(*^

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Mathematica as an Educational Tool for Signal Processing
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by Brian Evans, James McClellan, and Kevin West
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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 is the notebook version of the 1991 IEEE Southeastern Conference (Southeastcon '91) paper entitled ``Mathematica as an Educational Tool for Signal Processing''.
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Abstract
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 		This paper describes extensions for a symbolic mathematics program that make it an effective educational tool for courses in linear systems theory.  These extensions are in the form of packages and notebooks for the Mathematica environment. The new signal processing packages contain rule bases that symbolically compute convolutions and all the transforms common to signal processing:  Fourier, Laplace, z, etc.  In order to introduce a student to these concepts and how they are implemented in Mathematica, we have written four tutorials in Mathematica's notebook format.  Notebooks, a form of hypertext, allow students to access material randomly, to discover insights at their own pace, and to evaluate code provided in the examples.  Students can also use notebooks to solve homework problems in a self-documenting fashion.
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Initialization
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*)
Needs[ "SignalProcessing`SignalProcessing`" ]
(*
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Loading the signal processing support packages ...
Support module has been loaded.
Supporting routines for filter design are loaded.
The knowledge base of signal processing operators has been\
 
  loaded.
Region of convergence routines for the transform rule bases\
 
  are loaded.
Supporting routines for transform rule bases are loaded.
Loading in the digital signal processing packages ...
Supporting routines for the z-transform rule bases are\
 
  loaded.
The supporting functions are loaded for symbolic Fourier\
 
  transforms.
The forward z-transform rules have been loaded.
The inverse z-transform rules have been loaded.
The discrete-time Fourier transform (DTFT) rule bases are\
 
  loaded.
Correlation functions are now loaded.
ZSolve, a difference equation solver, is loaded.
The digital signal analyzer is now loaded.
The discrete Fourier transform (DFT) rule bases are loaded.
 
The digital signal processing extensions has been loaded.
Good starting points are the objects DSPAnalyze and\
 
  ZTransform.
Active packages are maintained in the variable\
 
  $ContextPath.
  New signal primitives are in SPsignals.
  New system primitives are in SPoperators.
  New Mathematica functions are in SPfunctions.
Note that all signal processing expressions will
be automatically simplified.
 
Loading the signal processing support packages ...
Loading in the analog signal processing packages ...
Piecewise convolution rules have been loaded.
The one-dimensional analog filter design objects are\
 
  loaded.
Supporting routines and objects for the Laplace transform\
 
  are loaded.
The forward Laplace transform rule base LaPlace has been\
 
  loaded.
The inverse Laplace transform rule base InvLaPlace has been\
 
  loaded.
The continuous Fourier transform (CTFT) functions are\
 
  loaded.
LSolve, a differential equation solver, is loaded.
The analog signal analyzer is now loaded.
 
The analog signal processing extensions has been loaded.
Good starting points are the objects ASPAnalyze and\
 
  LaPlace.
Active packages are maintained in the variable\
 
  $ContextPath.
  New signal primitives are in SPsignals.
  New system primitives are in SPoperators.
  New Mathematica functions are in SPfunctions.
Note that all signal processing expressions will
be automatically simplified.
 
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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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Digital::loaded: 
   Routines, objects, and rules for the digital signal
    processing package have been loaded successfully.
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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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Analog::loaded: 
   Routines, objects, and rules for the analog signal
    processing package have been loaded successfully.
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Introduction
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 		Several symbolic mathematical programs have recently become available on a wide variety of machines.  This paper discusses the use of one such environment, Mathematica, to supplement signal processing lectures and laboratories.  One attraction to using Mathematica in an educational setting is its notebook facility [1].  Not only can instructors write highly interactive tutorials in the notebook format, but students can use notebooks to solve homework problems in a self-documenting fashion.  Notebooks (a form of hypertext) arrange formatted text, graphs, and Mathematica code in a hierarchy of cells.  Students can discover insights at their own pace because they can access material randomly and ask for more information on a subject [2].  Most important, though, is the ability to evaluate and tweak existing code, as well as to write, edit, and run new code, all while perusing a notebook.
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		Since Mathematica is primarily a ``system for doing mathematics by computer'' [1], its knowledge of operations and structures common to signal processing is limited.  Therefore, we have developed new packages that enable Mathematica to understand signals, operators, and transforms [3,4].  These signal processing packages contain rule bases that symbolically compute convolutions and all the transforms common to signal processing:  Fourier, Laplace, z, DFT, etc.  The transforms can fully justify their answers so that students can view each step of the transformation process.
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 		Accompanying these signal processing packages are several notebooks.  One summarizes all of the new definitions, and another provides a brief overview of the new capabilities.  A third notebook is this paper itself.  The remaining notebooks are tutorials discussing piecewise convolution, analog filter design, the z-transform, and the Laplace transform.  We discuss all four tutorial notebooks in the sections following the introduction to Mathematica and the overview of the signal processing packages.
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Mathematica
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		The Mathematica environment consists of two distinct pieces: a front-end user interface and a back-end kernel [1].  On a Macintosh and a NeXT machine, the front-end interface is none other than the notebook format mentioned in the introduction.  Notebooks use the machine's windowing system and rely on a mouse to select menu options (keyboard short-cuts are also supported).  Notebooks also provide on-line help in the form of text mixed with useful examples, although the kernel does provide its own simple help facility.  For example,
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?*Transform
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CTFTransform       InvDFTransform     MultiDInvTransform
DFTransform        InvDTFTransform    MultiDTransform
DTFTransform       InvLaPlace         Transform
FilterTransform    InvZTransform      ZTransform
InvCTFTransform    LaPlace
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will return the names of all symbolic transforms in the signal processing packages,
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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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explains how to call the forward z-transform rule base properly, and
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Options[ZTransform]
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{Dialogue -> True}
;[o]
{Dialogue -> True}
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lists the default options for an object (routine).  On other machines, the front-end is nothing more than a line-by-line interface to the kernel, although notebook readers are available for these machines [2].  It is the kernel that crunches numbers, manipulates symbols, and generates graphics, but it is the front-end that displays answers and renders graphics.
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		The Mathematica kernel supports many programming paradigms, including procedural programming (like Fortran) and functional programming (like APL).  Mathematica even has its own version of object-oriented (like C++) and data-directed (like Lisp) programming.  These styles are certainly useful for implementing functions and operators common to signal processing, but we still need a convenient way to take transforms.  To this end, Mathematica provides a style of logic programming similar to Prolog and a powerful pattern matcher [5].
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		The key to the generality of the pattern matcher is that all data, functions, and expressions are represented in the same format--- as objects.  Each object contains a head, which serves as the object's tag and data type, and a collection of other objects.  For example, a list of the elements 1, a, 2.0, and c + d x would be represented as
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List[1, a, 2.0, c + d x]
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		The Mathematica kernel, of course, possesses much inherent knowledge about mathematical structures and operations.  It supports integer, rational, real, imaginary, and complex numeric data types, maintaining rational numbers and integers to an exact number of digits and representing floating point numbers to an arbitrary number of digits.  Besides addition, subtraction, multiplication, division, and exponentiation, Mathematica provides over 150 mathematical operators and functions.  For example, it can recognize, rearrange, factor, decompose, add, subtract, multiply, and divide polynomials.  In the signal processing packages, the transform rule bases rely on many of Mathematica's inherent abilities, especially the symbolic operations of partial fractions decomposition, series expansion, integration, and differentiation.
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Signal Processing Extensions
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		Mathematica provides many of computational functions common to linear systems and signal processing--  exponentials, sinusoids, Bessel functions, etc.  This environment, however, is missing functions that are basic components of signals, like step and impulse functions.  The signal processing packages introduce the continuous and discrete functions shown below in Table 1.  These new signal primitives are maintained the list SPsignals:
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SPsignals
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{"ASinc", "AliasSinc", CPulse, CStep, Delta, 
  Dirichlet, FIR, IIR, Impulse, LineImpulse, 
  Pulse, Sinc, Step, Unit}
;[o]
{ASinc, AliasSinc, CPulse, CStep, Delta, 
 
  Dirichlet, FIR, IIR, Impulse, LineImpulse, 
 
  Pulse, Sinc, Step, Unit}
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Table 1:   New Computational Functions
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ASinc        Dirichlet discrete kernel
AliasSinc    Dirichlet discrete kernel
CPulse       continuous pulse         
CStep        continuous step
Delta        Dirac delta
Dirichlet    Dirichlet discrete kernel
FIR          finite impulse response
IIR          infinite impulse response
Impulse      Kronecker delta
Pulse        discrete pulse
Sinc         Sine x over x
Step         discrete step
Unit[k]      order-continuous unit function
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( MORE INFORMATION )
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CFIR[t, {h0, h1, h2, ...}] represents the finite impulse
   response of an all-zero analog filter.  Input to the
   filter is specified as FIR[t, h] [ x[t] ], where x[t] is
   some symbolic signal processing expression.  For the
   generalized N-dimensional FIR structure, the argument t
   becomes a list of indices (like {t1, t2} for N=2) and
   the feedback coefficients are in an N x N matrix.
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CIIR[t, {a0, a1, a2, ...}] represents the infinite impulse
   response of an all-pole analog filter with gain 1, where
   a1, a2, ..., are the feedback coefficients and a0 is the
   scaling factor of the output. By default, the input to
   the filter structure is a Dirac delta function. An
   alternate input h can be specified by the code CIIR[t,
   {a0, a1, a2, ...}][h]. For the generalized N-dimensional
   IIR structure, the argument t becomes a list of indices
   (like {t1, t2} for N=2) and the feedback coefficients
   are specified by an N x N matrix.
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CPulse[l,t] defines a pulse which begins at t=0 and ends at
    t = l.  The CPulse has value 1 within the range (0,l),
   0 outside this range, and 1/2 at the points t=0 and t=l.
   A continuous-pulse center at the origin is written as
   CPulse[l, t + l/2] or Shift[-l/2, t][CPulse[l, t]].
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CStep[t], a.k.a. Unit[-1][t], is the unit step function
   which is 1 for t > 0, 0 for t < 0, and 1/2 at t = 0. It
   is commonly used for continuous expressions t. See also
   Step and Unit.
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Delta[expr] is the Dirac delta function.  The area under
   this functions is 1 but it only has value at the origin.
   That is, Integrate[ Delta[t] g[t], {t, t1, t2} ] is g[0]
   if t1 <= 0 <= t2, 0 otherwise.  It differs from the
   Kronecker  delta function Impulse[t].
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Dirichlet[N, w] is the aliased sinc function.  Its
   magnitude is of the form Sin[N w / 2] / ( N Sin [w / 2]
   ).
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FIR[n, {h0, h1, h2, ...}] represents the finite impulse
   response of an all-zero digital filter.  As a function
   of n, the response is actually equal to h0, h1, h2, ...,
   for n = 0, 1, 2, ....  Note that FIR[n, 2] will be
   simplified to 2 Impulse[n].  FIR can also represent an
   all-zero filter.  In this case, the input x[n] to the
   filter is specified as FIR[n, h] [ x[n] ], where x[n] is
   some symbolic signal processing expression.  For the
   generalized N-dimensional FIR structure, the argument n
   becomes a list of indices (like {n1, n2} for N=2) and
   the feedback coefficients are in an N x N matrix.
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IIR[n, {a0, a1, a2, ...}] represents the infinite impulse
   response of an all-pole filter with gain 1, where a1,
   a2, ..., are the feedback coefficients and a0 is the
   scaling factor of the output. In discrete time, the
   output of IIR is the solution to y[n] in the difference
   equation  a0 y[n] + a1 y[n-1] + ... + aN y[n - N] =
   impulse[n],  with y[n] = 0 for n < 0. By default, the
   input into the filter (a.k.a. the forcing function) is
   an impulse.  An alternate input h can be specified by
   the code IIR[n, {a0, a1, a2, ...}][h]. For the
   generalized N-dimensional IIR structure, the argument n
   becomes a list of indices (like {n1, n2} for N=2) and
   the feedback coefficients are specified by an N x N
   matrix.
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Impulse[t] is the unit Kronecker Delta function.  At t = 0,
   the function evaluates to 1.  Elsewhere, it evaluates to
   0.
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Sinc[t] is the unit sinc function which is the ratio of
   Sin[t] over t.  Note that Sinc[0] = 1.
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Step[n] is the unit step function which is 1 for n >= 0 and
   0 otherwise.  It is commonly used for discrete n. This
   function differs from CStep only at the origin.
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Unit[n][t] is the nth order continuous unit function in t.
   Unit[0][t] is the Dirac delta function (see Delta).
   Unit[-1][t] is the continuous step function (see CStep).
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New Operators
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		The signal processing packages also introduce common signal processing operators, such as reverse and summation [4].  When one of these operators is applied to a function, no evaluation takes place immediately.  Instead, the operation is represented symbolically, until it becomes convenient to compute it explicitly.  By deferring evaluation of operators, certain obvious simplifications can be carried out
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InvZ[z,n][ Z[n,z][ f[n] ] ]
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f[n]
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and code can be generated for the original operation (e.g., the equivalent TeX code).  The user can force an evaluation by applying to the expression, much as a numerical evaluation in Mathematica is triggered by the action of.  Thus, represents the z-transform of, but the transformation is not carried out.
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		The new signal processing operators are maintained in SPoperators:
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SPoperators
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{Aliasby, CConvolve, Convolve, DFT, DTFT, 
  Difference, Downsample, FIR, FT, IIR, 
  Interleave, InvDFT, InvDTFT, InvFT, InvL, 
  InvZ, L, Periodic, PolyphaseDownsample, 
  PolyphaseUpsample, Rev, ScaleAxis, Shift, 
  Summation, Upsample, Z}
;[o]
{Aliasby, CConvolve, Convolve, DFT, DTFT, 
 
  Difference, Downsample, FIR, FT, IIR, 
 
  Interleave, InvDFT, InvDTFT, InvFT, InvL, 
 
  InvZ, L, Periodic, PolyphaseDownsample, 
 
  PolyphaseUpsample, Rev, ScaleAxis, Shift, 
 
  Summation, Upsample, Z}
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Table 2: New Operators
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( More Information on Operators )
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Aliasby[m,w][x] represents the aliasing of x,
   a continuous-frequency signal, by
   replicating it every (2 Pi / m) in w and
   dividing the result by m.
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CConvolve[t][x1, x2, ...] represents the
   continuous convolution in t of functions
   x1, x2, .... CConvolve[{t1, t2, ...}][x1,
   x2, ...] represents the multidimensional 
   continuous convolution in t1, t2, ... of
   functions x1, x2, ...
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Convolve[n][x1, x2, ...] represents the
   discrete convolution in n of functions x1,
   x2, .... Convolve[{n1, n2, ...}][x1, x2,
   ...] represents the multidimensional 
   discrete convolution in n1, n2, ... of
   functions x1, x2, ...
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DFT[L, n, k] is the forward discrete Fourier
   transform operator. Here, L is the DFT
   length(s), n is the time-domain
   variable(s), and k is the frequency-domain
   variable(s). Multidimensional DFT's are
   supported. Applying TheFunction to the DFT
   operator will invoke the DFT rule base
   (object DFTransform) if loaded.
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DTFT[n, w] is the forward discrete-time
   Fourier transform operator. Here, n is the
   time-domain variable(s), and w is the
   frequency-domain variable(s).
   Multidimensional DTFT's are supported.
   Applying TheFunction to the DTFT operator
   will invoke the DTFT rule base DTFTransform
   if loaded.
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Difference[k,n] represents the k-th backward
   difference with respect to the variable n.
   The first-order difference of f equals f[n]
   - f[n-1], the second-order difference is
   f[n] - 2 f[n-1] + f[n-2], and so on.
   Applying this operator to the sequence x[n]
   would be written as Difference[k,n][x[n]].
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Downsample[m,n] represents the downsampling
   operator. Downsampling resamples a function
   at the points n = 0, m, 2m, 3m, ... In N
   dimensions, n is a list of N indices and m
   is an N x N matrix. Applying TheFunction to
   f[n] which is downsampled in one dimension,
   (written as Downsample[m, n][ f[n] ]) will
   produce f[m n].
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FIR[n, {h0, h1, h2, ...}] represents the
   finite impulse response of an all-zero
   digital filter. As a function of n, the
   response is actually equal to h0, h1, h2,
   ..., for n = 0, 1, 2, ....  Note that
   FIR[n, 2] will be simplified to 2
   Impulse[n]. FIR can also represent an
   all-zero filter. In this case, the input
   x[n] to the filter is specified as  FIR[n,
   h] [ x[n] ], where x[n] is some symbolic
   signal processing expression. For the
   generalized N-dimensional FIR structure,
   the argument n becomes a list of indices
   (like {n1, n2} for N=2) and the feedback
   coefficients are in an N x N matrix.
:[font = info; inactive; locked; ]
FT[t, w] is the forward continuous Fourier
   transform operator. Here, t is the
   time-domain variable(s) and w is the
   frequency-domain variable(s). Applying
   TheFunction to this operator will cause the
   invocation of the continuous-time Fourier
   transform rule base (object CTFTransform)
   if loaded.
:[font = info; inactive; locked; ]
IIR[n, {a0, a1, a2, ...}] represents the
   infinite impulse response of an all-pole
   filter with gain 1, where a1, a2, ..., are
   the feedback coefficients and a0 is the
   scaling factor of the output. In discrete
   time, the output of IIR is the solution to
   y[n] in the difference equation  a0 y[n] +
   a1 y[n-1] + ... + aN y[n - N] = impulse[n],
    with y[n] = 0 for n < 0. By default, the
   input into the filter (a.k.a. the forcing
   function) is an impulse. An alternate input
   h can be specified by the code IIR[n, {a0,
   a1, a2, ...}][h]. For the generalized
   N-dimensional IIR structure, the argument n
   becomes a list of indices (like {n1, n2}
   for N=2) and the feedback coefficients are
   specified by an N x N matrix.
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Interleave[n][x0, x1, ...] interleaves samples
   of signals x0, x1, ..., which are functions
   of n. This is only applicable to discrete
   signals.
:[font = info; inactive; locked; ]
InvDFT[L, k, n] is the inverse discrete
   Fourier transform operator. Here, L is the
   DFT length(s), n is the time-domain
   variable(s), and k is the frequency-domain
   variable(s). Multidimensional inverse DFT's
   are supported. Applying TheFunction to the
   InvDFT operator will invoke the inverse DFT
   rule base (object InvDFTransform) if
   loaded.
:[font = info; inactive; locked; ]
InvDTFT[w, n] is the inverse discrete-time
   Fourier transform operator. Here, n is the
   time-domain variable(s), and w is the
   frequency-domain variable(s).
   Multidimensional inverse DTFT's are
   supported. Applying TheFunction to the
   InvDTFT operator will invoke the inverse
   DTFT rule base InvDTFTransform if loaded.
:[font = info; inactive; locked; ]
InvL[s, t] is the inverse Laplace transform
   operator. Applying TheFunction to the InvL
   operator will invoke the bilateral inverse
   Laplace transform rule base InvLaPlace if
   loaded.
:[font = info; inactive; locked; ]
InvZ[z, n] is the inverse z-transform
   operator. Applying TheFunction to the InvZ
   operator will invoke the inverse
   z-transform rule base ZTransform if loaded.
:[font = info; inactive; locked; ]
L[t, s] is the forward Laplace transform
   operator. Applying TheFunction to the L
   operator will invoke the forward bilateral
   Laplace rule base LaPlace if loaded.
:[font = info; inactive; locked; ]
Periodic[k,n][f] indicates that f is a
   function of n which is periodic with period
   of k.
:[font = info; inactive; locked; ]
PolyphaseDownsample[m, n][x, h] is
   equivalent to downsampling x by m and then
   convolving the result with h.
:[font = info; inactive; locked; ]
PolyphaseUpsample[l, n][x, h] is
   equivalent to upsampling x by l and then
   convolving the result with h.
:[font = info; inactive; locked; ]
Rev[n][x] represents the operation of
   reversing the direction of x, with respect
   to the variables(s) n.
:[font = info; inactive; locked; ]
ScaleAxis[l,w][x] represents the compression
   by a factor of l (an integer) of the
   continuous w axis of x, which is usually a
   continuous-frequency signal.
:[font = info; inactive; locked; ]
Shift[m,v][x] represents the shifting of the
   expression x by m samples to the right in
   the v direction, i.e. x[v] --> x[v - m].
:[font = info; inactive; locked; ]
Summation[i][f], Summation[i, iend][f], and
   Summation[i, istart, iend, inc][f]
   represents the summation of f with respect
   to variable i:  i = (istart) to (iend) step
   (inc). The Summation operator is an
   abstraction of Mathematica's Sum object.
   Applying the object TheFunction to a
   Summation operator will invoke the Sum
   object if istart, iend, and inc are
   numbers.
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Upsample[k,n][f] represents the upsampling
   operation on f, which is either an
   expression of n or a squence (list) of
   values. Upsampling inserts k-1 zeroes
   between every two samples. In N dimensions,
   n is a list of N indices and k is an N x N
   matrix.
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Z[n, z] is the forward z-transform operator.
   Applying TheFunction to the Z operator will
   invoke the z-transform rule base (object
   ZTransform) if loaded.
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Mixing Systems (Operators) and Signals (Functions)
:[font = text; inactive; startGroup; ]
		An example of combining signals and systems is
:[font = input; ]
Downsample[2, n] [ IIR [n, {5/2, 3/4, 1/8}] ]
:[font = text; inactive; ]
This expression represents the downsampling by 2 of h[n], the impulse response of an all-pole digital IIR filter with gain 5/2 and feedback coefficients -3/4 and -1/8.   In this case,  h[n] = 5/2 delta[n] + 3/4 h[n-1] + 1/8 h[n-2],  with h[-1] = h[-2] = 0.  As Mathematica code, h[n] could be defined using three rules:
:[font = input; ]
h[-1] := 0;
h[-2] := 1;
h[n_] :=
	5/2 Impulse[n] + 3/4 h[n-1] + 1/8 h[n-2]
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The values of h[n] can be cached as they are computed by altering the last line to
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h[n_] := h[n] =
	5/2 Impulse[n] + 3/4 h[n-1] + 1/8 h[n-2]
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Piecewise Convolution
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		The linear convolution of two functions f(t) and g(t) is defined as:
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 Here, f(t) and g(t) are piecewise-continuous functions.  Piecewise convolution is a time-domain approach that finds the convolution by computing the integral over each interval where the two functions overlap as g(t-u) ``slides'' through f(u).  This is usually the first approach for solving convolution problems that electrical engineers learn.
:[font = text; inactive; ]
		The piecewise convolution notebook shows the student how to use the piecewise convolution package to solve one-dimensional continuous-time convolution problems.  In the introduction, the notebook informs the user that a piecewise function can be represented as a list of intervals.  Each interval is itself a list containing the start of the interval, the end of the interval, and the function defined on that interval.  For example, the function  , a unit triangle defined on  , can be represented as an expression  
:[font = input; ]
(1 + t) CPulse[1, 1 + t] + (1 - t) CPulse[1, t] 
:[font = text; inactive; ]
or in list form as
:[font = input; ]
{{1+t, -1, 0}, {1-t, 0, 1}}
:[font = text; inactive; ]
		The notebook gives several examples of piecewise convolution using the newly defined object PiecewiseConvolution, which requires three arguments--- f(t), g(t), and t.  For example, the Mathematica dialogue
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PiecewiseConvolution[ Delta[t - t0], f[t], t ]
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{{f[t - t0], DirectedInfinity[-1], DirectedInfinity[1]}}
;[o]
{{f[t - t0], -Infinity, Infinity}}
:[font = text; inactive; ]
illustrates the shifting property of the Dirac delta function.  In this example, we could have used the piecewise form of Delta[t - a] , which is {Area[1], a, a}. The notebook also gives an example of a convolution that produces an unstable function:  
:[font = input; startGroup; ]
PiecewiseConvolution[ CStep[t], CStep[t], t ]
:[font = message; inactive; startGroup; ]
Integrate::divg: Integral does not converge.
:[font = output; inactive; output; endGroup; endGroup; ]
{{t, 0, DirectedInfinity[1]}}
;[o]
{{t, 0, Infinity}}
:[font = text; inactive; endGroup; ]
Part of computing a piecewise convolution requires identifying the overlapping intervals by drawing pictures of g(t-u) and f(u).  So, for two simple functions, the notebook shows an animation of g(t-u) sliding through f(u).  Each frame shows an important interval over which to integrate, and the reader can control the frame rate.
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The Laplace Transform
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		The more common approach to solving convolution problems utilizes transform theory, so the signal processing packages implement bilateral Laplace and Fourier transforms.  The rule bases LaPlace and InvLaPlace implement the bilateral Laplace transform F(s) of a continuous-time function f(t):  [6]
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(Note that (2) is actually a contour integral.)  The forward transform tracks the region of convergence (ROC), which is a strip along the real s-axis.  The inverse transform uses ROC information to determine the corresponding time function.  In the inverse transform, the contour of integration includes the section of the s-plane whose real components are bounded by the constant a, i.e., either Re(s) < a or Re(s) > a.
:[font = text; inactive; ]
		The Laplace transform notebook states the definition and properties of the bilateral Laplace transform.  It discusses strategies for finding forward and inverse transforms and tests over 100 transforms.  The notebook also shows how to use the Laplace transform to solve linear constant coefficient differential equations with initial conditions.  The last section of the notebook demonstrates how to use the Laplace transform in analysis of signals using Stable and ASPAnalyze.  For example, exp(a t) u(t) is stable if the real part of is negative, as shown by the following Mathematica dialogue (note that means the output of the previous command):  
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LaPlace[ Exp[a t] CStep[t], t]
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LTransData[(-a + s)^(-1), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[Re[a]], Rplus[Infinity], 
           -a + s
 
  LVariables[s]]
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Stable[%]
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Re[a] < 0
;[o]
Re[a] < 0
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The general analysis of a 1-D or 2-D time domain expression consists of graphing the time-domain function, finding the function's Laplace transform and displaying its poles and zeroes, checking for stability, and computing the Fourier transform and plotting the frequency response.   Figure 1 shows the general analysis of the function t exp(- a t) cos(3 pi t / 16) u(t), one of the two ASPAnalyze examples in the notebook (the second involves a sum of Dirac delta functions).  This functions depends on the time variable t and the free parameter a.  Whenever a time-domain expression is a function of free parameters, ASPAnalyze will manipulate the entire expression including the free parameters and temporarily assign a numeric value of to the free parameters only when generating a graph.  The user can specify the numeric value (default is 1 for each free parameter) by specifying options to ASPAnalyze--  in this case, one of the form a -> value:
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ASPAnalyze[t Exp[- a t] Cos[3 Pi t / 16] CStep[t],
			t, 0, 10, a -> 2 ]
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Figure 1:  Analog Signal Analysis
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ASPAnalyze[t Exp[-a t] Cos[3 Pi t/16] CStep[t],
           t, 0, 10 ]
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For plotting only, these symbols will
be assigned a value of 1: {a}
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Real part of the function is shown as solid
  lines.
Imaginary part of the function is shown as
  dashed lines.
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               3 Pi t
t CStep[t] Cos[------]
                 16
----------------------
          a t
         E
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has the following Laplace transform:
:[font = print; inactive; ]
                2
      -2 (a + s)               1
-(------------------- + ----------------)
       2                    2
   9 Pi           2 2   9 Pi           2
  (----- + (a + s) )    ----- + (a + s)
    256                  256
:[font = print; inactive; ]
The strip of convergence is:
:[font = print; inactive; ]
-Re[a] < Re(s) < Infinity
:[font = print; inactive; ]
The system is stable if Re[a] > 0
:[font = print; inactive; ]

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The zeroes are:  {-1.58905, -0.410951}
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The poles are:   {-1. + 0.589049 I, 
 
   -1. - 0.589049 I, -1. + 0.589049 I, 
 
   -1. - 0.589049 I}
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Analog Filter Design
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		Linear filters are certainly the workhorse of signal processing.  The signal processing packages allow a user to design Butterworth, Chebyshev (type I and type II), elliptic, and Bessel filters using the object DesignFilter.  DesignFilter takes seven arguments:  Analog, filter class (Butterworth, ...), filter type (Lowpass, ...), time variable, delta1, delta2, w1, and w2.  For bandpass and bandstop filters, two additional frequencies w3 and w4 are needed.  Frequencies are in rad/sec and given in ascending order  w1 < w2 < w3 < w4 (for a lowpass filter, then, w1 = wp and w2 = ws).  The magnitude response of the passband resides on the interval [1 - delta1, 1] and all stopbands reside on the interval [0, delta2].  DesignFilter returns a time domain expression.  The Laplace and Fourier rule bases can transform such expressions, which means that the pole-zero diagram and frequency response of a filter can be plotted using PoleZeroPlot and MagnitudePhasePlot.
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?DesignFilter
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DesignFilter[domain, class, type, time-variable, deltap,
   deltas, w1, w2, w3, w4] returns a filter expression
   (involving IIR/FIR or CIIR/CFIR filter sections). 
   Domain is either Analog or Digital. Possibilities for
   class include Butterworth, ChebyshevI, ChebyshevII, and
   Elliptic.  Possible filter types are Lowpass, Highpass,
   Bandpass, and Bandstop.  For lowpass filters, the
   magnitude response of the passband resides on [1, 1 -
   deltap] and the magnitude response of the stopband
   occupies [0, deltas]. The last two arguments, w3 and w4,
   are only necessary for bandpass and bandstop filters. 
   Frequencies w1, w2, w3, and w4 are measured in rad/sec
   and should be given in ascending order.
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		 The notebook introduces the topic of filter design and guides the reader through several design examples.  These examples include a lowpass design example for each filter type, as well as a bandpass, bandstop, and highpass design example.  Using animation, the student gets a chance to visualize the changes in the filter's magnitude response as one design parameter varies (three animations are provided---  two for ripple control parameter and one for the filter order which is shown below).  The last section of the notebook examines the effects of precision in implementation--- the student can truncate the coefficients of the filter and observe the frequency response.
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		This animation demonstrates the improvement in transition region and stopband attenuation as the filter length increases.  The sequence of frames consists of magnitude responses of Butterworth filters of order 2 , 3, ... 10:
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The z-Transform
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		The objects ZTransform and InvZTransform compute the forward and inverse z-transform, respectively, according to the standard definitions: [7]
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The forward rule base works for 1-D and m-D signals, whereas the inverse rule base only inverts 1-D and separable m-D transforms (Mathematica simply cannot perform the multiple contour integrations required in the general m-D case).  Since the rule bases implement the bilateral form of the z-transform, ZTransform tracks the region of convergence (ROC) so that InvZTransform can uniquely determine the inverse [3].
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		The z-transform notebook first defines the bilateral z-transform and discusses the importance of the ROC in uniquely defining the transform. Discussion of the ROC leads naturally into introducing stability and casuality of a signal based on the location of its poles relative to the ROC.  The next section links the z-transform with the discrete-time Fourier transform (DTFT) and includes several animations relating pole-zero diagrams to magnitude frequency responses, a basic idea behind digital filter design [7, ch. 5].  Figure 2 shows one frame for three-pole, two-zero filter whose magnitude response is being evaluated at a frequency of pi rad.  The z-transform notebook concludes with examples of high-level digital signal analysis using DSPAnalyze [4], the discrete-time version of ASPAnalyze .
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Figure 2:  Animation of System with Multiple Poles and Zeroes
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		Animate the following cells by selecting the graphics cells to be animated and then choosing the Animation option from the Graph submenu. You can also set the desired animation speed by using the Preferences option in the main menu, or the options in the bottom right side of  the notebook border. 
		The black dot on the circle represents the point z = e^jw moving around the unit circle, while the symbol "X" represents a pole and the symbol "O" represents a zero.
		 Note that the frequency response becomes zero whenever the black dot crosses a zero located on the unit circle. Also, the resulting frequency response corresponds to a low-pass filter. The more poles we will have in the right part of the unit circle the flatter the response can become around the low frequencies if these poles are placed appropriately.
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(Code to generate the two animations)
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		This code was originally developed under Mathematica 1.2 which did not possess the ability to put two different plots side by side.
:[font = input; ]
Do[VectorPoleZeroPlot[
          (z^2 + 1)/((z-0.8)(z-0.9 Exp[I Pi/4])(z-0.9 Exp[-I  Pi/4])), 
           z, 0.9, Infinity, w, True,
                Dialogue -> False],
   {w,-Pi,Pi,Pi/4}] 
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Do[MagPhasePlot[(z^2 + 1)/((z-0.8)(z-0.9 Exp[I Pi/4])(z-0.9 Exp[-I  Pi/4])) 
                /. z -> Exp[I w], {w,-Pi-0.1,r}, PhaseRangeScale -> Null, 
                PlotRange -> {{-Pi,Pi},{0,20}}],
   {r,-Pi,Pi,Pi/4}]              
          
:[font = subsection; inactive; startGroup; Cclosed; ]
Figure 3:  Inverse z-Transform with Dialogue
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InvZTransform[ { -8 z / ((z - 3)(3 z - 1)), 1/2, 2 },
				z, n, Dialogue -> All ]
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  -1        -8 z
Z   {-------------------}
  z  (-3 + z) (-1 + 3 z)
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
  -1        -8 z
Z   {-------------------}
  z  (-3 + z) (-1 + 3 z)
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
  -1         -8
Z   {------------------}
  z       3
     (1 - -) (-1 + 3 z)
          z
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
  -1         -8
Z   {-------------------}
  z       3        1
     (1 - -) (1 - ---) z
          z       3 z
-------------------------
            3
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
( after the term
:[font = print; inactive; ]
          -8
  -------------------
       3        1
  (1 - -) (1 - ---) z
       z       3 z
:[font = print; inactive; ]
  has been broken up into its partial
:[font = print; inactive; ]
  fractions representation
:[font = print; inactive; ]
    -9       3
  ------ + ------ )
       1        3
  -3 + -   -1 + -
       z        z
:[font = print; inactive; ]
  -1   -9       3
Z   {------ + ------}
  z       1        3
     -3 + -   -1 + -
          z        z
---------------------
          3
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
  -1   -9        -1   3
Z   {------} + Z   {------}
  z       1      z       3
     -3 + -         -1 + -
          z              z
---------------------------
             3
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
                 -1   -9
               Z   {-------}
                 z       1
                    1 - ---
   -1   3               3 z
-Z   {-----} - -------------
   z      3          3
      1 - -
          z
----------------------------
             3
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
   n                   1 n
3 3  Step[-1 - n] + 3 (-)  Step[n]
                       3
----------------------------------
                3
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
   n                   1 n
3 3  Step[-1 - n] + 3 (-)  Step[n]
                       3
----------------------------------
                3
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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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Conclusion
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		This paper describes how Mathematica's notebook facility could be used by students to supplement classroom lectures and laboratory experience.  Not only can students solve homework problems using Mathematica, but they can document the solution as they go.  The signal processing packages extend Mathematica so that electrical engineering students can tackle rudimentary problems in signal processing and linear systems.  Notebooks can also serve as tutorials.  We have written four notebooks covering the topics of piecewise convolution, analog filter design, the Laplace transform, and the z-transform.  Although some schools are integrating Mathematica into their curricula, we hope that more schools encourage students to use it.
:[font = section; startGroup; Cclosed; ]
References
:[font = subsection; inactive; ]
[1] S. Wolfram.  Mathematica: A System for Doing Mathematics by Computer.  Addison-Wesley, 1988.
:[font = subsection; inactive; ]
[2] M. Slaney.  ``Interactive Signal Processing Documents''.  IEEE ASSP Magazine.  April, 1990.  vol. 7, no. 2, pp. 8-20.
:[font = subsection; inactive; ]
[3] B. Evans, J. McClellan, and W. McClure.  ``Symbolic z-Transforms Using DSP Knowledge Bases''.  IEEE International Conference on Acoustics, Speech, and Signal Processing.  April, 1990.  vol. D, pp. 1775-1778.
:[font = subsection; inactive; ]
[4] B. Evans and J. McClellan.  ``Symbolic Transforms with Applications to Signal Processing''.  Mathematica Journal.  Fall, 1990.  vol. 1, no. 2.  pp. 70-80.
:[font = subsection; inactive; ]
[5] R. Maeder.  Programming in Mathematica.  Addison-Wesley, 1989.
:[font = subsection; inactive; ]
[6] A. Oppenheim and A. Willsky.  Signals and Systems.  Prentice-Hall, 1983.
:[font = subsection; inactive; endGroup; ]
[7] A. Oppenheim and R. Schafer.  Discrete-Time Signal Processing.  Prentice-Hall, 1989.
:[font = smalltext; inactive; ]
This work was supported in part by the Joint Services Electronics Program
contract #DAAL-03-90-C-0004.  
^*)