(*^

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Symbolic Transforms
withApplications to
Signal Processing
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by Brian L. Evans and James H. McClellan
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Digital Signal Processing Laboratory
School of Electrical Engineering
Georgia Institute of Technology
Atlanta, GA  30332-0250

EMAIL:  evans@eedsp.gatech.edu
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		This notebook contains the examples presented in the paper entitled "Symbolic Transforms with Applications to Signal Processing," which appears the second issue of the Mathematica Journal.  This paper describes packages which perform basic operations common in both digital and analog signal processing.  Other notebooks besides this one describe the signal processing packages in more detail:  convolution, filter design, Laplace transforms, and z-transforms.  We have also included an additional section that summarizes the latest changes to the signal processing packages.
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		The major improvement of these packages since the Mathematica Journal paper was published is that the 10 symbolic transform rule bases can now justify their answers by displaying each step of the transform process (which is enabled by using the Dialogue -> All option).  Examples of this are included throughout this notebook (e.g., "Z-transform" subsection under the "DSP Examples" section).
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		Another significant improvement since the article is that users can give their own transform pairs using the TransformLookup option.  One use is to specify the transforms of abstract (undefined) signals, e.g. TransformLookup -> { x[n] :> X[s] } in the case of the forward Laplace transform rule base LaPlace.
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WARNING:  This notebook will automatically load in the signal processing packages (see the Initialization Section for the exact commands) which under Mathematica 1.2 may take as long 5 minutes.
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Initialization
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*)

Needs[ "SignalProcessing`Analog`" ]
(*
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Loading version 2.2 of the signal processing support\
 
  packages ...
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Support module has been loaded.
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The knowledge base of signal processing operators has\
 
  been loaded.
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Supporting routines for filter design are loaded.
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Region of convergence routines for the transform rule\
 
  bases are loaded.
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Piecewise convolution rules have been loaded.
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Supporting routines for transform rule bases are loaded.
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Support::loaded: 
   Supporting routines, objects, and rules for the signal
    processing package have been loaded successfully.
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Support::change: 
   The loading process has changed such that the files
    Tree.m and DataType.m are no longer automatically
    loaded (the signal processing packages no longer rely
    on the definitions in these files).
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Loading version 2.2 of the analog signal processing\
 
  packages ...
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The one-dimensional analog filter design objects are\
 
  loaded.
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Supporting routines and objects for the Laplace transform\
 
  are loaded.
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The forward Laplace transform rule base LaPlace has been\
 
  loaded.
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The inverse Laplace transform rule base InvLaPlace has\
 
  been loaded.
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The supporting functions are loaded for symbolic Fourier\
 
  transforms.
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The continuous Fourier transform (CTFT) functions are\
 
  loaded.
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LSolve, a differential equation solver, is loaded.
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The analog signal analyzer is now loaded.
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Analog::loaded: 
   Routines, objects, and rules for the analog signal
    processing packages have been loaded successfully.
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Analog signal processing extensions have been loaded.
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Good starting points are the objects ASPAnalyze and\
 
  LaPlace.
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Active packages are maintained in the variable\
 
  $ContextPath.
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  New signal primitives are in SPsignals.
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  New system primitives are in SPoperators.
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  New Mathematica functions are in SPfunctions.
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Note that all signal processing expressions will
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be automatically simplified.
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*)

Needs[ "SignalProcessing`Digital`" ]
(*
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Loading version 2.2 of the signal processing support\
 
  packages ...
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Support::loaded: 
   Supporting routines, objects, and rules for the signal
    processing package have been loaded successfully.
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Support::change: 
   The loading process has changed such that the files
    Tree.m and DataType.m are no longer automatically
    loaded (the signal processing packages no longer rely
    on the definitions in these files).
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Loading version 2.2 of the digital signal processing\
 
  packages ...
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Supporting routines for the z-transform rule bases are\
 
  loaded.
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The forward z-transform rules have been loaded.
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The inverse z-transform rules have been loaded.
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The discrete-time Fourier transform (DTFT) rule bases are\
 
  loaded.
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Correlation functions are now loaded.
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ZSolve, a difference equation solver, is loaded.
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The digital signal analyzer is now loaded.
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The discrete Fourier transform (DFT) rule bases are\
 
  loaded.
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Digital::loaded: 
   Routines, objects, and rules for the digital signal
    processing packages have been loaded successfully.
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Digital signal processing extensions have been loaded.
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Good starting points are the objects DSPAnalyze and\
 
  ZTransform.
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Active packages are maintained in the variable\
 
  $ContextPath.
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  New signal primitives are in SPsignals.
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  New system primitives are in SPoperators.
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  New Mathematica functions are in SPfunctions.
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Note that all signal processing expressions will
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be automatically simplified.
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Introduction
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Abstract
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		The examples illustrate the abilities of the signal processing packages.  These packages implement many of the basic operations in both digital and analog signal processing.  This hierarchical set of packages performs basic analyses of functions (signals) and operators (systems).  For the most part, these packages are based on transform theory, which is a powerful tool for analyzing linear, shift-invariant operators.  All the transforms were implemented using the same structure for encoding knowledge about transforms.
		The packages include the z, Laplace, Fourier, discrete-time Fourier, and discrete Fourier transforms, for both one-dimensional and multidimensional signals.  In the case of the z and Laplace transforms, the bilateral form of the transform is used, because that is the common practice in signal processing.  However, these packages should be useful in a number of fields where transform analysis is needed.
		These packages can perform a variety of different operations for symbolic, graphic, and numerical analysis of signals and systems.  Symbolic analyses include simplifying expressions and reasoning about properties of signals like stability.  Plotting capabilities include discrete-time plots, magnitude frequency responses, and phase frequency responses.  Another plotting feature is the presentation of pole-zero diagrams, including the region of  convergence, for z and Laplace transforms.
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For Further Information
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		The companion notebook to this one is entitled "LaPlaceTest".  It is an overview of the forward and inverse Laplace transform rule bases, complete with over 100 examples.  There is also a section on how to use the linear constant coefficient differential equation solver using LSolve, which relies on the Laplace transform rule bases.
		For those with a signal processing background, these signal processing packages are discussed in a book entitled {Symbolic and Knowledge-Based Signal Processing} which will be published by Prentice-Hall in the Spring of 1991.
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Digital Signal Processing Examples
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z-Transform Examples
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How to call the forward z-transform
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?ZTransform
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ZTransform[e,n] or ZTransform[e,n,z] gives the z-transform of the
   expression e, which is a function of n, by returning an object of
   three slots which is tagged by ZTransData:  <z-transform>,
   <rminus>, <rplus>, and <z-variables>.  The Region of Convergence
   (ROC) is <rminus> < |z| < <rplus>.  Note that the returned ROC is
   either the actual ROC or a subset of the actual ROC. In two
   dimensions, ZTransform[e, {n1, n2}, {z1, z2}], is the same as
   ZTransform [ ZTransform [e, n1, z1], n2, z2 ].  This notation
   extends naturally to higher dimensions.
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Forward z-Transform Examples
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ZTransform[ (1/3)^Abs[n], n, z, Dialogue -> True ]
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Rewriting the two-sided expression
 1 Abs[n]
(-)
 3
as a left-sided plus a right-sided sequence:
Step[-1 - n]    1 n
------------ + (-)  Step[n]
     1 n        3
    (-)
     3
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ZTransData[(1 - 1/(3*z))^(-1) + z/(3*(1 - z/3)), Rminus[1/3], 
  Rplus[3], ZVariables[z]]
;[o]
              1          z             1
ZTransData[------- + ---------, Rminus[-], Rplus[3], ZVariables[z]]
                1           z          3
           1 - ---   3 (1 - -)
               3 z          3
:[font = input; startGroup; ]
ZTransform[ (1/3)^Abs[n], n, z, Dialogue -> False ]
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ZTransData[(1 - 1/(3*z))^(-1) + z/(3*(1 - z/3)), Rminus[1/3], 
  Rplus[3], ZVariables[z]]
;[o]
              1          z             1
ZTransData[------- + ---------, Rminus[-], Rplus[3], ZVariables[z]]
                1           z          3
           1 - ---   3 (1 - -)
               3 z          3
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ZTransform[ (1/3)^Abs[n], n, z, Dialogue -> All ]
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     1 Abs[n]
Z  {(-)      }
  n  3
which becomes
( after the two-sided expression
   1 Abs[n]
  (-)
   3
  has been rewritten as a left-sided 
  plus a right sided sequence
  Step[-1 - n]    1 n
  ------------ + (-)  Step[n] )
       1 n        3
      (-)
       3
    Step[-1 - n]    1 n
Z  {------------ + (-)  Step[n]}
  n      1 n        3
        (-)
         3
which becomes
    Step[-1 - n]         1 n
Z  {------------} + Z  {(-)  Step[n]}
  n      1 n          n  3
        (-)
         3
which becomes
                                              Step[-n]
{Z  {Step[n]}}                        + z Z  {---------}
   n          z -> 3 z and scale ROC        n  1 -1 + n
                                              (-)
                                               3
which becomes
        Step[n]
z {Z  {---------}}                       + 
     n  1 -1 - n  z -> 1/z and flip ROC 
       (-)
        3
 
               1     1
  Transform[-------, -, Infinity]
                 1   3
            1 - ---
                3 z
which becomes
   {Z  {Step[n]}}
      n          z -> 3 z and scale ROC 
z {-------------------------------------}
                     3                   z -> 1/z
 
                                  1     1
                   + Transform[-------, -, Infinity]
     and flip ROC                   1   3
                               1 - ---
                                   3 z
which becomes
             1          z      1
Transform[------- + ---------, -, 3]
               1           z   3
          1 - ---   3 (1 - -)
              3 z          3
which becomes
             1          z      1
Transform[------- + ---------, -, 3]
               1           z   3
          1 - ---   3 (1 - -)
              3 z          3
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ZTransData[(1 - 1/(3*z))^(-1) + z/(3*(1 - z/3)), 
  Rminus[1/3], Rplus[3], ZVariables[z]]
;[o]
              1          z             1
ZTransData[------- + ---------, Rminus[-], Rplus[3], 
                1           z          3
           1 - ---   3 (1 - -)
               3 z          3
 
  ZVariables[z]]
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ZTransform[ a^n1 b^n2 Step[n1,n2], {n1,n2} ]
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ZTransData[1/((1 - a/z1)*(1 - b/z2)), Rminus[{Abs[a], Abs[b]}], 
  Rplus[{DirectedInfinity[1], DirectedInfinity[1]}], 
  ZVariables[{z1, z2}]]
;[o]
                   1
ZTransData[-----------------, Rminus[{Abs[a], Abs[b]}], 
                a        b
           (1 - --) (1 - --)
                z1       z2
 
  Rplus[{Infinity, Infinity}], ZVariables[{z1, z2}]]
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How to call the forward z-transform
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?InvZTransform
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InvZTransform[e,z] or InvZTransform[e,z,n] gives the inverse
   z-transform of e.  Here, e can be a function of z, a list, or a
   z-transform object (see ZTransData).  As a list, e should contain
   three elements:  F[z], rminus, and rplus, such that the region of
   convergence is rminus < |z| < rplus.   If e is a z-transform
   object, then the inverse z-transform is simply InvZTransform[e]
   because e contains all necessary information. If the returned
   expression is not the form you desire, then apply Simplify[],
   Cancel[], Together[], etc., to the expression.
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Inverse z-Transform Examples
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ZTransform[ (1/3)^Abs[n], n, z ]
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Rewriting the two-sided expression
 1 Abs[n]
(-)
 3
as a left-sided plus a right-sided sequence:
Step[-1 - n]    1 n
------------ + (-)  Step[n]
     1 n        3
    (-)
     3
:[font = output; inactive; output; endGroup; ]
ZTransData[(1 - 1/(3*z))^(-1) + z/(3*(1 - z/3)), Rminus[1/3], 
  Rplus[3], ZVariables[z]]
;[o]
              1          z             1
ZTransData[------- + ---------, Rminus[-], Rplus[3], ZVariables[z]]
                1           z          3
           1 - ---   3 (1 - -)
               3 z          3
:[font = input; startGroup; ]

InvZTransform[ ZTransform[ (1/3)^Abs[n], n, z, Dialogue -> False ] ]
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3^n*Step[-1 - n] + (1/3)^n*Step[n]
;[o]
 n                 1 n
3  Step[-1 - n] + (-)  Step[n]
                   3
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InvZTransform[ Log[1 + a z^-3], z, n, Dialogue -> All ]
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  -1         a
Z   {Log[1 + --]}
  z           3
             z
becomes
                -1         a
Upsample     [Z   {Log[1 + -]}]
        3,0,n   z          z
becomes
               n   -1         1
Upsample     [a  Z   {Log[1 + -]}]
        3,0,n      z          z
becomes
                  n  n
              (-1)  a  Step[-1 + n]
Upsample     [---------------------]
        3,0,n           n
becomes
                  n  n
              (-1)  a  Step[-1 + n]
Upsample     [---------------------]
        3,0,n           n
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Upsample[3, 0, n][((-1)^n*a^n*Step[-1 + n])/n]
;[o]
                  n  n
              (-1)  a  Step[-1 + n]
Upsample     [---------------------]
        3,0,n           n
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InvZTransform[ Exp[Tan[z]], z, n, Terms -> 6 ]
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Breaking up the expression
 Tan[z]
E
into its series representation:
         2    3      4       5
        z    z    3 z    37 z        6
1 + z + -- + -- + ---- + ----- + O[z]
        2    2     8      120
The inverse z-transform will now be applied to the first 6 terms.
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
Impulse[n] + Impulse[1 + n] + Impulse[2 + n]/2 + Impulse[3 + n]/2 + 
  (3*Impulse[4 + n])/8 + (37*Impulse[5 + n])/120
;[o]
                              Impulse[2 + n]   Impulse[3 + n]
Impulse[n] + Impulse[1 + n] + -------------- + -------------- + 
                                    2                2
 
  3 Impulse[4 + n]   37 Impulse[5 + n]
  ---------------- + -----------------
         8                  120
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DTFT Example
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?DTFTransform
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DTFTransform[function, in variable(s), out variable(s), options]
   returns the discrete-time Fourier transform of the input
   function. Note that this transform may rely on the z-transform
   and the substitution z = Exp[I w], which is not valid if the
   region of convergence does not include the unit circle.  Also
   note that only the first two arguments are required.
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DTFTransform[ Sinc[n], n, w ]
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DTFTData[Pi*(-CStep[-1 + w] + CStep[1 + w]), FVariables[w]]
;[o]
DTFTData[Pi (-CStep[-1 + w] + CStep[1 + w]), FVariables[w]]
:[font = input; startGroup; ]

DTFTransform[ 5 Sinc[n], n, w, Dialogue -> All ]
:[font = print; inactive; ]
DTFT  {5 Sinc[n]}
     n
becomes
5 DTFT  {Sinc[n]}
       n
becomes
5 Pi (-CStep[-1 + w] + CStep[1 + w])
becomes
5 Pi (-CStep[-1 + w] + CStep[1 + w])
:[font = output; inactive; output; endGroup; ]
DTFTData[5*Pi*(-CStep[-1 + w] + CStep[1 + w]), FVariables[w]]
;[o]
DTFTData[5 Pi (-CStep[-1 + w] + CStep[1 + w]), FVariables[w]]
:[font = input; startGroup; ]

InvDTFTransform[ DTFTransform[ Sinc[n], n, w ] ]
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Sinc[n]
;[o]
Sinc[n]
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DFT Example
:[font = input; startGroup; ]

?DFTransform
:[font = info; inactive; endGroup; ]
DFTransform[function, NumberOfPoints, TimeVariables,
   FourierVariables, options] returns the DFT of function, where
   function is a function of TimeVariables defined from 0 to
   NumberOfPoints - 1.  A side effect of this function is that the
   symbols in the arguments NumberOfPoints and TimeVariables will be
   associated with the data type Integer via AssociateDataType. 
   Note that only the first two arguments are required and that
   DFTransform calls DTFTransform.
:[font = input; startGroup; ]

DFTransform[ Exp[I 2 n], 10, n, k ]
:[font = output; inactive; output; endGroup; ]
DFTData[E^(9*I + (-9*I)/10*Pi*k)*Dirichlet[10, -2 + (Pi*k)/5], 
  Start[0], Finish[9], KVariables[k]]
;[o]
         9 I + (-9 I)/10 Pi k                    Pi k
DFTData[E                     Dirichlet[10, -2 + ----], Start[0], 
                                                  5
 
  Finish[9], KVariables[k]]
:[font = input; startGroup; ]

?InvDFTransform
:[font = info; inactive; endGroup; ]
InvDFTransform[function, NumberOfPoints, FourierVariables,
   TimeVariables, options] takes the inverse DFT of function, where
   function is a function of FourierVariables defined from 0 to
   NumberOfPoints - 1.  Note that only the first two arguments are
   required and that InvDFTransform calls InvDTFTransform.
:[font = input; startGroup; ]

InvDFTransform[ DFTransform[ Exp[I 2 n], 10, n, k ] ]
:[font = output; inactive; output; endGroup; ]
E^(9*I + 2*I*(-9/2 + n))*Pulse[10, n]
;[o]
 9 I + 2 I (-(9/2) + n)
E                       Pulse   [n]
                              10
:[font = text; inactive; ]
This expression is the same as Exp[I 2 n] for n = 0 ... 9:
:[font = input; startGroup; ]

MyCollectAll[ InvDFTransform[ DFTransform[Exp[I 2 n], 10, n, k] ], n ]
:[font = output; inactive; output; endGroup; endGroup; ]
E^(2*I*n)*Pulse[10, n]
;[o]
 2 I n
E      Pulse   [n]
             10
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Stability Analysis Example
:[font = input; startGroup; ]
?Stable
:[font = info; inactive; endGroup; ]
Stable[transexpr] returns True if the transform transexpr represents
   a stable time-domain signal.
:[font = input; startGroup; ]
Stable[ ZTransform[ a^n1 b^n2 Step[n1,n2], {n1,n2} ] ]
:[font = output; inactive; output; endGroup; endGroup; ]
Abs[a] < 1 && Abs[b] < 1
;[o]
Abs[a] < 1 && Abs[b] < 1
:[font = subsection; inactive; startGroup; Cclosed; ]
Solving a Difference Equation
:[font = input; startGroup; ]

?ZSolve
:[font = info; inactive; endGroup; ]
ZSolve[ diffequ == drivingfun, y[n] ] solves the difference equation
   diffequ = drivingfun, where diffequ is a linear constant
   coefficient difference equation and drivingfun is the driving
   function (a function of n).  Thus, diffequ has the form a0 y[n] +
   a1 y[n - 1] + ....  By using options, one can specify initial
   values; e.g., ZSolve[ y[n] + 3/2 y[n-1] + 1/2 y[n-2] == (1/4)^n
   Step[n], y[n], y[-1] -> 4, y[-2] -> 10 ].
:[font = input; initialization; startGroup; ]
*)

diffequ = y[n] - 3/2 y[n-1] + 1/2 y[n-2]
(*
:[font = output; inactive; output; endGroup; ]
y[-2 + n]/2 - (3*y[-1 + n])/2 + y[n]
;[o]
y[-2 + n]   3 y[-1 + n]
--------- - ----------- + y[n]
    2            2
:[font = input; initialization; startGroup; ]
*)

ZSolve[ diffequ == (1/4)^n Step[n],
		 y[n], y[-1] -> 4, y[-2] -> 10 ]
(*
:[font = message; inactive; startGroup; ]
ZSolve::redundant: 
   Same initial index is assigned two different values.
:[font = print; inactive; ]
Solving for y[n] in the difference equation
:[font = print; inactive; ]
augmented by the initial conditions:
:[font = print; inactive; ]
  y[-2 + n]   3 y[-1 + n]
  --------- - ----------- + y[n] = 
      2            2
 
                                            1 n
  -11 Impulse[1 + n] + 10 Impulse[2 + n] + (-)  Step[n]
                                            4
:[font = print; inactive; ]
After taking the z-transform of both sides and
:[font = print; inactive; ]
solving for the z-transform of y[n]:
:[font = print; inactive; ]
            30   108
      -80 - -- + ---
             2    z
            z
  -(-------------------)
      -5   7    14   8
    -z   + -- - -- + --
            4    3    2
           z    z    z
:[font = print; inactive; ]
Inverse transforming this gives y[n]:
:[font = output; inactive; output; endGroup; endGroup; ]
4*Impulse[1 + n] + 10*Impulse[2 + n] + (2*Step[n])/3 + 
  ((1/4)^n*Step[n])/3 + (1/2)^n*Step[n]
;[o]
                                       2 Step[n]
4 Impulse[1 + n] + 10 Impulse[2 + n] + --------- + 
                                           3
 
   1 n
  (-)  Step[n]
   4              1 n
  ------------ + (-)  Step[n]
       3          2
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y[n_] := ( 2 + (1/4)^n + 3 (1/2)^n ) / 3
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y[-1]
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4
;[o]
4
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y[-2]
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10
;[o]
10
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Table [ diffequ == (1/4)^n Step[n], {n, 0, 5} ]
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{True, True, True, True, True, True}
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{True, True, True, True, True, True}
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Digital Signal Analyzer
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?DSPAnalyze
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DSPAnalyze[f], DSPAnalyze[f,n], and DSPAnalyze[f,n,start,end] will
   plot f[n] from n = Floor[start] to n = Ceiling[end], print its
   z-transform, indicate conditions for stability, display the
   pole-zero diagram, and plot the magnitude and phase responses. 
   For the version of this routine which works for two-dimensional
   functions, n is a two-element list of variables (like {n1, n2})
   and start as well as end are also two-element lists indicating
   the range of the time-domain graph.
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DSPAnalyze[ Upsample[3,n][Sinc[n]], n, -20, 20 ]
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ZTransform::notvalid: The forward z-transform could not be found.
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 1 n1  4 n2
(-)   (-)   (n1 + n2)! Step[n1] Step[n2]
 2     5
----------------------------------------
                n1! n2!
has the following z-transform:
     -10
-------------
      5    8
-10 + -- + --
      z1   z2
The region of convergence is:
1
- < |z1| < Infinity
2
         1
4 Abs[--------]
           1
      1 - ----
          2 z1
--------------- < |z2| < Infinity
       5
                                 z1
                        8 Abs[---------]
                              -1 + 2 z1
The system is stable if ---------------- < 1
                               5
 
The zeroes are:  {}
                               I w
                    -5. 2.71828
The poles are:   {-------------------}
                                  I w
                  8. - 10. 2.71828
 
The zeroes are:  {}
                               I w
                    -8. 2.71828
The poles are:   {-------------------}
                                  I w
                  5. - 10. 2.71828
By substituting into the z-transform,
the frequency response is
           -10
-------------------------
         -I w1      -I w2
-10 + 5 E      + 8 E
 
 1 n1  4 n2
(-)   (-)   (n1 + n2)! Step[n1] Step[n2]
 2     5
----------------------------------------
                n1! n2!
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                1
---------------------------------
      -I w1             -I w2
     E               4 E
(1 - ------) (1 - --------------)
       2                  -I w1
                         E
                  5 (1 - ------)
                           2
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:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
ZTransData[-10/(-10 + 5/z1 + 8/z2), 
  Rminus[{1/2, (4*Abs[(1 - 1/(2*z1))^(-1)])/5}], 
  Rplus[{DirectedInfinity[1], DirectedInfinity[1]}], 
  ZVariables[{z1, z2}]]
;[o]
                                              1
                                     4 Abs[--------]
                                                1
                                           1 - ----
                -10               1            2 z1
ZTransData[-------------, Rminus[{-, ---------------}], 
                 5    8           2         5
           -10 + -- + --
                 z1   z2
 
  Rplus[{Infinity, Infinity}], ZVariables[{z1, z2}]]
:[font = section; inactive; startGroup; Cclosed; ]
Analog Signal Processing Examples
:[font = subsection; inactive; startGroup; Cclosed; ]
Laplace Transforms
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Forward Laplace Transforms
:[font = input; startGroup; ]

?LaPlace
:[font = info; inactive; endGroup; ]
LaPlace[e, t] or LaPlace[e, t, s] gives the two-sided LaPlace
   transform of the expression e, which is a function of t, by
   returning an object of four slots which is tagged by LTransData:
   <transform>, <rminus>, <rplus>, <laplace_variables>.  The Region
   of Convergence (ROC) is defined as <rminus> < Re{s} < <rplus>.
   Note that the returned ROC is either the actual ROC or a subset
   of the actual ROC. In two dimensions, LaPlace[e, {t1, t2}, {s1,
   s2}] is the same as LaPlace [ LaPlace[e, t1, s1], t2, s2 ]. 
   This notation extends naturally to higher dimensions.  Note that
   the right sided transform is specified by multiplying by
   CStep[t].
:[font = input; startGroup; ]

LaPlace[ A t Exp[- (a + b) t / 2]
		 ( BesselI[0, (a - b) t / 2] -
		   BesselI[1, (a - b) t / 2] ) CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(A*b + A*s)/((a + s)^(3/2)*(b + s)^(3/2)), 
  Rminus[Abs[Re[(a - b)/2]] - Re[(a + b)/2]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                 A b + A s
LTransData[---------------------, 
                  3/2        3/2
           (a + s)    (b + s)
 
                a - b        a + b
  Rminus[Abs[Re[-----]] - Re[-----]], Rplus[Infinity], 
                  2            2
 
  LVariables[s]]
:[font = text; inactive; ]

Unfortunately, Mathematica does not quite simplify this one all of the way.  The multiplication by the constant A prevents the simplification somehow.  Without the constant A, the transform is simplified properly:
:[font = input; startGroup; ]

LaPlace[ t Exp[- (a + b) t / 2]
		 ( BesselI[0, (a - b) t / 2] -
		   BesselI[1, (a - b) t / 2] ) CStep[t], t, s ]
:[font = input; ]

:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[1/((a + s)^(3/2)*(b + s)^(1/2)), 
  Rminus[Abs[Re[(a - b)/2]] - Re[(a + b)/2]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                     1
LTransData[----------------------, 
                  3/2
           (a + s)    Sqrt[b + s]
 
                a - b        a + b
  Rminus[Abs[Re[-----]] - Re[-----]], Rplus[Infinity], 
                  2            2
 
  LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Inverse Laplace Transforms
:[font = input; startGroup; ]

?InvLaPlace
:[font = info; inactive; endGroup; ]
InvLaPlace[f, s] and InvLaPlace[f, s, t] gives the multidimensional
   bilateral inverse Laplace transform of f.  A region of
   convergence can be specified by using InvLaPlace[{f, rm, rp}, s,
   t], where rm is Rminus and rp is Rplus in the region (strip) of
   convergence: Rminus < Re(s) < Rplus.
:[font = input; startGroup; ]

InvLaPlace[ LaPlace[ t Exp[- (a + b) t / 2]
					 ( BesselI[0, (a - b) t / 2] -
					   BesselI[1, (a - b) t / 2] ) CStep[t], t, s ] ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
(t*(BesselI[0, ((a - b)*t)/2] - BesselI[1, ((a - b)*t)/2])*
    CStep[t])/E^(((a + b)*t)/2)
;[o]
              (a - b) t               (a - b) t
t (BesselI[0, ---------] - BesselI[1, ---------]) CStep[t]
                  2                       2
----------------------------------------------------------
                       ((a + b) t)/2
                      E
:[font = subsection; inactive; startGroup; Cclosed; ]
Fourier Transforms
:[font = input; startGroup; ]

?CTFTransform
:[font = info; inactive; endGroup; ]
CTFTransform[f, t] or CTFTransform[f, t, w] computes the
   continuous-time Fourier transform of f.  It has time argument(s)
   t and frequency argument(s) w.
:[font = input; startGroup; ]

?InvCTFTransform
:[font = info; inactive; endGroup; endGroup; ]
InvCTFTransform[f, w, t] computes the inverse continuous time
   Fourier transform of f with frequency arguments w and time
   arguments t.
:[font = subsection; inactive; startGroup; Cclosed; ]
Stability Analysis Example
:[font = input; startGroup; ]

Stable [ LaPlace[ A t Exp[- (a + b) t / 2]
				 ( BesselI[0, (a - b) t / 2] -
				   BesselI[1, (a - b) t / 2] ) CStep[t], t, s ] ]
:[font = output; inactive; output; endGroup; ]
Abs[Re[(a - b)/2]] - Re[(a + b)/2] < 0
;[o]
       a - b        a + b
Abs[Re[-----]] - Re[-----] < 0
         2            2
:[font = text; inactive; endGroup; ]

If a > b, a is real-valued, and b is real-valued, then this stability condition reduces to a > b > 0.
:[font = subsection; inactive; startGroup; Cclosed; ]
Solving a Differential Equation
:[font = input; startGroup; ]

?LSolve
:[font = info; inactive; endGroup; ]
LSolve[ diffequ == drivingfun, y[t] ] solves the difference
   equation diffequ = drivingfun, where diffequ is a linear
   constant coefficient differential equation and drivingfun is the
   driving function (a function of t).  Thus, diffequ has the form
   a0 y[t] + a1 y'[t] + ....  By using options, one can specify
   initial values; e.g., LSolve[ y''[t] + 3/2 y'[t] + 1/2 y[t] ==
   Exp[a t] CStep[t], y[t], y[0] -> 4, y'[0] -> 10 ].
:[font = input; startGroup; ]

LSolve[ y''[t] + 3/2 y'[t] + 1/2 y[t] == Exp[a t] CStep[t],
		y[t], y'[0] -> 1/2, y[0] -> 4 ]
:[font = output; inactive; output; endGroup; ]
((-10 - 36*a)*CStep[t/2])/(2*E^(t/2)*(-1 - 2*a)) + 
  ((3 + 5*a)*CStep[t])/(E^t*(-1 - a)) - 
  (2*E^(a*t)*CStep[t])/(-1 - 3*a - 2*a^2)
;[o]
                   t
(-10 - 36 a) CStep[-]                           a t
                   2    (3 + 5 a) CStep[t]   2 E    CStep[t]
--------------------- + ------------------ - ---------------
     t/2                    t                              2
  2 E    (-1 - 2 a)        E  (-1 - a)       -1 - 3 a - 2 a
:[font = text; inactive; ]

By removing the step functions, we can convert the bilateral result to a unilateral one.  This conversion makes the verification of answer easier.
:[font = input; initialization; startGroup; ]
*)

ysol = LSolve[ y''[t] + 3/2 y'[t] + 1/2 y[t] == Exp[a t] CStep[t],
				y[t], y'[0] -> 1/2, y[0] -> 4 ] /.
		{ CStep[a_] :> 1 }
(*
:[font = print; inactive; startGroup; ]
Solving for y[t] in the differential equation
:[font = print; inactive; ]
  y[t]   3 y'[t]             a t
  ---- + ------- + y''[t] = E    CStep[t]
   2        2
:[font = print; inactive; ]
  subject to the initial conditions
:[font = print; inactive; ]
            1
  {y'[0] -> -, y[0] -> 4}
            2
:[font = print; inactive; ]
After taking the Laplace transform of both sides
:[font = print; inactive; ]
and solving for the transform of y[t]:
:[font = print; inactive; ]
  13           1
  -- + 4 s + -----
  2          s - a
  ----------------
    1   3 s    2
    - + --- + s
    2    2
:[font = print; inactive; ]
Inverse transforming this gives y[t]:
:[font = output; inactive; output; endGroup; endGroup; ]
(-3 + 5*E^(t/2) + 2*E^(t + a*t) - 11*a + 23*E^(t/2)*a - 
    10*a^2 + 18*E^(t/2)*a^2)/(E^t*(1 + a)*(1 + 2*a))
;[o]
         t/2      t + a t              t/2         2
(-3 + 5 E    + 2 E        - 11 a + 23 E    a - 10 a  + 
 
        t/2  2      t
    18 E    a ) / (E  (1 + a) (1 + 2 a))
:[font = input; ]


:[font = input; initialization; startGroup; ]
*)

ysolprime = D[ ysol, t]
(*
:[font = output; inactive; output; endGroup; ]
-((-3 + 5*E^(t/2) + 2*E^(t + a*t) - 11*a + 
       23*E^(t/2)*a - 10*a^2 + 18*E^(t/2)*a^2)/
     (E^t*(1 + a)*(1 + 2*a))) + 
  ((5*E^(t/2))/2 + (23*E^(t/2)*a)/2 + 9*E^(t/2)*a^2 + 
     2*E^(t + a*t)*(1 + a))/(E^t*(1 + a)*(1 + 2*a))
;[o]
           t/2      t + a t              t/2         2
-((-3 + 5 E    + 2 E        - 11 a + 23 E    a - 10 a  + 
 
           t/2  2      t
       18 E    a ) / (E  (1 + a) (1 + 2 a))) + 
 
     t/2       t/2
  5 E      23 E    a      t/2  2      t + a t
  ------ + --------- + 9 E    a  + 2 E        (1 + a)
    2          2
  ---------------------------------------------------
                  t
                 E  (1 + a) (1 + 2 a)
:[font = input; startGroup; ]

ysol2prime = D[ ysolprime, t ];

Simplify[ ysol2prime + 3/2 ysolprime + 1/2 ysol ]
:[font = output; inactive; output; endGroup; ]
E^(a*t)
;[o]
 a t
E
:[font = input; startGroup; ]

Simplify[ ysol /. t -> 0 ]
:[font = output; inactive; output; endGroup; ]
4
;[o]
4
:[font = input; startGroup; ]

Simplify[ ysolprime /. t -> 0 ]
:[font = output; inactive; output; endGroup; endGroup; ]
1/2
;[o]
1
-
2
:[font = subsection; inactive; startGroup; Cclosed; ]
Analog Signal Analyzer
:[font = input; startGroup; ]

?ASPAnalyze
:[font = info; inactive; endGroup; ]
ASPAnalyze[f], ASPAnalyze[f, t], and ASPAnalyze[f, t, start, end]
   will plot f(t) as a one-dimensional, continuous-time function.
   It will also print the strip of convergence, indicate stability
   criteria, display the pole-zero diagram, and plot the magnitude
   and phase responses.  For the two-dimensional version, t is a
   list of two variables (like {t1, t2}) and start/end are optional
   two-element lists specifying the range of the time-domain plot.
:[font = input; startGroup; ]

ASPAnalyze[ t Exp[- a t] Cos[3 Pi t / 16] CStep[t], t, 0, 3 Pi ]
:[font = print; inactive; ]
These variables were assigned a value of 1: {a}
               3 Pi t
t CStep[t] Cos[------]
                 16
----------------------
           t
          E
has the following Laplace transform:
-256 (-16 + 3 Pi - 16 s) (16 + 3 Pi + 16 s)
-------------------------------------------
                 2                2 2
      (256 + 9 Pi  + 512 s + 256 s )
The strip of convergence is:
-1 < Re(s) < Infinity
The system is stable.
 
The zeroes are:  {-1.58905, -0.410951}
The poles are:   {-1. + 0.589049 I, -1. - 0.589049 I, 
 
   -1. + 0.589049 I, -1. - 0.589049 I}
               3 Pi t
t CStep[t] Cos[------]
                 16
----------------------
           t
          E
has the following frequency response:
-256 (-16 + 3 Pi + -16 I w) (16 + 3 Pi + 16 I w)
------------------------------------------------
                  2                  2 2
       (256 + 9 Pi  + 512 I w - 256 w )
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        3                 1
Delta[-(-) + t] + Delta[-(-) + t]
        2                 2
has the following Laplace transform:
      s
 1 + E
--------
 (3 s)/2
E
The strip of convergence is:
-Infinity < Re(s) < Infinity
The system is stable.
        3                 1
Delta[-(-) + t] + Delta[-(-) + t]
        2                 2
has the following frequency response:
 (-3 I)/2 w       I w
E           (1 + E   )
:[font = message; inactive; ]
ASPAnalyze::notinteresting: 
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LTransData[(1 + E^s)/E^((3*s)/2), Rminus[DirectedInfinity[-1]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                 s
            1 + E
LTransData[--------, Rminus[-Infinity], Rplus[Infinity], 
            (3 s)/2
           E
 
  LVariables[s]]
:[font = section; inactive; startGroup; Cclosed; ]
Bug Fixes to the Signal Processing Packages
:[font = text; inactive; ]
		The following problems have been reported by various users.  The ones which are starred have been fixed.   If you find any other problems with the signal processing packages, please email  this information to evans@eedsp.gatech.edu.
:[font = subsection; inactive; startGroup; Cclosed; ]
Operators and Signals
:[font = text; inactive; ]
    *1. Shift[-7,n][Sinc[Pi/10] ] these fail to get to the magnitude plot
    without the shift it's OK;
:[font = text; inactive; ]
    *2. Pulse[9,0,n+4] fails, but Pulse[9,-4,n] works w.r.t. stability (discrete
    pulses are Pulse; continuous pulses are CPulse; pulses now take two
    arguments only --  length and t).
:[font = text; inactive; ]
   *3. Shift works in the opposite direction;
:[font = text; inactive; endGroup; ]
   *4. Integrating delta functions over a finite interval does not work.
:[font = subsection; inactive; startGroup; Cclosed; ]
Transforms
:[font = text; inactive; ]
    *1. The transforms do not justify their answers (now use Dialogue -> All
	 and the rule base will display each step of the transform process), and
:[font = text; inactive; ]
	*2.  The transforms do not handle the convolution operator.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
LaPlace/InvLaPlace
:[font = text; inactive; ]
    *1. The inverse Laplace transform of Log[s] / s is incorrect;
:[font = text; inactive; ]
	*2. The forward Laplace transform of (t -1) CStep[t] was incorrect; and
:[font = text; inactive; ]
	*3. The inverse Laplace transform uses MyApart to work around two
	drawbacks to Apart:  (a) it cannot handle repeated complex-valued roots
	that arise from rational polynomials, and (b) it cannot find the approximate
	roots when one or more the polynomial coefficients is real-valued.  If the
	user would like the inverse to use approximate roots, then the option
	Apart -> All must be specified.  The default is Apart -> Rational.
:[font = text; inactive; endGroup; ]
	*4. The inverse Laplace transform was not always correctly inverting
	terms that correspond to integration in the time domain.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
DTFTransform/InvDTFTransform
:[font = text; inactive; ]
   *1. The DTFT of Sinc squared fails.....  (Sinc[Pi/5 n])^2;
:[font = text; inactive; ]
    *2. The forward DTFT:  certain transform pairs are incorrect
	 like Cos[w0 n + phi] and Sin[w0 n + phi];
:[font = text; inactive; ]
    *3. The multidimensional inverse DTFT does not work at all;
:[font = text; inactive; ]
    *4. The inverse DTFT could not invert pulses or Dirichlet functions;
:[font = text; inactive; ]
    *5. The DTFT of a step function is missing a pi in the second term;
:[font = text; inactive; endGroup; ]
    *6. The interaction between the forward DTFT and the forward z-transform
          does not work for unresolved z-transforms.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
InvZTransform
:[font = text; inactive; ]
   *1. the following example does weird things:

        InvZTransform [ { (1 - b z^-1)/( z^-1 - b ), Abs[1/b], Infinity },
			z, n, Dialogue -> True ]
			
	It gives a phony partial fraction expansion, has trouble with b
	as a constant, and has the impulse at n=0 problem (seems to be
	hung up on "b"--doesn't really think it is a constant);
:[font = text; inactive; ]
    *2. The inverse z-transform could not handle common transform
    functions with a denominator of (1 - 2 r Cos[w0] z^-1 + r^2 z^-2),
    even when r = 1;
:[font = text; inactive; ]
    *3. The inverse z-transform required a ROC of |z| > 1 to invert a
	 pulse instead of |z| > 0;
:[font = text; inactive; endGroup; ]
    *4. The inverse z-transform of the z-transform of a two-sided
	 decaying exponential put over a common denominator.  That is,

	 InvZTransform[ Together[ZTransform[(1/3)^Abs[n], n, z]], z, n ] ;
:[font = subsubsection; inactive; startGroup; Cclosed; ]
ZTransform
:[font = text; inactive; ]
    1. Transforms of 3-D line impulses;
:[font = text; inactive; ]
   *2. impulses at n=0 are not handled properly in inverse z;
:[font = text; inactive; ]
   *3. ZTransform[ (1/3)^Abs[n], n ] gave

             1           1
	 -------- + -----------
	       1                 1
     1 - ---      3 (1 - -) z
           3 z              z

	instead of

             1          z
	 -------- + --------- ;
	       1                1
     1 - ---     3 (1 - -)
           3 z             z
:[font = text; inactive; ]
    *4. the left-sided case still has a problem; x[n] = (1/2)^n Step[-n] has 
    a z-transform of 1/(1 - 2z), BUT the inverse transform of 1/(1 - 2z) is
    wrong !  It gives 1/(1/2)^n Step[-n], i.e., 2^n Step[-n].  I think the
    problem can be solved by removing the n -> -n  substitution in the left-
    sided rule;
:[font = text; inactive; ]
    *5. A ROC check is missing:

         3^Abs[n]  has no legitimate z-transform, but the ZTransform
	 package gives one with Rminus = 3 and Rplus = 1/3 !!!!

	 SO we need to check that Rminus < Rplus;
:[font = text; inactive; ]
    *6. The ROC is incorrect for the forward transform of a pulse;
:[font = text; inactive; endGroup; endGroup; ]
    *7. The forward z-transform of Step[n] / ( n + 1 ) is off by a
	 minus sign;
:[font = subsection; inactive; startGroup; Cclosed; ]
MagPhasePlot
:[font = text; inactive; ]
    *1. Example #1 in the paper is not right. The frequency response of an upsampled
    signal should repeat every 2 Pi / L where L is the upsampling factor;
:[font = text; inactive; ]
   *2. Limit magnitude plots to the range -80 dB to infinity (this is now
   adjustable via PlotRange option to MagPhasePlot);
:[font = text; inactive; ]
   *3. Plotting delta functions on a MacIntosh;
:[font = text; inactive; ]
   *4. Properly labeling logarithmic frequency scale;
:[font = text; inactive; ]
    *5. There is no way to disable the magnitude or phase plot when
	 calling MagPhasePlot.  (Now, set PhaseMagRange -> Null to disable
	 the phase plot or set MagPhaseRange -> Null to disable the
	 magnitude plot);
:[font = text; inactive; endGroup; ]
	*6. MagPhasePlot can not directly plot the frequency response of DTFT, DFT,
	or CTFT objects.
:[font = subsection; inactive; startGroup; Cclosed; ]
PoleZeroPlot
:[font = text; inactive; ]
	*1. PoleZeroPlot does not shade the region of convergence;
:[font = text; inactive; ]
    *2. PoleZeroPlot prints out the poles and zeroes before they plot
	 them (poles and zeroes are plotted only if the Dialogue option is True or All).
:[font = text; inactive; ]
    *3. PoleZeroPlot does not always correctly report zeroes at z = 0.
:[font = text; inactive; endGroup; ]
	*4. Two-dimensional version of PoleZeroPlot displays two pole-zero
	diagrams (or root maps) for a symmetric transfer function even though
	only one is necessary;
:[font = subsection; inactive; startGroup; Cclosed; ]
DSPAnalyze
:[font = text; inactive; ]
    *1. DSPAnalyze[ Sinc[n] ]  fails to plot at all..... it bombs it also gives
    a message of power::undefined 1/0;
:[font = text; inactive; ]
    *2. For the DSPAnalyze[ n a^n Step[n], n], I get the message
    PoleZeroPlot:: not rational, even though it gives a rational
    function for the answer;
:[font = text; inactive; ]
    *3. DSPAnalyze[ Pulse[10,0,n] ] says it is not stable (which is wrong
    I think it is missing the pole-zero cancellation, except that the list of
    poles and zeros is correct--there are 9 of them);
:[font = text; inactive; ]
   *4. Sinc[Pi/10 (n-7)] through DSPAnalyze;
:[font = text; inactive; ]
	*5.  DSPAnalyze breaks down when the discrete-time function is complex
	valued, so we only plot the real values of the sequence; and
:[font = text; inactive; endGroup; ]
	*6. DSPAnalyze of IIR and FIR objects crashes and burns---  cannot plot their 
	time-domain values and PoleZeroPlot complains about poles occurring in the
	region of convergence.
:[font = subsection; inactive; startGroup; Cclosed; ]
ASPAnalyze
:[font = text; inactive; endGroup; endGroup; ]
    *1. ASPAnalyze[ Sinc[ Pi t ]]  has a glitch in the plot; without the Pi,
    it works OK;
:[font = section; inactive; startGroup; Cclosed; ]
Signal Processing Packages 2.4
:[font = text; inactive; ]
		Version 2.4 of the signal processing packages is the first version that works simultaneously under version 1.2 and version 2.0 of Mathematica.  The major changes are reported below.
:[font = subsection; inactive; startGroup; Cclosed; ]
New Objects
:[font = subsubsection; inactive; startGroup; Cclosed; ]
IIRFunction
:[font = text; inactive; ]
		IIR objects can be converted into a computational form called IIRFunction, and FIR objects can be rewritten as a formula (this conversion is triggered by the TheFunction object).  Their values can now be plotted.  Values are calculated as they are needed, and IIRFunction remembers (caches) previous calculations.
:[font = input; endGroup; ]
	TheFunction[ IIR[n, {1, 1/2, 1/4}] ]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
MyApart
:[font = text; inactive; ]
		In Mathematica 1.2, the Apart function is not robust because it does not handle repeated complex-valued roots in the denominator that arise when factoring rational polynomials.  For example, Apart does not properly decompose F(s) = ( s^2 - 1 ) / (s^2 + 2 s + 2)^2.  A new function MyApart does.
:[font = input; startGroup; ]
Apart[ (s^2 - 1) / ( s^2 + 2 s + 2 )^2, s ]
:[font = output; inactive; output; endGroup; ]
(-3 - 2*s)/(2 + 2*s + s^2)^2 + (2 + 2*s + s^2)^(-1)
;[o]
   -3 - 2 s            1
--------------- + ------------
            2 2              2
(2 + 2 s + s )    2 + 2 s + s
:[font = text; inactive; ]
But this is really not the complete partial fractions decomposition.  Rather, this is:
:[font = input; startGroup; ]
MyApart[ (s^2 - 1) / ( s^2 + 2 s + 2 )^2, s ]
:[font = output; inactive; output; endGroup; ]
(1/4 + I/2)/(1 - I + s)^2 + -I/4/(1 - I + s) + 
  (1/4 - I/2)/(1 + I + s)^2 + I/4/(1 + I + s)
;[o]
   1   I          -I          1   I           I
   - + -          --          - - -           -
   4   2          4           4   2           4
------------ + --------- + ------------ + ---------
           2   1 - I + s              2   1 + I + s
(1 - I + s)                (1 + I + s)
:[font = text; inactive; endGroup; ]
		Apart does not have an option to handle polynomials with real-valued coefficients.  MyApart takes a third argument which is a filter placed on the output of the Solve command which find the roots of the denominator.  If it is N, then the approximate roots will be used.
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RootLocus
:[font = text; inactive; ]
		RootLocus plots root loci for one varying parameter in a function of one variable.
:[font = input; endGroup; ]
	RootLocus[ 1 / ( s^2 + K s + K ), s, {K, 0, 5, 0.1} ]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
SignalPlot
:[font = text; inactive; ]
		SignalPlot that plots the real and imaginary components of a continuous function of one variable on the same plot, as well as Dirac delta functions.  It also plots functions of two variables.
:[font = input; endGroup; ]
	SignalPlot[ Cos[t] + I Sin[t], {t, 0, 4 Pi} ]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
SequencePlot
:[font = text; inactive; ]
		SequencePlot now provides a standard way to plot discrete-time expressions. It can plot either functions of one variable or functions of two variables.  The one-variable plotting appears in the conventional lollipop style.  This is the discrete-time version of SignalPlot.
:[font = input; endGroup; endGroup; ]
	SequencePlot[ IIR[n, {1, 1/2, 1/4}], {n, -5, 15} ]
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New Capabilities
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Justification
:[font = text; inactive; ]
		As mentioned in the opening paragraphs, all 10 transform rule bases can fully justify their answers.  The value of the option Dialogue indicates the level of justification:  False (none), True (only indicate when strategies are used), and All (show each step of the transformation process).
:[font = input; ]
	LaPlace[ A Exp[a t] Cos[b t] CStep[t], t, s,
			 Dialogue -> All ]
:[font = text; inactive; endGroup; ]
The linear constant coefficient difference and differential equation solvers (ZSolve and LSolve) also support three levels of justification (none, some, and complete) indicated by the value of the dialogue option (False, True, All).
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Transforms of Abstract Signals
:[font = text; inactive; ]
		The various transform rule bases allows users to specify their own transform pairs.  This can allow a user to transform general functions, like x[n], without ever defining them.  For example,
:[font = input; startGroup; Cclosed; ]
	ZTransform[ Shift[L,n][x[n]], n, z,
				TransformLookup -> { x[n] :> X[z] } ]
:[font = output; inactive; output; endGroup; ]
ZTransData[X[z]/z^L, Rminus[0], 
  Rplus[DirectedInfinity[1]], ZVariables[z]]
;[o]
           X[z]
ZTransData[----, Rminus[0], Rplus[Infinity], 
             L
            z
 
  ZVariables[z]]
:[font = text; inactive; ]
In this example, the abstract signal x[n] is shifted by L to produce another abstract signal.  The resulting z-transform is z^(-L) X(z), where x[n] <--> X[z].  Now, the z-transform can be manipulated as an algebraic expression.
:[font = text; inactive; ]
		In N dimensions, the transform rule bases apply a one-dimensional rule base N times.  For example, in the two-dimensional case,
:[font = input; startGroup; Cclosed; ]
ZTransform[ x[n1,n2], {n1,n2}, {z1,z2},
			TransformLookup -> { x[n1,n2] :> X[z1, z2] } ]
:[font = message; inactive; startGroup; ]
Transform::incomplete: 
   The rule base could not compute the 
    forward z-transform of 1 with respect to n2.
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MakeZObject[SignalProcessing`Digital`ZTransform`Private\
    `MultiDROC[MakeZObject[ScaleT[ZForm, 
     SignalProcessing`Digital`ZTransform`Private`forwar\
       dz[1, n2, z2, {True, True, True, False, False}, 
      {n1, n2}, {z1, z2}, {Dialogue -> True}], 
     X[z1, z2]], z2], ZTransData[X[z1, z2], Rminus[0], 
    Rplus[DirectedInfinity[1]], ZVariables[z1]]]]
;[o]
-Incomplete z-Transform-
:[font = text; inactive; ]
so a user cannot specify a two-dimensional transform of an abstract function in this way.  Instead, a two-dimensional transform pair would be specified using a partial transform X1[z1, n2] which represents the z-transform of x[n1,n2] w/r to n1:
:[font = input; startGroup; Cclosed; ]
ZTransform[ x[n1,n2], {n1,n2}, {z1,z2},
			TransformLookup -> { x[n1,n2] :> X1[z1, n2],
								 X1[z1,n2] :> X[z1, z2] } ]
:[font = output; inactive; output; endGroup; endGroup; ]
ZTransData[X[z1, z2], Rminus[{0, 0}], 
  Rplus[{DirectedInfinity[1], DirectedInfinity[1]}], 
  ZVariables[{z1, z2}]]
;[o]
ZTransData[X[z1, z2], Rminus[{0, 0}], 
 
  Rplus[{Infinity, Infinity}], ZVariables[{z1, z2}]]
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LaPlace transforms
:[font = text; inactive; ]
	Mathematica 1.2 has a problem performing partial fractions decomposition on real-valued polynomials even if you do all the factoring yourself.  So, we have added a new option to the InvLaPlace object called Apart.  If set to All, then we kludge around the Apart deficiency so that a partial fractions decomposition is formed.  Otherwise, we inverse transform the function as an IIR filter whose input is the inverse transform of the numerator.
:[font = input; endGroup; ]
	LaPlace[ 1 / ( s^2 + 2.0 s + 1), t, s, Apart -> All ]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Fourier Transforms
:[font = text; inactive; ]
		We have added many transform pairs to the continuous-time Fourier transform rule bases CTFTransform and InvCTFTransform
:[font = text; inactive; startGroup; ]
		The Fourier transform rule bases now handle the convolution operator:
:[font = input; endGroup; endGroup; ]
	CTFTransform[ Convolve[t] [	CPulse[1, t + 1/2],
								CPulse[1, t + 1/2] ], t, w ]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
z-Transform
:[font = text; inactive; ]
		The z-transform of IIR structures returns a valid region of convergence.		
:[font = input; endGroup; endGroup; ]
	ZTransform[ IIR[n, {1, 1/2, 1/4}], n, z ]
:[font = subsection; inactive; startGroup; Cclosed; ]
Miscellaneous Improvements
:[font = text; inactive; ]
		The functional form of the Upsample function did not return values when the sample index was not an integer.  This was a problem when the sample index was real-valued integer like 1.0.  This disrupted the plotting of upsampled sequences.  No longer.
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	SequencePlot[Upsample[3,n][Cos[Pi n/8] Step[n]], {n, 0, 48}] 
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The Unformatted text for this cell was not generated.
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when Unformatted text is generated.
;[o]
-Graphics-
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Another problem with SequencePlot was plotting 2-D signals.  It did not sample the function at integer indices, so the plot below 
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SequencePlot[ (1/2)^n1 (4/5)^n2 Multinomial[n1,n2] Step[n1,n2],
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		The Piecewise Convolution package for convolving piecewise-continuous functionsis now more consistent in that either argument to the PiecewiseConvolution function can be in list or expression form.  See the PiecewiseConvolution notebook for examples.
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		CFIR and CIIR structures have an optional third argument: Roots -> roots.  See the AnalogFilters notebook for examples.
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Initialization with Stubs
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		Mathematica 2.0 allows the association of a package with a set of symbols without having to load the package.  This means that initialization now occurs very rapidly even for a large set of extensions and that only packages will be automatically loaded as needed.  Stubs are now defined for all  of the objects in the signal processing packages.  This is what the initialization files "Analog.m", "Digital.m", and "Support.m" do under Mathematica 2.0.
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Impact
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		Because of the changes, we can recreate (for the most part) the pictorial dictionary of Fourier transforms in chapter 19 of Bracewell's book entitled "The Fourier Transform and Its Applications".
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		Because of the changes and because of subtle modifications to the ExtractVariables object, DSPAnalyze now works correctly for IIR and FIR structures.
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		We have updated all eight of the notebooks accompanying the signal processing packages to reflect the above enhancements.
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This work was supported in part by the Joint Services Electronics Program
contract #DAAL-03-90-C-0004.  
^*)