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]
(10.4) 
where is the input signal at time , and . In this section, we will show how the DFT can be computed exactly from a bank of 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 perfectreconstruction filter bank. The STFT is then an extension of the DFT filter bank to include nonrectangular analysis windows and a downsampling factor .
The RunningSum Lowpass Filter
Perhaps the simplest FIR lowpass filter is the socalled runningsum lowpass filter [175]. The impulse response of the length runningsum lowpass filter is given by
Figure 9.10 depicts the generic operation of filtering by to produce , where is the impulse response of the filter. The output signal is given by the convolution of and :
In this form, it is clear why the filter (9.5) is called ``running sum'' filter. Dividing it by , it becomes a ``moving average'' filter, averaging the most recent input samples.
The transfer function of the runningsum filter is given by [263]
(10.6) 
so that its frequency response is
Recall that the term is a linear phase term corresponding to a delay of samples (half of the FIR filter order). This arises because we defined the runningsum lowpass filter as a causal, linear phase filter.
We encountered the ``aliased sinc function''
(10.7) 
previously in Chapter 5 (§3.1.2) and elsewhere as the Fourier transform (DTFT) of a sampled rectangular pulse (or rectangular window).
Note that the dc gain of the length running sum filter is . We could use a moving average instead of a running sum ( ) to obtain unity dc gain.
Figure 9.11 shows the amplitude response of the runningsum lowpass filter for length . The gain at dc is , and nulls occur at and . These nulls occur at the sinusoidal frequencies having respectively one and two periods under the 5sample ``rectangular window''. (Three periods would need at least samples, so doesn't ``fit''.) Since the passband about dc is not flat, it is better to call this a ``dcpass filter'' rather than a ``lowpass filter.'' We could also call it a dc sampling filter.^{10.1}
Modulation by a Complex Sinusoid
Figure 9.12 shows the system diagram for complex demodulation.^{10.2}The input signal is multiplied by a complex sinusoid to produce the frequencyshifted result
(10.8) 
Given a signal expressed as a sum of sinusoids,
(10.9) 
then the demodulation produces
(10.10) 
We see that frequency is downshifted to . In particular, frequency (the ``center frequency'') is downshifted to dc.
Making a Bandpass Filter from a Lowpass Filter

Figure 9.13 shows how a bandpass filter can be made using a lowpass filter together with modulation. The input spectrum is frequencyshifted by , lowpass filtered, then frequencyshifted by , thereby creating a bandpass filter centered at frequency . From our experience with rectangularwindow transforms (Fig.9.11 being one example), we can say that the bandpassfilter bandwidth is equal to the mainlobe width of the aliased sinc function, or radians per sample (measured from zerocrossing to zerocrossing).
Uniform RunningSum Filter Banks
Using a length runningsum filter, let's make bandpass filters tuned to center frequencies
(10.11) 
Since the bandwidths, as defined, are , the filter passbands overlap by 50%. A superposition of the bandpass frequency responses for is shown in Fig.9.14. Also shown is the frequencyresponse sum, which we will show to be exactly constant and equal to . This gives our filter bank the perfect reconstruction property. We can simply add the outputs of the filters in the filter bank to recreate our input signal exactly. This is the source of the name FilterBank Summation (FBS).
System Diagram of the RunningSum Filter Bank
Figure 9.15 shows the system diagram of the complete
channel filter bank
constructed using length
FIR runningsum lowpass filters. The
th channel computes:
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 filterbank output , , is precisely the DFT of the input signal when , i.e.,
(10.14) 
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:
(10.15) 
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 a CooleyTukey 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 discretetime Short Time Fourier Transform (STFT). The STFT normally also uses a window function other than the rectangular window used in this development (the runningsum lowpass filter).
Inverse DFT and the DFT Filter Bank Sum
The Length inverse DFT is given by [264]
(10.16) 
This suggests that the DFT Filter Bank can be inverted by simply remodulating the baseband filterbank signals , summing over , and dividing by for proper normalization. That is, we are led to conjecture that
(10.17) 
This is in fact true, as we will later see. (It is straightforward to show as an exercise.)
Next Section:
FBS Window Constraints for R=1
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STFT Filter Bank