# Connexions

You are here: Home » Content » Digital Signal Processing: A User's Guide » DFT-Based Filterbanks

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

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

# DFT-Based Filterbanks

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

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

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(j2πkLn) 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 0kN1 0 k N 1 . Consider also a lowpass filter hn h n , Hω{1  if  |ω|<πN0  otherwise   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(j2πknN) 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=0M1xnmhme(j2πkmN)= 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 MN 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 xnmhm 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)

#### 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.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks