LR Rabiner and RW Schafer, June 3

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DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 436CHAPTER 8. THE CEPSTRUM AND HOMOMORPHIC SPEECH PROCESSING unit circle, and the terms (1−b −1 k z) correspond to zeros outside the unit circle.3 The representation of Eq. (8.19) suggests the factorization into a product of a minimum-phase and a maximum-phase signal [15] as in X(z) = Xmin(z) · z −Mo Xmax(z), (8.20a) where the minimum-phase component has z-transform Xmin(z) = Mi A (1 − akz −1 ) k=1 Ni (1 − ckz −1 ) k=1 (8.20b) with all poles and zeros inside the unit circle, and the z-transform of the maximum-phase component is Mo Xmax(z) = (−b −1 k ) Mo (1 − bkz) (8.20c) k=1 with all the zeros lying outside the unit circle. The product representation in Eq. (8.20a) suggests that x[n] can be represented by the convolution x[n] = xmin[n] ∗ xmax[n − Mo] where the minimum-phase component is causal (i.e., xmin[n] = 0 for n < 0) and the maximum-phase component is anti-causal (i.e., xmax[n] = 0 for n > 0). The factor z −Mo represents simply the shift in the time origin by Mo samples that is required to make the overall sequence, x[n], causal. This factor is generally omitted in computing the complex cepstrum because it is often irrelevant or because it can be dealt with by simply noting the value of Mo. This is equivalent to assuming that x[n] = xmin[n] ∗ xmax[n] and therefore the complex cepstrum is ˆx[n] = ˆxmin[n] + ˆxmax[n]; i.e. it is the sum of a part due to the minimum-phase component of the input and a part due to the maximum-phase component. Under the assumption of Eq. (8.18), the complex logarithm of X(z) is M0 ˆX(z) = log |A| + log |b −1 k | + log[z−Mo Mi ] + log(1 − akz −1 ) k=1 k=1 k=1 k=1 Mo Ni + log(1 − bkz) − log(1 − ckz −1 ) (8.21) k=1 Mo Observe that in Eq. (8.21) we have used only the magnitude of the term A (−b −1 k ) when taking the complex logarithm of X(z) in Eq. (8.19). This is because A 3 To simplify our discussion, we do not allow zeros or poles to be precisely on the unit circle of the z-plane. This restriction is not a serious practical limitation, and it does not limit the conclusions the we draw about the general properties of the cepstrum and complex cepstrum. Further, we restrict our attention to stable, realizable systems whose poles are all inside the unit circle of the z−plane. k=1
DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009 8.2. HOMOMORPHIC SYSTEMS FOR CONVOLUTION 437 times the product of the zeros outside the unit circle (which occur in complex conjugate pairs) will always be real if the signal x[n] is real. The algebraic sign of the product can be determined if necessary, but it is usually not used in computing the complex cepstrum. When Eq. (8.21) is evaluated on the unit circle, it can be seen that the term log[e −jωMo ] will contribute only to the imaginary part of the complex logarithm. Since this term only carries information about the time origin, it is generally removed in the process of computing the complex cepstrum [20]. Thus, we will also neglect this term in our discussion of the properties of the complex cepstrum. Using the fact that each of the logarithmic terms can be written as a power series expansion, based on the well known power series ∞ Z log(1 − Z) = − n n n=1 |Z| < 1, (8.22) it is relatively straightforward to show that the complex cepstrum has the form ⎧ M0 log |A| + log |b −1 k | n = 0 k=1 M0 ⎪⎩ k=1 ⎪⎨ Ni c ˆx[n] = n k n − Mi k=1 b −n k n k=1 a n k n n > 0 n < 0. (8.23) Equation (8.23) shows a number of important properties of the complex cepstrum, i.e., 1. we see that, in general, the complex cepstrum is nonzero and of infinite extent for both positive and negative n, even though x[n] may be causal, or even of finite duration (when X(z) has only zeros). 2. it is seen that the complex cepstrum is a decaying sequence that is bounded by |ˆx[n]| < β α|n| |n| for |n| → ∞, (8.24) where α is the maximum absolute value of the quantities ak, bk, and ck, and β is a constant multiplier. 4 4 In practice, we generally deal with finite-length signals, which are represented by polynomials in z −1 ; i.e., the numerator in Eq. (8.19). In many cases, the sequence may be hundreds or thousands of samples long. A somewhat remarkable result is that for such sequences, almost all of the zeros of the polynomial tend to cluster around the unit circle, and as the sequence length increases, the roots move closer to the unit circle [5]. This implies that for long, finite-length sequences, the decay of the complex cepstrum is due primarily to the factor 1/|n|.
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LR Rabiner and RW Schafer, June 3

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