The DFT Filter Bank

To obtain insight into the operation of filter banks implemented using an FFT, this section will derive the details of the DFT Filter Bank. More general STFT filter banks are obtained by using different windows and hop sizes, but otherwise are no different from the basic DFT filter bank.

The Discrete Fourier Transform (DFT) is defined by [264]

$$X(\omega_k) = \sum_{n=0}^{N-1} x(n) e^{-j\omega_k n}$$ (10.4)

where $x(n)$ is the input signal at time $n$ , and $\omega_k\isdef 2\pi k/N$ . In this section, we will show how the DFT can be computed exactly from a bank of $N$ FIR bandpass filters, where each bandpass filter is implemented as a demodulator followed by a lowpass filter. We will then find that the inverse DFT is computed by remodulating and summing the output of this filter bank. In this way, the DFT filter bank is shown to be a perfect-reconstruction filter bank. The STFT is then an extension of the DFT filter bank to include non-rectangular analysis windows $w$ and a downsampling factor $R$ .