5.5 Dft-based filterbanks

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

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)

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.