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Signal Processing Packages and Notebooks for the Mathematica Environment Brian Evans and James McClellan Digital Signal Processing Group School of Electrical Engineering Georgia Institute of Technology Atlanta, GA 30332-0250 EMAIL: evans@eedsp.gatech.edu Since the version of the signal processing packages released by The Mathematica Journal, I have made changes to just about every file. The major changes implemented by the new version of the signal processing packages are discussed in the SignalProcessingExamples notebook, but they are summarized here: -------------------------- Version 2.0 ---------------------------------- (1) added object SignalPlot that plots the real and imaginary components of a 1-D function simultaneously on the same plot, as well as Dirac delta functions. It also plots 2-D signals. (The signal processing packages already plot 1-D discrete-time functions in lollipop style via DiscreteGraphics.) (2) the option Dialogue -> All now works for the forward bilateral Laplace transform (which means that all transform rule bases can completely justify their answers) (3) can plot upsampled sequences again (4) unified piecewise convolution (either function can be in list or expression form) (5) added many transform pairs to the continuous-time Fourier transform rule bases CTFTransform and InvCTFTransform (6) the Fourier transform rule bases now handle the convolution operator: CTFTransform[ Convolve[t][CPulse[1, t + 1/2], CPulse[1, t + 1/2]], t, w ] (7) RootLocus plots a root locus for one varying parameter (8) CFIR and CIIR structures have an optional third argument: Roots -> roots. Change is reflected in AnalogFilters notebook. (9) Computational objects for IIR structures, called IIRFunction, now exist. Also, FIR structures can be rewritten as a formula. Their values can now be plotted. Values are calculated as they are needed, and IIRFunction remembers (caches) previous calculations. (10) SequencePlot now provides a standard way to plot discrete-time expressions (11) Mathematica 1.2 has a problem performing partial fractions decomposition on real-valued polynomials even if you do all the factoring yourself. So, we have added a new option to the InvLaPlace object called Apart. If set to All, then we kludge around the Apart deficiency so that a partial fractions decomposition is formed. Otherwise, we inverse transform the function as an continuous-time IIR filter whose input is the inverse transform of the numerator. This is made possible by a new function called MyApart (see below). (12) The z-transform of IIR structures returns a valid region of convergence. (13) RealQ[z] returns True if and only if the head (tag) of z is Real. We have added RealValuedQ[z] which returns True if the imaginary component of z is 0 or 0.0 (this behaves the way RealQ used to). -------------------------- Version 2.1 ---------------------------------- (14) ZSolve, a difference equation solver, is more robust. ZSolve can also justify its answers in the same way as the transforms do. (15) DTFT of Step[n] is now correct (was missing a pi term). (16) LSolve can also justify its answers in the same way as the transforms do. (17) Convolution is now represented by more than one operator. Linear convolution in the discrete domain is Convolve[n] and linear convolution in the continuous domain is CConvolve[t]. -------------------------- Version 2.2 ---------------------------------- (18) Discrete-time convolution is now implemented. (19) New notebook called "SignalProcessingIntroduction" serves as an introduction to Mathematica as well as DSP. -------------------------- Version 2.21 --------------------------------- (20) SequencePlot for 2-D signals now samples functions at integer indices. -------------------------- Version 2.22 --------------------------------- (21) Stable will resolve stability of non-separable signals more often (e.g., Stable will return False when applied to the z-transform of (1/2)^n1 (4/5)^n2 Multinomial[n1,n2] Step[n1,n2] instead of an unresolved expression.) -------------------------- Version 2.23 --------------------------------- (22) Cleaned up descriptions and implementation of parameterized operators like Shift, Upsample, and Z. -------------------------- Version 2.3 ---------------------------------- (23) The TransformLookup option for the transform rule bases allows one to specify additional transform pairs. This is useful when a trans- form rule base does not contain a pair that you need. It is also useful for transforming abstract functions (i.e., saying that x[n] becomes X[z]). For example, In:= ZTransform[a^n x[n], n, z, TransformLookup -> { x[n] :> X[z] }] Out= ZTransData[ X[a z], Rminus[0], Rplus[Infinity], ZVariables[z]] The N-dimensional case is not as straightforward because each transform rule base applies a one-dimensional rule base N times in the order that the N time variables are given. Therefore, users must specify intermediate transform pairs: In:= LaPlace[ x[t1,t2], {t1,t2}, {s1,s2}, TransformLookup -> { x[t1,t2] :> X1[s1,t2], X1[s1,t2] :> X[s1,s2 } ] Out= LTransData[ X[s1, s2], Rminus[{0, 0}], Rplus[{Infinity, Infinity}], LVariables[{s1, s2}] ] Here, X1[s1, t2] denotes the Laplace transform of x[t1,t2] with respect to the first variable t1. It turns out that you could use X instead of X1 and the transform will still work out correctly. (This has been made simpler--- see Version 2.69). -------------------------- Version 2.31 --------------------------------- (24) Debugged the TeX forms of the signals (functions) and systems (operators) introduced by the signal processing packages, esp. their multidimensional forms. (25) Added AllSubsets function which returns all of the possible subsets of a set. For example, AllSubsets[ {v1,v2,v3} ] returns { v1, v2, v3, {v1, v2}, {v1, v3}, {v2, v3}, {v1, v2, v3} }. Similarly, AllSubsets[ {v1, v2, v3}, Plus] returns { v1, v2, v3, v1 + v2, v1 + v3, v2 + v3, v1 + v2 + v3 }. ------------------------- Version 2.32 -------------------------------- (26) The Laplace transform of (t - 1) CStep[t] was incorrect. It returned the same transform as (t - 1) CStep[t - 1] did. One rule was patched in "LaPlace.m". (27) The inverse Laplace transform cannot handle complex-valued roots of a rational polynomial because of limitations of 1.2 Mathematica's Apart function. We have encoded a new function MyApart which works around the problem but it is very, very slow. Now, inverse transforms of functions like "s / ( s^2 + 4 )^2" are possible. ------------------------- Version 2.33 -------------------------------- (28) The inverse Laplace transform was not always correctly inverting terms that corresponded to integration in the time domain. Patched one rule and added another to the post-processing rules. (29) The primitive MyApart did not normalize the coefficient of the highest power term in the denominator before rooting it. ------------------------- Version 2.34 -------------------------------- (30) The inverse Laplace transform was still not always correctly inverting terms that corresponded to integration in the time domain. Patched one rule and added another to the post-processing rules. ------------------------- Version 2.4 -------------------------------- (31) Integrating Delta functions over a finite interval did not work when the coefficient on the variable was one. Fixed. (36) The routines supporting the rules for multirate multidimensional signal processing (i.e., lattice and integer matrix theory) have been gathered in "Multirate.m" in the Support directory, including "DistinctCosetVectors", "SmithNormalForm", and "SmithReducedForm". (33) Inverse z-transforms of affine log functions are fixed. (34) WORKS UNDER ALL VERSIONS OF MATHEMATICA NOW!!!!! In making sure that the signal processing packages work for BOTH Mathematica 1.2 and 2.0, the following changes have taken place: (a) aliases have been discontinued so the former aliased objects AliasedSinc, AliasSinc, ASinc, InverseLaPlaceTransform, LaPlaceTransform, and MagnitudePhasePlot are now set equal to Dirichlet, Dirichlet, Dirichlet, InvLaPlace, LaPlace, and MagPhasePlot, respectively. Pretty much a transparent change. Added new alias: RootLocusPlot for RootLocus. (b) Mathematica 2.0 has a new undocumented function called ListQ which replaces the one provide by us (our definition for ListQ is still used for Mathematica 1.2 and lower). (c) Mathematica 2.0 represents collections (headless expressions like "(0,2,1,0)") using the head (data tag) Sequence. So, ToCollection has been modified to return a collection with the proper syntax under both Mathematica 1.2 and 2.0. As an indirect result, the functions GetAllExponents and GetAllFactors have been rewritten to return lists instead of collections. We have removed the function ExtractVariables, but GetVariables was kept so use it instead. (d) The evaluation strategy for the Which construct has changed. Now, each conditional statement in the construct MUST evaluate to True or False. We now force all of our Which conditional statements to evaluate to True or False by using TrueQ. (35) Mathematica 2.0 offers the DeclarePackage which allows one to specify the objects defined by a package. That way, when a user invokes a new primitive, Mathematica 2.0 will automatically load the package that defines it and then continue evaluating the expression. This works for most objects but does not handle the case where additional packages build up a definition incrementally. The initialization files "Analog.m", "Digital.m", and "Support.m" now only define the ``stubs'' for the objects in the analog, digital, and support signal processing packages, respectively. This means that you can place Needs[ "SignalProcessing`SignalProcessing`" ] in the initialization file without noticeably slowing down the initialization process. That way, students can go about their business without having to know which packages to load in beforehand. (36) One warning about Mathematica 2.0: it now assumes that variables represent complex (instead of real) numbers, so Sqrt[x^2] remains unsimplified. Although this is usually beneficial, this assumption sometimes clashes when you want to treat a variable as real-valued. For example, Plot[ x^(1/3), {x, -5, 5} ] cannot handle x in [-5, 0) because it takes the wrong branch cut at the origin which can be seen by plotting the real part of x^(1/3). It does plot the correct curve over x in [0, 5]. (37) For the most part, the notebook interface in 2.0 is upwardly compatible with that of Mathematica 1.2. However, 2.0 includes the preamble with all PostScript graphics now (an extra 20 kb of information per graphics cell) so some of our notebooks, like the piecewise convolution, will double in size. This change caused us to break up the z-transform notebook into three notebooks. ------------------------- Version 2.41 -------------------------------- (38) MagPhasePlot no longer generates a "Set::raw" warning message. and can plot Delta functions that appear in the frequency response (including impulse trains). (39) The inverse Laplace transform finally works identically under Mathematica 1.2 and 2.0. The main problem was that the partial fractions primitive Apart no longer normalized polynomials in denominators so the resulting denominators failed to match any of the transform pairs. (40) Added SPSimplify which applies simplification rules for signal processing expressions. It supports a dialogue option. (41) SequencePlot displays the "lollipops" more boldly and adapts their widths according to the number of samples plotted. ------------------------- Version 2.42 -------------------------------- (42) New tutorial notebook on the discrete-time Fourier transform. (43) SPSimplify failed to return if Dialogue was enabled. (44) Apart in Mathematica 2.0 does not handle non-rational polynomials (45) SignalPlot, SequencePlot, and MagPhasePlot are more robust under Mathematica 2.0: (46) ASPAnalyze and DSPAnalyze uses more descriptions for the analyses that they perform. They are also immune to changes in the Global environment. ------------------------- Version 2.43 -------------------------------- (47) CTFTransform cannot find SPSimplify. Fix to BeginPackage. (48) MagPhasePlot did not handle discontinuities well. Fixed. (49) Integral of Delta[t] over a finite interval. Fixed. (50) Continuous-time piecewise convolution works again. ------------------------- Version 2.44 -------------------------------- (51) Made the forward and inverse continuous-time Fourier transforms use the same set of rules by exploiting the duality property. (52) Fixed the procedure that normalizes expressions in the inverse Laplace transform rule base ------------------------- Version 2.45 -------------------------------- (53) Discrete convolution has been fixed. ------------------------- Version 2.5 --------------------------------- (54) Discrete and continuous Convolution has been fixed to work when one or more endpoints is infinite. (55) For SequencePlot of 1-D signals, the integer axis now is guaranteed to have integer ticks. (56) RationalGCD accepts the same style as Mathematica's GCD primitive. Specifically, RationalGCD no longer accepts a list of elements but instead has the form RationalGCD[e1, e2, ...]. (57) Added the Multirate supporting functions ColumnHermiteForm and RowHermiteForm as well as LCLM, LCRM, GCRD, and GCLD. (58) Automatic simplification of expressions is no longer performed, but syntax checking is still automatic. Simplification is now handled by SPSimplify which augments the built-in routine Simplify. There are new routines that perform extensive rewriting of signal processing expressions as discussed below. (59) The evaluation of Needs[ "SignalProcessing`ObjectOriented`" ] will enable to attachment of properties to (parameterized) operators using the routine DefSystem. The default properties (ASSOCIATIVE, LINEAR, SHIFTINVARIANT, etc.) are kept in the variable SPproperties. They will have already been attached to Mathematica's built-in operators (Re, Im, Conjugate, Plus, and Times) and to the new operators (Shift, Upsample, etc.). Users can attach properties to their own or other existing operators. (60) Syntax checking and simplification rules have already been developed for the new signals (functions) and systems (operators). We have also developed a comprehensive set of rearrangement rules for signals and systems based on their properties. These are accessible by the routine SPRecursiveRewrite. Heuristic techniques will be added soon. ------------------------- Version 2.51 -------------------------------- (61) SignalPlot supports exactly the same options as do Plot and Plot3D. Signal plots look better, too. (62) The Laplace transform rule bases now handle the Summation operator. ------------------------- Version 2.52 -------------------------------- (63) Cleaned up definition of the Summation operator and Delta function. (64) Cleaned up SignalPlot's display of delta functions and a train of delta functions. Updated DeltaPlot, Arrow2D, and SignalPlot. ------------------------- Version 2.53 -------------------------------- (65) Integration rules for Delta functions are now covered by DeltaIntegrate. Users can customize them (but first you will need to unprotect it). ------------------------- Version 2.54 -------------------------------- (66) The inverse z-transform can now inverse the forward transform of a downsample operation. (67) The forward and inverse z-transforms handle more cases involving the Summation operator. (68) The forward z-transform handles affine step functions and the inverse z-transform more cleanly works around the shortcomings of Mathematica's Apart primitive by relying on MyApart to invert those transforms than Apart cannot handle. (69) Integration is now more robust for expressions involving continuous step (CStep) functions and continuous pulse (CPulse) forms. (70) A minor adjustment was made to DistinctCosetVectors function. ------------------------- Version 2.55 -------------------------------- (71) Provided a "master" file for the signal processing extensions as to be compatible with the convention used by Mathematica's standard packages: use Needs["SignalProcessing`Master`"] (72) Inverse z-transform of the forward z-transform of Sin[b n] Step[n] returns Sin[b n] Step[n] as it used to do. The sequence of steps for the inverse z-transform has changed. OLD ORDER: rational pairs, non-rational pairs, partial fractions, pick off constants, additivity, delays, partial fractions (again), logarithm properties, downsampled sequence, multiplication by a^n, .... NEW ORDER: rational pairs, non-rational pairs, downsampled sequence, additivity in form of summation operator, multiplication by a^n, pick off constants, delays, additivity, normalize denominator, partial fractions, additivity in numerator, logarithm properties, ... (73) The routine TheFunction that converts signal processing expressions to formulas has been extended. (74) GetAllFactors, the function that helps determine if a variable in an expression has been scaled, is more robust. This helps the inverse z-transform properly handle the z-transform of functions like a^n Sin[b n] Step[n]. I have also improved ScalingFactor and UpsampleFactor so that the factors returned are independent of the variable being checked. ------------------------- Version 2.6 -------------------------------- (75) Introduced a heuristic to apply rewrite rules intelligently called SPHeuristicRewrite. (76) CirclePS, used by the pole-zero plotting routine, now uses the built-in Graphics primitive Circle. (77) The inverse Laplace transform produces right-sided transforms more often. The problem was in Mathematica 2.0 because Apart (and therefore MyApart) does not normalize the denominator(s) after partial fractions. Also, the Similarity Property is correctly implemented as only to allow f(cs) to be rewritten when c > 0. (78) Regardless of the value of the Dialogue option, the transform rule bases will keep track of the assumptions made on the free parameters. They can be retrieved at any time by evaluating Assuming[All]. The set of conditions can be cleared by evaluating Assuming[]. (79) The transform rule bases now print out assumptions made on parameters if the Dialogue option is set to All. (80) SignalPlot and SequencePlot handle periodic functions better, so Summation[i, -Infinity, Infinity, 1][CStep[1, t + 2 i]], or alternately Periodic[2,t][CStep[1,t]], plots correctly. (81) The TransformLookup option is more robust for all transforms. (82) You can now specify a list of symbols to treat as variables to SPSimplify by using the Variables option. (83) Improved DTFT Notebook that introduces Fourier analysis of sequences, the DTFT, and the DFT. ------------------------- Version 2.61 -------------------------------- (84) The user-specified transform pairs are now put at the beginning of the rule base so that they override the built-in knowledge and they appear in proper order during dialogue. As a consequence, the forward and inverse Fourier transforms now properly handle the TransformLookup option for specifying new transform pairs. (85) The inverse z-transform can handle downsampled and upsampled signals again. (86) The assumptions made on free parameters during the taking of a linear transforms are displayed in a more meaningful way. (87) New operator called CircularShift which is an abstract form of the primitive RotateRight. It was introduced because of the new DFT transform pairs written into the DFT rule bases. (88) I have idiot-proofed the transform rule bases. So, if you typed something like LaPlace[ CStep[t], t s ] it would complain about t s because you meant t, s. This should help new users. ------------------------- Version 2.62 -------------------------------- (89) The MagPhasePlot routine now takes the same options as Plot does. Furthermore, it works better under Mathematica 2.0. (90) The notation for PolyphaseDownsample and PolyphaseUpsample has changed to include the filter as a parameter. These operators are now single input, single output operators. (91) The rearrangement rules contained in SystemRewriteRules have been debugged, including the m-D versions of Crochiere and Rabiner's rules for 1-D multirate signal processing. Some rules were added. ------------------------- Version 2.63 -------------------------------- (92) All of the system rearrangement and simplification rules listed in Appendix D of Myers' MIT thesis have been encoded in their original 1-D forms. The multidimensional analogs have been developed and encoded as well. (93) ConvertToList, which converts an algebraic form of a (piecewise) function into a list of F-intervals, is more robust. It now properly converts functions like the trapezoid function defined by (t+1) u(t+1) - t u(t) - (t-1) u(t - 1) + (t-2) u(t-2) where u(v) is the continuous step function. Before, it would have claimed that this function had infinite extent when in fact its extent is from t=-1 to t=2 inclusive. (94) Renamed "Multirate.m" as "LatticeTheory.m" and made the package independent from other signal processing packages. (95) added the number theoretic EuclidFactors to complement the routine BezoutNumbers. BezoutNumbers finds integers mu and lambda given integers a and b such that a mu + b lambda == gcd(a,b) (BezoutNumbers calls the Mathematica primitive ExtendedGCD.) EuclidFactors finds the integers mu and lambda given relatively prime integers p and q and any integer k so that p mu + q lambda == k EuclidFactors also works when p and q are integer matrices (in which case mu, lambda, and k become integer vectors). ------------------------- Version 2.64 -------------------------------- (96) One of the simplification rules that handles the conjugation of complex exponentials dropped terms. It is fixed. Also, more simplification rules for complex exponentials have been added. (97) Under Mathematica 2.0, SPSimplify did not support the Trig option that Simplify supports. (98) For 1-D signals and sequences, SequencePlot and SignalPlot will plot a list of them on the same graph. (99) ZSolve switched the driving function and the difference equation when there were no initial conditions. (100) The continuous Fourier transform can handle filter sections again and even returns the proper region of convergence. ------------------------- Version 2.65 -------------------------------- (101) Made RootLocus faster and more robust. (102) 2-D pole-zero root locus plots handle non-separable polynomials. (103) The DFT of impulses is faster (104) GetShiftFactor computes shift factors in an expression for a given variable. (105) The inverse Laplace transform rule base better handles shifts in the Laplace variable. For example, s is shifted by b below: InvLaPlace[ Sqrt[Pi] Exp[-2 Sqrt[a] Sqrt[b + s] / Sqrt[b + s], s ] -(a/t) - b t E CStep[t] ---------------------- Sqrt[t] (106) The forward and inverse DFT, DTFT, and Laplace transform rule bases support an option called Definition. If set to True, then the rule base will apply the definition of the transform if all other attempts at the transform have failed. ------------------------- Version 2.66 -------------------------------- (107) The convolution routines simplify expressions involving Delta and Impulse functions. (108) The Definition option has been implemented for the z- and Fourier transforms so now all of the rule bases have this option. ------------------------- Version 2.67 -------------------------------- (109) When the Fourier transform cannot find a transform, it will try to apply the definition before it calls the Laplace transform rule bases. Also, the extra comma introduced in Version 2.66 at the end of the list of the Fourier transform rules has been removed. ------------------------- Version 2.68 -------------------------------- (110) The forward z-transform recognizes common shifts in the discrete- time variable, e.g. (n + 1) f[n + 1]. (111) Introduced ZPolynomial[m, n] which is an mth degree polynomial in n. It implements the property that the inverse z-transform of the mth derivative of F(z) is ZPolynomial[m, n] times f[n] shifted by m. (112) Simplification of conjugated polynomials carried out by SPSimplify has been enhanced. It now will show that H(z) = H*(z*) when H(z) is a rational polynomial having real-valued poles and zeroes. (113) PoleZeroPlot shades regions of convergence for the s-domain. Also, the DisplayFunction option is properly support so that you can redirect the output of the pole-zero plot to file via PoleZeroPlot[ s / ((s+1)(s+2)), s, -3, -2, False, DisplayFunction -> ( Display["pzplot1.mps", #1]& ) ] (114) SignalPlot for 1-D signals was generating the correct graphics but not plotting them on the screen. Fixed. ------------------------- Version 2.7 -------------------------------- (115) Using the TransformLookup option for multidimensional transforms is now as straightforward as saying x[n1,n2] :> X[z1,z2], i.e. intermediate transform pairs (one per dimension) do not have to be specified. (116) The forward z-transform now transforms multidimensional upsampled and downsampled functions and properly adjusts the region of convergence. It also handles transforms of expressions like Upsample[l1,n1][ Upsample[l2,n2][ x[n1,n2 ] ]: In:= ZTransform[ Upsample[l1,n1][ Upsample[l2,n2][ x[n1,n2] ] ], {n1,n2}, {z1, z2}, TransformLookup -> { x[n1,n2] :> X[z1,z2] } ] l1 l2 Out= ZTransData[ X[z1 , z2 ], Rminus[{0, 0}], Rplus[{Infinity, Infinity}], ZVariables[{z1, z2}] ] (117) The forward DTFT handles multidimensional upsampled and downsampled signals. It also inverse transforms upsampled signals such as the diamond tile CPulse[2 Pi, w1 + w2 + Pi] CPulse[2 Pi, w1 - w2 + Pi] which is Upsample[{{1,1},{-1,1}}. {n1,n2}][ Sinc[Pi n1] Sinc[Pi n2] ] in the discrete-time domain. (118) Discontinued the functions MyTogether, NegExponent, and NormalizedQ. (119) GetAllFactors, the function that helps determine if a variable has been scaled in an expression, has been made more robust (again). It is used by the z and Laplace transforms. (120) Added the System property SEPARABLE (121) The best news is that many of the multidimensional capabilities have been decoupled from the signal processing packages so that only 1-D and simple m-D operations are initially loaded. The packages will load in m-D extensions to existing operators (e.g., z-transforms of multidimensional resampled signals) as needed. This strikes a balance in its use by researchers and students. That is, the SPP initial load the facilities useful to undergraduates. Additional features are loaded as needed. (122) SPSimplify works around an infinite loop in Mathematica's Simplify command. This error in Simplify causes both LSolve and DSolve to hang when solving some first-order DE's. LSolve no longer hangs because it uses SPSimplify. (123) Analog filters can now be specified completely by their poles. CIIR[t, Roots -> {-1, -2, -3}] would compute the missing coefficient list from the poles: CIIR[t, {6, 11, 6, 1}]. Similiarly, CFIR[t, Roots -> {-1, 0, 1}] gives CFIR[t, {0, -1, 0, 1}]. (124) The time responses of analog filters can be plotted. (125) By default, MyApart will now root denominators of higher than fifth order numerically. ------------------------- Version 2.71 -------------------------------- (126) When an improper region of convergence is given to PoleZeroPlot, it will now display a union of all possible combinations. (127) All linear symbolic transforms (z, LaPlace, etc.) are faster--- the efficiency increase is proportional to the size of the expression being transformed. The speed of the transformation is now independent of the size of local information associated with each sub-expression. (128) The symbolic DFT is more robust. ------------------------- Version 2.72 -------------------------------- (129) The signal processing Notebooks are finally compatible with the two versions of Mathematica (1.2/2.0). (130) Inverse DTFT's of transforms that are upsampled in the time domain now work under 2.0. (131) Values for signals that are upsampled in multidimensions can now be computed and plotted, e.g. SequencePlot[ Upsample[{{1, 1},{-1, 1}}, {n1,n2}][ Abs[n1 + n2] ], {n1, 0, 9}, {n2, 0, 9} ] (132) ASPAnalyze does a better job analyzing simple functions like sin(t). (133) The continuous-time Fourier transform communicates better with the Laplace transform rule base so more transforms can be resolved. Also, many CTFT transform pairs were fixed and enhanced. (134) Magnitude responses involving Dirac delta functions are properly displayed again. ------------------------- Version 2.73 -------------------------------- (135) The plotting routines are now uniform in the way they convert Delta Delta functions, infinite summations, periodic operators, and other oddities into standard Mathematica expressions. The Extent1D routine has been improved and the SignalCleanup routine has been introduced. (136) The Aliasby operator is now always rewritten in terms of the Periodic operator. (137) The discrete-time Fourier transform (DTFT) has been made "mathematically correct" in that the forward transforms track the inherent periodicity. Normally, DTFT's are periodic with a period of 2 Pi, but resampled signals exhibit different periodicities. (138) The filter design routines automatically load additional routines when necessary. (139) PlotList has now been aliased to PiecewisePlot so as to reduce the confusion with ListPlot. ------------------------- Version 2.74 -------------------------------- (140) There is a new global variable called $DeltaFunctionScaling that controls how the height of Delta functions are to be plotted. The heights are either scaled to be proportional to the areas under the Delta functions (Scaled) or are set to maximum value of the rest of the plot. The default option is Scaled which is the way students learn to plot Delta functions. (141) Arrows representing Delta functions are drawn in a more uniform way. Downward and upward arrows are drawn according to the sign of the fourth parameter to Arrow2D. (142) Sampling with infinite impulse trains is now implemented as a part of the forward and inverse Fourier transform rule bases. Infinite impulse trains in continuous time are represented either as Summation[n, -Infinity, Infinity, 1][Delta[t + T n]] or as Periodic[T,t][Delta[t]] where T is the sampling period, n is an integer (index) variable, and t is the continuous-time variable. ------------------------- Future Plans -------------------------------- (143) The old way to convert signal processing expressions to formulas was to use TheFunction. That is still supported, but the functionality of TheFunction has been transferred to Mathematica's Normal routine. (144) Upsample and Downsample should take an optional third argument, namely a reindexing symbol. (145) Generating C/SPOX code for algorithms (146) Providing an algorithm design environment for linear multidimensional multirate systems (at the level of saying "write thesis") Because of changes (1), (5), and (6), we can recreate (for the most part) the pictorial dictionary of Fourier transforms in chapter 19 of Bracewell's book entitled "The Fourier Transform and Its Applications". Because of changes (9) and (12) and because of subtle modifications to the ExtractVariables object, DSPAnalyze now works correctly for IIR and FIR structures. I have included a release of all of the files that compose the signal processing packages. We have updated all of the notebooks because of the above changes. We have included all eight notebooks. The signal processing packages alter these standard Mathematica functions: (a) Integrate so that it can handle generalized functions like the Dirac delta (Delta) and functions with discontinuities like continuous-time step (CStep) and pulse (CPulse) functions. (b) Simplify applied to an And expression removes redundant conditions (c) Det of a number is that number (d) Dot product of two numbers is the product of the two numbers (e) The TeX forms of Re and Im primitives are Re and Im instead of R and I. (f) Under Mathematica 2.0, the default options for the Limit operator have been changed to { Analytic -> True, Direction -> Automatic }.