DFT Filter Bank
Recall that the Length
Discrete Fourier Transform (DFT) is
defined as
![]() |
(10.13) |
Comparing this to (9.12), we see that the filter-bank output



![]() |
(10.14) |
In other words, the filter-bank output at time










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:
![]() |
(10.15) |
When






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).
Next Section:
Inverse DFT and the DFT Filter Bank Sum
Previous Section:
System Diagram of the Running-Sum Filter Bank