(*^

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Signal Processing Packages:
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Implementing Linear Systems Theory in
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MATHEMATICA
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by
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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

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EMAIL:  evans@eedsp.gatech.edu

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		What you are reading is a Mathematica notebook.  It is a collection of cells containing text, graphics, animation, equations, tables, and "live" Mathematica code arranged a tree structure.  You can read about subjects in a notebook like you would in a book.  The advantage over a book is that you can interact with the "text" by tinkering with the mathematics underlying the subject.  And because the information is presented in a hierarchy, you can select more information on a topic either by searching for more information on a subject via the Find menu option or by opening up a closed section of cells.  A closed section is displayed with a downward pointing half arrow, such as the one shown below:
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Example of a section with more information in it
(Open by clicking twice on the half arrow)
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Here is more information
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and more information.
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and more information.
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		Fonts are indicative of what the cell is.  For example, this cell is normal-sized text.  The next cell is "live" Mathematica code, and the cell after that contains the output of the "live" code:
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2 + 2
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4
;[o]
4
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You can have the Mathematica kernel evaluate code by selecting the cell with the code in it and then hitting either SHIFT-RETURN or ENTER.  The RETURN key by itself is only useful for formatting text.
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		Note that we have written this notebook so that each closed groups of cells will expand to fill one screen lengthwise.  So, please expand this notebook window so that it fills the screen lengthwise.
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Table of Contents

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Introduction
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		This notebook provides an overview of packages that implement aspects of linear systems theory.  The packages can take transforms, perform convolution, etc.  Several notebooks accompany the packages.  Three teach the z-transform, the mechanics of convolution, and analog filter design.  The other signal processing notebooks serve as user guides and reference manuals. 
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		Because you are reading this notebook, you probably already have access to the other notebooks and the signal processing packages.  If not, you can order them from Wolfram Research Inc. (for the cost of the distribution medium, shipping, and handling) or you can download them from InterNet at no cost via anonymous ftp (which is the topic of the first subsection below).
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Obtaining the Signal Processing Packages
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		To download the extensions, perform the following commands (begin at Unix prompt):
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		unix>  ftp vax.eedsp.gatech.edu
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		Connected to vax.eedsp.gatech.edu
		220 vax.eedsp.gatech.edu FTP server ...
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		Name:  anonymous
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		331 Guest login ok, send ident as password
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		Password:  <your email address goes here>
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		ftp> cd Mathematica
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		250 CWD command successful
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		ftp> binary
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		200 Type set to I
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		ftp> get SigProc2.0.tar.Z
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		ftp> quit
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		unix> uncompress SigProc2.0.tar.Z
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		unix> tar xf SigProc2.0.tar
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(The "get" command should take 2 minutes or so.)  These commands should produce a README and four sub-directories:  Macintosh, Notebooks, Packages, and doc.  Packages and Notebooks contain the signal processing packages and notebooks, respectively, which are discussed in detail in this notebook.  A reference manual and a user's guide to the packages reside in the doc sub-directory as two LaTeX files.  The notebooks document the extensions more thoroughly than do the LaTeX files.
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Loading the Signal Processing Packages
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		Once you have acquired the signal processing packages, you can load all or part of them as desired.   To load all of the packages, simply execute the following command:
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Needs[ "SignalProcessing`SignalProcessing`" ]
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To load in only the digital signal processing packages, execute
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Needs[ "SignalProcessing`Digital`" ]
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To load in only the digital signal processing packages, execute
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Needs[ "SignalProcessing`Analog`" ]
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The loading process for all of the packages may take up to five minutes.  You the reader do not have to execute any of the above commands to make sense out of this document, but the drawback is that you will not be able to evaluate any of the Mathematica expressions in the two signal processing sections.
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Getting a Print Out of this Notebook
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		You can also obtain a print out of this document by:
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1.  select all cells in the notebook by clicking on the first cell and dragging
      the mouse down so as to select all of the cells in the notebook
2.  use the "Open All Subgroups" option under the Cell menu (or
      hit COMMAND-{ as keyboard shortcut)
3.  use the Print... option (or hit COMMAND-p).
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The result will be a 54-page document.  In the Print menu, you have the option of previewing the document instead of printing it.
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Introduction to Mathematica
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		Mathematica is really two distinct programs:  the kernel and the user interface.  The kernel is available for a dozen computer systems.  All extensions to the kernel written in the Mathematica programming language are 100% portable across machines.  The user interface varies from machine to machine.  On the NeXT, Macintosh, and 386 machines (under MicroSoft Windows), this notebook interface is available.  A notebook is an ASCII file which is compatible on all three machines.  You are reading such a notebook.
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Capabilities of the Kernel
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		All of the built-in Mathematica functions and variables begin with a capital letter.  Therefore, in order to avoid conflict with the built-in objects, users can begin their own functions and variables with lower case letters.
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Numeric Computations
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		Mathematica is a powerful symbolic calculator, but it can crunch
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2 + 2
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4
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4
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		to exact precision or to arbitrary precision.   For example, the exact
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40!
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815915283247897734345611269596115894272000000000
;[o]
815915283247897734345611269596115894272000000000
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		can be approximated as a double precision floating point number
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N[40!]
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8.15915283247898*10^47
;[o]
          47
8.15915 10
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N[40!, 23]
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8.1591528324789773434561*10^47
;[o]
                           47
8.1591528324789773434561 10
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3 4 5
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60
;[o]
60
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		Mathematica can handle higher mathematical functions as well:
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N[ BesselI[1, 2 + 3 I] ]
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-1.260982060238848 + 0.7801488485792538*I
;[o]
-1.26098 + 0.780149 I
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Algebraic  Manipulation
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		Mathematica can factor polynomials
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Factor[ 6 + 11 x + 6 x^2 + x^3 ]
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(1 + x)*(2 + x)*(3 + x)
;[o]
(1 + x) (2 + x) (3 + x)
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		or generate new ones
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ChebyshevT[5, x]
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5*x - 20*x^3 + 16*x^5
;[o]
          3       5
5 x - 20 x  + 16 x
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		or decompose a ratio of polynomials into partial fractions.
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Apart[ ( s^3 + 27 s^2 + 242 s + 720 ) /
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84/(1 + s) - 168/(2 + s) + 105/(3 + s) - 20/(4 + s)
;[o]
 84      168     105     20
----- - ----- + ----- - -----
1 + s   2 + s   3 + s   4 + s
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		Mathematica can solve simultaneous equations, even non-linear ones.
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N[ Solve[ { a b^2 + Pi == 0, a + b == 2 },
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{{a -> -0.5019, b -> 2.502}, 
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;[o]
{{a -> -0.5019, b -> 2.502}, 
 
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		Here, we get three solutions to two equations.  Three solutions makes
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Integral and Differential Calculus
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		Mathematica can perform symbolic differentiation
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D[ x^2 + x Log[x], x ]
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1 + 2*x + Log[x]
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1 + 2 x + Log[x]
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		and indefinite integration
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Integrate[ 1 + 2 x + Log[x], x ]
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x^2 + x*Log[x]
;[o]
 2
x  + x Log[x]
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		and numerical integration (even in two dimensions).
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NIntegrate[x^2 + y^2, {x, -1, 1}, {y, -1, 1}]
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2.666666666666668
;[o]
2.66667
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		Mathematica can compute contour integrations.  For example, the 
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NIntegrate[1/z, {z, -1, -I, 1, I, -1}]
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6.28318530717959*I
;[o]
6.28319 I
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Graphics and Plotting
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		Mathematica handles all of the details of plotting badly behaved functions
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Programming Paradigms
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		Mathematica resembles many other programming languages because
		it supports
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1.	the writing of procedures by providing Block and Function
	constructs
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2.	the alteration of arguments by changing an object's attributes
	via SetAttributes
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3. the extending of operators by attaching functions to operators
	based on the data type (tag) of one of the inputs. 		
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4.	the ability to chain conditional rules in one direction via Replace,
	ReplaceAll, and ReplaceRepeated commands,
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5.	the pattern matching of expressions via MatchQ, TagSetDelayed,
	and TagSet, and
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6.	the immediate and delayed evaluation of expressions via = and :=.
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		Mathematica is a superset of many common modern programming styles.
		One drawback, however, is that it is not an expert system.
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Getting Help On-Line
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		The kernel offers a help function called Help (abbreviated ?) which can
		give usage information on a particular object (function)
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?Integrate
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Integrate[f, x] gives the indefinite integral of f
   with respect to x. Integrate[f, {x, xmin,
   xmax}] gives the definite integral.
   Integrate[f, {x, xmin, xmax}, {y, ymin, ymax}]
   gives a multiple integral.
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		or match a collection of object names to a pattern.
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?*Solve*
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DSolve      MainSolve   Solve       SolveAlways
LinearSolve
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		The notebook interface offers kinds of help to you the home viewer.
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Educational Uses of the Notebook Interface
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		The interactive nature of Mathematica and its multimedia capabilities make it a good platform in which to write tutorials.  Teachers can also use them to create problem sets and their solutions.  Students can use them to solve problems and document their solution as they go.   At the very least, such a facility can provide decent on-line help.
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Calculus Courseware
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		For the last three years,  Don Brown, Horacio Porta, and Jerry Uhl at the University of Illinois at Urbana-Champaign have been developing a one-year Calculus course in the form of tutorial notebooks.  They have adapted a conventional one-year freshmen Calculus course into a laboratory setting where the teacher only lecture 30 minutes per week.  The lessons are broken down into 60 notebooks, which students complete at a rate of two per week.  The developers faced two major obstacles.  First, how does one take advantage of the multimedia capabilities so that one does not simply dump the contents of a text book into a notebook format?  So, they split each notebook into six sections:
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1. Introduction													2. Basic Problems
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3. Tutorial problems										4. "Give It a Try"
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5. Student feedback											6. "Literacy Sheet"
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The second issue was grading.  They decided to embed problems in the notebook which had been partially completed.  It was then up to the students to fill in the missing pieces and document why they chose certain procedures.  Students also worked problems from scratch.  The primary source of their grades was conventional paper-and-pencil tests.
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Current Classroom Uses
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		Besides the Calculus courseware discussed in the previous subsubsection, other educators are using Mathematica to enhance the learning process.  The list below is only representative and by no means exhaustive.   The first two lines identify the departments and the school.  The third line indicates the purpose.  The remaining lines list the classes in which Mathematica without any extensions is being used.  (The next section entitled "Signal Processing Notebooks" lists some of the schools that are using the signal processing packages and notebooks as educational tools).
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Engineering, Mathematics, Physics, and Chemistry Departments
Rose-Hulman Institute of Technology
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Joint effort by above department to teach the
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		Integrated Freshmen Curriculum
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Mathematics Department
Georgia Institute of Technology
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Lecture from notebooks with laboratory 
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		Advanced Engineering Mathematics I & II
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		Mathematical Programming
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Music Department
Stanford University
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Plotting windows and their spectra
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		The Discrete Fourier Transform 
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		Applications of the Short-Time Fourier Transform
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For More Information
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		Mathematica comes with several notebooks.  The notebook entitled "TourOfMathematica" in the /NextLibrary/Mathematica/Notebooks directory provides an overview of the system. 		
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		Several books published by Addison-Wesley are listed below.  The first is more of a reference manual than a user's guide.  The 1988 edition documents Mathematica 1.0 and the 1991 edition documents Mathematica 2.0.   The second book is an introduction on how to encode knowledge in Mathematica.  The other book are less general.
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S. Wolfram, Mathematica:  A System for Doing Mathematics by Computer,  1988.
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R. Maeder, Programming in Mathematica,  1990.
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S. Skiena, Implementing Discrete Mathematics:  Combinatorics and Graph Theory with Mathematica, 1990.
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T. Gray and J. Glynn, Exploring Mathematics with Mathematica, 1991.
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R. Crandall, Mathematica for the Sciences, 1991.
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I. Vardi, Computational Recreations with Mathematica, 1991.
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Introduction to Signal Processing
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Except for the sub-section and sub-sub-section headings, the following is taken from "Symbolic Transforms with Applications to Signal Processing" by B. L. Evans, J. H. McClellan, and W. B. McClure which appeared in the Fall 1990 issue of "The Mathematica Journal" pages 70-80.  The reference section lists this article as [Evans, 1990].
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		Transform theory is a powerful tool employed by scientists
[Oberhettinger and Badii 1973], mathematicians [Churchill 1958], and engineers [Lathi 1983; Muth 1978; Oppenheim and Schafer 1989].  One important use of this theory is to transform equations involving integrals and derivatives into algebraic equations that are easier to solve.  The solutions to the original equations are obtained by inverse transforming the algebraic solutions.  This is the case when the Laplace transform is applied to linear
differential equations with constant coefficients, or when the z-transform is applied to linear difference equations with constant coefficients.
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Signal Processing
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Signals
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		This process is one of the staples of signal processing, the area
of engineering that deals with the design and analysis of systems that act on signals in prescribed ways.  A signal is essentially a function of one or more variables; often, in the one-variable case, the variable is time.  The variable can be continuous (for an analog signal), or discrete (for a digital, or sampled, signal).  Thus a one-dimensional discrete-time signal is a finite or infinite sequence of values, or samples.
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Systems
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		A system is a mathematical operator that transforms one signal into
another, according to characteristic rules.  Filters, or linear time-invariant  systems, form a particularly important class of systems, eminently suited to
the application of transform methods.  A filter is characterized by a function, its impulse response, in terms of which the filter's action on any signal can be determined.  The output signal is the convolution of the input signal with the filter's impulse response; the operation of convolution involves an integral for continuous signals and a summation for discrete ones.  Using transforms, one can reduce the task of convolving these two functions to that of multiplying two algebraic expressions--- the functions' transforms--- and then inverse transforming the product.
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Analog Signal Processing (ASP)
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		For an example of analog system design, consider the transmission of AM radio signals.  Each AM radio station broadcasts at a specific frequency fo which is between 550 kHz and 1600 kHz and is a multiple of 10 kHz [Lathi 1983].   In order to avoid overlap with other stations, a station must transmit its signal between the frequencies fo - 5 kHz and fo + 5 kHz.  Before transmission, a station filters its signal so that frequencies above 5 kHz are rejected and frequencies below 5 kHz are passed.  This filtered signal is then shifted by fo in the frequency domain and transmitted.  The entire design and analysis of this system can be carried out in the frequency domain:  the Laplace transform would be used for the design and implementation of the 5 kHz low-pass filter, and the Fourier transform would be used to analyze the transmission system.
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Digital Signal Processing (DSP)
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Discrete Signals and Systems
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		By contrast, digital signal processing (DSP) deals with discrete-time signals (sequences) and systems that act on them [Oppenheim and Schafer 1989].  Primarily, digital signals come from sampling and quantizing
analog signals.  In many cases, implementing a system in digital hardware is faster, more flexible, and more cost-effective than building it from
analog components.  One reason for this is that increasing precision in discrete systems is as simple as adding bits, whereas more expensive components may be required in analog systems.  Another reason is that the time to prototype a system is often shorter for discrete systems. Apart from that, some DSP operators, like those that alter the sampling rate, have little meaning in the continuous domain, and may be very difficult to implement with analog components.
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Commonly Used Transforms
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		The most commonly used transforms in DSP are the z-transform, the discrete-time Fourier transform (DTFT) and the discrete Fourier transform (DFT).  Just as the Laplace transform is a generalization of the Fourier transform, the z-transform is a generalized form of the DTFT.  The z-transform is most useful in the implementation of digital filters as digital hardware or programs for DSP chips, while the DTFT is preferred when the frequency-domain characteristics of a system must be defined.
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Multidimensional Signal Processing
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		Neither digital nor analog signal processing is confined to a single dimension.  Many spatial array and imaging applications require two- and
three-dimensional processing.  This is the case for detecting edges in monochrome digital images, or finding angles of arrival with an array of sensors.  A digitized image can be thought of as a function defined on an N x M grid, where each grid square is assigned an intensity value known as a grey level (usually a positive integer).  In such a digitized image, human eyes perceive an edge as a sharp transition between grey levels in adjacent regions.  Such a sharp transition gives rise to high-frequency components in
the image.  Therefore, filtering the image to boost its high-frequency components and decrease its low-frequency components would sharpen edges in the image.  This operation on the image is a two-dimensional high-pass filter; it corresponds to convolution in the two-dimensional ``time'' domain.
:[font = subsection; inactive; startGroup; Cclosed; ]
Computer Implementations of DSP
:[font = text; inactive; ]
		Digital signal processing has always been tied closely to computer implementations, where the signals are viewed as a stream of numbers.  The machine provides an efficient way to compute the DSP operator that transforms one stream of numbers into another.  On the other hand, the design of a signal processing system treats the signals as functions (in the mathematical sense).
:[font = text; inactive; endGroup; ]
		Over the last decade, special attention has been devoted by several researchers to the field of symbolic signal processing [Kopec 1980, 1985;
Myers 1986; Covell 1989].  In symbolic processing, the signal is represented in a computer as a formula, rather than as a sequence of numbers.  Thus, the value of a signal might only be known in terms of a formula, instead of a number.  In a similar manner, signal processing operators, the building blocks for systems, are maintained in symbolic form.  This enables a machine to simplify, rearrange, and rewrite symbolic expressions until they take a desired form.  For example, the desired form might be an optimal implementation of an expression in which the number of additions and multiplications is minimized.
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Rest of the Paper [Notebook]
:[font = text; inactive; ]
		The remainder of this paper [notebook] describes a set of packages that provide a general platform for implementing symbolic transforms in Mathematica and that represent a first step toward the building of a comprehensive signal processing environment in Mathematica.  The packages are divided into three groups, corresponding to the contexts
:[font = input; center; ]
SignalProcessing`Support`
:[font = input; center; ]
SignalProcessing`Digital`
:[font = input; center; ]
SignalProcessing`Analog` 
:[font = text; inactive; endGroup; endGroup; ]
(Loading either of the last two automatically causes the first to load as well.)
:[font = section; inactive; startGroup; Cclosed; ]
Signal Processing Packages
:[font = text; inactive; ]
		The signal processing packages are extensions to the Mathematica kernel.  These packages add knowledge representations for functions, operators, and structures common in signal processing and implement the bilateral forms of the usual linear transforms: 
:[font = special3; inactive; center; ]
discrete signals
:[font = network; inactive; center; ]
 z-transform
discrete-time Fourier transform
discrete Fourier transform
:[font = special3; inactive; center; ]
continuous signals
:[font = network; inactive; center; ]
Laplace transform
Fourier transform
:[font = text; inactive; ]
The transforms can fully justify their answers so that students can learn the mechanics of how to compute transform. [Evans, 1990]
:[font = text; inactive; ]
		The packages also perform convolution and solve difference/differential equations.  The solvers can tackle linear constant coefficient (lcc) difference and differential equations subject to initial conditions.  Like the transform rule base, these solvers can fully justify their answers. [Evans, 1990]
:[font = special3; inactive; center; ]
discrete signals
:[font = network; inactive; center; ]
 one-dimensional convolution
solving lcc difference equations
:[font = special3; inactive; center; ]
continuous signals
:[font = network; inactive; center; ]
one-dimensional piecewise convolution
solving lcc differential equations
one-dimensional analog filter design
:[font = subsection; inactive; startGroup; Cclosed; ]
New Signals and Systems
:[font = text; inactive; ]
		Here are the new signals (functions) introduced by the packages:
:[font = network; inactive; center; ]
Kronecker impulse, Dirac delta, sinc,
continuous and discrete step functions,
continuous and discrete pulse functions,
aliased sinc, filter representations
:[font = text; inactive; ]
		Here are the new systems (operators) introduced by the packages:
:[font = network; inactive; endGroup; center; ]
aliasing, convolution, DFT, DTFT, backward
difference, downsampling, Fourier transform,
interleave samples, Laplace transform,
periodic, reverse, scale axis, shift,
summation, upsampling
:[font = subsection; inactive; startGroup; Cclosed; ]
New Graphics Capabilities
:[font = text; inactive; ]
		Of course, no implementation of signal processing operations would be complete without special graphical representations for signals (functions) and systems (operators):
:[font = special3; inactive; center; ]
general plotting
:[font = network; inactive; endGroup; center; ]
1-D and 2-D signals and sequences
frequency responses
pole-zero diagrams
root loci

:[font = subsection; inactive; startGroup; Cclosed; ]
Laplace Transform Examples
:[font = text; inactive; ]
		These packages can compute transforms as well as fully justify the answers.  This is useful for showing the mechanics of  taking transforms to students.   Since the justification (dialogue) capabilities are pretty much the same for all of the transforms,we have included only Laplace transform examples because the Laplace transform is probably the most familiar of the implemented linear transforms.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Forward Transform with Complete Justification
:[font = text; inactive; ]
		For the example below, note that  CStep  is the continuous step function.  The justification is enabled by the Dialogue -> All option.  Other possible values for Dialogue are True and False.   True, the default value, tells the Laplace transform object LaPlace to report only strategies used in finding the transform (like partial fractions or series expansion).
:[font = input; startGroup; ]
LaPlace[ A t Exp[a t] CStep[t],
		  t, s, Dialogue -> All ]
:[font = print; inactive; startGroup; ]
       a t
L  {A E    t CStep[t]}
  t
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
       a t
A L  {E    t CStep[t]}
    t
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
A {L  {t CStep[t]}}
     t             s -> -a + s and shift ROC
:[font = print; inactive; ]
which becomes
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[A/(-a + s)^2, Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
               A
LTransData[---------, Rminus[Re[a]], 
                   2
           (-a + s)
 
  Rplus[Infinity], LVariables[s]]
:[font = text; inactive; endGroup; ]
Here, the transform is A / (s - a)^2 with a region of convergence of
Re{a} < Re{s}.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Inverse Transform with Justification
:[font = text; inactive; ]
		The justification is enabled by the Dialogue -> Trueoption.  Other possible values for Dialogue are All and False.   True, the default value, tells the Laplace transform object LaPlace to report only strategies used in finding the transform (like partial fractions or series expansion).
:[font = input; startGroup; ]
InvLaPlace[ ( s^3 + 27 s^2 + 242 s + 720 ) /
			 ( s^4 + 10 s^3 + 35 s^2 + 50 s + 24 ),
			 s, t, Simplify -> False ]
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( After performing partial fraction expansion on
:[font = print; inactive; ]
                       2    3
     720 + 242 s + 27 s  + s
  ------------------------------
                  2       3    4
  24 + 50 s + 35 s  + 10 s  + s
:[font = print; inactive; ]
  in order to break it up into first and
:[font = print; inactive; ]
  second order sections:
:[font = print; inactive; ]
   84      168     105     20
  ----- - ----- + ----- - ----- . )
  1 + s   2 + s   3 + s   4 + s
:[font = output; inactive; output; endGroup; endGroup; ]
(-20*CStep[t])/E^(4*t) + (105*CStep[t])/E^(3*t) - 
  (168*CStep[t])/E^(2*t) + (84*CStep[t])/E^t
;[o]
-20 CStep[t]   105 CStep[t]   168 CStep[t]
------------ + ------------ - ------------ + 
     4 t            3 t            2 t
    E              E              E
 
  84 CStep[t]
  -----------
       t
      E
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Here, the inverse transform is a sum of four decaying exponentials.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Providing Your Own Transform Pairs
:[font = text; inactive; ]
		All ten transform rule bases allow you to specify your own transform pairs through the TransformLookup option.  These transform pairs are evaluated after the rule base has finished applying its knowledge to find the transform.  Thus, the transform pairs that the user specifies will not override the transform pairs encoded into the rule bases.
:[font = text; inactive; ]
		Besides augmenting the knowledge of an existing rule base, another advantage to being able to provide your own transform pairs is transforming general functions, e.g.  x(t) <---> X(s). 
:[font = input; startGroup; ]
LaPlace[ t x[t], t, s,
         TransformLookup -> { x[t] :> X[s] } ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
LTransData[-Derivative[1][X][s], 
  Rminus[DirectedInfinity[-1]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
LTransData[-X'[s], Rminus[-Infinity], 
 
  Rplus[Infinity], LVariables[s]]
:[font = subsection; inactive; startGroup; Cclosed; ]
Signal Analysis
:[font = text; inactive; ]
		The signal processing packages can perform high-level signal analysis on digital signals (via DSPAnalyze) and analog signals (via ASPAnalyze).  The analysis mimics that expected of a junior or senior electrical engineering student.   For 1-D and 2-D signals, DSPAnalyze and ASPAnalyze 
:[font = special3; inactive; ]
	1.	plot the time-domain signal,
:[font = special3; inactive; ]
	2.	take its transform (z-transform for digital signals and Laplace
		transform for analog signals),
:[font = special3; inactive; ]
	3.	determine conditions on the free parameters for the signal
		to be stable,
:[font = special3; inactive; ]
	4.	display the pole-zero diagram, and
:[font = special3; inactive; ]
	5.	compute and plots the frequency response (if it exists).
:[font = text; inactive; ]
		Since the digital analysis performed by DSPAnalyze is similar to the analog analysis by ASPAnalyze, we have included one example.  aHere is the analysis of a two-sided decaying one-dimensional discrete-time exponential signal:
:[font = input; startGroup; Cclosed; ]
DSPAnalyze[ a^Abs[n], n, -10, 10, a -> 4/5 ]
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For plotting only, these symbols will
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will be assigned default values:
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      4
{a -> -}
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( after rewriting the two-sided expression
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   Abs[n]
  a
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  Step[-1 - n]    n
  ------------ + a  Step[n] )
        n
       a
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 Abs[n]
a
:[font = print; inactive; ]
has the following z-transform:
:[font = print; inactive; ]
  1       a z
----- + -------
    a   1 - a z
1 - -
    z
:[font = print; inactive; ]
The region of convergence is:
:[font = print; inactive; ]
                 1
Abs[a] < |z| < ------
               Abs[a]
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                              1
The system is stable if 1 < ------ && Abs[a] < 1
                            Abs[a]
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The zeroes are:  {0.}
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Solving Difference and Differential Equations
:[font = text; inactive; ]
		The signal processing packages can solve linear constant coefficient (lcc) difference (via via ZSolve) and differential (LSolve) equations subject to initial conditions.  ZSolve and LSolve need to know the difference / differential equation and the function to be solved for.  Like the transform rule bases, ZSolve and LSolve can fully justify their answers.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Second-Order Difference Equation
:[font = special3; inactive; startGroup; Cclosed; center; ]
--  Without Justification  --
:[font = input; startGroup; ]
yofn =
  ZSolve[ y[n-2] + 1/2 y[n-1] + 1/4 y[n] == 0,
          y[n], y[1] -> 1, Dialogue -> False ]
:[font = output; inactive; output; endGroup; endGroup; ]
(2^n*Sin[(2*Pi*n)/3]*Step[n])/3^(1/2)
;[o]
 n     2 Pi n
2  Sin[------] Step[n]
         3
----------------------
       Sqrt[3]
:[font = special3; inactive; startGroup; Cclosed; center; ]
--  With Partial Justification   --
:[font = input; startGroup; ]
yofn =
  ZSolve[ y[n-2] + 1/2 y[n-1] + 1/4 y[n] == 0,
          y[n], y[1] -> 1 ]
:[font = print; inactive; startGroup; ]
Solving for y[n] in the difference equation
:[font = print; inactive; ]
augmented by the initial conditions:
:[font = print; inactive; ]
              y[-1 + n]   y[n]   Impulse[-1 + n]
  y[-2 + n] + --------- + ---- = ---------------
                  2        4            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; ]
        1
  --------------
       4    2
  (1 + -- + -) z
        2   z
       z
:[font = print; inactive; ]
Inversing transforming this gives y[n]:
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
(2^n*Sin[(2*Pi*n)/3]*Step[n])/3^(1/2)
;[o]
 n     2 Pi n
2  Sin[------] Step[n]
         3
----------------------
       Sqrt[3]
:[font = special3; inactive; startGroup; Cclosed; center; ]
--  Verifying the Result  --
:[font = text; inactive; ]
		Assuming that the value of y[n] determined by ZSolve has been
		assigned to the variable yofn, we can verify that yofn is indeed the
		solution.  First, we substitute yofn back into the original difference
		equation to obtain a function of n in closed form called expdiffequ:
:[font = input; startGroup; ]
expdiffequ = (yofn /. n -> n - 2) +
             1/2 (yofn /. n -> n - 1) +
             1/4 yofn
:[font = output; inactive; output; endGroup; ]
(2^(-2 + n)*Sin[(2*Pi*(-2 + n))/3]*Step[-2 + n])/3^(1/2) + 
  (2^(-1 + n)*Sin[(2*Pi*(-1 + n))/3]*Step[-1 + n])/
   (2*3^(1/2)) + (2^n*Sin[(2*Pi*n)/3]*Step[n])/(4*3^(1/2))
;[o]
 -2 + n     2 Pi (-2 + n)
2       Sin[-------------] Step[-2 + n]
                  3
--------------------------------------- + 
                  1/2
                 3
 
   -1 + n     2 Pi (-1 + n)
  2       Sin[-------------] Step[-1 + n]
                    3
  --------------------------------------- + 
                     1/2
                  2 3
 
   n     2 Pi n
  2  Sin[------] Step[n]
           3
  ----------------------
             1/2
          4 3
:[font = text; inactive; ]
		We can then ask for the values of the difference equation, which has
		been assign to expdiffequ, for n = 0 to n = 10.  Here, y[1] = 1/4 and
		y[0] = 0 which agrees with the initial conditions.  The difference
		equation should produce 0 for all values of n greater than 1:
:[font = input; startGroup; ]
Table[expdiffequ, {n, 0, 10} ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
{0, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0}
;[o]
    1
{0, -, 0, 0, 0, 0, 0, 0, 0, 0, 0}
    4
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Second-Order Differential Equation
:[font = special3; inactive; startGroup; Cclosed; center; ]
-- With Full Justification  --
:[font = input; startGroup; ]
yoft =
  LSolve[ y''[t] + 3/2 y'[t] + 1/2 y[t] ==
          Exp[a t] CStep[t], y[t], y[0] -> 4,
          y'[0] -> 4, Dialogue -> All ]
:[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; ]
  {y[0] -> 4, y'[0] -> 4}
:[font = print; inactive; ]

:[font = print; inactive; ]
The Laplace transform of the left side is:
:[font = print; inactive; ]
              1   3 s    2    3 y[0]
  L { y[t] } (- + --- + s ) - ------ - s y[0] - y'[0]
              2    2            2
:[font = print; inactive; ]
( In the general case, the bilateral Laplace
:[font = print; inactive; ]
  transform of the nth derivative of y[t] is:
:[font = print; inactive; ]
                                    -1 + n
      (n)                    n     s       y[t0]
  L {y   [t] } = L { y[t] } s  + -(-------------) + 
                                        s t0
                                       E
 
     -2 + n                    (-1 + n)
    s       y'[t0]            y        [t0]
  -(--------------) + ... + -(-------------)
         s t0                      s t0
        E                         E
:[font = print; inactive; ]
  where t=t0 is the initial condition. )
:[font = print; inactive; ]

:[font = print; inactive; ]
The Laplace transform of the right side is:
:[font = print; inactive; ]
    1
  -----
  s - a
:[font = print; inactive; ]
Solving for the unknown transform yields
:[font = print; inactive; ]
    1     3 y[0]
  ----- + ------ + s y[0] + y'[0]
  s - a     2
  -------------------------------
           1   3 s    2
           - + --- + s
           2    2
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
               1
  10 + 4 s + -----
             s - a
  ----------------
    1   3 s    2
    - + --- + s
    2    2
:[font = print; inactive; ]
Inverse transforming this gives y[t]:
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
((-24 - 64*a)*CStep[t])/(2*E^(t/2)*(-1 - 2*a)) + 
  ((10 + 12*a)*CStep[t])/(E^t*(-1 - a)) - 
  (2*E^(a*t)*CStep[t])/(-1 - 3*a - 2*a^2)
;[o]
(-24 - 64 a) CStep[t]   (10 + 12 a) CStep[t]
--------------------- + -------------------- - 
     t/2                     t
  2 E    (-1 - 2 a)         E  (-1 - a)
 
     a t
  2 E    CStep[t]
  ---------------
                2
  -1 - 3 a - 2 a
:[font = special3; inactive; startGroup; Cclosed; center; ]
--  Interpreting and Verifying the Solution  --
:[font = text; inactive; ]
		Assuming that the value of y(t) determined by LSolve has been
		assigned to the variable yoft, we can verify that yofn is indeed the
		solution.  First, we interpret the solution for t >= 0+ so we can
		replace all of the continuous step functions of by 1:
:[font = input; startGroup; ]
yoftpos = yoft /. { CStep[t] -> 1 }
:[font = output; inactive; output; endGroup; ]
(-24 - 64*a)/(2*E^(t/2)*(-1 - 2*a)) + 
  (10 + 12*a)/(E^t*(-1 - a)) - 
  (2*E^(a*t))/(-1 - 3*a - 2*a^2)
;[o]
                                         a t
   -24 - 64 a        10 + 12 a        2 E
----------------- + ----------- - ---------------
   t/2               t                          2
2 E    (-1 - 2 a)   E  (-1 - a)   -1 - 3 a - 2 a
:[font = text; inactive; ]
		We can now plot y(t) for positive values of t.  However, we must
		give a value to the parameter a (which controls the damping of the
		driving function in the differential equation).  Here, we have chosen
		to set a to 2:
:[font = input; startGroup; ]
Plot[ yoftpos /. a -> -2, {t, 0, 10} ]
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;[o]
-Graphics-
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		We can also verifying that y(0+) = 4:
:[font = input; startGroup; ]
Simplify[ yoftpos /. t -> 0 ]
:[font = output; inactive; output; endGroup; endGroup; ]
4
;[o]
4
:[font = text; inactive; startGroup; ]
		We can also verifying that y'(0+) = 4.  This boils down to taking
		the derivative of y(t) and substituting t = 0+:
:[font = input; startGroup; ]
D[ yoftpos, t ]
:[font = output; inactive; output; endGroup; ]
-(-24 - 64*a)/(4*E^(t/2)*(-1 - 2*a)) - 
  (10 + 12*a)/(E^t*(-1 - a)) - 
  (2*E^(a*t)*a)/(-1 - 3*a - 2*a^2)
;[o]
                                        a t
  -(-24 - 64 a)      10 + 12 a       2 E    a
----------------- - ----------- - ---------------
   t/2               t                          2
4 E    (-1 - 2 a)   E  (-1 - a)   -1 - 3 a - 2 a
:[font = input; startGroup; ]
Simplify[ % /. t -> 0 ]
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4
;[o]
4
:[font = section; inactive; startGroup; Cclosed; pageBreak; ]
Signal Processing Notebooks
:[font = text; inactive; ]
		We have written several notebooks to accompany the signal processing packages, including tutorials on convolution, the z-transform, and analog filter design.   The tutorials are designed to accompany junior, senior, and first-year graduate engineering courses.  The other notebooks provided on-line help and training.  The LaplaceTest notebook demonstrates how-to-use the forward and inverse Laplace transform objects as well as their limitations.  Two notebooks, SignalProcessingExamples and EducationalTool, provide an overview of the signal processing packages and notebooks.  The last notebook, SignalProcessingUsage, contains an on-line reference to all of the new Mathematica objects introduced by the signal processing packages. [Evans, 1991]
:[font = subsection; inactive; startGroup; Cclosed; ]
Tutorial Notebooks
:[font = text; inactive; ]
		These tutorials present the usual text book information, but they allow the reader to tinker with mathematical models and to visualize ideas through animation.  For example, the analog filter design notebook animates the effect of filter order on filter roll-off .  The z-transform notebook illustrates the effect on a filter's magnitude response when the location of a pair of poles is slightly disturbed.  The convolution notebook animates the flip-and-slide approach to piecewise convolution.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Animation of the Flip-and-Slide View of Convolution
:[font = text; inactive; ]
		Teaching the mechanics of the flip-and-slide approach to piecewise convolution is difficult.  The following animation illustrates the method of "sliding" one function against another by showing the resulting function as it is generated in time.  We will convolve x with h.
:[font = input; startGroup; ]
x = ConvertFromList[{{2 t,0,2}},t]
:[font = output; inactive; output; endGroup; ]
2*t*CPulse[2, t]
;[o]
2 t CPulse  [t]
           2
:[font = input; startGroup; ]
h = ConvertFromList[{{2,0,2},{1,2,3}},t]
:[font = output; inactive; output; endGroup; ]
CPulse[1, -2 + t] + 2*CPulse[2, t]
;[o]
CPulse  [-2 + t] + 2 CPulse  [t]
       1                    2
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Overview of the z-Transform Notebook
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		The z-transform notebook defines the bilateral z-transform and discusses the importance of the region of convergence (ROC).  This ROC coupled with the location of poles characterizes the stability and casuality of the signal.  The notebook then explores the relationship between the z-transform and the discrete-time Fourier transform (DTFT), namely the fact that not every function with a z-transform has a DTFT and vice-versa.  The notebook comes stocked with built-in exercises and many animations, including two that show the effect on the magnitude response of slight perturbations of poles and several that show how the locations of poles and zeroes influence the magnitude response.
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Overview of the Filter Design Notebook
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		The analog filter design notebook discusses the characteristics of the classical Bessel, Butterworth, Chebyshev I, Chebyshev II, and Elliptic analog filters as well as lowpass, bandpass, bandstop, and highpass magnitude responses.  It shows how to manipulate filter representations (namely how to take transforms and plot the results).  It demonstrates the effects of coefficient truncation and how to convolve an input function with a filter symbolically.  The notebook contains two animations.  One shows how the magnitude filter response varies with filter order and the other how the ripple varies with the ripple gain parameter epsilon.
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Current Uses of Signal Processing Notebooks
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University of Pennsylvania
Department of Civil Engineering
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		Teaching convolution  water flow (input) through soil (linear system) to find contribution to local water level (output) in graduate-level hydrology class.
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 Georgia Institute of Technology
 School of Electrical Engineering
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		Teaching convolution and Fourier transforms in an introductory communications class.
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		Generating problems z-transform sets and their solutions for an introductory digital signal processing class.
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Summary
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		Several schools (Rose-Hulman, University of Illinois, etc.) are experimenting with integrating Mathematica into the freshmen curriculum.   The signal processing packages and notebooks are an attempt at developing the extensions for classes at a higher level, namely those that teach linear systems, signal processing, and transform methods.  This means that engineering, mathematics, chemistry, and physics students can benefit from these extensions.
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Signal Processing Packages
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    	 The signal processing packages implement a wide range of topics from linear systems theory from an engineering perspective.  These packages provide many functions and operators common in linear systems theory and signal processing that are not defined in Mathematica: [Evans, 1990]
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discrete signals
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 z-transform
discrete-time Fourier transform
discrete Fourier transform
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 one-dimensional convolution
solving lcc difference equations
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 signal analysis
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continuous signals
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Laplace transform *
Fourier transform *
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one-dimensional piecewise convolution
solving lcc differential equations
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 signal analysis
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one-dimensional analog filter design
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general plotting
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1-D and 2-D signals and sequences
frequency responses
pole-zero diagrams
root loci
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*  Although Mathematica 2.0 comes with some Laplace  and  Fourier
     transform capabilities (written from a mathematicians point of view), 
     it will not be able to transform expressions common  in  signal
     processing because Mathematica 2.0 manipulates mathematical
     formulas not signal processing expressions.  Besides, Mathematica's
     implementation is for one-dimensional, one-sided functions, whereas
     ours handles two-sided and multidimensional functions.
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Signal Processing Notebooks
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		So far, seven notebooks (besides the one you are presently reading) describe the signal processing packages.  Three serve as tutorials on the z-transform, convolution, and analog filter design.  The others serve as on-line help.  Together, these notebooks allow students to read about signal processing, tinker with equations, and visualize ideas through animation [Evans, 1991].  With Mathematica 2.0, students will also be able to learn through hearing.  As a multimedia tool, Mathematica can enhance the learning process and make it more enjoyable.
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Tutorial Notebooks
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Convolution
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z-Transform
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Analog Filter Design
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Demonstration Notebooks
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Laplace Transforms
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On-Line Documentation
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Usage Information
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Overview with Examples
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Southeastcon Paper
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References
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[Clements and Pease 1989]
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M. A. Clements and J. P. Pease. 1989.  ``On Causal Linear Phase IIR Digital Filters,''  {IEEE Transactions on Acoustics, Speech, and Signal Processing},  vol.  37 (4), pp. 479-484.
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[Churchill 1978]
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R. Churchill.  1958. {Operational Mathematics}.  McGraw-Hill, New York.
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[Covell 1989]
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M. M. Covell. 1989.  {An Algorithm Design Environment for Signal Processing}.  Ph.  D. Thesis and RLE Tech. Rep. #549.  MIT, Cambridge (MA).
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[Dudgeon & Mersereau 1984]
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D. E. Dudgeon and R. M. Mersereau. 1984.  {Multidimensional Digital Signal Processing}.  Prentice Hall, Englewood Cliffs (NJ).
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[Evans 1990]
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B. L. Evans, J. H. McClellan, and W. B. McClure.  1990.  "Symbolic Transforms with Application to Signal Processing."  {The Mathematica Journal}.  Fall, 1990.  vol. 1, issue #2, pp. 70-80.
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[Evans 1991]
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B. L. Evans, J. H. McClellan, and K. A. West.  "Mathematica As An Educational Tool for Signal Processing".  {Proc. of IEEE Southeastern Conference}.  April, 1991.  pp. 1172-1176.
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[Kopec 1980]
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G. Kopec. 1980.  {The Representation of Discrete-Time Signals and Systems in Programs.}  Ph. D. Thesis. MIT, Cambridge (MA).
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[Kopec 1985]
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G. Kopec. 1985.  ``The Signal Representation Language SRL.''  {IEEE Transactions on Acoustics, Speech, and Signal Processing}, vol.  33 (4), pp. 921-932.
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[Lathi 1983]
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B. P. Lathi. 1983.  {Modern Digital and Analog Communication Systems}.  Holt, Rinehart, and Winston, New York.
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[Muth 1978]
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E. J. Muth. 1978.  {Transform Methods}.  Prentice Hall, Englewood Cliffs (NJ).
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[Myers 1986]
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C. S. Myers. 1986.  {Signal Representation for Symbolic and Numeric Processing}.  Ph.  D. Thesis and RLE Tech. Rep. #521.  MIT, Cambridge (MA).
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[Oberhettinger & Badii 1973]
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F. Oberhettinger and L. Badii. 1973.  {Tables of Laplace Transforms}.  Springer-Verlag, New York.
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[Oppenheim & Schafer 1989]
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A. V. Oppenheim and R. W. Schafer. 1989.  {Discrete-Time Signal Processing}.  Prentice Hall, Englewood Cliffs (NJ).
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[Oppenheim & Willsky 1983]
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A. V. Oppenheim and A. Willsky. 1983.  {Signals and Systems}.  Prentice Hall, Englewood Cliffs (NJ).
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This work was supported in part by the Joint Services Electronics Program
contract #DAAL-03-90-C-0004.  
^*)