NeXT Education Software Sampler 1992 Fall/NeXT Education Software Sampler 1992 Fall.iso/Mathematics/Notebooks/SigProc2.0/CHANGES.SUMMARY

Text File | 1992-08-18 | 9.7 KB | 195 lines

                Signal Processing Packages and Notebooks
                    for the Mathematica Environment

                    Brian Evans and James McClellan
                    Digital Signal Processing Group
                    School of Electrical Engineering
                    Georgia Institute of Technology
                        Atlanta, GA  30332-0250

                    E-MAIL: [e-mail]


     Since the version of the signal processing packages released by
The Mathematica Journal, I have made changes to every file.  The
following highlights those changes.  The packages work simultaneously
under Mathematica 1.2 and 2.0.  Under Mathematica 2.0, evaluating
Needs[ "SignalProcessing`Master`" ] tells Mathematica where to find all
of the new routines so that they can be automatically loaded when
referenced.
     When routines load, their 1-D definitions are evaluated.
If multidimensional features are requested, then those definitions
would be automatically loaded.  This decoupling means that undergraduates
using the extensions would need less memory.  For example, one-dimensional
and separable multidimensional z-transforms require 1/3 less memory.


--> Integration

    o Integration rules for Delta functions are now covered by DeltaIntegrate.
      Users can customize them by attaching rules to DeltaIntegrate
      (but first you will need to unprotect it).
    o Integration is now more robust for expressions involving continuous
      step (CStep) functions and continuous pulse (CPulse) forms.

--> Expression Simplification

    o Added SPSimplify which applies simplification rules for signal
      processing expressions.  It supports a dialogue option:
        Dialogue -> False  no justification
        Dialogue -> True   some justification
        Dialogue -> All    complete justification
    o You can specify a list of symbols to treat as variables to
      SPSimplify by using the Variables option.
    o SPSimplify supports the same options as Simplify does

--> New Plotting Routines

    o SignalPlot plots the real and imaginary components of a 1-D function
      simultaneously on the same plot, as well as Dirac delta functions.
      The variable $DeltaFunctionScaling directs how Dirac delta functions
      are to be plotted.  A value of None means that the height of all Dirac
      delta functions will be equal to the maximum value of the plot.  A
      value of Scaled means that the height of Dirac delta functions will
      be proportional to their area.  SignalPlot also plots 2-D signals.
    o SequencePlot which plots 1-D and 2-D discrete-time expressions
    o RootLocus plots a root locus for one varying parameter
    o All new plotting routines support the same options as does Plot.
    o PoleZeroPlot shades the region of convergence for s-domain plots.
    o When an invalid region of convergence (ROC) is given to PoleZeroPlot,
      it will display the union of all possible ROC's.

--> Convolution

    o Either "function" argument can be a list of F-intervals or a
      function in algebraic form
    o Discrete and continuous convolution is performed by PiecewiseConvolution
      which calls either CTPiecewiseConvolution or DTPiecewiseConvolution
      depending on the value of the Domain option (Discrete or Continuous).
      The default domain is the value of $ConvolutionDomain.
    o The discrete linear convolution operator is represented by Convolve[n]
      and the continuous linear convolution operator is denoted CConvolve[t].
      These are merely representations of operations used by the transform
      rules bases; that is, they do not actually perform convolution.

--> Transforms

    o All transform rule bases can justify their answers.  The level
      of justification is set by the Dialogue option (see above).
    o The transform rule bases recognize convolution and summation operators
    o The inverse z- and Laplace transforms rely on Mathematica's
      partial fractions routine Apart which is not robust.  I have
      written MyApart which will be used by the transform rule bases
      whenever Apart does not finish the decomposition.  By default, MyApart
      will root denominators higher than 5th-order numerically.
    o The forward z-transform handles multidimensional downsampling and
      upsampling operations.
    o The inverse DTFT recognizes transforms that are multidimensionally
      upsampled in the discrete-time domain.
    o The forward and inverse continuous-time Fourier transforms
      use the same set of rules by exploiting the duality property.
    o The Fourier transforms handle infinite impulse trains (represented as
      either Summation[n,-Infinity,Infinity,1][Delta[t + n T]] or
      Periodic[T,t][Delta[t]]).  Aspects of sampling theory can now be shown.
    o The forward discrete-time Fourier transforms explicitly track
      periodicities of frequency responses.
    o All of the transform rule bases support an option called Definition.
      If True, then the rule base will apply the definition of the transform
      (using Mathematica's built-in routines for symbolic integration or
      symbolic summation) if all other attempts at the transform have failed.
    o Regardless of the value of the Dialogue option, the transform
      rule bases will keep track of the assumptions made on the free
      parameters.  They can be retrieved at any time by evaluating
      Assuming[All].  The set of conditions can be cleared by
      evaluating Assuming[].
    o New operator called CircularShift which is an abstract form of
      the primitive RotateRight.  It was introduced because of the
      new DFT transform pairs written into the DFT rule bases.
    o I have idiot-proofed the transform rule bases.  So, if you typed
      something like LaPlace[ CStep[t], t s ] it would complain about
      t s because you meant t, s.  This should help new users.
    o The TransformLookup option for the transform rule bases allows one
      to specify additional transform pairs.  This is useful when a trans-
      form rule base does not contain a pair that you need.  It is also
      useful for transforming abstract functions (i.e., saying that x[n]
      becomes X[z]).  For example,

      In:= ZTransform[a^n x[n], n, z, TransformLookup -> { x[n] :> X[z] }]
      Out= ZTransData[ X[a z], Rminus[0], Rplus[Infinity], ZVariables[z] ]

      In:= ZTransform[a^n x[n], n, z,
                      TransformLookup -> { x[n] :> { X[z], rm, rp } } ]
      Out= ZTransData[ X[a z], Rminus[rm], Rplus[rp], ZVariables[z] ]

      These transform pairs override the pairs in the transform rule bases.
      The multidimensional case is a natural extension:

      In:= ZTransform[ Upsample[l1,n1][ Upsample[l2,n2][ x[n1,n2] ] ],
                       {n1,n2}, {z1, z2},
                       TransformLookup -> { x[n1,n2] :> X[z1,z2] } ]
                           l1    l2
      Out= ZTransData[ X[z1  , z2  ], Rminus[{0, 0}],
                       Rplus[{Infinity, Infinity}], ZVariables[{z1, z2}] ]

--> Filters

    o CFIR and CIIR structures have an optional third argument: Roots -> roots.
      This argument can be used in lieu of a coefficient list so that a user
      may specify filters in terms of their poles and zeroes.  For example,
      CIIR[t, Roots -> {-1, -2, -3}] gives CIIR[t, {6, 11, 6, 1}].
    o The time responses of analog filters can now be plotted
    o Computational objects for IIR structures, called IIRFunction, now exist.
      Also, FIR structures can be rewritten as a formula.  Their values
      can now be plotted.  Values are calculated as they are needed, and
      IIRFunction remembers (caches) previous calculations.

--> Difference/Differential Equation Solvers

    o ZSolve and LSolve solve difference and differential equations,
      respectively.  They can justify their answers in the same way
      as the transforms do according to the level of Dialogue (see above).

--> Stability Checking

    o Stable will resolve stability of non-separable signals more often
      (e.g., Stable will return False when applied to the z-transform of
             (1/2)^n1 (4/5)^n2 Multinomial[n1,n2] Step[n1,n2]
      instead of an unresolved expression.)

--> Lattice Theory

    o The routines supporting the rules for multirate multidimensional
      signal processing (i.e., lattice and integer matrix theory) have
      been gathered in "Multirate.m" in the Support directory, including
      "DistinctCosetVectors", "SmithNormalForm", and "SmithReducedForm".
    o Added the functions ColumnHermiteForm and RowHermiteForm as well as
      LCLM, LCRM, GCRD, and GCLD.
    o Added the number theoretic EuclidFactors to complement the routine
      BezoutNumbers.  BezoutNumbers finds integers mu and lambda given
      integers a and b such that

                     a mu + b lambda == gcd(a,b)

      (BezoutNumbers calls the Mathematica primitive ExtendedGCD.)
      EuclidFactors finds the integers mu and lambda given relatively
      prime integers p and q and any integer k so that

                       p mu + q lambda == k

      EuclidFactors also works when p and q are integer matrices (in
      which case mu, lambda, and k become integer vectors).
    o Added UpsampledSystem and ReorderResampling to help detect
      resampled relationships among variables in an expression

--> Changes to Mathematica Primitives

    o Integrate so that it can handle generalized functions like the
      Dirac delta (Delta) and functions with discontinuities like
      continuous-time step (CStep) and pulse (CPulse) functions.
    o Simplify applied to an And expression removes redundant conditions
    o Det of a number is that number
    o Dot product of two numbers is the product of the two numbers
    o The TeX forms of Re and Im primitives are Re and Im instead of
      R and I.