DFT Filter Bank

Recall that the Length Discrete Fourier Transform (DFT) is defined as

Comparing this to (9.4), we see that the filterbank output , , is precisely the DFT of the input signal when , i.e.,

In other words, the filterbank output at time (the set of samples for ), equals the DFT of the first samples of (, ). That is, taking a snapshot of all filterbank channels at time yields the DFT of the input data from time 0 through .

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 for any integer , the Sliding DFT coincides with the DFT filter bank. At other times, they differ by a linear phase term. (Exercise: find the linear phase term.) The Sliding DFT redefines the time origin every sampling period (each modulation term within the DFT starts at time 0 for each transform), while the DFT Filter Bank does not redefine the time origin (modulation terms are ``free running'' as they would be in an analog filter bank). Since ``DFT time'' repeats every samples, the two treatments coincide every samples (i.e., for every integer ).

When is a power of 2, the DFT can be implemented using the 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 the FFT. Uniform FIR filter banks are very often implemented in practice using the FFT.

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).

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Order a Hardcopy of Spectral Audio Signal Processing

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Next: Inverse DFT and the DFT Filter Bank Sum

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.