LR Rabiner and RW Schafer, June 3

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DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 438CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING 3. we see that the zero-quefrency value of the complex cepstrum (and the cepstrum) depends on the gain constant. Setting ˆx[0] = 0 (c[0] = 0) is equivalent to normalizing the log magnitude spectrum to a gain constant of Mo A (−b −1 ) = 1 (8.25) k=1 4. if X(z) has no zeros outside the unit circle (i.e., all bk = 0), 5 then k ˆx[n] = 0 for n < 0 (8.26) Such signals are called minimum phase signals [15]. 5. if X(z) has no poles or zeros inside the unit circle (i.e., all ak = 0 and all ck = 0) then ˆx[n] = 0 for n > 0 (8.27) Such signals are called maximum-phase signals [15]. 8.2.4 Some Examples of Complex Cepstrum Analysis In this section, we give a few examples of complex cepstra of simple signals. We base our work on the power series expansion of Eq (8.22), which is the basis for the general formula in Eq. (8.23). Example 8.1 (Decaying Exponential Sequence) Determine the complex cepstrum of the minimum-phase sequence Solution x1[n] = a n u[n] |a| < 1. We first determine the z-transform of x1[n] as follows: X1(z) = ∞ n=0 a n z −n = Next we solve for ˆ X1(z) as: 1 1 − az −1 |z| > |a|. (8.28a) ˆX1(z) = log[X1(z)] = − log(1 − az −1 ) ∞ n a = z n −n . (8.28b) n=1 5 We tacitly assume no poles outside the unit circle.
DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 8.2. HOMOMORPHIC SYSTEMS FOR CONVOLUTION 439 The complex cepstrum value ˆx1[n] is simply the coefficient of the term z −n in Eq. (8.28b); i.e., ˆx1[n] = an u[n − 1]. (8.28c) n Example 8.2 (Single Zero Outside Unit Circle) Determine the complex cepstrum of the maximum-phase sequence Solution x2[n] = δ[n] + bδ[n + 1] |b| < 1. In this case the z-transform of x2[n] is easily shown to be X2(z) = 1 + bz = bz(1 + b −1 z −1 ). (8.29a) That is, X2(z) is a single zero outside the unit circle. Next we determine ˆ X2(z) obtaining ˆX2(z) = log[X2(z)] = log(1 + bz) ∞ (−1) = n+1 n n=1 Again picking the coefficient of z −n for ˆx2[n] we obtain Example 8.3 (Simple Echo) b n z n . (8.29b) ˆx2[n] = (−1)n+1bn u[−n − 1]. (8.29c) n Determine the complex cepstrum of the sequence x3[n] = δ[n] + αδ[n − Np]. Discrete convolution of any sequence x1[n] with this sequence produces a scaled-by-α echo of the first sequence; i.e., x1[n] ∗ (δ[n] + αδ[n − Np]) = x1[n] + αx1[n − Np]. Solution The z-transform of x3[n] is X3(z) = 1 + αz −Np . (8.30a)
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LR Rabiner and RW Schafer, June 3

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