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).