PROBLEMS
8.1. In this problem, we will show that STFT can be interpreted as a bank of subband filters. Given the STFT, X(n, Ωk), of the input signal, x(n),
where w(n) is the sliding analysis window. Give a filter-bank realization of the STFT for a discrete frequency variable Ωk = k(ΔΩ), k = 0, 1, … , 7 (i.e., 8 bands). Choose ΔΩ such that the speech band (20–4000 Hz) is covered. Assume that the frequencies, Ωk, are uniformly spaced.
8.2. The mother wavelet function, ξ(t), is given in Figure 8.13. Determine and sketch carefully the wavelet basis functions, ξυ,τ(t), for υ = 0, 1, 2 and τ = 0, 1, 2 associated with ξ(t),
where υ and τ denote the dilation (frequency scaling) and translation (time shift) indices, respectively.
8.3. Let 
and 
. Compute the scaling and wavelet functions, ϕ(t) and ξ(t). Using ξ(t) as the mother wavelet and generate the wavelet basis functions, ξ0,0(t), ξ0,1(t), ξ1,0(t), and ξ1,1(t).
Figure 8.13. An example wavelet function.
Hint: From the DWT theory, the Fourier ...
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