Question

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.

   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.

Audio Signal Processing and Coding

Andreas Spanias, Ted… 1st Edition

Chapter 8, Problem 1

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Step 1

Step 1: **Understand the STFT Expression** The Short-Time Fourier Transform (STFT) of a signal $ x(n) $ is given by: \[ X(n, \Omega_k) = \sum_{m=-\infty}^{\infty} x(m) w(n-m) e^{-j \Omega_k m} \] This can be interpreted as the convolution of the signal \( x(n) Show more…

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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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