Audio Signal Processing and Coding

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, Ω_k), 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 Ω_k = 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, Ω_k, 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, ξ_υ,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, ξ_0,0(t), ξ_0,1(t), ξ_1,0(t), and ξ_1,1(t).

Figure 8.13. An example wavelet function.

Hint: From the DWT theory, the Fourier ...