# 6.5 Dft-based filterbanks

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One common application of multirate processing arises in multirate, multi-channel filter banks ( [link] ).

One application is separating frequency-division-multiplexed channels. If the filters are narrowband, the output channelscan be decimated without significant aliasing.

Such structures are especially attractive when they can be implemented efficiently. For example, if the filters are simplyfrequency modulated (by $e^{-(i\frac{2\pi k}{L}n)}$ ) 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 $M$ channels), even though the output of each channel experiences aliasing in the decimation step. These types of filterbankshave 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 $N$ frequency bands, uniformly spaced at center frequencies ${\omega }_{k}=\frac{2\pi k}{N}$ , for $0\le k\le N-1$ . Consider also a lowpass filter $h(n)$ , $H(\omega )\approx \begin{cases}1 & \text{if \left|\omega \right|< \frac{\pi }{N}}\\ 0 & \text{otherwise}\end{cases}$ . Bandpass filters can be constructed which have the frequency response ${H}_{k}(\omega )=H(\omega +\frac{2\pi k}{N})$ from ${h}_{k}(n)=h(n)e^{-(i\frac{2\pi kn}{N})}$ The output of the $k$ th bandpass filter is simply (assume $h(n)$ are FIR)

$(x(n), {h}_{k}(n))=\sum_{m=0}^{M-1} x(n-m)h(m)e^{-(i\frac{2\pi km}{N})}={y}_{k}(n)$
This looks suspiciously like a DFT, except that $M\neq N$ , in general. However, if we fix $M=N$ , then we can compute all ${y}_{k}(n)$ outputs simultaneously using an FFT of $x(n-m)h(m)$ : The $\text{kth FFT frequency output}={y}_{k}(n)$ ! So the cost of computing all of these filter banks outputs is $O(N\lg N)$ , rather than $N^{2}$ , per a given $n$ . This is very useful for efficient implementation of transmultiplexors (FDM to TDM).

How would we implement this efficiently if we wanted to decimate the individual channels ${y}_{k}(n)$ by a factor of $N$ , to their approximate Nyquist bandwidth?

Simply step by $N$ time samples between FFTs.

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

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 $\frac{N}{2}$ . This is efficiently done by time-stepping by the appropriate factor.
2. Design better (and thus longer) filters, say of length $LN$ . These can be efficiently computed by producing only $N$ (every $L$ th) FFT outputs using simplified FFTs.

How might one convert from $N$ input channels into an FDM signal efficiently? ( [link] )

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

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

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Source:  OpenStax, Digital signal processing: a user's guide. OpenStax CNX. Aug 29, 2006 Download for free at http://cnx.org/content/col10372/1.2
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