The user may transform the original integer data from the A/D
converter by means of his own program into floating-point data which can then
be read by this routine.  The floating-point input data files must conform to
the structure set forth in section 2.  The routine then skips the
transformations described above and proceeds to the calculation of the mean and
root-mean-square (rms) values of each data record and averaged values for the
file.

     Summarizing then, the spectrum routine begins by reading in an input data
file.  If this file consists of integers, then these integers are transformed
to voltages using a voltage transformation factor (VOFST) selected by the
user.  Optionally, these voltages can be further transformed into some other
dimensional quantity.  If the input data file contains floating-point numbers,
then the routine assumes that any transformation of the input data has already
been applied by the user.  As the input data is read in by the spectrum
routine, record by record, the mean and rms values of each record are
calculated and displayed on the screen at run-time.  After all records of the
input file have been read, the mean and rms values for each record are averaged
to obtain values which are representative of the entire file.  These averaged
mean and rms quantities are also displayed on the screen at run-time.  The
spectrum routine then proceeds to the various data preparation operations
discussed in the next section.  Note that the averaged mean and rms values for
the file are calculated a second time later on in the program as a result of
the spectral computation and are also displayed on the screen.  Theoretically,
these two sets of values should be identical and serve as a check on the
accuracy of the spectral computation.  However, if any of the data preparation
operations to be discussed next are applied to the data to improve the accuracy
of the spectral computation, the two sets of mean and rms values may not
exactly match.  This is due to the fact that the data preparation operations
change the input data.  The first set of mean and rms values are calculated
prior to the data preparation operations, and the second set are computed from
spectral density calculations which occur subsequent to the data preparation
procedures.

3.2 Mean Removal

     For calculation of spectral quantities, it is common to remove the mean
value from the input data.  Both this operation and the trend removal operation
described in the next section may be used for the removal of the mean from the
sample data.  Trend removal is not always desirable; for that reason a separate
operation to remove the mean is needed and is described here.  The mean value
for each record of data in the input file is calculated from


[EQUATION "center x bar #3#3 = #3#3 1 over N #3 sum below [n=1] above N #3 x
sub



where  [EQUATION "x sub n", Position:In Text]  is defined as in equation 1.1;
for convenience, and to reduce the algebra in the derivations in this section,
[EQUATION "n", Position:In Text]  is defined such that
[EQUATION "n #3#3 = #3#3 1,2, ... , N ", Position:In Text] , and the initial
point of the record  [EQUATION "t sub 0", Position:In Text]  is set to zero.
If  [EQUATION "y sub n", Position:In Text]  is used to represent the data with
zero mean, then this quantity is computed from


[EQUATION "center y sub n #3#3 = #3#3 x sub n #3#3 - #3#3 x bar right #R(3.4)"]
[PAGE BREAK]

3.3 Removal of Linear Trend

     As recommended by Bendat and Piersol (1986), a linear trend removal
operation has been included in this routine.  This operation is designed to
remove low frequency components in the data with a period longer than the
record length  [EQUATION "T #3 = #3 N Delta t", Position:In Text] .  These low
frequency trends can significantly distort spectral data; however, trend
removal should only be performed if instrumentation drift or other sources of
trends are expected.  Use of trend removal when no trends are present can
introduce small contributions to the spectral density at frequencies on the
order of the reciprocal of the record length.

     This operation is accomplished by fitting a straight line through the data
using the least squares fit method.  The values calculated from this fit are
then subtracted from the original data to yield data with zero mean and with no
frequency components smaller than  [EQUATION "1 #1 / #1 T", Position:In Text]
where  [EQUATION "T", Position:In Text]  is the record length.  Let
[EQUATION "x tilde", Position:In Text]  be a first degree polynomial fit to the
original data  [EQUATION "x sub n", Position:In Text]  (defined as in equation
1.1) where  [EQUATION "x tilde", Position:In Text]  is given by


[EQUATION "center x tilde sub n #3#3 = #3#3 b sub 0 #3#3 + #3#3 b sub 1 #1 (n
De



and with  [EQUATION "b sub 0", Position:In Text]  and
[EQUATION "b sub 1", Position:In Text]  defined as


[EQUATION "center #^ b sub 0 #3#3 = #3#3 [2 #1 (2N #3 + #3 1) #1 sum below
[n=1]



Let  [EQUATION "y sub n", Position:In Text]  represent the detrended data;
these data are obtained from the following difference


[EQUATION "center y sub n #3#3 = #3#3 x sub n #3#3 - #3#3 x tilde sub n right
#R



3.4 Filtering

     The filtering operations incorporated in this routine were added to allow
the user to pass only desired frequencies in the input data on to the spectral
calculation.  Four recursive (infinite impulse response) Butterworth digital
filters for filtering in the time domain have been included for this purpose:
lowpass, highpass, bandpass and bandstop (notch).  A short discussion of
digital filtering in the time domain including a summary of the particular
filters used in this routine and instructions for their use follows.  Note
however that filtering may be applied not only in the time domain but also in
the frequency domain.  When filtering in the frequency domain, the input data
record is Fast Fourier Transformed, then the FFT output is multiplied by a
filter function, [EQUATION "H(f)", Position:In Text] ; if required, an inverse
FFT could be performed to get the filtered data back into the time domain.
Frequency domain filtering may perhaps be implemented in a future version of
this program.

     Consider a set of discrete input values
[EQUATION "x sub n", Position:In Text]  defined as in equation 1.1.  Digital
filtering involves a computing algorithm in which a set of filtered values
[EQUATION "y sub n", Position:In Text]  is produced as a result of operating on
the inputs  [EQUATION "x sub n", Position:In Text] .  For the filters included
in this routine, this is a linear operation.  Digital filters can generally be
categorized as recursive or nonrecursive; if the formula for
[EQUATION "y sub n", Position:In Text]  contains only input terms
[EQUATION "x sub n", Position:In Text] , then the filter is termed
nonrecursive.  The general form of the linear nonrecursive algorithm is given
by


[EQUATION "center y sub n #3#3 = #3#3 sum below [m #3 = #3 - #1 M] above M b
sub



where  [EQUATION "b sub m", Position:In Text]  are the coefficients of the
transformation.  For values of  [EQUATION "m #3 < #3 0", Position:In Text] ,
the above algorithm requires future values of the signal
[EQUATION "x(t)", Position:In Text] ; this can cause a realizibility problem in
analog systems, but can be accomplished in digital systems where, in many
cases, the future values of  [EQUATION "x(t)", Position:In Text]  can be stored
in advance.  A recursive filter is one in which previously filtered values such
as  [EQUATION "y sub [n-1]", Position:In Text] ,
[EQUATION "y sub [n-2]", Position:In Text] , etc. appear explicitly in the
algorithm for  [EQUATION "y sub n", Position:In Text] .  For recursive filters,
the above equation is modified to include past values of the filtered output
[EQUATION "y(t)", Position:In Text]  as follows:


[EQUATION "center y sub n #3#3 = #3#3 sum below [m #3 = #3 - #1 M] above M b
sub



The transfer function  [EQUATION "H(f)", Position:In Text]  for the filter is
related to the coefficients  [EQUATION "a sub m", Position:In Text]  and
[EQUATION "b sub m", Position:In Text]  by


[EQUATION "center H(f) #3#3 = #3#3 [sum below [m #3 = #3 - #1 M] above M b sub
m



This equation tells how to determine  [EQUATION "H(f)", Position:In Text]  from
the coefficients  [EQUATION "a sub m", Position:In Text]  and
[EQUATION "b sub m", Position:In Text] .  However, filter design requires the
inverse problem: how to obtain a suitable set of coefficients based upon
knowledge of a desired filter transfer function.  A variety of techniques exist
to solve this problem, and the interested reader is referred to Stearns (1975)
for a lengthy discussion of the matter.  The transfer function discussed above
is more commonly expressed in terms of the variable
[EQUATION "z", Position:In Text]  where


[EQUATION "center z #3#3 == #3#3 e super [i 2 pi f Delta t] right #R(3.12)"]


For recursive filters,  [EQUATION "H(z)", Position:In Text]  is a rational
function of  [EQUATION "z", Position:In Text] .  It seems that rational
functions are particular good at fitting functions with sharp edges; most
filter functions are in this category.  The filters contained in this routine
are modified versions of FORTRAN subroutines which originally appeared in
Stearns (1975).  The order of the filter is determined by the number of
cascaded stages that are used; a filter of order
[EQUATION "P", Position:In Text]  will require
[EQUATION "P #1 / #1 2", Position:In Text]  cascaded stages.  The transfer
functions,  [EQUATION "H sub k #1 (z)", Position:In Text] , for the
[EQUATION "kth", Position:In Text]  stage for each filter, and the recursive
algorithms required to implement those transfer functions are given below.



[EQUATION "center #^ H sub k #1 (z) #3#3 = #3#3 [A sub k #1 (z super 2 #3 + #3
2



Note that the quantity  [EQUATION "K", Position:In Text]  in equation 3.16
above is a constant which is the same for all stages of the notch filter.  The
other coefficients,  [EQUATION "A sub k", Position:In Text]  through
[EQUATION "E sub k", Position:In Text] , are a more compact notation for the
[EQUATION "a sub m", Position:In Text]  and
[EQUATION "b sub m", Position:In Text]  values for each stage
[EQUATION "k", Position:In Text] .  In general,
[EQUATION "A sub k", Position:In Text]  through
[EQUATION "E sub k", Position:In Text]  change from stage to stage and are
evaluated from complicated expressions which the interested reader may find in
the source code.

     To use these filters, the user selects the desired filter type and enters
the critical (-3 dB) frequency for the low and highpass filters.  For the
bandpass and bandstop filters, two critical frequencies must be specified.  The
number of filter sections to cascade must also be specified.  The order of the
filter is twice the number of cascaded sections.  As the order of the filter is
increased, the transfer function for the filter becomes sharper and more
closely approximates the ideal; however, the amount of computation time is also
significantly increased.  A good starting point may be to use five cascaded
sections (tenth order); this is a reasonable trade-off between accuracy and
computational efficiency.  These filters work well when the input data is not
deterministic; however, unlike nonrecursive filters, recursive filters can
become unstable and produce an output which grows exponentially.  This has been
observed in a few rare instances during testing of this program and, so far,
has been found to occur only when the input data has been a perfect
computer-generated sinusoid.  The user should be aware of this limitation.
[PAGE BREAK]

3.5 Tapering

     The tapering option has been added to the routine as a grooming operation
in order to improve the resulting spectral density estimates.  A wide variety
of window functions may be found in the literature including extended
discussions concerning their performance characteristics.  See, for example,
Bendat and Piersol (1986) and Press, et al. (1986).  Rather than repeat these
long discussions here, a brief summary of the need for tapering will be
followed by a description of the window functions available for use with this
routine.

     As discussed earlier in regard to equation 1.1, the data processed by this
routine is in the form of a finite set of samples
[EQUATION "x sub n", Position:In Text]  of the continuous signal
[EQUATION "x(t)", Position:In Text]  which exists for a time
[EQUATION "T #3 = #3 N #1 Delta t", Position:In Text] .  Now, suppose that
[EQUATION "x(t)", Position:In Text]  is a finite portion of a continuous
signal  [EQUATION "v(t)", Position:In Text]  which is of unlimited extent in
time.  Then,  [EQUATION "x(t)", Position:In Text]  can be considered to be the
product of  [EQUATION "v(t)", Position:In Text]  and a rectangular window
function  [EQUATION "w(t)", Position:In Text]

[EQUATION "center x(t) #3#3 = #3#3 w(t) #3 v(t) right #R(3.17)"]


where

[EQUATION "center w(t) #3#3 = #3#3 #( { stack left [[ 1 #4#4 0 <= t <= T] [ 0
#4



A plot of the rectangular (square) window function
[EQUATION "w(t)", Position:In Text]  is shown in figure 3.1 where, for
convenience,  [EQUATION "T", Position:In Text]  is chosen to equal one.  The
computation of spectral density estimates for
[EQUATION "x(t)", Position:In Text]  involves the calculation of the Fourier
transform of  [EQUATION "x(t)", Position:In Text]  which is equivalent to the
Fourier transform of the product of  [EQUATION "w(t)", Position:In Text]  and
[EQUATION "v(t)", Position:In Text] .  A theorem from Fourier analysis relates
the Fourier transform of a product of two functions in time to the convolution
of the transforms of these functions.  In other words, the following is a
Fourier transform pair:


[EQUATION "center w(t) #3 v(t) #3#3 <==> #3#3 W(f) #3 * #3 V(f) right
#R(3.19)"]



where

[EQUATION "center W(f) #3 * #3 V(f) #3#3 = #3#3 int sub [#3 - infinity] super
[i



The spectral density of interest would be exact if the transform
[EQUATION "V(f)", Position:In Text]  could be calculated.  Instead,
[EQUATION "X(f) #3 = #3 W(f) #3 * #3 V(f)", Position:In Text]  is the quantity
which is, in fact, calculated.  This problem then is seen to be directly
related to the finite length of the data to be transformed and to the fact that
the data is sharply truncated at each end of the time record.  A plot of the
modulus of the Fourier transform of the rectangular window function,
[EQUATION "W(f)", Position:In Text] , may be found in figure 3.2 and is defined
as[PAGE BREAK]

                  [WHITESPACE 0.25 in, Place All on Next Page]


                        [PICTURE FIG31.CGM, Scale:0.800]

                 Figure 3.1  Window functions available for tapering the data




                        [PICTURE FIG32.CGM, Scale:0.800]

               Figure 3.2  Normalized amplitude of Fourier transform of window
                             functions[PAGE BREAK]


[EQUATION "center W(f) #3#3 = #3#3 [sin #1 pi f T] over [pi f] #3#3 e super [-i



Again, for plotting convenience,  [EQUATION "T", Position:In Text]  is chosen
to equal one.  This function is characterized by a large main lobe with smaller
side lobes which slowly decrease in size.  The ideal window function would be a
delta function; that is, an infinitely thin main lobe with no side lobes at
all.  As  [EQUATION "W(f)", Position:In Text]  is convolved with
[EQUATION "V(f)", Position:In Text] , information at frequencies well separated
from the main lobe can leak into the calculation and significantly distort the
resulting spectra.  The large amplitude sidelobes need to be reduced to
suppress this problem.

     This is accomplished by multiplying the original data by a window function
which tapers the time history at the beginning and end of the record to remove
the discontinuities there.  The tapering functions included in this routine are
the most commonly used ones, and the choice of a particular function requires a
trade-off between the thickness of the main lobe (resolving power) and the size
of the side lobes.  The following window functions (the subscripts merely
indicate the name of the window) may be selected:




[EQUATION "center #^ w sub h (t) #3#3 = #3#3 #( { stack left [[ 1 over 2 #3 (1
#
[PAGE BREAK]

[EQUATION "center #^ w sub p (t) #3#3 = #3#3 #( { stack left [[ #4 1 #3 - #3 (|




These functions are plotted in figure 3.1.  The moduli of the Fourier
transforms for each of the above windows are plotted in figure 3.2 and are
normalized by their value at  [EQUATION "f #3 = #3 0", Position:In Text] .  The
fact that these transforms do not equal one for
[EQUATION "f #3 = #3 0", Position:In Text]  implies that the resulting spectral
estimates would be reduced in magnitude upon application of one of these
windows.  To avoid this problem, the appropriate scale factor is computed by
the routine when any of these windows are selected and automatically applied to
the data before computing the spectral estimate in order to preserve the
correct amplitudes.  The Fourier transform for each of the above windows is
given below.



[EQUATION "center W sub h (f,T) #3#3 = #3#3 {#3#3 1 over 2 ( [sin #3 pi f T]
ove




[EQUATION "center W sub t (f,T) #3#3 #^ = #3#3 {#3#3 1 over 2 ( [sin #3 pi f T]




[EQUATION "center W sub w (f,T) #3#3 = #3#3 2 over [pi f T] #3#3 ( 1 over [ pi
f




[EQUATION "center W sub p (f,T) #3#3 = #3#3 1 over [pi f T] #3#3 ( 1 over [ pi
f




In equations 3.26 and 3.27 above
[EQUATION "f sub 1 #3 = #3 1 #1 / #1 T", Position:In Text] .  Recall that if
none of these windows are selected (no tapering), then the square window
defined in equation 3.18 and with a Fourier transform as defined in equation
3.21 is implicitly used.  The most commonly used window function is the Hanning
window; the transform for this function has the smallest side lobes but also
has the widest main lobe.  Also shown is a window which provides a cosine taper
for the first and last tenths of the data record, is one for the remainder of
the data record and is denoted by the name Tukey (also known as a 10% cosine
taper).  This window has been used by many researchers in the hopes that it
will throw away less of the data.  Examination of the modulus of the Fourier
transform for this window shows that it manages to narrow the main lobe a small
amount relative to the other windows, but it does this at the expense of large
side lobes that are nearly as large as that of the square window (no tapering
at all).

     A better way to throw away less of the data is to use any of the windows
available above and then to use overlapped processing techniques.  Before
describing overlapped processing, it is useful to review the procedure for the
calculation of spectra when no overlap is used.  The data are prepared
(detrended, filtered, etc.), and then the data are tapered.  The data are
tapered record by record and transformed to get the necessary spectra.  When
50% overlapped processing is used, the tapering operation proceeds a bit
differently.  First, the initial record of the data is tapered.  Then, the
second half of the first record is grouped together with the first half of the
second record to form a new second record, and then this new record of data is
tapered.  Then, the two halves of the old second record form the new third
record, and this is then tapered.  Then, the second half of old second record
is grouped together with the first half of the old third record to form a new
fourth record which is then tapered.  This proceeds throughout the file until
[EQUATION "M", Position:In Text]  old records are processed creating exactly
[EQUATION "2M", Position:In Text]  new records.  The last grouping pairs the
second half of the old last record with the first half of the old first record
to form the new  [EQUATION "2Mth", Position:In Text]  record.  This last record
was included in order to simplify the code for this procedure.  Because the
[EQUATION "2Mth", Position:In Text]  record has a discontinuity at the
midpoint, this record should not be transformed.  However, simply removing this
record would require that an odd number of records be submitted to the FFT
code.  As pointed out earlier, the FFT routine has been designed such that it
transforms two records of data in one pass as a time-saving measure; this
requires (in the one channel case) that an even number of records be processed
by the FFT routine.  Therefore, the overlapping operation has not been
modified;  [EQUATION "2M", Position:In Text]  records are created from
[EQUATION "M", Position:In Text]  original records for both the one and two
channel cases.  These  [EQUATION "2M", Position:In Text]  records are then
submitted to the FFT subroutine and are transformed; but the results of the
[EQUATION "2Mth", Position:In Text]  record are discarded after the
transformation.  Summarizing, when overlapping is employed,
[EQUATION "2M", Position:In Text]  records are created and are then
transformed, but only the results from
[EQUATION "2M #3 - #3 1", Position:In Text]  records are actually used in the
final calculations.  To avoid discontinuities resulting from the use of
overlapping in any of the other  [EQUATION "2M #3 - #3 1", Position:In Text]
records, the  [EQUATION "M", Position:In Text]  original records must be
contiguous in time.  The data need not be contiguous from record to record if
overlapping is not employed.  The use of 50% overlapped processing doubles the
number of tapering and FFT operations; however, this will retrieve about 90% of
the stability lost due to the tapering operation and will decrease the
variability of the resulting estimate.

     Finally, tapering is best used when the input data are essentially random
data -- that is, not deterministic.  When this program was tested, input data
(both deterministic and random) with known spectral densities were used.  When
a sinewave was used for input, the power and amplitude spectra exhibited a
lower noise floor when no tapering was used.  On the other hand, when colored
noise (bandwidth limited white noise) was used for input, a substantial
improvement was evident in the output spectra when any of the four window
functions described above was used to taper the data.

                  ^SÿSpectrum Computation

     This section introduces the definitions of the discrete Fourier transform
and the various spectral density functions computed by this routine.  The means
by which these estimates are computed as well as the errors associated with the
calculation are discussed.  The need for additional code to adjust the value of
the computed phase angle of the cross-spectral density function is explained by
means of an example, and an explanation of the coherence function calculated by
this routine is given.

4.1 Fast Fourier Transform

     Using the notation of Bendat and Piersol (1986), the Fourier transform of
a function  [EQUATION "x(t)", Position:In Text] , where
[EQUATION "x(t)", Position:In Text]  may be real or complex-valued, is given by


[EQUATION "center X(f) #3#3 = #3#3 int super [infinity] sub [- infinity] #3
x(t)



This definition is not particularly useful for experimentally derived data
which exist for only a finite time interval.  For this reason, one usually
employs a finite-range transform over the interval
[EQUATION "0 <= t <= T", Position:In Text] , and this transform is defined by


[EQUATION "center X (f,T) #3#3 = #3#3 int super T sub [#3 0] #3 x (t) #3 e
super



As in equation 1.1, assume that each record of the input data file consists of
[EQUATION "N", Position:In Text]  samples of
[EQUATION "x(t)", Position:In Text]  which have been obtained at equally spaced
points in time with spacing  [EQUATION "Delta t", Position:In Text] .  Note
that  [EQUATION "t sub 0", Position:In Text]  has been set equal to zero for
convenience.  Then, each record contains the values
[EQUATION "x sub n", Position:In Text]  where


[EQUATION "center x sub n #3#3 = #3#3 x(n Delta t) #4#4 #R[for] #4 n #3 = #3
0,1



Now, the finite-range transform defined in equation 4.2 can be written in
discrete form as


[EQUATION "center X(f,T) #3#3 = #3#3 Delta t #3#3 sum below [n=0] above [N-1]
#3



Since only  [EQUATION "N", Position:In Text]  input values of
[EQUATION "x sub n", Position:In Text]  are used in the computation of the
transform,  only  [EQUATION "N", Position:In Text]  output values of
[EQUATION "X(f,T)", Position:In Text]  may be specified.  These are typically
chosen at the following discrete frequency values


[EQUATION "center f sub k #3#3 = #3#3 k over T #3#3 = #3#3 k over [N Delta t]
#3



Thus, the discrete form of the finite-range Fourier transform becomes


[EQUATION "center X(f sub k #1,T) #3#3 = #3#3 Delta t #3#3 sum below [n=0]
above



Although  [EQUATION "N", Position:In Text]  output values have been specified,
results are unique only out to
[EQUATION "k #3 = #3 N #1 / #1 2", Position:In Text] ; this is a result of the
symmetry of the Fourier transform for real input values.  Thus, the highest
frequency that can be resolved for a sampling rate of
[EQUATION "1 #1 / #1 Delta t", Position:In Text]  samples per second is the
Nyquist frequency given by


[EQUATION "center f sub [N over 2] #3#3 = #3#3 f sub c #3#3 = #3#3 1 over [2 #1



which occurs for  [EQUATION "k #3 = #3 N #1 / #1 2", Position:In Text] .  To
compute the  [EQUATION "X sub k", Position:In Text]  values in equation 4.6
requires  [EQUATION "N super 2", Position:In Text]  complex multiply-add
operations for each record of the data file; note that 1 complex multiply-add
operation is equivalent to 4 real multiply-adds.  This can be quite a
formidable computational task as  [EQUATION "N", Position:In Text]  becomes
large.  Fortunately, several algorithms for reducing the complexity and
increasing the speed of the computation exist and are known collectively as
fast Fourier transforms (FFT).  Rather than discuss these algorithms in detail,
the reader is referred to the discussions that may be found in the references.
The FFT algorithm used here employs a technique known as time decomposition
with input bit reversal.  The kernel of this algorithm was taken from
Stearns (1975).  The simplest implementation of this algorithm calls for
[EQUATION "N", Position:In Text]  to be a power of 2.  This is required in this
routine, and if necessary, each record of the input data file should be padded
with zeros at the end to satisfy this requirement.

     The FFT algorithm used in this routine is designed to handle the general
case where  [EQUATION "x(t)", Position:In Text]  is a complex-valued function;
that is, the input data which is passed to the FFT subroutine consists of two
arrays: one containing the real parts and the other containing the imaginary
parts.  However, experimentally sampled random time series data are
real-valued.  When the FFT of a purely real-valued function is computed,
certain symmetries in the resulting transform allow one to compute the FFTs for
two real-valued functions simultaneously by inserting one into the input array
of real parts and the other into the array of imaginary parts.  The trick is to
unscramble the individual transforms from the resulting computation.  This
speed improvement has been incorporated into this routine, but it places a
restriction on the input data files.  If the input data file to be processed by
SPECTRUM consists of one channel of sampled data, then the data file must
consist of an even number of records.  The reason for this is that input data
records are submitted to the FFT algorithm two at a time to obtain the speed
improvement discussed above.  If the input data file consists of two channels
of sampled data, then each record of the file is first decomposed into the two
separate channels, and these are then submitted to the FFT algorithm to be
simultaneously processed.  So, for the case of two channels of sampled data,
the input data file may have an even or odd number of records.

4.2 Autospectral (Power) Density

     The one-sided autospectral density function, defined for positive
frequencies only, is usually expressed as twice the amplitude of the Fourier
transform of the autocorrelation function as follows:


[EQUATION "center G sub [xx] (f) #3#3 = #3#3 2 #3 int super infinity sub [#3 -
i



where  [EQUATION "R sub [xx] (tau)", Position:In Text]  is the autocorrelation
function defined as


[EQUATION "center R sub [xx] (tau) #3#3 = #3#3 E #[ #3 x(t) #3 x(t + tau) #3 #]



and the notation E[ ] denotes the expected value operation.
[EQUATION "G sub [xx] (f)", Position:In Text]  is a real function of
[EQUATION "f", Position:In Text]  regardless of whether
[EQUATION "x(t)", Position:In Text]  is real or complex.  This calculation is
implemented in this routine using digital computing techniques employing finite
fast Fourier transforms of the original data records.  Using this method, the
autospectral density may be obtained from


[EQUATION "center G sub [xx] (f) #3#3 = #3#3 2 #3#3 lim below [T -> infinity]
#3



[EQUATION "| #3#3 |", Position:In Text]  denotes the modulus of a complex
quantity, and  [EQUATION "X (f,T)", Position:In Text]  is the finite Fourier
transform which is valid for the interval
[EQUATION "0 <= t <= T", Position:In Text]  and is calculated as in equation
4.6.  If the input data file contains two channels of data, denoted as
[EQUATION "x(t)", Position:In Text]  and  [EQUATION "y(t)", Position:In Text] ,
then selection of the power spectral density option results in the computation
of  [EQUATION "G sub [xx] (f)", Position:In Text]  and
[EQUATION "G sub [yy] (f)", Position:In Text] .

     Specifically, the calculation proceeds by computing the FFT for each
record of the input data file, determining the modulus of the complex result,
squaring it and then saving the results to an array.  This is repeated for each
record; upon completion, the results are ensemble averaged to approximate the
expected value operation required in the definition.  The computation of an
expected value is needed to reduce the random error of the estimated
autospectral density function.  The number of records in the input data file
determines the number of members in the ensemble which will be averaged.  If
the number of records in the data file (before tapering) is denoted by
[EQUATION "n sub r", Position:In Text] , then the normalized random error of
the autospectral density estimate is given by


[EQUATION "center G sub [xx] (f) #3 : #4#4 epsilon #3#3 = #3#3 1 over [root[n
su



where the normalized random error is defined as a fractional quantity by
dividing the standard deviation of the estimate by the actual value estimated.
If the tapering operation is used, it will increase the variability of the
resulting estimate by an amount which depends upon the particular window
function chosen.  This increase in the variance of the estimate is, in the
worst case (Hanning), about two; but if the 50% overlap technique is also
employed, it will counteract this increase and will recover about 90% of the
variability caused by tapering.  Thus, the normalized random error calculated
from equation 4.11 will serve as a reasonably accurate estimate for both the no
taper and taper with overlap cases.  If tapering is used but overlapping is
not, then the variance is doubled and the normalized random error defined in
equation 4.11 should be multiplied by the square root of two.  The routine
computes the appropriate value of the normalized random error and displays it
on the screen at run-time.  As an example, consider an autospectral density
estimate computed from a data file with 100 original records which are then
overlapped and tapered to produce 199 records; this estimate will have a
normalized random error of 10%.  To reduce this error to 5%, the input data
file would have to contain 400 records.  There is no specific numerical limit
to the number of records that an input data file may contain; the spectrum
routine has successfully computed spectral densities for input data files with
10,000 records.  However, data files with more than 100 records are usually
quite large. Furthermore, the spectrum routine must create temporary files
during computation, and the size of these files (depends upon the size of the
input data file) is often very large.  Reserving a sufficient amount of space
on the hard disk or on a RAM disk for these files is typically the limiting
factor on the number of records.

     The computation of a fast Fourier transform and power spectral density
function allows the calculation of the mean and root-mean-square (rms) values
of each record of the input time series.  The equation for the mean value of
each data record can be obtained by setting
[EQUATION "k #3 = #3 0", Position:In Text]  in equation 4.6.  One then obtains


[EQUATION "center x bar #3#3 = #3#3 X sub 0 over N #3#3 = #3#3 1 over N #3 sum
b



The mean-square value of each record can be computed from


[EQUATION "center R sub [xx] (0) #3#3 = #3#3 E #[ #3 x super 2 (t) #3 #] #3#3 =



The rms value is just the square root of this number.  The spectrum routine
calculates the mean and rms values of the input time series using this
procedure and reports them on the screen at run-time along with the appropriate
value of the normalized random error.

4.3 Cross-spectral Density

     4.3.1 Calculation Procedure

     A one-sided cross-spectral density function
[EQUATION "G sub [xy] #1 (f)", Position:In Text]  may be defined in a manner
similar to equation 4.8; this quantity is, in general, complex and is given by


[EQUATION "center G sub [xy] (f) #3#3 = #3#3 2 #3 int super infinity sub [#3 -
i



The quantity  [EQUATION "R sub [xy] (tau)", Position:In Text]  is the cross
correlation function defined by


[EQUATION "center R sub [xy] (tau) #3#3 = #3#3 E #[ #3 x(t) #3 y(t + tau) #3 #]



The cross-spectral density may also be computed using fast Fourier transform
procedures and is the method adopted here.  The equivalent expression to
equation 4.10 for the one-sided cross-spectral density function is


[EQUATION "center G sub [xy] (f) #3#3 = #3#3 2 #3#3 lim below [T -> infinity]
#3



where the notation  [EQUATION "X super * (f,T)", Position:In Text]  is used to
represent the complex conjugate.  Alternatively, this can be expressed in polar
notation as


[EQUATION "center G sub [xy] #1 (f) #3#3 = #3#3 | G sub [xy] #1 (f) | #3 e
super



Note that this option may be selected only when the input data file contains
two channels of data, then, the magnitude and phase,
[EQUATION "| G sub [xy] #1 (f) |", Position:In Text]  and
[EQUATION "theta sub [xy] #1 (f)", Position:In Text]  respectively, are the
quantities which are calculated and then saved in the output file.

     As described above in the section on power spectral density calculation,
it is possible to compute the value of the autocorrelation function for
[EQUATION "tau #3 = #3 0", Position:In Text]  (the mean-square value) by
calculating the integral of the autospectral density function.  In a similar
manner one may compute the cross correlation function evaluated at
[EQUATION "tau #3 = #3 0", Position:In Text]  from


[EQUATION "center R sub [xy] #1 (0) #3#3 = #3#3 E #[ #3 x(t) #3 y(t) #3 #] #3#3



The spectrum routine calculates this quantity and displays it on the screen at
run-time.

     4.3.2 Adjustment of Phase Angle

     In order to clarify the need for additional code to adjust the values of
the phase angle  [EQUATION "theta sub [xy] (f)", Position:In Text] , an example
taken from Hess (1990) will be given which also serves to illustrate the use of
the cross spectral density function in general.  Specifically, a point
measurement of the vertical displacement of a compliant surface, responding to
the unsteady forcing of a fluid flow above the surface, was acquired; call this
displacement signal  [EQUATION "x(t)", Position:In Text] .  As
[EQUATION "x(t)", Position:In Text]  was measured, a second displacement signal
was simultaneously measured; call this second signal
[EQUATION "y(t)", Position:In Text] .  The locations on the compliant surface
of the measurement of  [EQUATION "x(t)", Position:In Text]  and
[EQUATION "y(t)", Position:In Text]  were as follows:  both were obtained on
the centerline of the rectangular compliant slab, and
[EQUATION "y(t)", Position:In Text]  was located 11.4 cm downstream of
[EQUATION "x(t)", Position:In Text] .  The fluid flow was turbulent; the
instantaneous streamwise component of velocity above the compliant surface
consisted of a small fluctuating quantity superimposed upon a larger mean
value.  A disturbance in the fluid flow caused a displacement of the compliant
surface which was measured at the upstream location and was recorded at time
[EQUATION "t", Position:In Text]  as  [EQUATION "x(t)", Position:In Text] .  As
the flow disturbance was convected downstream by the mean flow, the
displacement of the compliant surface also travelled downstream and was
measured at the downstream location as  [EQUATION "y(t+tau)", Position:In Text]
  where  [EQUATION "tau", Position:In Text]  was the time delay for the
disturbance to travel 11.4 cm.  Quantities of interest, then, are the
convection speed of the displacement fluctuation on the compliant surface as
well as the decrease in amplitude of this fluctuation as it decays.

     For this particular situation, the decay of the displacement field may be
modeled by the time delay problem, see Bendat and Piersol (1986).  This model
provides a method for determining the convection speed of the displacement
fluctuation.  Consider two zero-mean stationary random signals
[EQUATION "x(t)", Position:In Text]  and  [EQUATION "y(t)", Position:In Text]
where  [EQUATION "y(t)", Position:In Text]  is defined by



[EQUATION "center y(t) #3#3 = #3#3 alpha #1 x(t-tau sub 0) #3#3 + #3#3 n(t)
#3#3



The value  [EQUATION "alpha", Position:In Text]  is a constant attenuation,
[EQUATION "tau sub 0", Position:In Text]  is a constant time delay and
[EQUATION "n(t)", Position:In Text]  is uncorrelated, zero mean value noise at
the output.  This process is illustrated in figure 4.1.

                        [PICTURE FIG41.CGM, Scale:0.800]

                  Figure 4.1  Schematic for time delay problem


The cross-correlation of these two signals can be computed to be


[EQUATION "center R sub [xy] #1 (tau) #3#3 = #3#3 alpha #3 R sub [xx] #1 (tau -



Thus, the cross-correlation is an attenuated replica of the autocorrelation,
which is displaced in time by the amount
[EQUATION "tau sub 0", Position:In Text] .  Without going through all of the
details, a one-sided cross-spectral density function
[EQUATION "G sub [xy] #1 (f)", Position:In Text]  may also be computed; this
quantity is, in general, complex and for this problem is given by


[EQUATION "center G sub [xy] #1 (f) #3#3 = #3#3 alpha #3 G sub [xx] #1 (f) #3 e



Since the time delay  [EQUATION "tau sub 0", Position:In Text]  appears only in
the phase angle,  [EQUATION "theta sub [xy] #1 (f)", Position:In Text] , it can
be determined from measurement of the phase angle and by noting that it is a
linear function of the frequency with slope
[EQUATION "2 pi tau sub 0", Position:In Text] .  Summarizing then, a
cross-spectral density is computed for  [EQUATION "x(t)", Position:In Text]
and  [EQUATION "y(t)", Position:In Text] .  If the process can be modelled by
the time delay problem, then the modulus will be an attenuated replica of the
autospectral density,  [EQUATION "G sub [xx] (f)", Position:In Text] , and the
phase angle will be a linear function of the frequency with a slope given by
[EQUATION "2 pi tau sub 0", Position:In Text] .  A convection speed may then be
calculated from
[EQUATION "U sub c #3#3 = #3#3 11.4 over [tau sub 0]", Position:In Text]
  cm/sec.

     An example of this procedure is shown in figure 4.2.  The top graph shows
the autospectral density curve  [EQUATION "G sub [xx] (f)", Position:In Text]
and below it the modulus  [EQUATION "| G sub [xy] (f) |", Position:In Text]
for the corresponding cross-correlation; indeed, the bottom curve does appear
to be an attenuated copy of the first.  Note the substantial additional scatter
in the cross-spectral density estimate relative to that for the autospectral
density estimate at the higher frequencies.  The normalized random error

                        [PICTURE FIG42.CGM, Scale:0.800]

                     Figure 4.2  Cross-spectral density modulus example


for the cross-spectral density estimate turns out to be a function of frequency
unlike that for the autospectral density estimate.  A discussion of the means
for estimating this error is given in a later section.

     Figure 4.3 displays the computed phase angle expressed in radians.  With
the data presented in this form, it is difficult to fit the data with a least
squares fit in order to determine the slope, and therefore,
[EQUATION "tau sub 0", Position:In Text] .  This difficulty may be traced to
the fact that an inverse tangent must be computed in order to calculate the
value of the phase angle.  The FORTRAN intrinsic function ATAN2 returns a value
for the inverse tangent, but forces the result to lie in the range
[EQUATION "- pi #3 <= #3 result #3 <= #3 pi", Position:In Text] .  However, the
phase angle function is periodic with period
[EQUATION "2 pi", Position:In Text] .  Using this fact, additional code may be
used to straighten out the curve by adding integer multiples of
[EQUATION "2 pi", Position:In Text]  as required.  Figure 4.4 shows the
adjusted values of the phase angle and a least squares fit to the slope.  The
calculated values of  [EQUATION "tau sub 0", Position:In Text]  and
[EQUATION "[U sub c] #1 / #1 [U sub infinity]", Position:In Text] , where
[EQUATION "U sub [infinity]", Position:In Text]  is the mean flow speed far
above the compliant surface, are given on the plot.  The scatter in the phase
angle data becomes large at a frequency for which the energy in the signal is
about two decades below the peak value.  Computed values of the convection
speed using this method vary by less than 2% from those obtained using an
alternative method.

     The phase angle data can be adjusted manually, but this is a tedious and
time-consuming procedure.  Instead, the spectrum routine automatically computes
both the original values of the phase angle as well as the adjusted values of
the phase angle when the cross spectral density and the coherence function are
calculated.  Note that the relationships derived in equations 4.21 apply only
when equation 4.19 may be used as a model for the data.  Therefore, both sets
of phase angle data, unadjusted and adjusted values, are stored to the output
file and made available to the user.  The task of adjusting these phase angle
values, while conceptually simple, is somewhat tricky to program.  A
description of the technique follows.

     The standard deviation of the phase angle estimate as a function of
frequency is computed when the coherence function is computed (this will be
explained in the section on error estimates).  The average of a moving window
of  [EQUATION "NSD", Position:In Text]  values of this standard deviation is
computed and checked against a threshold value given by
[EQUATION "SDLIM", Position:In Text] .  If the average standard deviation is
too high, then the phase angle points are considered to be essentially random
and no further attempts to adjust the values are made.  Thus, when this
condition occurs, the remaining phase angle values are kept the same as the
original values (not adjusted).  When the moving average of the standard
deviation is less than  [EQUATION "SDLIM", Position:In Text] , then adjustment
may take place as follows.  A straight line is fit to a moving window of
[EQUATION "NPTS", Position:In Text]  values of the original phase angle data
using a least squares fit.  The coefficients of this fit are used to guess the
next value of the phase angle; if the actual value of the next phase angle data
point differs from the guessed value by an amount greater than
[EQUATION "pi", Position:In Text]  radians, then multiples of
[EQUATION "2 pi", Position:In Text]  radians are added to (or subtracted from)
the actual value until the difference between the actual and guessed values is
less than  [EQUATION "pi", Position:In Text]  radians.  The procedure continues
to adjust subsequent values of the phase angle until the moving average of the
standard deviation exceeds  [EQUATION "SDLIM", Position:In Text] .  This
portion of the code contains three adjustable parameters:
[EQUATION "NSD #3 = #3 7", Position:In Text] ,
[EQUATION "NPTS #3 = #3 7", Position:In Text]  and
[EQUATION "SDLIM =70.0 degree", Position:In Text] .  These values have been
selected by trial and error to yield the best response for the data most often
processed by the author.  Other values may be chosen by changing the numbers in
the appropriate PARAMETER statement at the beginning of the code.  Note that
[EQUATION "NSD", Position:In Text]  and [EQUATION "NPTS", Position:In Text]
must be chosen such that


[EQUATION "center NSD #3#3 <= #3#3 2 #3 NPTS #3#3 - #3#3 1 right #R(4.22)"]


is satisfied.  This adjustment procedure seems to work reasonably well for most
situations encountered by the author; however, some manual adjustment of values
may still be required in some circumstances.[PAGE BREAK]

                  [WHITESPACE 0.4 in, Place All on Next Page]

                        [PICTURE FIG43.CGM, Scale:0.800]

                               Figure 4.3  Phase angle example

                  [WHITESPACE 0.25 in, Place All on Next Page]

                        [PICTURE FIG44.CGM, Scale:0.800]

                    Figure 4.4  Phase angle - adjusted values[PAGE BREAK]

     4.3.3 Coherence Function

     The coherence function, also known as the coherency squared function is
defined by


[EQUATION "center gamma super 2 sub [xy] (f) #3#3 = #3#3 [ | G sub [xy] (f) |
su



where
[EQUATION "0 #3 <= #3 gamma super 2 sub [xy] (f) #3 <= #3 1", Position:In Text]
  for all  [EQUATION "f", Position:In Text] .  This function is analogous to
the square of the more well known correlation coefficient function given by


[EQUATION "center rho sub [xy] (tau) #3#3 = #3#3 [R sub [xy] (tau)] over [ #[ R



where  [EQUATION "-1 #3 <= #3 rho sub [xy] (tau) #3 <= #3 1", Position:In Text]
  for all values of  [EQUATION "tau", Position:In Text] .  As described by
Bendat and Piersol (1986), the function
[EQUATION "rho sub [xy] (tau)", Position:In Text]  measures the degree of
linear dependence between  [EQUATION "x(t)", Position:In Text]  and
[EQUATION "y(t)", Position:In Text]  for a displacement of
[EQUATION "tau", Position:In Text]  in  [EQUATION "y(t)", Position:In Text]
relative to  [EQUATION "x(t)", Position:In Text] .  Similarly, for linear
systems, the function  [EQUATION "gamma super 2 sub [xy] (f)", Position:In
Text]
  measures the fractional portion of the mean square value at the output
[EQUATION "y(t)", Position:In Text]  that is contributed by the input
[EQUATION "x(t)", Position:In Text]  at frequency
[EQUATION "f", Position:In Text] .  On the other hand, the quantity
[EQUATION "#[ 1 #3 - #3 gamma super 2 sub [xy] (f) #]", Position:In Text]  is a
measure of the mean square value of  [EQUATION "y(t)", Position:In Text]  not
accounted for by  [EQUATION "x(t)", Position:In Text]  at frequency
[EQUATION "f", Position:In Text] .  If  [EQUATION "x(t)", Position:In Text]
and  [EQUATION "y(t)", Position:In Text]  are completely unrelated, the
coherence function will be zero.  Furthermore, Bendat and Piersol (1986)
indicate that if the coherence function is greater than zero but less than
unity, one or more of the following three possible physical situations exist.


          1.   Extraneous noise is present in the measurements.

          2.   The system relating
               [EQUATION "x(t)", Position:In Text]  and
               [EQUATION "y(t)", Position:In Text]  is not linear.

          3.   [EQUATION "y(t)", Position:In Text]  is an output due
               to an input  [EQUATION "x(t)", Position:In Text]  as
               well as to other inputs.


The calculation of this function requires not only the computation of the cross
spectral density function but also the computation of power spectral densities
for each channel of data.  The code has been optimized to minimize the amount
of additional computational effort; however, in anticipation of the possibility
that a slow computer might be employed, the routine prompts the user for this
option at run-time.

     4.3.4 Error Estimates

     The definition of the coherence function makes possible at this point a
discussion of the error associated with estimates of the magnitude and phase of
the cross spectral density function and of the coherence function itself.  This
information comes from Bendat and Piersol (1986).  The normalized random error
of the estimate of the magnitude of the cross spectral density function is
given by


[EQUATION "center   | G sub [xy] (f) | #3 : #4#4 epsilon (f) #3#3 = #3#3 1 over



where  [EQUATION "| gamma sub [xy] (f) |", Position:In Text]  is the positive
square root of the coherence function evaluated at
[EQUATION "f", Position:In Text] , and  [EQUATION "n sub r", Position:In Text]
is the number of records in the input data file.  The error approaches that for
the power spectral density estimate (see equation 4.11) as
[EQUATION "| gamma sub [xy] (f) |", Position:In Text]  goes to one, and the
error is higher for values less than one.  Unlike the error estimate for the
power spectral density function, all of the error estimates in this section are
a function of frequency.

     Since the phase angle may assume the value zero, a normalized random error
cannot be specified; instead the standard deviation (s.d.) of the phase angle
estimate is given by


[EQUATION "center theta sub [xy] (f) #3 : #4#4 s. #3 d. #1 (f) #3#3 = #3#3 [ #[



The standard deviation is seen to approach zero as
[EQUATION "| gamma sub [xy] (f) |", Position:In Text]  goes to one and does so
independent of  [EQUATION "n sub r", Position:In Text]  for
[EQUATION "n sub r #3 > #3 1", Position:In Text] .  Continuing, the normalized
random error of the coherence function itself can also be calculated and is
defined as


[EQUATION "center gamma super 2 sub [xy] (f) #3 : #4#4 epsilon (f) #3#3 = #3#3
[



where the estimated value of the coherence function is substituted as an
approximation to the unknown true values of
[EQUATION "gamma super 2 sub [xy] (f)", Position:In Text]  and
[EQUATION "| gamma sub [xy] (f) |", Position:In Text]  called for in equation
4.27.  Note that the normalized random error approaches zero as either
[EQUATION "n sub r #3 -> #3 infinity", Position:In Text]  or
[EQUATION "gamma super 2 sub [xy] #3 -> #3 1", Position:In Text] .  This
implies that coherence function estimates can be more accurate than either
autospectra or cross spectra estimates (which are used to calculate the
coherence function) when
[EQUATION "gamma super 2 sub [xy] (f)", Position:In Text]  is close to unity.
If the coherence function is calculated, the spectrum routine automatically
computes the three error estimates defined in equations 4.25 to 4.27 as a
function of frequency and saves the results in a separate output file.  The
file name and structure of this file will be discussed in the section on output
files.  Equations 4.25 to 4.27 are valid for the no taper and taper with
overlap cases.  If tapering is used without overlapping, then the error
estimates in equations 4.25 to 4.27 must each be increased by multiplying by
the square root of two, and this is automatically performed by the spectrum
routine.

     It is perhaps useful to know the number of records required in the input
data file to attain a specified normalized random error or standard deviation
for each of the estimates mentioned above.  This has been calculated for
various values of  [EQUATION "gamma super 2 sub [xy] (f)", Position:In Text]
for the cases:
[EQUATION "epsilon #3 #[ #3 | G sub [xy] (f) | #3 #] #3 = #3 0.1", Position:In
T
 ,
[EQUATION "epsilon #3 #[ #3 gamma super 2 sub [xy] (f) #3 #] #3 = #3 0.1",
Posit
  and
[EQUATION "s. #3 d. #3 #[ #3 theta sub [xy] (f) #3 #] #3 = #3 0.052 #3 rad #3
~~
 , and the results are given in table 4.1 below.



        [EQUATION "gamma super 2 sub [xy] (f)", Position:In
Text]
0.3    0.4    0.5    0.6    0.7    0.8    0.9

        [EQUATION "epsilon #3 #[ #3 | G sub [xy] (f) | #3 #] #3 = #3 0.1",
Position:In Text]
334    250    200    167    143    125    112

        [EQUATION "s. #3 d. #3 #[ #3 theta sub [xy] (f) #3 #] #3 ~~ #3 3
degree",Position:In Text]
426    274    183    122     79     46     21

        [EQUATION "epsilon #3 #[ #3 gamma super 2 sub [xy] (f) #3 #] #3 = #3
0.1",Position:In Text]
327    180    100     54     26     10     2

                Table 4.1  Number of records for specified error

4.4 Amplitude Spectral Density

     The amplitude spectrum option was added to allow the user to examine the
Fourier coefficients directly.  The amplitude spectrum is defined as


[EQUATION "center F sub x (f) #3#3 = #3#3 lim below [T -> infinity] #3#3 E #3
#[



where  [EQUATION "| X (f,T) |", Position:In Text]  is the modulus of the
quantity calculated in equation 4.6.  If the input data file contains two
channels of data, denoted as  [EQUATION "x(t)", Position:In Text]  and
[EQUATION "y(t)", Position:In Text] , then selection of this option results in
the computation of  [EQUATION "F sub x (f)", Position:In Text]  and
[EQUATION "F sub y (f)", Position:In Text] .

     A simple example taken from Stearns (1975) is used to demonstrate the
utility of this option.  Consider the following function which was
computer-generated and then saved as a set of digital values to simulate the
sampling process:


[EQUATION "center x(t) #3#3 = #3#3 e super [-t] #3 sin #3 t right #R(4.29)"]


A plot of a portion of this function is shown in figure 4.5.  The Fourier
transform of this function can be derived using the definition in equation 4.1
as


[EQUATION "center X(f) #3#3 = #3#3 1 over [[(i #3 2 pi f #3 + #3 1)] super [#3
2



and the modulus of this quantity becomes


[EQUATION "center |X(f)| #3#3 = #3#3 1 over [ #[ #3 [(2 pi f)] super [#3 4] #3
+

[PAGE BREAK]

                  [WHITESPACE 0.25 in, Place All on Next Page]

                        [PICTURE FIG45.CGM, Scale:0.800]

                   Figure 4.5  Plot of example input function

                  [WHITESPACE 0.25 in, Place All on Next Page]

                        [PICTURE FIG46.CGM, Scale:0.800]

         Figure 4.6  Plot of amplitude spectrum calculation[PAGE BREAK]

This input function is deterministic, and therefore only one record of data
would normally be required; however, because an even number of records are
required by the spectrum routine for one channel of input data, two identical
records of this function were generated.
[EQUATION "N #3 = #3 2048", Position:In Text]  samples per record were created
with a time interval of  [EQUATION "Delta t #3 = #3 0.01", Position:In Text]
seconds between successive samples.  A portion of the results is shown in
figure 4.6 with the symbols representing the values computed by the spectrum
routine and the continuous line the values generated by equation 4.31.
[PAGE BREAK]

                    ^SU' Output Data Files

     For every input data file processed, the spectrum routine creates one (or
sometimes two) ASCII output files containing the results.  A second output file
is created only if the user processes two channels of input data and computes
the cross-spectral density and the coherence function; the second output file
then contains error estimates of cross-spectral magnitude and phase and of the
coherence function.

5.1 Naming Convention

     The output file name will consist of nine characters.  The last four
characters specify the file extension; the default file extension, [.PRN], will
be used unless specified otherwise in the configuration file.  The first four
characters of the output file name will match the first four characters of the
corresponding input data file.  The fifth character is used to designate the
type of file that is being written.  The four possibilities are:

                     power spectrum                 P

                     cross spectrum                 C

                     amplitude spectrum             A

                     error estimates (if computed)  E

The output file will be written to the same location as the input data file;
this will, by default, be the location where the executable version of the
spectrum routine resides.  This location can be changed, however, in the
configuration file.

5.2 File Structure

     These output data files consist of ASCII data as opposed to binary data.
They may be easily edited by any ASCII editor, imported into spreadsheet
programs for further manipulation or imported into plotting programs to be
graphed.  These files contain numbers only; no labels are included to describe
the various columns of data in the output.  While somewhat inconvenient for the
user, this was done to prevent the confusion that sometimes occurs when data
containing both numbers and labels are imported into other software programs.
In the sections that follow, a thorough description of the output to be
expected for each of the four types of output files is given, and this should
allow the user to interpret the results without any difficulty.

     5.2.1 Power Spectrum Output Files

     The first line of the output contains the mean and rms values computed as
described in equations 4.12 and 4.13 and then ensemble-averaged over all
records of the file.  Note that the mean and rms values reported here are
calculated subsequent to any cosmetic operations that may have been applied to
the data to improve the quality of the spectral density computation.  Removal
of the mean value, detrending, filtering and tapering operations may modify the
mean and rms so that they do not match the original values for the file which
were calculated and reported at run-time prior to the application of these
operations.  Continuing, the next line of the output file is blank.  The
calculated results begin on the third line and consist of four or six columns
of data depending on whether the input file contains one or two channels.  The
quantities are stored in the following order:


     One channel:

[EQUATION "center #^ #R[mean] #4#4 #R[rms] #4#4#4#4#4#4#4#4#4#4#4#4#4 end
center



     Two channels:

[EQUATION "center #^ #R[mean] #4#4 #R[rms] #4#4#4#4#4#4#4#4#4#4#4#4#4 end
center



In either case the output file contains
[EQUATION "N #1 / #1 2", Position:In Text]  rows of data after the blank line
with each row calculated for values of  [EQUATION "f", Position:In Text]
ranging from  [EQUATION "1 #1 / #1 N Delta t", Position:In Text]  to
[EQUATION "1 #1 / #1 2 Delta t", Position:In Text]  with a resolution of
[EQUATION "Delta f #3 = #3 1 #1 / #1 N Delta t", Position:In Text] .  This
structure is typical for all of the output files created by the spectrum
routine, and in the following descriptions of the output only the column labels
will be identified.

     5.2.2 Cross Spectrum Output Files

     A cross spectrum may only be calculated when the input file contains two
channels of data.  However, the output file is somewhat different depending
upon whether or not the coherence function is calculated.  In either case, the
file begins with the value of the cross correlation function evaluated at
[EQUATION "tau #3 = #3 0", Position:In Text]  and calculated according to
equation 4.18.  The next line is skipped, and the following record is the first
of  [EQUATION "N #1 / #1 2 ", Position:In Text]  rows of data arranged in five
or seven columns as described below.


     Coherence function not calculated:


[EQUATION "center #^ R sub [xy] (0) #4#4#4#4#4#4#4#4#4#4#4#4#4#4#4#4 end center



     Coherence function calculated:


[EQUATION "center #^ R sub [xy] (0) #4#4#4#4#4#4#4#4#4#4#4#4#4#4#4#4 end center



     5.2.3 Amplitude Spectrum Output Files

     When the amplitude spectrum option is chosen, the structure of the output
file is slightly different than for the other files.  No mean or rms value is
reported.  Instead, the file contains
[EQUATION "N #1 / #1 2 #3 + #3 1", Position:In Text]  rows of data arranged in
the columns shown below.  The first row consists of data computed for
[EQUATION "f #3 = #3 0", Position:In Text] ; data for this value of
[EQUATION "f", Position:In Text]  is not stored when other options are
selected, but is useful when examining the Fourier coefficients.  Actually, the
value  [EQUATION "f #3 = #3 epsilon", Position:In Text]  is used; this is
necessary since  [EQUATION "log #1 (0)", Position:In Text]  is undefined.  The
value of  [EQUATION "epsilon", Position:In Text] is defined in a PARAMETER
statement at the beginning of the code and is currently set at
[EQUATION "epsilon #3 = #3 1.0 #3 x #3 10 super [-20]", Position:In Text] .


     One channel:

[EQUATION "center matrix [[ [#^ f] [log #3 f] [F sub [x] (f)] #4#4#4#4#4#4 ] [
[

[PAGE BREAK]

     Two channels:

[EQUATION "center matrix [[ [#^ f] [log #3 f] [F sub [x] (f)] [F sub [y] (f)]
#4



     5.2.4 Error Estimate Output Files

     When both the cross-spectral density and the coherence function options
are selected, the spectrum routine calculates the errors associated with these
estimates as a function of frequency using equations 4.25 through 4.27.  These
error data are then stored in a separate file from the other results.  This
file contains  [EQUATION "N #1 / #1 2", Position:In Text]  rows of data
arranged in the columns identified below.  The columns containing normalized
random errors consist of dimensionless numbers, and the standard deviation of
the phase angle is expressed in degrees.


[EQUATION "center matrix [[ [#^ f] [log #3 f] [ epsilon #3 #[ #3 | G sub [xy]
(f



                 ^Sm" Run-time Considerations

     Two topics are discussed in this section which assume importance when the
program is executed.  The creation of a configuration file allows the user to
bend some of the rules regarding file name creation and the location of data
files as well as temporary files created by the spectrum routine.  The second
topic concerns the manner in which the size of the temporary files created by
the routine may be determined.

6.1 Configuration File

     The presence of an ASCII file with the name SPECTRUM.CFG in the same
directory as the SPECTRUM.EXE file allows optional settings to take effect
which may have a favorable impact on the execution speed and overall ease of
use of the routine.  This ASCII file allows the user to specify three
parameters which will become default values when the program is executed.

1.      The first item is the drive and path information for the location of
        input, output and coefficient data files.  If the user wishes to
        specify that these files may be found in (and written to) the same
        location as the SPECTRUM.EXE file, then the word ROOT should be entered
        on a line by itself.  Otherwise, the desired drive and path information
        should be entered in the following format:  C:\SPECTRUM\DATA\
        Note that the last backslash is required.

2.      The second item in the file is the drive and path information which
        specifies where temporary data files created by the spectrum routine
        will be stored.  If a RAM disk is present on the user's computer and is
        large enough to hold the temporary files, then the execution speed of
        the routine will be greatly enhanced if these files are located on this
        RAM disk.  A means for estimating the size of these temporary files is
        discussed below.  Again, the user specifies ROOT or the desired drive
        and path information using a format such as:  D:\TEMP\

3.      The last item allows the user to change the file extension for output
        data files from its default value of [.PRN] to some other desired
        value.  The user should place the extension on a line by itself in the
        following format:  .OUT
        Note that the leading period is required.

Comment lines may be included in the configuration file; these lines must start
with an asterisk character.  The routine is not case-sensitive -- the options
may be specified in upper or lower case.  The spectrum routine does not require
the presence of the configuration file; if this file is absent, then the
following default values are used:  1. root  2. root  3. [.prn]  If a
configuration file is used, then it must reside in the same directory as the
SPECTRUM.EXE file and all three options must be specified.  An example
configuration file follows.[PAGE BREAK]

*    This is a sample configuration file for Spectrum.  Comment lines
*    and blank lines must begin with an asterisk.
*
*    The first item in the file is the path where the input data
*    files are found and to which the output data will be written.
*    Acceptable values are: root, c:\ , g:\ , c:\test\bin\ , etc.  The
*    path must end with a backslash if root is not used.
root
*    The second item in the file is the drive on which the temporary
*    files are to be created.  Program execution speed will increase
*    dramatically if this is a RAM disk.
*    Acceptable values are: root, c:\ , g:\ , c:\test\bin\ , etc.  The
*    path must end with a backslash if root is not used.
d:\
*    The third item in the file is the default file extension to use
*    for the output data files.  If not specified in this file, then
*    the .PRN extension will be used in order that the data may be
*    imported into LOTUS for plotting.  The extension must consist of
*    a period followed by three letters.
.prn

6.2 Temporary Files

     The input data files that are to be processed by the spectrum routine are
typically very large; therefore, the data is analyzed one record at a time.
Furthermore, the final result of the calculation is not arrived at in one pass;
instead, intermediate results are generated and stored in temporary files which
are then used to obtain the final results.  These temporary files are
automatically deleted upon completion; however, sufficient space must be
reserved on the hard disk or on a RAM disk for temporary use by these files.
The following discussion explains how these files are created and allows the
user to estimate their size based upon the size of the input data file.  Note
that if several input files are to be processed by the routine, only the
temporary files for the input data file currently being processed reside on the
hard disk at any one time; the user needs to allow sufficient space based upon
the size of the largest input data file.

     Two temporary files are created for each channel of input data.  After a
record of the input data file is read in by the routine, the integer data is
sorted into one or two arrays depending on the number of input channels; each
channel is then transformed into floating point data (if necessary), the mean
value or a linear trend is removed and any desired filtering takes place.
These data are then stored to temporary files with the extensions : [.TRF] -
channel 1 and [.TR2] - channel 2 (if required).  This process is repeated until
all records of the input data file are read.  If the input data file consisted
of 2-byte integers, then the [.TRF] file will be twice the size of the input
[.DAT] file since the [.TRF] file contains 4-byte floating point numbers.  If
the input data file consists of two channels of integer data, then the [.TRF]
and [.TR2] files will each be the same size as the [.DAT] file (the [.TRF] and
[.TR2] files each contain half as many numbers as the input [.DAT] file, but
each number is a 4-byte value).  If the input data file consists of one channel
of 4-byte real numbers, the [.TRF] file will be the same size as the [.DAT]
file, and finally, for two channels of 4-byte values in the input file, the
[.TRF] and [.TR2] files will each be half the size of the [.DAT] file.  After
creating the [.TRF] and [.TR2] (if required) files, the spectrum routine
proceeds to the tapering operation.  If tapering is selected, the routine reads
a record of data at a time from the [.TRF] and [.TR2] files, creates two new
records for each channel (if overlapping is used) and stores the results in two
temporary files with the extensions [.TAP] and [.TP2].  The last step in the
process is to transform the data stored in the [.TAP] and [.TP2] files and
perform the required spectral calculation.  The final data is written to one
output file regardless of the number of channels of input data and the
extension [.PRN], or alternatively an extension specified in the configuration
file is used.  If tapering is not selected, then the [.TAP] and [.TP2] files
are not created, and storage space does not need to be allocated. If the [.TAP]
and [.TP2] files are created, each will be twice the size of the previous
[.TRF] or [.TR2] file, since two tapered records are written for every input
record read to implement the 50% overlap technique.  (If tapering is employed
but overlapping is not, then the [.TAP] and [.TP2] files will be the same size
as the [.TRF] or [.TR2] file.  The table belows gives the worst case scenario
and assumes overlapping is used when tapering is used.)  At the end of the
calculation, only the original [.DAT] file and the final [.PRN] file will
remain.  If no alternative is specified in the configuration file, the
temporary files will be created in the same directory as the input data files.
Several examples follow which show the sizes of the various files based upon an
input data file size of 1 Mb.




      Two Channels ?      No        Yes         No        Yes         No
        Integers ?       Yes        Yes         No         No        Yes
        Tapering ?       Yes        Yes        Yes        Yes         No

          [.DAT]         1 Mb       1 Mb       1 Mb       1 Mb       1 Mb
          [.TRF]         2 Mb       1 Mb       1 Mb      500 kb      2 Mb
          [.TR2]          --        1 Mb        --       500 kb       --
          [.TAP]         4 Mb       2 Mb       2 Mb       1 Mb        --
          [.TP2]          --        2 Mb        --        1 Mb        --
          [.PRN]        64 kb      93 kb      48 kb      83 kb      64 kb

        Total Temp
       space needed      6 Mb       6 Mb       3 Mb       3 Mb       2 Mb



                 Table 4.2  Memory required for temporary files

                        ^So' Examples

     Three simple examples are considered that serve to illustrate various
features of the spectrum routine.  The example using colored noise demonstrates
the use of tapering and allows the calculated power spectrum to be compared
with a known power spectrum.  The computation of the amplitude spectrum of a
sinewave reveals the importance that sampling parameters can have on the
computed result, and the last example shows the effects of the application of a
variety of digital filters prior to the computation of power spectra.  A few
comments regarding the considerations necessary for the acquisition of
experimental data files are given at the end.

7.1 Colored Noise

     A routine to produce bandwidth-limited white noise (colored noise) has
been written and used extensively during the testing phase of the development
of the spectrum routine.  This program prompts the user for a few parameters,
then generates the colored noise sequence and stores the output data in a
manner that simulates the sampling process.  This data may then be input into
the spectrum program, and the output power spectrum estimate may be compared
with the known power spectrum for colored noise.  This handy routine is a
useful utility program and has been included on the diskette with the name
COLOR.

     The procedure for producing colored noise begins by generating random
numbers that vary from 0 to 1.  This sequence is uniformly distributed over the
interval from 0 to 1.  Let  [EQUATION "x sub n", Position:In Text]  be the nth
random number, then  [EQUATION "x sub n #3 - #3 1 #1 / #1 2", Position:In Text]
  will be uniformly distributed over the interval from
[EQUATION "- 1 #1 / #1 2", Position:In Text]  to
[EQUATION "1 #1 / #1 2", Position:In Text] .  Now, recall that the mean and
variance of a uniformly distributed random variable is given by

[EQUATION "center mu sub x #3#3 = #3#3 [a #3 + #3 b] over 2 #4#4 #R[and] #4#4
si



where  [EQUATION "a", Position:In Text]  and  [EQUATION "b", Position:In Text]
are the endpoints of the interval.  Using these definitions, we find that the
mean and variance of the sequence
[EQUATION "x sub n #3 - #3 1 #1 / #1 2", Position:In Text]  is


[EQUATION "center mu sub x #3#3 = #3#3 0 #4#4 #R[and] #4#4 sigma sub x super 2
#



If a power spectrum for this sequence were calculated, then one would obtain
the mean value,  [EQUATION "mu sub x", Position:In Text] , the mean-square
value,  [EQUATION "psi sub x super 2", Position:In Text]  and the function
[EQUATION "G sub [xx] (f)", Position:In Text] .  Note that
[EQUATION "psi sub x super 2 #3 = #3 sigma sub x super 2 #3 + #3 mu sub x super
 , so that for this particular sequence


[EQUATION "center psi sub x super 2 #3#3 = #3#3 sigma sub x super 2 #3#3 = #3#3



If these data are considered to be samples of a continuous signal
[EQUATION "x(t)", Position:In Text]  with sampling interval
[EQUATION "Delta t", Position:In Text] , then the power spectrum should look
like that shown in figure 7.1.


                        [PICTURE FIG71.CGM, Scale:0.500]

                  Figure 7.1  White noise spectrum with total power P


For convenience, it is useful to adjust the total power in the sequence in
order that the power density
[EQUATION "G sub [xx] (f) #3 = #3 1", Position:In Text]  for
[EQUATION "0 #3 <= #3 f #3 <= #3 f sub c", Position:In Text]  where
[EQUATION "f sub c #3 = #3 1 #1 / #1 2 Delta t", Position:In Text]  is the
Nyquist frequency.  To accomplish this, consider the sequence

[EQUATION "center y sub n #3#3 = #3#3 K #3 (x sub n #3#3 - #3#3 1 over 2) right



where  [EQUATION "x sub n", Position:In Text]  is a random number on the
interval from 0 to 1 as before and  [EQUATION "y sub n", Position:In Text]
ranges from  [EQUATION "- K #1 / #1 2", Position:In Text]  to
[EQUATION "K #1 / #1 2", Position:In Text] .  For this sequence, the mean and
variance are


[EQUATION "center mu sub x #3#3 = #3#3 0 #4#4 #R[and] #4#4 sigma sub x super 2
#



Now, choose  [EQUATION "P #3 = #3 1 #1 / #1 2 Delta t", Position:In Text], so
that


[EQUATION "center psi sub x super 2 #3#3 = #3#3 sigma sub x super 2 #3#3 = #3#3



From this, we find that
[EQUATION "K #3 = #3 [(6 over [Delta t]) super [1/2]]", Position:In Text] , and
for this value of  [EQUATION "K", Position:In Text] , the power density for the
sequence defined in equation 7.4 satisfies


[EQUATION "center G sub [yy] (f) #3#3 = #3#3 #( { stack left [[ 1 #4#4 0 <= f
<=


A plot of this power spectrum is shown in figure 7.2.


                        [PICTURE FIG72.CGM, Scale:0.500]

                   Figure 7.2  White noise spectrum with total power
               [EQUATION "1 #1 / #1 2 Delta t", Position:In Text]


The final step is to create a new sequence
[EQUATION "z sub n", Position:In Text]  by passing the values
[EQUATION "y sub n", Position:In Text]  through a bandpass filter with transfer
function  [EQUATION "H(f)", Position:In Text] .  The power density for this new
sequence is given by


[EQUATION "center G sub [zz] (f) #3#3 = #3#3 | #1 H (f) #1 | super 2 #3#3 G sub



A recursive digital bandpass filter with
[EQUATION "| #1 H(f) #1 | #3 = #3 1", Position:In Text]  and described in
equation 3.15 has been used in the COLOR routine.  The user specifies the two
(-3 dB) points,  [EQUATION "f sub 1", Position:In Text]  and
[EQUATION "f sub 2", Position:In Text] , and the number of cascaded filter
stages to use.  The order of the filter is twice the number of cascaded
stages.  The output of the program is a sequence
[EQUATION "z sub n", Position:In Text]  with a power spectrum as shown in
figure 7.3.


                        [PICTURE FIG73.CGM, Scale:0.500]

                  Figure 7.3  Colored noise spectrum with total power
            [EQUATION "(f sub 2 #3 - #3 f sub 1)", Position:In Text]


     The color routine was used to create the sequence
[EQUATION "z sub n", Position:In Text]  (simulated sampling of the continuous
signal  [EQUATION "z(t)", Position:In Text] ) which was stored in a data file
that could be read by the spectrum routine.  The file contained 100 records
with 2048 data points per record.  The spacing between data points was 0.01
seconds.  The parameters for the bandpass filter were:
[EQUATION "f sub 1 #3 = #3 20 #3 Hz", Position:In Text] ,
[EQUATION "f sub 2 #3 = #3 30 #3 Hz", Position:In Text]  and five cascaded
stages for a tenth order filter.  A plot of a portion of the simulated time
series appears in figure 7.4.  The data file was processed by the spectrum
routine several times using various choices of tapering functions, and the
power spectra are plotted in figure 7.5.  When computing these spectra, no mean
removal or detrending was used, and no further filtering was performed.  As can
be seen,  [EQUATION "G sub [zz] (f) #3 = #3 1", Position:In Text]  in the
passband and rolls off to zero very rapidly elsewhere.  (Note that the base 10
logarithm of these numbers are plotted in order to magnify the rolloff
region.)  This plot clearly shows the advantages gained with the use of
tapering and overlapped processing as opposed to no tapering; however, in this
case, the choice of tapering function makes little difference.  Table 7.1 shows
the rms values reported for each case; all of these are within 0.6 % of the
theoretical value, and the minor variations are due to the fact that the
rolloff is not perfect.


                        [PICTURE FIG74.CGM, Scale:0.800]

          Figure 7.4  Simulated time series  [EQUATION "z(t)", Position:In
Text]




                   Theory    No Taper   Hanning     Parzen     Tukey      Welch

   rms values     3.16228    3.14463    3.14665    3.14617    3.14315
3.14429

    error (%)        --        0.56       0.49       0.51       0.60       0.57



               Table 7.1  RMS values reported by spectrum routine

            [PAGE BREAK][WHITESPACE 0.25 in, Place All on Next Page]

                        [PICTURE FIG75.CGM, Scale:0.800]

               Figure 7.5  Power spectra for colored noise; total power =
                    [EQUATION "root[10]", Position:In Text]



                        [PICTURE FIG76.CGM, Scale:0.800]

                Figure 7.6  Effect of detrend on power spectra[PAGE BREAK]

For data generated by the computer, no linear trend in the data is to be
expected, and the detrending operation should not be used.  The indiscriminate
use of this operation can lead to added power at
[EQUATION "f #3 = #3 1 #1 / #1 N Delta t", Position:In Text]  since this
operation is applied record by record; this can only occur if the power at this
frequency is lower than the small amount of noise generated by the use of this
operation.  In almost every case where the random data originates from an
experiment (vis-a-vis computer generated perfect random data) this problem will
not occur, but the user should be aware of the possibility.  A plot of the
added power that occurs with the use of the detrending operation is shown in
figure 7.6.

     The following is an example session showing the various prompts and
responses that were used with the color routine for this example.


color

Enter the low frequency cutoff : 20.0
                                                      (Defines passband)
Enter the high frequency cutoff : 30.0

Enter delta T seconds. (1.0/Sample Rate) : 0.01       (Sampling rate = 100 Hz)

Enter # of filter sections to cascade : 5             (Tenth order filter)

Enter # of points per record (N) : 2048               (Must be power of two)

Enter # of records (NUMREC) : 100                     (No limit on # records)

Enter a negative integer : -5                          (Required seed value
for)
                                                      (random number generator)
Enter output file name (4 chars) : COLR

              C O L O R E D   N O I S E   U T I L I T Y

                        Creating COLR.DAT now.

                        Writing record 0100

              Use voltage transformation factor = 20.0 / 4096.0

                   Program terminated successfully.


     Next is an example session showing the prompts and responses needed to
process the COLR.DAT file using the spectrum routine.


spectrum colr

Transform into (V)oltages or into (O)ther quantity
or read (R)eal data which is already transformed : v

Enter VOFST 1) 10/4096, 2) 20/4096, 3) 2/4096 or 4) other : 2

Remove (M)ean value, (D)etrend or (N)either : n

Filter the data using (L)owpass, (H)ighpass
                   (B)andpass, Band(S)top or (N)one : n

How many channels of data (1 or 2) : 1

(A)mplitude, (P)ower or (C)ross spectrum : p

Taper the data (Y/N) : y

(H)anning, (P)arzen, (T)ukey or (W)elch window : h

Use overlap (Y/N) : y

                        Processing COLR.DAT now.

               Record 0100  Ave = .15974E-03   Rms = 3.1035

               File Totals : Ave = .26703E-05  Rms = 3.1446

                        Taper applied = Hanning.
                        50% overlap used.
                        0200 records will be written.

                        Tapering record 0200.

                        Transforming records 0199 and 0200.

                        Record 0200 was discarded.

                        Writing COLRP.PRN now.

                             mean         rms
                         2.18332E-06     3.1467

                        norm. random error = .10000

                        starting time :  09:15:01.71
                        ending time   :  09:16:06.68
                        elapsed time  :  00:01:04.97

                        Program terminated successfully.


     Note that the maximum number of points per record is controlled by the
value of NMAX.  The value of NMAX is currently set to 16384.  (If the user
should ever desire to make a change to the value of NMAX, the change must be
made both in the COLOR routine and also in the SPECTRUM routine.  NMAX must be
the same number in both routines at all times.)

7.2 Sinusoidal Input

     When developing the spectrum routine, it was felt that a simple test would
be to attempt to determine the Fourier coefficient (amplitude) of a sinewave by
computing its amplitude spectrum.  Although this can be done, it is not a
trivial procedure and is therefore repeated here as an example.  A sinewave
generation routine was written and is included on the diskette as SINE; this
routine creates a data file that can be read by the spectrum routine.  Each
record of this data file contains a portion of the time series


[EQUATION "center x(t,k) #3#3 = #3#3 A #3 sin #3 #[ 2 pi f sub 0 #1 t #3#3 +
#3#



where  [EQUATION "A", Position:In Text]  is the amplitude,
[EQUATION "f sub 0", Position:In Text]  is the frequency and
[EQUATION "theta sub k", Position:In Text]  is a phase angle which changes from
record to record and is a function therefore of the record number
[EQUATION "k", Position:In Text] .  The presence of a changing phase angle from
record to record is merely to simulate sampling of noncontiguous chunks of the
continuous function.  This phase angle will not alter the value of the
amplitude spectrum and is therefore left out of the following derivation for
simplicity.

     The finite-range Fourier transform defined in equation 4.2 can be computed
for this function  [EQUATION "x(t)", Position:In Text]  and is given by


[EQUATION "center X(f,T) #3#3 = #3#3 [AT] over [2i] #3#3 [#[ #3 [sin #3 pi #3
(f



This function must now be evaluated at
[EQUATION "f #3 = #3 f sub 0", Position:In Text] , so taking the limit gives


[EQUATION "center lim below [f -> f sub 0] #3#3 X(f,T) #3#3 = #3#3 [AT] over
[2i



where the first term was evaluated using l'Hôpital's rule.  The modulus of this
quantity is a function of the record length  [EQUATION "T", Position:In Text] ,
and is given by


[EQUATION "center | #3 lim below [f -> f sub 0] #3#3 X(f,T) #3 | #3#3 = #3#3
[AT



To further examine this dependence upon the record length, one can express
[EQUATION "T", Position:In Text]  in terms of  [EQUATION "n", Position:In Text]
 , the number of cycles of the sinewave, using


[EQUATION "center T #3#3 = #3#3 N Delta t #3#3 = #3#3 n over [f sub 0] right
#R(



With this value for the record length, equation 7.12 becomes


[EQUATION "center | #3 lim below [f -> f sub 0] #3#3 X(f,T) #3 | #3#3 = #3#3
[AT



The function  [EQUATION "f #1 (#1 n)", Position:In Text]  is plotted in figure
7.7 and is shown to oscillate about the value one.  The extent of the
oscillation decreases with increasing  [EQUATION "n", Position:In Text] , and
in the limit as the record length goes to infinity, the function equals one.


                        [PICTURE FIG77.CGM, Scale:0.800]

                            Figure 7.7  Oscillatory function
                   [EQUATION "f #1 (#1 n)", Position:In Text]


Due to the finite nature of the signal, one can obtain the limiting value for
the Fourier coefficient only by paying careful attention to the choice of
record length.  Specifically, if the record length is chosen to contain an
integral number of cycles of the sinewave, then as can be seen from
figure 7.7,  [EQUATION "f #1 (#1 n) #3 = #3 1", Position:In Text]  for integer
[EQUATION "n", Position:In Text] .  Summarizing then, the limiting value of the
amplitude spectrum of the sinewave can be computed if the record length is
chosen to contain an integral number of cycles of the sinewave; when this
criterion is satisfied, then the modulus is given by


[EQUATION "center | #3 lim below [f -> f sub 0] #3#3 X(f,T) #3 | #3#3 = #3#3
[AT



Now, using the definition given in equation 4.28, the value of the amplitude
spectrum reported by the spectrum routine at the frequency
[EQUATION "f sub 0", Position:In Text]  will be the ensemble average over all
records of the data file of the modulus defined in equation 7.15.  Three
further constraints are placed upon the record length in addition to the
requirement of integral values of  [EQUATION "n", Position:In Text] :
[EQUATION "N", Position:In Text] must be a power of two,
[EQUATION "N", Position:In Text]  should be large for better accuracy and
[EQUATION "Delta t #3 <= #3 1 #1 / #1 [2 f sub 0]", Position:In Text]  which is
a statement of the fact that the frequency of the sinewave must be less than or
equal to the Nyquist frequency determined by the sampling interval.  These
requirements are satisfied if
[EQUATION "Delta t #3 = #3 1 #1 / #1 [2 super j #1 f sub 0]", Position:In Text]
 ,  [EQUATION "N #3 = #3 2 super k", Position:In Text]  and
[EQUATION "k >> j", Position:In Text]  for integer values of
[EQUATION "j", Position:In Text]  and  [EQUATION "k", Position:In Text] ; for
these choices  [EQUATION "n #3 = #3 2 super [#3 k-j]", Position:In Text] .
Generally, the best accuracy is obtained when
[EQUATION "n #3 = #3 N #1 / #1 4", Position:In Text]  (
[EQUATION "j #3 = #3 2", Position:In Text] ).

     As an example of this technique, the sinewave


[EQUATION "center x(t,k) #3#3 = #3#3 1.25 #3#3 sin #3#3 #[ 2 pi #3 (128.0) #3 t



was created using the SINE routine.  The file contained ten records with 8192
points per record.  The sampling interval was chosen to be (using
[EQUATION "n #3 = #3 N #1 / #1 4", Position:In Text] )


[EQUATION "center Delta t #3#3 = #3#3 1 over [(4)(128)] #3#3 = #3#3 0.001953125



so that the record contained exactly 2048 cycles of the sinewave.  The spectrum
should have a peak at a frequency of 128 Hz where the amplitude is given by


[EQUATION "center F sub x (128) #3#3 = #3#3 lim below [T -> infinity] #3#3 E #3



The actual value reported by the spectrum routine was 9.9656 at a frequency of
128.0082 Hz which was the closest discrete frequency value.  A plot of the
amplitude spectrum is shown in figure 7.8.  When computing this spectrum, no
mean removal or detrending was used, and no filtering was performed.
Furthermore, application of a tapering function would significantly change the
expected amplitude spectrum and therefore no tapering was used.


                        [PICTURE FIG78.CGM, Scale:0.800]

                        Figure 7.8  Amplitude spectrum of sinewave


The required inputs to the SINE routine follow.


sine

Enter the amplitude of the sinewave : 1.25            (gives a simple answer)

Enter the frequency of the sinewave : 128.0           (can be any desired
number)

Enter the phase (in degrees) : 32.8                   (choose any initial
value)

Enter DC value : 0.0                                   (no DC for this example)

Enter delta T secs. (1.0/Sample Rate) : 0.001953125   (according to eq. 7.17)

Enter # of points per record (N) : 8192               (must be power of 2)

Enter # of records (ASIZE) : 10                       (OK for this example)

Enter output file name (4 chars) : SINE

              S I N E W A V E   C R E A T I O N   U T I L I T Y

                        Creating SINE.DAT now.

                        Writing record 0010

              Use voltage transformation factor = 20.0 / 4096.0

                   Program terminated successfully.


The required inputs to the spectrum routine to create the amplitude spectrum
given in figure 7.8 follow.


spectrum sine

Transform into (V)oltages or into (O)ther quantity
or read (R)eal data which is already transformed : v

Enter VOFST 1) 10/4096, 2) 20/4096, 3) 2/4096 or 4) other : 2

Remove (M)ean value, (D)etrend or (N)either : n

Filter the data using (L)owpass, (H)ighpass
                   (B)andpass, Band(S)top or (N)one : n

How many channels of data (1 or 2) : 1

(A)mplitude, (P)ower or (C)ross spectrum : a

Taper the data (Y/N) : n

                        Processing SINE.DAT now.

                   Record 0010 : Ave = .00000  Rms = .88208

                   File Totals   : Ave = .00000  Rms = .88090

                        No tapering applied.
                        0010 records will be used.

                        Transforming records 0009 and 0010.

                        Writing SINEA.PRN now.

                        starting time :  20:24:53.77
                        ending time   :  20:25:13.21
                        elapsed time  :  00:00:19.44

                        Program terminated successfully.


     Note that the maximum number of points per record is controlled by the
value of NMAX.  The value of NMAX is currently set to 16384.  (If the user
should ever desire to make a change to the value of NMAX, the change must be
made both in the SINE routine and also in the SPECTRUM routine.  NMAX must be
the same number in both routines at all times.)[PAGE BREAK]

7.3 Experimental Noise Input

     The input data file for this example was derived from an experiment
employing an electrodynamic shaker table.  This device is essentially a large
solenoid mated to an electronic feedback system which is used to accurately
control the oscillatory motion of the core.  The displacement amplitude and
frequency can be set to desired levels and then maintained very accurately.
This instrument was used in an experiment designed to test the frequency
response of an optical system which was built to measure displacement of a
surface at a point.  Prior to the actual frequency response test, the optical
system was used to measure the displacement of the core of the solenoid with
the shaker table energized but undergoing no motion.  The computation of a
power spectrum of this displacement noise can then be a useful diagnostic tool
for the identification of noise sources in subsequent displacement spectrum
measurements.

     The signal was digitized using a 12-bit A/D converter with an input range
of -10 to 10 V.  The sampling rate was 2 kHz which resulted in a sampling
interval of  [EQUATION "Delta t #3 = #3 0.0005", Position:In Text]  seconds.
80 records of data were acquired with 2048 points per record.  The power
spectrum was computed for a range of frequencies up to the Nyquist frequency
which was 1 kHz; the resolution was given by
[EQUATION "Delta f #3 = #3 1 #1 / #1 N Delta t #3 = #3 0.97656 #3 Hz",
Position:
 .  All power spectra shown in this section were computed by first detrending,
then tapering the input time records with the Welch taper and employing 50%
overlap.  The resulting power spectrum is shown in figure 7.9; logarithmic
scales are used for greater clarity.  The figure reveals that most of the power
in the signal lies below 100 Hz.  A 60 Hz contribution attributable to the
power supply may be identified as well as other resonances and their harmonics.


                        [PICTURE FIG79.CGM, Scale:0.800]

                      Figure 7.9  Power spectrum of displacement noise


     This input file allows a useful demonstration of the filtering
capabilities of the spectrum routine.  Shown in Figure 7.10 are power spectra
of the same input file for which figure 7.9 was obtained; however, prior to the
spectral calculation, each of the four types of filters available for use were
applied to the data.  In each case five cascaded filter stages were used
providing a tenth order digital Butterworth filter.  As can be seen, the
rolloff is extremely rapid with power in the stopband 3 to 4 decades below that
in the passband.  This is an effective means for removing undesirable frequency
components from the data prior to spectral computation.  This can also be
useful for quickly estimating the amount of power contributed by specific peaks
or regions of the spectrum.  The rms value, calculated as the square root of
the area under the spectral density curve (according to eq. 4.13), is given in
table 7.2 for each of the filter cases shown in figure 7.10 and for the
no-filter case plotted in figure 7.9.



                    No filter   Lowpass    Highpass   Bandpass   Bandstop

      rms values     0.014803   0.013319   0.009529   0.005894   0.010116



      Table 7.2  RMS values reported after application of each filter type


The table indicates that about 10% of the contribution to the root-mean-square
value of the signal may be found above 100 Hz, about a third lies below 30 Hz
and just under half may be found between 50 and 100 Hz.  However, adding the
rms value for the bandpass case to the rms value for the bandstop case does not
equal the rms value for the no filter case.  This discrepancy is due to the
fact that some power is in the stopband for the bandpass and bandstop cases,
and this contributes a small amount to the rms values indicated.  Therefore,
this method should only be used for rough estimates; the most accurate method
is to integrate the spectral density function over frequency bands of interest.
[PAGE BREAK]

                   [WHITESPACE 4 in, Place All on Next Page]

               Substitute Figure 7.10 for this page.[PAGE BREAK]

7.4 Sampling Considerations

     A full discussion of sampling procedures will not be reiterated here; a
complete discussion of these techniques can be found in the references, and the
interested reader is urged to look there.  Instead, this section attempts to
present some of the considerations necessary when making choices among the
various sampling parameters, and the manner in which these choices affect the
spectral density results.

     The most important parameter is probably the choice of sampling interval
[EQUATION "Delta t", Position:In Text] .  In general, one attempts to choose
this quantity to be of the order of the smallest time scale of interest in the
physical phenomenon under investigation.  The selection of
[EQUATION "Delta t", Position:In Text]  determines the largest frequency
(bandwidth) of the resulting spectral analysis.  This critical frequency
[EQUATION "f sub c", Position:In Text]  is known as the Nyquist frequency and
is given by


[EQUATION "center f sub c #3#3 = #3#3 1 over [2 Delta t] right #R(7.19)"]


Another quick way for determining the Nyquist frequency is to note that
[EQUATION "f sub c #3 = #3 f sub s #1 / #1 2", Position:In Text]  where
[EQUATION "f sub s #3 = #3 1 #1 / #1 Delta t", Position:In Text]  is the
sampling rate at which data are acquired.  If energy exists in the signal above
the Nyquist frequency determined by the desired sampling rate, then this energy
must be removed by means of an analog lowpass filter prior to A/D conversion.
This is necessary to prevent aliasing of the high frequency components in the
data.  Aliasing is essentially confusion among the high and low frequency
components in the data that occurs whenever the sampling rate is set too low.
A very good discussion of aliasing may be found in Bendat and Piersol (1986).

     Another parameter of interest is the number of data points per record per
channel  [EQUATION "N", Position:In Text] .  This quantity combined with the
choice of  [EQUATION "Delta t", Position:In Text]  determines the record
length  [EQUATION "T #3 = #3 N Delta t", Position:In Text]  and the resolution
bandwidth of the spectral analysis,


[EQUATION "center Delta f #3#3 = #3#3 1 over [N Delta t] right #R(7.20)"]


A large value of  [EQUATION "N", Position:In Text]  for a given value of
[EQUATION "Delta t", Position:In Text]  gives a smaller
[EQUATION "Delta f", Position:In Text]  and can give a more detailed spectral
density estimate.  On the other hand, if the record length
[EQUATION "T", Position:In Text]  is very long, realizability problems may
occur: is the physical phenomenon stationary for these long time periods?  Is
there a limitation on maximum file size?  Can the resulting massive amounts of
data be adequately stored?

     Another important parameter is the number of records
[EQUATION "n sub r", Position:In Text]  of length
[EQUATION "T", Position:In Text]  to sample for each data file.  These records
should be contiguous in time (in order to use overlapping) and constitute an
ensemble of statistically stationary records; the expected value operation
contained in the definitions of the spectral density estimates is approximated
when an ensemble average is calculated for this set.  Recall, that it is
[EQUATION "n sub r", Position:In Text]  (the number of records before
overlapping) which is used in the determination of the errors associated with
each of the spectral density estimates and not  [EQUATION "N", Position:In
Text]
  or  [EQUATION "Delta t", Position:In Text] .  Increasing
[EQUATION "n sub r", Position:In Text]  rapidly decreases the normalized random
error  [EQUATION "epsilon", Position:In Text]  up to about
[EQUATION "n sub r #3 = #3 100", Position:In Text]  or so; for larger values
of  [EQUATION "n sub r", Position:In Text]  the error decreases much more
slowly.  There may be certain situations in which, for a given overall file
size, it is better to increase  [EQUATION "n sub r", Position:In Text]  and
decrease  [EQUATION "N", Position:In Text]  for a given
[EQUATION "Delta t", Position:In Text]  in order to increase accuracy at the
expense of resolution bandwidth  [EQUATION "Delta f", Position:In Text] .

                    ^S▒" Utility Routines

     Four utility programs are included on the diskette.  The SINE and COLOR
routines have been discussed in the previous section concerning examples.
MAKDAT is a routine designed to read a data file in ASCII format and then to
write the necessary file header and data to an unformatted file.  MAKCOEF is
similar; it reads a user-prepared ASCII format coefficient data file and then
writes the file header and coefficients to an unformatted file.  It is easiest,
of course, to add the necessary code to create the unformatted files directly
to the routine that is used to create (or collect) the data in the first place;
however, if the user is unfamiliar with unformatted FORTRAN data files or the
FORTRAN language in general, then these routines attempt to solve the problem.

8.1 MAKDAT routine

     In order to use MAKDAT, the data to be analyzed by the spectrum routine
must be generated, or collected by an A/D converter, and then stored to a file
in ASCII format.  In this format the file may be viewed by any ASCII editor,
can be displayed on the screen and can be printed on a printer.  This is not an
optimum way to store data; these files tend to be extremely large.  In
anticipation of this fact other types of data files were explored for possible
use, and unformatted FORTRAN files appeared to be the simplest solution.
Unformatted files are typically a factor of five smaller in size than the
equivalent ASCII file.  An example will be given to illustrate the means for
transferring the data to unformatted files which can then be used by SPECTRUM.

     Suppose that 100 records of one channel of data saved as 2-byte integers
and sampled at 10000 samples per second with 2048 data points per record have
been acquired.  These data should be placed in an ASCII file with the file
extension [.ASC] and with a file name conforming to the various naming
conventions discussed earlier.  The file must contain numbers only; the numbers
may be written one per line or many per line (but this does not shorten the
length of the file).  Blank lines may be added if desired.  The first number in
the file should be the first data point of the first record; the 2049th number
in the file should be the first data point of the second record, and the last
number in the file should be the 2048th data point of the 100th record.  A
session with MAKDAT to accomplish this transformation would proceed as follows.


makdat

(I)nteger (2-byte) or (F)loating-point (4-byte) data : i

Enter # of channels (1 or 2) : 1

One channel  : Total # points per record <= 16384.
Two channels : Total # points per record per channel <= 8192.

Enter # of points per record (power of two) : 2048

One channel  : Must be EVEN # of records.
Two channels : May be EVEN or ODD # of records.

Enter # of records in the data file : 100

One chan    : Delta t is spacing between data points.
Two chans : Delta t is spacing between data pts - SAME channel
              Delta t divided by 2 is spacing between data pts
              - different channels.

Enter sampling interval delta t (secs) : 0.0001

Enter ASCII input file name (4 chars) : test

              D A T A   F I L E   C R E A T I O N   U T I L I T Y

                       Creating TEST.DAT now.

                       # channels = 1
                       record size = 4096 bytes
                       # of records = 100
                       delta t = 100 microseconds

                       Record 0100

                       Program terminated successfully.


The MAKDAT routine reads the user-prepared TEST.ASC file and calculates the
values needed for the file header.  The file header would consist of


[EQUATION "center 1 #4#4 4096 #4#4 100 #4#4 100 #4#4"]


The routine then transfers the data to the unformatted file with the name
TEST.DAT.  Finally, to process this file with the spectrum routine the user
would enter: spectrum test.

     For two channels of data, it is necessary that the numbers be stored in
each record of the ASCII file in the following manner:


[EQUATION "center #R[one] #4 #R[channel] #^ #R[:] #4#4 x sub 0 #3 x sub 1 #3 x
s



using the notation defined earlier.  Therefore, the first number in the file
will be the first data point of the first record for channel 1; the second
number will be the first data point of the first record for channel 2; the
third number in the file will be the second number for the first record of
channel 1, and so on.  Also, care should be taken to enter correct responses to
the prompts when two channels of data are to be transferred.  In response to
the number of points per record per channel prompt, the user would enter 1024
points if the data file consisted of two channels of data with 2048 points per
record and therefore 1024 points for each channel in each record.  Note that
the maximum number of points per record is controlled by the value of NMAX; the
maximum number of points per record per channel is then NMAX / 2.  The value of
NMAX is currently set to 16384.  (If the user should ever desire to make a
change to the value of NMAX, the change must be made both in the MAKDAT routine
and also in the SPECTRUM routine.  NMAX must be the same number in both
routines at all times.)  Similarly, when responding to the prompt for the
sampling interval, the user would enter 0.0002 seconds for the sampling
interval between data points of the same channel for the case in which the data
were sampled at 10000 samples per second.  Although, 0.0001 seconds is the
sampling interval between adjacent data points in the record, the sampling
interval for data points of the same channel is a number twice as large.

8.2 MAKCOEF routine

     The primary reason for using unformatted coefficient data files with the
spectrum routine is to allow the user a means for preparing all of the
coefficient data for the transformation of voltages into other dimensional
quantities for several files which will then be processed by SPECTRUM in a
batch.  The MAKCOEF utility allows the user to prepare this coefficient data in
an ASCII file; the utility will write the file header and then transfer the
data to an unformatted coefficient data file which can be read by the spectrum
routine.

     Suppose that it was desired to process the data files D061.DAT thru
D086.DAT in a batch by the spectrum routine.  Each of these data files
consisted of two channels of 2-byte integers taken from an experiment in fluid
mechanics, say.  Furthermore, suppose that the quantities sampled were two
components of a fluid velocity, which were transduced into voltages by some
means and then sampled by an A/D converter.  The user knows the calibration
data for the transducer, and the required transformation from voltages back
into velocities can be accomplished for each velocity component by using a
polynomial of up to fourth degree in the form:


[EQUATION "center u sub n #3#3 = #3#3 b sub 0 #3 + #3 b sub 1 v sub n #3 + #3 b



where  [EQUATION "v sub n", Position:In Text]  is the nth data point expressed
as a voltage,
[EQUATION "b sub 0 #1 , #3 b sub 1 #1 , ... , b sub 4", Position:In Text]  are
the coefficients of the velocity transformation and
[EQUATION "u sub n", Position:In Text]  is the nth data point which has been
converted to a velocity.  Now, suppose further that the data in each of the
files were taken over a period of time in which the calibration coefficients
changed; thus, a different set of coefficients are required for each of the 26
files to be processed.  Following the instructions in the section on
coefficient data files, this scenario would require the creation of two
unformatted coefficient data files, DCON.DAT and DCON2.DAT, each containing the
26 sets of five coefficients
[EQUATION "b sub 0 #1 , #3 b sub 1 #1 , ... , b sub 4", Position:In Text]  for
the velocity transformation for channel 1 and channel 2, respectively.

     The user should create each ASCII data file with a four character name
followed by the extension [.ASC].  The names should conform to the various
naming conventions specified earlier.  The ASCII file must contain numbers
only, although blank lines may be inserted into the file if desired.  The
numbers will be transferred to 4-byte floating point quantities in the
unformatted data file thereby limiting the coefficients to single precision (6
or 7 decimal places).  The coefficients may be stored one per line or many per
line in the ASCII file.  The first number in the file will be
[EQUATION "b sub 0", Position:In Text]  for channel 1 of data file D061.DAT;
the fifth number in the file should be  [EQUATION "b sub 4", Position:In Text]
for channel 1 of data file D061.DAT; the sixth number in the file will be
[EQUATION "b sub 0", Position:In Text]  for channel 1 of data file D062.DAT,
and so on.  The 130th and last number in the file should be
[EQUATION "b sub 4", Position:In Text]  for channel 1 of data file D086.DAT.
The user should then create a second ASCII file in a similar manner specifying
all of the coefficients for channel 2 of the 26 files to be processed.  The
utility MAKCOEF should then be executed twice, first transferring the contents
of the first ASCII file (for channel 1) to DCON.DAT and then for the second
ASCII file (for channel 2) to DCON2.DAT.  A session using MAKCOEF would proceed
as follows.


makcoef

Enter first letter of data file to
which these coefficients will be associated : d

Are these coefficients for channel (1 or 2) : 1

Enter # of sets of coefficients : 26

Enter starting file number : 61

Enter ASCII input file name (4 chars) : chn1

        C O E F F I C I E N T   F I L E   C R E A T I O N   U T I L I T Y

                       Creating DCON.DAT now.

                       # sets of coeffs = 26
                       # coeffs in each set = 5
                       # of starting file = 61

                       Program terminated successfully.


The MAKCOEF routine will read the ASCII file CHN1.ASC and will write the
following file header to the unformatted file DCON.DAT:


[EQUATION "center 26 #4#4 61"]


The coefficients for channel 1 are then transferred to the unformatted file.
Note that 5 coefficients per set are required; therefore, this number is not
written to the file header.  If, at a later time, the user wishes to reprocess
the files D074.DAT through D080.DAT (perhaps specifying different options in
the spectrum routine), it is not necessary to re-create the two coefficient
data files; as long as the data files to be reprocessed are chosen from among
the 26 original members of the set, the spectrum routine will use only the
coefficients that it needs.  Similarly, if the user wishes to process the data
files in an arbitrarily numbered order (not consecutively), the coefficient
data files do not need to be changed as long as the data files come from the 26
original members of the set.  Thus, D061.DAT, D064.DAT, D069.DAT and D086.DAT
could be processed by SPECTRUM without changing the coefficient data files.
The converse is also true, however.  If the user desires at the outset to
process only the four arbitrarily ordered files above, two coefficient data
files must be created which contain not 4, but 26 sets of 5 coefficients for
the data files D061.DAT through D086.DAT.  The spectrum routine requires that
sets of coefficients in the coefficient data file be listed for consecutively
numbered files even if not all of the sets of coefficients are used.  With this
in mind, it is perhaps best to create the coefficient data files for an entire
set of input data files; then members of the set can be processed by the
spectrum routine either consecutively or in an arbitrarily numbered order at
any subsequent time.  Of course, the user could transform the voltages back
into dimensional quantities using his/her own code and store the resulting
floating-point numbers into ASCII data files (then use MAKDAT) or directly into
unformatted data files to be processed by SPECTRUM; then, all of this mess
concerning coefficient data files could be avoided altogether.

     Note that the maximum number of points per record is controlled by the
value of NMAX.  The value of NMAX is currently set to 16384.  The maximum
number of files that may be processed by SPECTRUM in a batch is controlled by
the setting of IFMAX; this is currently set to 100.  (If the user should ever
desire to make a change to the value of NMAX or IFMAX, the change must be made
both in the MAKCOEF routine and also in the SPECTRUM routine.  NMAX and IFMAX
must be the same numbers in both routines at all times.)

                           ^SReferences

[WHITESPACE 0.1 in, Place All on Next Page]

Bendat, J.S. and Piersol, A.G. 1986 Random Data, 2nd ed., John Wiley & Sons,
     Inc., New York.


Hess, D.E. 1990 An experimental investigation of a compliant surface beneath a
     turbulent boundary layer. Ph.D. thesis, Johns Hopkins University.


Microsoft Fortran Optimizing Compiler, Version 5.00, Microsoft Corporation,
     Redmond, Washington, 1989.


Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. 1986
     Numerical Recipes, Cambridge University Press, New York.


Sirivat, A. 1985 Statistical Package for the Analysis of Data (SPAD), private
     communication.


Stearns, S.D. 1975 Digital Signal Analysis, Hayden Book Co., Rochelle Park, New
     Jersey.