Chapter 8, SUBBAND CODERS Video Solutions, Audio Signal Processing and Coding

Problem 1

In this problem, we will show that STFT can be interpreted as a bank of subband filters. Given the STFT, $X\left(n, \Omega_k\right)$, of the input signal, $x(n)$,
$$
X\left(n, \Omega_k\right)=\sum_{m=-\infty}^{\infty} x(m) w(n-m) e^{-j \Omega_k m}=w(n) * x(n) e^{-j \Omega_k n},
$$
where $w(n)$ is the sliding analysis window. Give a filter-bank realization of the STFT for a discrete frequency variable $\Omega_k=k(\Delta \Omega), k=0,1, \ldots, 7$ (i.e., 8 bands). Choose $\Delta \Omega$ such that the speech band $(20-4000 \mathrm{~Hz})$ is covered. Assume that the frequencies, $\Omega_k$, are uniformly spaced.

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Problem 2

The mother wavelet function, $\xi(t)$, is given in Figure 8.13. Determine and sketch carefully the wavelet basis functions, $\xi_{v, t}(t)$, for $v=0,1,2$ and $\tau=0,1,2$ associated with $\xi(t)$,
$$
\xi_{v, \tau}(t) \triangleq 2^{-v / 2} \xi\left(2^{-v} t-\tau\right),
$$
where $v$ and $\tau$ denote the dilation (frequency scaling) and translation (time shift) indices, respectively.

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Problem 3

Let $h_0(n)=[1 / \sqrt{2}, 1 / \sqrt{2}]$ and $h_1(n)=[1 / \sqrt{2},-1 / \sqrt{2}]$. Compute the scaling and wavelet functions, $\phi(t)$ and $\xi(t)$. Using $\xi(t)$ as the mother wavelet and generate the wavelet basis functions, $\xi_{0,0}(t), \xi_{0,1}(t), \xi_{1,0}(t)$, and $\xi_{1,1}(t)$.
FIGURE CANT COPY
Figure 8.13. An example wavelet function.

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Problem 4

Let $H_0\left(e^{j \Omega}\right)$ and $H_1\left(e^{j \Omega}\right)$ be ideal lowpass and highpass filters with cutoff frequency, $\pi / 2$, as shown in Figure 8.14. Sketch $\Phi(\Omega), \xi(\Omega)$, and the wavelet basis functions, $\xi_{0,0}(t), \xi_{0,1}(t), \xi_{1,0}(t)$, and $\xi_{1,1}(t)$.

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Problem 5

Show that if both $H_0\left(e^{j \Omega}\right)$ and $H_1\left(e^{j \Omega}\right)$ are causal FIR filters of order $N$, then the wavelet basis functions, $\xi_{v, \tau}(t)$, will have finite duration of $(N+1) 2^v$.

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Problem 6

Using equations (8.7) and (8.8), prove the following: 1) $\Phi(\Omega / 2)=\prod_{p=2}^{\infty}$ $H_0\left(e^{j \Omega / 2^p}\right)$, and 2$)|\Phi(\Omega)|^2+|\xi(\Omega)|^2=|\Phi(\Omega / 2)|^2$.

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Problem 7

From problem 8.6 we have, $\Phi(\Omega)=\frac{1}{\sqrt{2}} H_0\left(e^{j \Omega / 2}\right) \Phi(\Omega / 2)$ and $\xi(\Omega)=$ $\frac{1}{\sqrt{2}} H_1\left(e^{j \Omega / 2}\right) \Phi(\Omega / 2)$. Show that $\phi(t)$ and $\xi(t)$ can be obtained
$$
\text { Figure 8.14. Ideal lowpass and highpass filters with cutoff frequency, } \pi / 2 \text {. }
$$FIGURE CANT COPY
Figure 8.15. (a) The scaling function, $\phi(t)$, (b) the mother wavelet function, $\xi(t)$, and (c) input signal, $x(t)$.
recursively as,
$$
\begin{aligned}
& \phi(t)=\sqrt{2} \sum_n h_0(n) \phi(2 t-n) \\
& \xi(t)=\sqrt{2} \sum_n h_1(n) \phi(2 t-n)
\end{aligned}
$$

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Problem 8

Let the scaling function, $\phi(t)$, and the mother wavelet function, $\xi(t)$, be as shown in Figure 8.15(a) and Figure 8.15(b), respectively. Assume that the input signal, $x(t)$, is as shown in Figure 8.15(c). Given the wavelet series expansion,
$$
x(t)=\sum_{\tau=-\infty}^{\infty} \alpha(\tau) \phi(t-\tau)+\sum_{v=0}^{\infty} \sum_{\tau=-\infty}^{\infty} \beta(v, \tau) \xi_{v, \tau}(t),
$$
where both $v$ and $\tau$ are integers and denote the dilation and translation indices, respectively, $\alpha(\tau)$ and $\beta(v, \tau)$ are the wavelet expansion coefficients. Solve for $\alpha(\tau)$ and $\beta(v, \tau)$.

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