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[WS 1 in] SPECTRUM CALCULATION ROUTINE Version 2.0 [WS 4 in] David E. Hess Fluid Flow Group Process Measurements Division Chemical Science and Technology Laboratory National Institute of Standards and Technology Email: HESS@ENH.NIST.GOV Phone: (301) 975-5937 [WS 0.5 in] [April 1992] ▒New Features * Two additional window functions are available for tapering the data. The first, referred to as the Minimum Sidelobe window, has a maximum sidelobe level of -93 dB; the sidelobes then decay at a rate of 18 dB/octave. The second window, the Maximum Decay window, has a maximum sidelobe level of -61 dB, but these sidelobes decay at a rate of 42 dB/octave. * The structure of the data files to be processed has been modified to allow inclusion of information about the voltage transformation factor for each channel of data. The file format has been changed to BINARY in order to permit these data files to be written by programming languages other than FORTRAN. * Memory for data arrays is now allocated dynamically at run-time, and only the memory needed for the current file being processed is allocated at any particular time. Storage for these arrays can be quite large and dynamic allocation of memory reduces the amount of storage required within the SPECTRUM executable file. This modification has reduced the size of the executable file by 75%. * All internal temporary files created by the SPECTRUM routine are now written in BINARY format. This saves an additional small amount of memory at run-time. * When an input data file consisted of only one channel of data to be processed, the first version of SPECTRUM required that this data file have an even number of records. This restriction has been removed. * The manual included with the first version of SPECTRUM provides a detailed account of the manner in which the amount of space on the hard disk (or Ram disk) required by the routine for storage of temporary files may be estimated. Version 2 of SPECTRUM now calculates at run-time the amount of temporary space required to process the current file, and then checks the hard disk (or Ram disk) to determine if that space is available. * The code has been restructured by removing portions of the code from the Main module and placing these segments into functions or subroutines. The subsequent reduction in size of the Main module has permitted additional optimization procedures contained within the Microsoft Fortran Optimizing Compiler, V 5.1 to be employed. Also, the code for the Main module is now more readable. * All of the utility files included on the diskette have been modified to use the new input data file format. Each of these routines should be used only with SPECTRUM V 2.0. * The SPECTRUM routine requires a computer equipped with an 80286 or better microprocessor. An 80287 or better numeric coprocessor is recommended but is no longer required. All information contained within this document supersedes the information contained within the manual which refers to version 1.0 of the SPECTRUM routine. Any information found in the manual which is not countermanded by statements made here is considered to still apply. ▒Diskette Contents The following files are contained on the diskette. SPECTRUM FOR 66165 04-22-92 8:16a Main program source code. SPECTRUM FI 1852 04-22-92 7:59a Interface statements. SPECTRUM FN 2406 04-22-92 8:04a Data declaration & common areas. SPECTRUM CFG 1197 04-23-92 3:50p Optional setup file. SPECTRUM EXE 111592 04-23-92 3:03p Works with or w/o a coprocessor. SPEC387 EXE 103800 04-23-92 2:43p Works only with a coprocessor. COLOR FOR 8314 04-22-92 8:05a Source for noise generator. COLOR EXE 57452 04-23-92 3:07p Executable for noise generator. COLOR2 FOR 9073 04-23-92 3:13p Source for 2-chan noise gen. COLOR2 EXE 57872 04-23-92 3:13p Executable for 2-chan noise gen. SINE FOR 4371 04-22-92 8:06a Source for sinewave generator. SINE EXE 50490 04-23-92 3:09p Executable for sinewave gen. MAKDAT FOR 8960 04-22-92 8:06a Ascii to Binary converter MAKDAT EXE 51564 04-23-92 3:11p source. MAKCOEF FOR 6296 04-22-92 8:07a Asc to Bin converter executable. MAKCOEF EXE 44388 04-23-92 3:11p Coefficient file creation README 1ST 1594 04-28-92 2:54p source. Coeff. file creation executable. Cover letter. The README.1ST file is simply a reminder that this is the second version of SPECTRUM and that an update to the manual exists and should be read prior to the first use of the program. All files with an .FOR, .FI or .FN extension are Ascii files containing FORTRAN source code for the respective program. These files are not required when using SPECTRUM and need not be kept on the hard disk. If the SPECTRUM routine is recompiled, then three files are required during compilation: SPECTRUM.FOR, SPECTRUM.FI and SPECTRUM.FN. The SPECTRUM.FI file contains interface statements for the functions and subroutines used in SPECTRUM, and the SPECTRUM.FN file contains data declaration statements and COMMON statements which permit data areas to be shared by both the Main module and other program modules. These two files are compiled along with the main program file using the INCLUDE statement in appropriate locations in the main program source code. This program was compiled using the Microsoft FORTRAN Optimizing Compiler, V 5.1. The command used to compile the code and to generate the SPECTRUM.EXE file, which may be used with or without a coprocessor, is given by FL /G2 /W2 /FPi SPECTRUM.FOR In addition, an executable version of SPECTRUM has been provided which will work only with a numeric coprocessor installed (80287, 80387 or 80486). This executable version is called SPEC387.EXE and consists of more streamlined code which executes somewhat faster. This version was compiled using the command FL /G2 /W2 /FPi87 SPECTRUM.FOR The use of the SPECTRUM.CFG file is optional and is described in the manual which accompanied version 1 of SPECTRUM. The COLOR, COLOR2 and SINE utility routines generate data with known spectral densities; the data files produced by these routines are compatible with SPECTRUM and may be used for testing the operation of the program. The use of these utility routines is reviewed in detail in the version 1 manual. Finally, the MAKDAT and MAKCOEF routines are used to convert Ascii data files into the BINARY data files required by the SPECTRUM routine. These utilities are also explained in the version 1 manual. ▒Additional Windows Two additional window functions for tapering the input data are available in SPECTRUM V 2.0. These windows have been added as possible alternatives to the more commonly used windows made available in V 1.0. The choice of a window function depends upon such factors as the highest sidelobe level (dB), the rate of decay of the sidelobes (dB/octave) and the halfwidth of the mainlobe (f T). In general, as the width of the mainlobe is decreased, thereby increasing the resolving power, the amplitude of the sidelobes is increased which intensifies the problem of leakage. In many cases it is useful to allow the width of the mainlobe to increase somewhat in order to decrease the height of the sidelobes and / or to reduce the rate of decrease in the sidelobe level. With this end in mind, two windows were selected from Nuttall (1981) which are highly optimized members of a family of windows discussed by Harris (1978). The first window, which will be denoted here as the Minimum Sidelobe window, is the 4-term Blackman window with continuous first derivative. This window minimizes the maximum height of the sidelobes and also has a reasonably good rate of decay of the sidelobes at the expense of a widened mainlobe. The second window will be referred to here as the Maximum Decay window; this is the 4-term Blackman window with continuous fifth derivative which maximizes the rate of decay of the sidelobes and which also has a low maximum sidelobe amplitude. As expected, the mainlobe for this window is also wider than any of the windows contained in the first version. These windows are defined by [EQN "center #^ w (t) #3#3 = #3#3 #( { #3#3 stack left [[ a sub 0 #3#3 - #3#3 a where the coefficients for the Minimum Sidelobe window are given by [EQN "center a sub 0 #3#3 = #3#3 0.355768 #4#4 a sub 1 #3#3 = #3#3 0.487396 #4#4 and those for the Maximum Decay window by [EQN "center a sub 0 #3#3 = #3#3 10 over 32 #4#4 a sub 1 #3#3 = #3#3 15 over 32 These functions are plotted in figure 1 with the Hanning window shown for comparison. Notice that the Hanning window is a member of this family of windows where the first two coefficients are 1/2 and the remaining coefficients are zero.[PB] The Fourier transforms for these windows are found to be [EQN "center W #1 (f,T) #3#3 = #3#3 { #3#3 a sub 0 #3#3 #[ [sin #3 pi f T] over where [EQN "f sub 1 #3 = #3 1 #1 / #1 T"] . The moduli of the Fourier transforms for each of the above windows as well as for the Hanning window, shown for comparison, are plotted in figures 2, 3 and 4. In order to clearly reveal the maximum sidelobe height and the rate of decay of these sidelobes, the vertical axes of the plots are magnified by using a logarithmic scale; the power response of the windows normalized by the peak response at [EQN "f #3 = #3 0"] and converted to decibels is defined by [EQN "center #R[dB] #3#3 = #3#3 10 #3 #R[log] #3 [| W(f) #1 / #1 W(0)|] super 2" The plots clearly show the optimal features for each window function. A list of the performance parameters for all of the windows contained within SPECTRUM is included below for completeness. Maximum Maximum Halfwidth of Window Sidelobe Decay Mainlobe Type (dB) (dB/octave) (f T) Square -13 6 1.0 Tukey -13 18 1.1 Hanning -31 18 2.0 Parzen -27 12 2.0 Welch -21 12 1.4 Min. Sidelobe -93 18 4.0 Max. Decay -61 42 4.0 Table 1 Window performance parameters [PIC FIG1.CGM] Figure 1 Additional window functions for tapering the data [WS 0.5 in] [PIC FIG2.CGM] Figure 2 Power response for Hanning window[PB] [PIC FIG3.CGM] Figure 3 Power response for minimum sidelobe window [WS 0.5 in] [PIC FIG4.CGM] Figure 4 Power response for maximum decay window Input Data File Structure The author has obtained a new A/D converter which has the capability of altering the input voltage range to be digitized. The selection of an input voltage range may be made via software as opposed to changing jumpers on the board itself. After data has been sampled by the A/D converter, the author has, in the past, written certain information to the first record of each data file which describes the digitized information that follows. With the added capability offered by the new A/D converter, it was felt that the input voltage range used by each channel in the input data file would be prudent pieces of information to add to the file header. The A/D converter has the ability to convert up to eight differential inputs sequentially, and each channel may use a different input voltage range. Therefore, data files from this A/D converter may contain up to eight channels of digitized information, and the second record of the data file is now used to record the input voltage range used by each channel. These input data files are then used by a variety of routines written by the author to perform various aspects of random data analysis. The SPECTRUM program is just one member of this family of routines. Since the SPECTRUM program can only process one or two channels of data, placing information on the voltage range used by up to eight channels of data leads to unnecessary information being included for six channels which do not exist. As silly as this may seem, it turns out to be useful for the author to adopt a single universal input data file structure which can then be used by all of the routines which examine this data file. Therefore, this input data file structure, while not optimal for the SPECTRUM program, is the most useful universal setup for the author at this time. All input data files must be sequential BINARY files. These files contain no internal formatting information, and thus, are the most useful type of files to use when dealing with large quantities of data. Sequential unformatted FORTRAN data files were used as the file type for input data files in the first version of SPECTRUM. The drawback with unformatted FORTRAN data files is that they may be written only by FORTRAN programs. Using BINARY data files allows the data to be written using other programming languages. Furthermore, unformatted FORTRAN data files do, in fact, use a small amount of internal formatting information, whereas BINARY files do not. Thus, BINARY data files are slightly smaller than unformatted FORTRAN data files. The first two records of the data file contain descriptive information about the digitized data to follow and form the file header. Following the file header is the actual digitized data. The first record of the file header is changed only slightly from version 1.0 of the SPECTRUM routine, and the text describing this first record is taken from the manual issued with the first version. Changes are noted in boldface text as required. 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 must be 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 4-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 first record of the data file would contain the four values: [EQN "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 first record for this data file should consist of the following four numbers: [EQN "center 2 #4#4 32768 #4#4 40 #4#4 32766"] The second record of the data file consists of eight 2-byte integer values which encode the input voltage range used by channels 1 to 8. Acceptable values are [EQN "center matrix [[[# ] [# ] [# ] [# ] [# ] [# ] [#R[Value]] [# ] [#R[Range]] The user should write a 2-byte integer value from 0 to 3 to indicate the voltage range used by the first channel and the second channel. Six additional zero values must be entered for channels 3 through 8 which are not used by the SPECTRUM routine. For example, to indicate that the input voltage range for the first channel was -5V to 5V and for the second channel was -1.25V to 1.25V, the user would write the following to the second record of the data file. [EQN "center 1 #3#3 3 #3#3 0 #3#3 0 #3#3 0 #3#3 0 #3#3 0 #3#3 0 #3#3"] When the SPECTRUM program is run, the routine will ask if an alternate voltage transformation factor should be used. If the user responds with yes, then the user must type in the voltage per digital level conversion factor to be used for that channel, and this factor will be used in place of the factor stored in record 2 of the data file. After reading in the data file, the SPECTRUM routine will convert the digitized data back into voltages using the conversion factors shown above. If none of the above voltage ranges are appropriate for a particular data file, then the user should proceed as follows. Store eight zero values (2-byte integers) on the second record of the data file. Then, execute the SPECTRUM routine and enter the appropriate alternate conversion factor when requested. Following the first two records 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. In the first version of SPECTRUM, the program required an even number of records of data in the input data file when the file consisted of one channel of sampled data. This is no longer required. When the program submits the data to the FFT module to be transformed, 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 in the input data file in the case of one channel of data, then the program automatically appends an additional record of data containing a copy of the data values of the previous record. After transformation, the results of the added record are discarded so that the output is not corrupted by the internal addition of a record of data. When two channels of data are sampled, an even or odd number of records is not a factor because one record from each channel is submitted to the FFT module for simultaneous transformation. One or two separate time series (channels) may be stored per record. The method of storing the data values in each record follows the standard procedure used by analog to digital converters: [EQN "center #R[one] #4 #R[channel] #^ #R[:] #4#4 x sub 0 #3 x sub 1 #3 x sub 2 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 BINARY 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 BINARY data files required by the spectrum routine. See the section on utility files in the manual included with version 1.0. 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, NUMREC INTEGER*4 IRSIZE, IDELTMS, N, I INTEGER*2 NDATA (NMAX), GAIN (0 : 7) OPEN (2, FILE = 'A001.DAT', STATUS = 'UNKNOWN', ACCESS = 'SEQUENTIAL', FORM = 'BINARY') WRITE (2) ICHANS, IRSIZE, NUMREC, IDELTMS WRITE (2) (GAIN (I), I = 0, 7) 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 file is closed. INTEGER*2 ICHANS, NUMREC INTEGER*4 IRSIZE, IDELTMS, N, I INTEGER*2 NDATA (NMAX), GAIN (0 : 7) OPEN (2, FILE = 'A001.DAT', STATUS = 'OLD', ACCESS = 'SEQUENTIAL', FORM = 'BINARY') READ (2) ICHANS, IRSIZE, NUMREC, IDELTMS READ (2) (GAIN (I), I = 0, 7) 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') MAKCOEF Utility Routine The use of this utility routine for the generation of coefficient data files for use with SPECTRUM is essentially the same. The output of the program is now a BINARY coefficient data file vis a vis the unformatted FORTRAN coefficient data file generated in the first version. The SPECTRUM routine now allocates memory dynamically for all internal arrays at run-time. This improvement in the main program has unfortunately introduced an additional restriction on the generation of coefficient data files. The first set of coefficients contained in the coefficient data file must correspond to the first file to be processed by SPECTRUM, and the last set of coefficients contained in the coefficient data file must correspond to the last file to be processed by SPECTRUM. The total number of sets of coefficients contained in the coefficient data file should be given by [EQN "center #R[numsets] #3#3 = #3#3 #R[last] #3#3 - #3#3 #R[first] #3#3 + #3#3 The total number of files processed by SPECTRUM need not match the total number of sets of coefficients contained in the coefficient data file as long as the first and last sets of coefficients correspond to the first and last files processed, and the remainder of the files to be processed correspond to sets of coefficients that are contained between the first and last sets in the coefficient data file. This restriction will be clarified below using the example given on pp 70-71 from the manual included with the first version. 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: [EQN "center u sub n #3#3 = #3#3 b sub 0 #3 + #3 b sub 1 v sub n #3 + #3 b sub 2 where [EQN "v sub n"] is the nth data point expressed as a voltage, [EQN "b sub 0 #1 , #3 b sub 1 #1 , ... , b sub 4"] are the coefficients of the velocity transformation and [EQN "u sub n"] 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 BINARY coefficient data files, DCON.DAT and DCON2.DAT, each containing the 26 sets of five coefficients [EQN "b sub 0 #1 , #3 b sub 1 #1 , ... , b sub 4"] 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 [EQN "b sub 0"] for channel 1 of data file D061.DAT; the fifth number in the file should be [EQN "b sub 4"] for channel 1 of data file D061.DAT; the sixth number in the file will be [EQN "b sub 0"] for channel 1 of data file D062.DAT, and so on. The 130th and last number in the file should be [EQN "b sub 4"] 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 BINARY file DCON.DAT: [EQN "center 26 #4#4 61"] The coefficients for channel 1 are then transferred to the BINARY 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 necessary to re-create the two coefficient data files; the new data files will contain 7 sets of coefficients beginning with those corresponding to D074.DAT and ending with those corresponding to D080.DAT. If the user wishes to process the data files in an arbitrarily numbered order (not consecutively), the coefficient data files may need to be changed. For example, if the files D064.DAT, D069.DAT and D086.DAT were to be processed by SPECTRUM, this would necessitate a change in the coefficient data files requiring 23 sets of coefficients in each file: the first set corresponding to D064.DAT and the last set to D086.DAT. On the other hand, the files D061.DAT, D064.DAT, D069.DAT and D086.DAT could be processed by SPECTRUM without changing the coefficient data files since the first and last files correspond to the first and last sets of coefficients. The converse is also true, however. If the user desires at the outset to process only the four arbitrarily ordered files in the latter case 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 store all of the coefficients for an entire set of input data files in a master ASCII file; then members of the set can be processed by the spectrum routine either consecutively or in an arbitrarily numbered order at any subsequent time and the BINARY coefficient data files can be easily created by editing the master ASCII file and using MAKCOEF. 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 BINARY data files to be processed by SPECTRUM; then, all of this mess concerning coefficient data files could be avoided altogether. *References Harris, F.J. 1978 On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings of the IEEE, 66, 51-83. Nuttall, A.H. 1981 Some windows with very good sidelobe behavior, IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP-29, 84-91.