### TABLE I EXPERIMENTAL SETTING (W IS THE INVERSE DISCRETE WAVELET TRANSFORM).

### Table 2: Comparison of row BAS Function and its inverse Fourier Transform with various T values.

"... In PAGE 8: ... The plot shows the rapid decay of the spectral coefficients which becomes zero after n=60. Table2... ..."

Cited by 1

### Table 2.1: Discrete Short-Time Fourier Transform Relations

### Table 2: E ects of correcting the inverse ltering method on information content. discrete Fourier space is periodic, and this convolution can \wrap around quot; producing some undesirable border e ects. These can be seen in Figure 7(c). Zero-padding can be employed to ameliorate this. Figure 7(e) shows the result when this is done. The improvement to the deconvolution is shown at the borders of gure 7(f). Finally, this image ( gure 6) has been collected through a 128 128 rectangular window. This is equivalent to convolving the Fourier transform with a small sinc function ( gure 2). This has the e ect of \smearing out quot; the power spectrum of the image. To improve this, the original signal is multiplied by a Hamming window before it is deconvolved. The result of this windowing is the image in gure 7(g). The improvement is shown in gure 7(h). The di erence is most noticable at the high frequency sections of the image. The focus has enhanced because there is less leakage in the Fourier domain. Figure 7(g) could be considered the \correct quot; deconvolution.

"... In PAGE 14: ... Figure 7(g) could be considered the \correct quot; deconvolution. Table2 shows the information content of the deconvolutions shown in gure 7. These values are determined from the correlation coe cient of the image as desrcribed by Vollath ([29]).... ..."

### Table 1. A set of modules after HW/SW partitioning and mapping of real-life applications

"... In PAGE 5: ... malization, and feature extraction to generate the features from the speech. After manually partitioning and mapping of the above applications, Table1 depicts a set of on-chip modules for the audio decoder and the speech recognition system. The modules are two PowerPC processors, inverse modified discrete cosine transformation (IMDCT), compact flash interface, speech processor, fast Fourier transformation (FFT), and an audio buffer for streaming.... ..."

### Table 5.1: For Inverse Discrete Cosine Transformation Number of RF Number of Registers

2003

### Table 1: A list of three-dimensional Fourier transforms of various integrable functions used in meshless methods. with the exception of that of the Gaussian, the inverse multiquadric, and the Sobolev spline [28, Theorems 6.10, 6.13, and Page 133], the Fourier transform of each function was computed using (6).

"... In PAGE 6: ...Examples of integrable radial basis functions are given in Table1 . Non-integrable radial basis functions and polynomial terms are sometimes used, examples of which include the multiquadric and the thin-plate spline.... ..."

### Table 1: Some properties of the Fourier transform

"... In PAGE 16: ...imension of u is 1=time, i.e., frequency). In the case of image processing, the signal is a function of space rather than time, and in that case the domain of the Fourier transform is called spatial frequency. Table1 summarizes some important properties of the one-dimensional Fourier transform. The symbol is used to denote convolution:... ..."

### Table 1: Fast Fourier and other transform factorizations.

1997

"... In PAGE 2: ... 2 The Language of Factorizations A fast transform algorithm can be seen as a sparse factorization of the transform matrix. Table1 displays di erent factorizations for the DFT matrix as well as other transforms that arise in signal processing. The abbreviation Bn;m = (Fn Im)Dn;m is used in the FFT, with D a diagonal matrix of weights, Pn;p is the stride permutation matrix.... In PAGE 10: ... In general, the same transformation strategy can han- dle all problems of the same dimension, di erent transformation strategies are needed for multidimensional problems. bbb We successfully transform all fast Fourier and other transform and transposition algo- rithms shown on Table1 using the above rules for determining the loop structure and high level assignments. We generated the radix code, the straight-line code that computes an FFT in the loop nest by symbolically capturing the assignments and operations of an FFT calculation of appropriate length using the prime factor FFTs as the base cases.... ..."

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### Table 1: Signal objects de ned in SignalProcessing`Support`. For the syntax of CPulse, Dirichlet, FIR, IIR, LineImpulse and Pulse, consult the on-line documentation (e.g., ?CPulse).

"... In PAGE 5: ... We now examine them in more detail. Table1 lists the twelve new functions. There are discrete and continuous versions of the impulse (Impulse and Delta), step (Step and CStep), and pulse functions (Pulse and CPulse).... In PAGE 8: ... (However, certain obvious simpli cations are carried out: for example, InvZ[z,n][Z[n,z][f]] reduces to f.) Similarly, the functions of in Table1 are not reduced to Mathematica built-in objects until they appear as arguments to TheFunction. One may wish to reduce them, for example, in order to use Mathematica apos;s built-in plotting routines to plot them.... In PAGE 8: ... One may wish to reduce them, for example, in order to use Mathematica apos;s built-in plotting routines to plot them. Naturally, some functions in Table1 , like Delta and Unit, cannot be expressed in terms of Mathematica built-in objects, so TheFunction leaves them alone. Other Features Another facility provided by SignalProcessing`Support` is the plotting of signals and transforms.... In PAGE 18: ... Transforms of exponentials in the time domain are inverse-transformed by the exponential property rule, not by table lookup. Some strategies for inverting z-transforms ( Table1 0) are similar to those applied in taking forward z-transforms, but some new ones are also needed. Two such strategies are partial fractions and power series expansion.... In PAGE 19: ... complex cepstrum: Z?1flog X(z)g ! ? 1 nZ?1 ( z X(z) d dz X(z)) *9. apply the inverse z-transform to the rst N terms of a series expansion about z = 0 Table1 0: Strategies for inverse z-transforms. An asterisk means that once the rule is applied to an expression, it will no longer be applied to any part of that expression.... In PAGE 33: ...designing/analyzing 1-D analog lters DTFT discrete Fourier analysis EducationalTool interactive version of a conference paper describ- ing educational impact of Mathematica LaPlaceTest testing procedure for Laplace transforms PiecewiseConvolution tutorial on discrete/continuous convolution README brief introduction SignalProcessingExamples interactive version of paper in the The Mathemat- ica Journal SignalProcessingIntroduction introduction to Mathematica, signal processing, and the signal processing packages SignalProcessingUsage usage information about every new object de ned by the signal processing packages zTransformI z-transform tutorial, part I zTransformII z-transform tutorial, part II zTransformIII z-transform tutorial, part III Table1 1: List of the signal processing Notebooks transforms as long as the options are set properly. The default options are biased toward DTFT apos;s: Domain - gt; Continuous, DomainScale - gt; Linear, MagRangeScale - gt; Linear, PhaseRangeScale - gt; Degree, and PlotRange - gt; All.... In PAGE 40: ...Possible Values Meaning Apart Rational, All Partial fraction decomposition only applies to polynomials with real or rational coe cients Definition True, False Use the transform de nition if all else fails to nd the transform (does not apply to the inverse z or Laplace transforms) Dialogue False, True, All Ascending levels of justi cation Simplify True, False Apply SPSimplify to result Terms False or integer Number of terms in series expansion (False means none) TransformLookup list of rules Users can specify their own transform pairs, like {x[n] : gt; X[z]} or {y[t1,t2] : gt; Y[s1,s2]} Table1 2: Meaning of the Options for the Transform Rule Bases... In PAGE 41: ...Option Default Value CTFTransform Dialogue False Simplify True DFTransform Dialogue False InvDFTransform Dialogue False Terms False DTFTransform Dialogue False LaPlace Dialogue True Simplify True InvCTFTransform Apart Rational Dialogue False Simplify True Terms False InvDTFTransform Dialogue False Terms False InvLaPlace Apart Rational Dialogue True Simplify True Terms 10 InvZTransform Dialogue True Terms 10 ZTransform Dialogue True Table1 3: Options for the Transform Rule Bases. Definition always defaults to False and TransformLookup always defaults to an empty list.... ..."