DISCRETE-SIGNAL ANALYSIS AND DESIGN- P22

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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P22:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with.

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MULTIPLICATION AND CONVOLUTION 91 (c) The convolution of (a) and (b) using the basic convolution equation (5-4), y(n) = x (m) ∗ y(m). We see the familiar smoothing and stretching operation that the convolution performs on x (m) and h(m). Convolution needs the additional time region for correct results, as noted in the equation in part (c) and in Eq. (5-6). (d) This step gets the convolution of x (m) and h(m) and also the spectrum (DFT) of the convolution in one step using the double summation of Eq. (5-6). (e) See step (f). (f) Steps (e) and (f). These steps get the DFT spectrum X (k ) of x (m) and spectrum H (k ) of h(m) using the DFT in Eq. (1-2). (g) The product X (k )H (k ) is the spectrum Z (k ) of the “output”. This product is the sequence multiplication described in Eq. (5-1). The additional factor N will be explained below. Note that the spectrum of the output in part (g) is identical to the spectrum of the output found in part (d). That is, X (k )H (k ) = DFT of x (m) ∗ h(m), and the IDFT of X (k ) ∗ H (k ) = x (m)h(m), not shown here. (h) The IDFT in Eq. (1-8) produces the sequence z (n), which is iden- tical to the convolution x (m) ∗ h(m) that we found in part (c). In part (g) we introduced the factor N . If we look at the equation in part (d) we see a single factor 1/N . But the product of X (k )H (k ) in part (g) produces the factor (1/N )2 . A review of parts (e) and (f) verify this. This produces an incorrect scale factor in part (h), so we correct part (g) to Þx this problem. There are different conventions used for the scale factors for the various forms of the DFT and IDFT. The correction used here takes this problem into account for the Bracewell (see Chapter 1) conventions that we are using. This action does not produce an error. it makes the “bookkeeping” correct, and the Mathcad worksheet takes the correct action as needed. These mild “discrepancies” can show up, and need not create concern. In Fig. 5-7 the convolution in parts (c) and (h) has a sharp peak at location n = 8. When the signal is mildly contaminated with noise, this is a good location to place a detector circuit that creates a synchronizing pulse. Convolution is frequently employed in this manner: for example, in radar receivers [Blinchikoff and Zverev, 1971, Chap. 7].
92 x(m) := 0 N := 32 m := −N, −N + 1.. N − 1 h(m) := 0 1 if m ≥ 0 k := −N, −N + 1.. N − 1 exp −m if m ≥ 0 0 if m ≥ 8 n := −N, −N + 1.. N − 1 6 1 1 0.5 x(m) h(m) 0.5 0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 m m (a) (b) DFT of convolution of x(m) and h(m) Convolution of x(m) and h(m) N−1 N−1 N−1 1 Q(k) := · (x(m)·h(n − m))·exp −j·2·π· n ·k y(n) := (x(m)⋅h(n − m)) N ∑∑ N ∑ m=0 n=0 m = −N 6 2 5 1.2 4 y(n) = x(n) * h(n) 0.4 3 0.4 DFT of convolution 2 1 1.2 0 2 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 n k (c) (d ) Figure 5-7 Convolution and spectra of two sequences.
Spectrum of X(k) Spectrum of H(k) N−1 N−1 1 X(k) : = 1 ⋅ x(m)⋅exp −j⋅2⋅π⋅k⋅ m H(k) : = ⋅ h(m)⋅exp −j⋅2⋅π⋅k⋅ m N ∑ N ∑ N m =−N N m=−N 0.2 Spectrum of X(k) 0.2 Spectrum of H(k) 0 0 −0.2 −0.2 0 5 10 15 20 25 30 0 5 10 15 20 25 30 k k (e) (f ) Product of X(k) and H(k) IDFT of product of X(k) and H(k) N−1 Z(k) := N⋅X⋅(k)⋅H(k) z(n) : = z(k)⋅exp −j⋅2⋅π⋅k⋅ n ∑ N k=0 2 6 5 Product of X(k) and H(k) 4 0 3 z(n), same as y(n) 2 −2 1 0 5 10 15 20 25 30 35 0 0 5 10 15 20 25 30 k n (g) (h) 93 Figure 5-7 (Contined ).
94 DISCRETE-SIGNAL ANALYSIS AND DESIGN DECONVOLUTION The inverse of convolution is deconvolution, where we try to separate the elements that are convolved into their constituent parts. There are some interesting uses for this, including de-reverberation, estimation of speech parameters, restoration of acoustic recordings, echo removal, video anal- ysis, and nonlinear systems. This introductory book cannot get involved in this advanced subject, but the references, in particular [Oppenheim and Schafer, 1999, Chap. 10] and [Wikipedia] provide readable introductions. In the example of Fig. 5-7, the results in parts (c) and (h) are related to the inputs x (m) and h(m) in parts (a) and (b), but it may be difÞcult or impos- sible to say that these particular x (m) and h(m) are the only two functions that can produce the results in parts (c) and (h). When the input functions include random properties, the difÞculty is compounded, and this is a major problem with deconvolution. An effort is made to Þnd something that is not random that can be a basis for deconvolution. One example is the dynamics of an antique phonograph recording machine. Advanced statistical methods try to perform deconvolution in a noise-contaminated environment [Wikipedia]. REFERENCES Carlson, A.B., 1986, Communications Systems, 3rd ed., McGraw-Hill, New York. Schwartz, M., 1980, Information Transmission, Modulation and Noise, 3rd ed., McGraw-Hill, New York. Blinchikoff, H.J., and A.I. Zverev, 1976, Filtering in the Time and Frequency Domains Wiley & Sons, New York. Oppenheim, A.V., and R.W. Schafer, Digital Signal Processing, 1976, McGraw-Hill, New York. Wikipedia [http://en.wikipedia.org/wiki/Deconvolution].
6 Probability and Correlation The ideas to be described and illustrated in this chapter are introduced initially in terms of deterministic, discrete, eternal steady-state sequences, conforming to the limited goals of this introductory book. Then small amounts of additive noise (as we saw in Chapter 4) are added for further analysis. Some interesting and useful techniques will be introduced, and these ideas are used in communication systems analysis and design. This material, plus the References, will help the reader to get started on more advanced topics, but Þrst we want to review (once more) brießy an item from Chapter 1 that we will need. Where we perform a summation from 0 to N − 1, we assume that all of the signiÞcant signal and noise energy that we are concerned with lies within, or has been conÞned to, those boundaries. This also validates our assumptions about the steady-state repetition of sequences. For this reason we stipulate that all signals are “power signals” (repetitive) and not “energy signals” (nonrepetitive). In Chapter 3 we looked at aliasing and spectral leakage and ways to deal with them, and that helps to assure our reliance on 0 to N −1. We can increase N by 2M (M = 2, 3, 4, · · ·) and Discrete-Signal Analysis and Design, By William E. Sabin Copyright  2008 John Wiley & Sons, Inc. 95