Chapter 6, TIME-FREQUENCY ANALYSIS: FILTER BANKS AND TRANSFORMS Video Solutions, Audio Signal Processing and Coding

Problem 1

Prove the identities shown in Figure 6.10.

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Consider the analysis-synthesis filter bank shown in Figure 6.1 with input signal, $s(n)$. Show that $\hat{s}(n)=\frac{1}{M} \sum_{m=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} \sum_{k=0}^{M-1} s(m)$ $h_k(l M-m) g_k(l-M n)$, where $M$ is the number of subbands.

Lucas Finney

Numerade Educator

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Problem 3

For the down-sampling and up-sampling processes given in Figure 6.26, show that
$$
\begin{aligned}
& \qquad \begin{aligned}
S_d(\Omega) & =\frac{1}{M} \sum_{l=0}^{M-1} S\left(\frac{\Omega+2 \pi l}{M}\right) H\left(\frac{\Omega+2 \pi l}{M}\right) \\
\text { and } S_u(\Omega) & =S(\Omega M) G(\Omega)
\end{aligned}
\end{aligned}
$$

Victor Salazar

Numerade Educator

Problem 4

Using results from Problem 6.3, Prove Eq. (6.5) for the analysis-synthesis framework shown in Figure 6.1.

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Problem 5

Consider Figure 6.27,
Given $s(n)=0.75 \sin (\pi n / 3)+0.5 \cos (\pi n / 6), n=0,1, \ldots, 6, H_0(z)=1-$ $z^{-1}$, and $H_1(z)=1+z^{-1}$
a. Design the synthesis filters, $G_0(z)$ and $G_1(z)$, in Figure 6.27 such that aliasing distortions are minimized.
b. Write the closed-form expression for $v_0(n), v_1(n), y_0(n), y_1(n)$, $w_0(n), w_1(n)$, and the synthesized waveform, $\hat{s}(n)$. In Figure 6.27, assume $y_i(n)=\hat{y}_i(n)$, for $i=0,1$.
c. Assuming an alias-free scenario, show that $\hat{s}(n)=\alpha s\left(n-n_0\right)$, where $\alpha$ is the QMF bank gain, $n_0$ is a delay that depends on $H_i(z)$ and $G_i(z)$. Estimate the value of $n_0$.
d. Repeat steps (a) and (c) for $H_0(z)=1-0.75 z^{-1}$ and $H_1(z)=1+$ $0.75 z^{-1}$.
FIGURE CANT COPY
Figure 6.27. A two-band maximally decimated analysis-synthesis filter bank.

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Problem 6

In this problem, we will compare the two-band QMF and CQF designs. Given $H_0(z)=1-0.9 z^{-1}$.
a. Design a two-band (i.e., $M=2$ ) QMF [Use Eq. (6.7)].
b. Design a two-band CQF for $L=8$ [Use Eq. (6.8)].
c. Consider the two-band QMF and CQF banks in Figure 6.28 with input signal $\quad s(n)=0.75 \sin (\pi n / 3)+0.5 \cos (\pi n / 6), n=0,1, \ldots, 6$. Compare the designs in (a) and (b) and check for the alias-free reconstruction in case of CQF. Give the delay values $d_1$ and $d_2$.
d. Extend the two-band QMF design in part (a) to polyphase factorization [use Equations (6.9) and (6.10)]. What are the advantages of employing polyphase factorization?

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Problem 7

In this problem, we will design and analyze a four-channel uniform treestructured QMF bank.
a. Given $H_{00}(z)=1+0.1 z^{-1}$ and $H_{10}(z)=1+0.9 z^{-1}$. Complete the tree-structured QMF bank (use Figure 6.8) for four channels.
b. Using the identities given in Figure 6.10 (or Eq. (6.11)), construct a parallel analysis-synthesis filter bank. The parallel analysis-synthesis filter bank structure must be similar to the one shown in Figure 6.1 with $M=4$.
c. Plot the frequency response of the resulting parallel filter bank analysis filters, i.e., $H_0(z), H_1(z), H_2(z)$, and $H_3(z)$. Comment on the pass-band and stopband structures of the magnitude responses associated with these filters.
d. Plot the impulse response $h_1(n)$. Is $h_1(n)$ symmetric?

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Problem 8

Repeat Problem 6.7 for a four-channel uniform tree-structured CQF bank with $L=4$.

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Problem 9

A time-domain plot of an audio signal is shown in Figure 6.29. Given the flexibility to encode the regions A through E with varying frame sizes. Which of the following choices is preferred?

Choice I:
Long frames in regions B and D.
Choice II:
Long frames in regions $\mathrm{A}, \mathrm{C}$, and $\mathrm{E}$.
FIGURE CANT COPY
Figure 6.28. A two-band QMF and CQF design comparison.
Figure 6.29. An example audio segment with harmonics (region A), a transient (region B), background noise (region C), exponentially weighted harmonics (region D), and background noise (region E) segments.
Choice III:
Short frames in region B only.
Choice IV:
Short frames in regions B and D.
Choice V:
Short frames in region A only.

Explain how would you assign frequency-resolution (high or low) among the regions $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$, and $\mathrm{E}$.

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Problem 10

A pure tone at $f_0$ with $P_0 \mathrm{~dB} \mathrm{SPL}$ is encoded such that the quantization noise is masked. Let us assume that a 256-point MDCT produced an in-band signal-to-noise ratio of $S N R_A$ and encodes the tone with $b_A$ bits/sample. And, a 1024-point MDCT yielded $S N R_B$ and encodes the tone with $b_B$ bits/sample.
Figure 6.30. Audio frames x1(n), x2(n), x3(n), and x4(n) for Problem 6.11.
FIGURE CANT COPY
In which of the two cases we will require the most bits/sample (state if $b_A>b_B$ or $b_A<b_B$ ) to mask the quantization noise.

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Problem 11

Given the signals, $x_1(n), x_2(n), x_3(n)$, and $x_4(n)$ as shown in Figure 6.30. Let all the signals be of length 1024 samples. When transform coded using a 512-point MDCT, which of the signals will result in pre-echo distortion?

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Problem 12

n this problem, we will study the filter banks that are based on the DFT. Use Eq. (6.39a) to implement a $2 M$-point DFT of $x(n)$ given in Figure 6.31. Assume $M=8$.
a. Give the plots of $|X(k)|$.
b. Plot the frequency response of the second- and third-channel analysis filters that are associated with the basis vectors $h_1(n)$ and $h_2(n)$.
c. State whether the DFT filter bank is evenly stacked or oddly stacked.

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Problem 13

In this problem, we will study the filter banks that are based on the DCT. Use Eq. (6.41a) to implement a $M$-point DCT of $x(n)$ given in Figure 6.31. Assume $M=8$.
a. Give the plots of $|X(k)|$.
b. Also plot the frequency response of the second and third channel analysis filters that are associated with the basis vectors $h_1(n)$ and $h_2(n)$.
c. Plot the impulse response of $h_1(n)$ and see if it is symmetric.
d. Is the DCT filter bank evenly stacked or oddly stacked?

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Problem 14

In this problem, we will study the filter banks based on the MDCT.
a. First, design a sine window, $w(n)=\sin [(2 n+1) \pi / 4 M]$ with $M=8$.
b. Check if the sine window satisfies the generalized perfect reconstruction conditions, i.e., Eqs. (6.28a) (6.28b).
c. Next, design a MDCT analysis filter bank, $h_k(n)$, for $0<k<7$.
Figure 6.31. Input signal, $x(n)$, for Problems 6.12, 6.13, and 6.14.
FIGURE CANT COPY
d. Plot both the impulse response and the frequency response of the analysis filters, $h_1(n)$ and $h_2(n)$.
e. Compute the MDCT coefficients, $X(k)$, of the input signal, $x(n)$, shown in Figure 6.31.
f. Is the impulse response of the analysis filter, $h_1(n)$, symmetric?

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08:09

Problem 15

Show analytically that a DCT can be implemented using FFTs. Also, use $x(n)$ given in Figure 6.32 as your test signal and verify your software implementation.

Khoobchandra Agrawal

Numerade Educator

Problem 16

Give expressions for DCT-I, DCT-II, DCT-III, and DCT-IV orthonormal transforms (e.g., see [Rao90]). Use the signals, $x_1(n)$ and $x_2(n)$, shown in Figure 6.33 to study the differences in the 4-point DCT coefficients obtained from different types of DCT. Describe, in general, whether choosing a particular type of DCT affects the energy compaction of a signal.

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Problem 17

In this problem, we will design a two-band $(M=2)$ cosine-modulated PQMF bank with $L=8$.
a. First, design a linear phase FIR prototype lowpass filter (i.e., $w(n)$ ), with normalized cutoff frequency $\pi / 4$. Plot the frequency response of this window. Use fir 2 command in MATLAB to design the lowpass filter.
b. Use Eq. (6.13) and (6.14) to design the PQMF analysis and synthesis filters, respectively.
Figure 6.33. Test signals to study the differences among various types of orthonormal
DCT transforms.FIGURE CANT COPY
Figure 6.34. A two-band PQMF design.
c. In Figure 6.34, use $s(n)=0.75 \sin (\pi n / 3)+0.5 \cos (\pi n / 6), n=0$, $1, \ldots, 6$, and generate $s_3(n)$. Compare $s(n)$ and $s_3(n)$ and comment on any type of distortion that you may observe.
d. What are the advantages of employing a cosine modulated filter bank over the two-band QMF and two-band CQF.
e. List some of the key differences between the CQF and the PQMF in terms of 1) the analysis filter bank frequency responses, 2) phase distortion, 3) impulse response symmetries.
f. Use sine window, $w(n)=\sin [(2 n+1) \pi / 4 M]$ with $M=8$, and repeat steps (b) and (c).

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Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17