Recall that the Length Discrete Fourier Transform (DFT) is defined as
More generally, for all , we will call Fig.9.15 the DFT filter bank. The DFT filter bank is the special case of the STFT for which a rectangular window and hop size are used.
The sliding DFT is obtained by advancing successive DFTs by one sample:
When is a power of 2, the DFT can be implemented using a Cooley-Tukey Fast Fourier Transform (FFT) using only operations per transform. By keeping track of the linear phase term (an modification), a DFT Filter Bank can be implemented efficiently using an FFT. Uniform FIR filter banks are very often implemented in practice using FFT software such as fftw.
Note that the channel bandwidths are narrow compared with half the sampling rate (especially for large ), so that the filter bank output signals are oversampled, in general. We will later look at downsampling the channel signals to obtain a ``hopping FFT'' filter bank. ``Sliding'' and ``hopping'' FFTs are special cases of the discrete-time Short Time Fourier Transform (STFT). The STFT normally also uses a window function other than the rectangular window used in this development (the running-sum lowpass filter).