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DFT-Based Filterbanks

Module by: Douglas L. Jones. E-mail the author

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One common application of multirate processing arises in multirate, multi-channel filter banks (Figure 1).

Figure 1
Figure 1 (imag001.png)
One application is separating frequency-division-multiplexed channels. If the filters are narrowband, the output channels can be decimated without significant aliasing.

Such structures are especially attractive when they can be implemented efficiently. For example, if the filters are simply frequency modulated (by ⅇ-ⅈ2πkLn 2 k L n ) versions of each other, they can be efficiently implemented using FFTs!

Furthermore, there are classes of filters called perfect reconstruction filters which are of finite length but from which the signal can be reconstructed exactly (using all MM channels), even though the output of each channel experiences aliasing in the decimation step. These types of filterbanks have received a lot of research attention, culminating in wavelet theory and techniques.

Uniform DFT Filter Banks

Suppose we wish to split a digital input signal into NN frequency bands, uniformly spaced at center frequencies ω k =2πkN ω k 2 k N , for 0≤k≤N−1 0 k N 1 . Consider also a lowpass filter hn h n , Hω≈1if|ω|<πN0otherwise H ω 1 ω N 0 . Bandpass filters can be constructed which have the frequency response H k ω=Hω+2πkN H k ω H ω 2 k N from h k n=hnⅇ-ⅈ2πknN h k n h n 2 k n N The output of the kkth bandpass filter is simply (assume hn h n are FIR)

xn* h k n=∑m=0M−1xn−mhmⅇ-ⅈ2πkmN= y k n x n h k n m 0 M 1 x n m h m 2 k m N y k n (1)
This looks suspiciously like a DFT, except that M≠N M N , in general. However, if we fix M=N M N , then we can compute all y k n y k n outputs simultaneously using an FFT of xn−mhm x n m h m : The kth FFT frequency output= y k n kth FFT frequency output y k n ! So the cost of computing all of these filter banks outputs is ONlogN O N N , rather than N2 N 2 , per a given nn. This is very useful for efficient implementation of transmultiplexors (FDM to TDM).

Exercise 1

How would we implement this efficiently if we wanted to decimate the individual channels y k n y k n by a factor of NN, to their approximate Nyquist bandwidth?

Solution

Simply step by NN time samples between FFTs.

Exercise 2

Do you expect significant aliasing? If so, how do you propose to combat it? Efficiently?

Solution

Aliasing should be expected. There are two ways to reduce it:

  1. Decimate by less ("oversample" the individual channels) such as decimating by a factor of N2 N 2 . This is efficiently done by time-stepping by the appropriate factor.
  2. Design better (and thus longer) filters, say of length LN L N . These can be efficiently computed by producing only NN (every LLth) FFT outputs using simplified FFTs.

Exercise 3

How might one convert from NN input channels into an FDM signal efficiently? (Figure 2)

Figure 2
Figure 2 (imag002.png)

Note:

Such systems are used throughout the telephone system, satellite communication links, etc.

Solution

Use an FFT and an inverse FFT for the modulation (TDM to FDM) and demodulation (FDM to TDM), respectively.

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