Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.1 System notation for the mapping and inverse mapping between a signal and its complex cepstrum.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.2 Determination of the complex cepstrum for minimum-phase signals.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.3 Cascade of three systems implementing the computation of the complex cepstrum operation D ∗ [ ].

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.4 Approximate realization using the DFT of (a) D ∗ [ ・ ] and (b) D − ∗ 1 [ ・ ].

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.5 (a) Samples of arg[X(e jω )]. (b) Principal value of part (a). (c) Correction sequence for obtaining arg from ARG.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.6 Canonic form for homomorphic systems where inputs and corresponding outputs are combined by convolution.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.7 Deconvolution of a sequence into minimum-phase and allpass components using the cepstrum.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.8 The use of homomorphic deconvolution to separate a sequence into minimum-phase and maximum- phase components.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure 13.9 Pole-zero plot of the z -transform X(z) = V(z)P(z) for the example signal of Figure

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure The sequences: (a) v[n], (b) p[n], and (c) x[n] corresponding to the pole–zero plot of Figure 13.9.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure The sequences: (a) v[n], (b) p[n], and (c) x[n]. ˆ ˆˆ

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure The sequences: (a) c v [n], (b) c p [n], and (c) c x [n].

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Fourier transforms of x[n] in Figure (a) Log magnitude. (b) Principal value of the phase. (c) Continuous “unwrapped” phase after removing a linear-phase component from part (b). The DFT samples are connected by straight lines.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure (a) Complex cepstrum xp[n] of sequence in Figure 13.10(c). (b) Cepstrum c x [n] of sequence in Figure 13.10(c). ˆ

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure (a) System for homomorphic deconvolution. (b) Time-domain representation of frequency-invariant filtering.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Time response of frequency-invariant linear systems for homomorphic deconvolution. (a) Lowpass system. (b) Highpass system. (Solid line indicates envelope of the sequence [n] as it would be applied in a DFT implementation. The dashed line indicates the periodic extension.)

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Lowpass frequency-invariant linear filtering in the system of Figure (a) Real parts of the Fourier transforms of the input (solid line) and output (dashed line) of the lowpass system with N 1 = 14 and N 2 = 14 in Figure 13.16(a). (b) Imaginary parts of the input (solid line) and output (dashed line). (c) Output sequence y [n] for the input of Figure 13.10(c).

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Illustration of highpass frequency-invariant linear filtering in the system of Figure (a) Real part of the Fourier transform of the output of the highpass frequency-invariant system with N 1 = 14 and N 2 = 512 in Figure 13.16(b). (b) Imaginary part for conditions of part (a). (c) Output sequence y [n] for the input of Figure

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure (a) Complex cepstrum of x[n] = x min n ∗ x ap [n]. (b) Complex cepstrum of x min [n]. (c) Complex cepstrum of x ap [n].

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure (a) Minimum-phase output. (b) Allpass output obtained as depicted in Figure 13.7.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure (a) Minimum-phase output. (b) Maximum-phase output obtained as depicted in Figure 13.8.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Discrete-time model of speech production.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Homomorphic deconvolution of speech. (a) Segment of speech weighted by a Hamming window. (b) High quefrency component of the signal in (a). (c) Low quefrency component of the signal in (a).

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Complex logarithm of the signal of Figure 13.23(a): (a) Log magnitude. (b) Unwrapped phase.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure Complex cepstrum of the signal in Figure 13.23(a) (inverse DTFT of the complex logarithm in Figure 13.24).

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure (a) System for cepstrum analysis of speech signals. (b) Analysis for voice speech. (c) Analysis for unvoiced speech.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure (a) Cepstra and (b) log spectra for sequential segments of voiced speech.

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P13.10

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P13.12

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P13.14

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P13.17

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W. Schafer Figure P13.31