LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
494 BIBLIOGRAPHY
8.8 Consider the sequence
x[n] = δ[n] + αδ[n − Np]
(a) Find the complex cepstrum of x[n]. Sketch your result.
(b) Sketch the cepstrum, c[n], for x[n].
(c) Now suppose that the approximation ˆxp[n] is computed using
Eq. (8.57c). Sketch ˆxp[n] for 0 ≤ n ≤ N − 1, for the case Np = N/6.
What if N is not divisible by Np?
(d) Repeat (c) for the cepstrum approximation cp[n] for 0 ≤ n ≤
N − 1, as computed using Eq. (8.60).
(e) If the largest impulse in the cepstrum approximation, cp[n], is
used to detect Np, how large must N be in order to avoid confusion?
8.9 In order to smooth the log magnitude spectrum of a signal, its
cepstrum is often windowed and Fourier transformed, as shown in
Fig. P8.9.
Figure P8.9: Cepstral signal processing system.
(a) Write an expression relating ˜ X(e jω ), to log |X(e jω )| and L(e jω )
where L(e jω ) is the Fourier tranform of l[n].
(b) To smooth log |X(e jω )| what type of cepstral window, l[n], should
be used?
(c) Compare the use of a rectangular cepstral window, and a Hamming
cepstral window.
(d) How long should the cepstral window be? Why?
8.10 Consider a segment of voiced speech to be represented as
where
x[m] = p[m] ∗ hv[m]
p[m] =
∞
r=−∞
δ[m − rNp]
In computing the complex cepstrum (or cepstrum) the first step is
to multiply s[m] by a window, w[m], thereby obtaining the segment