# Connexions

You are here: Home » Content » DSPA » Quadrature Mirror Filterbanks (QMF)

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Janko Calic. E-mail the author

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

Although the DFT filterbanks are widely used, there is a problem with aliasing in the decimated channels. At first glance, one might think that this is an insurmountable problem and must simply be accepted. Clearly, with FIR filters and maximal decimation, aliasing will occur. However, a simple example will show that it is possible to exactly cancel out aliasing under certain conditions!!!

Consider the following trivial filterbank system, with two channels. (Figure 1)

Note x ^ n=xn x ^ n x n with no error whatsoever, although clearly aliasing occurs in both channels! Note that the overall data rate is still the Nyquist rate, so there are clearly enough degrees of freedom available to reconstruct the data, if the filterbank is designed carefully. However, this isn't splitting the data into separate frequency bands, so one questions whether something other than this trivial example could work.

Let's consider a general two-channel filterbank, and try to determine conditions under which aliasing can be cancelled, and the signal can be reconstructed perfectly (Figure 2).

Let's derive x ^ n x ^ n , using z-transforms, in terms of the components of this system. Recall (Figure 3) is equivalent to Yz=HzXz Y z H z X z Yω=HωXω Y ω H ω X ω and note that (Figure 4) is equivalent to Yz=m=xmz(Lm)=xzL Y z m x m z L m x z L Yω=XLω Y ω X L ω and (Figure 5) is equivalent to Yz=1Mk=0M1Xz1M W M k Y z 1 M k 0 M 1 X z 1 M W M k Yω=1Mk=0M1XωM+2πkM Y ω 1 M k 0 M 1 X ω M 2 k M Yz Y z is derived in the downsampler as follows: Yz=m=xMmzm Y z m x M m z m Let n=Mm n M m and m=nM m n M , then Yz=n=xnp=δnMpznM Y z n x n p δ n M p z n M

Now

xnp=δnMp=IDFTxω*2πMk=0M1δω2πkM=IDFT2πMk=0M1Xω2πkM=1Mk=0M1Xn W M - n k | W M =ej2πM x n p δ n M p IDFT x ω 2 M k 0 M 1 δ ω 2 k M IDFT 2 M k 0 M 1 X ω 2 k M W M 2 M 1 M k 0 M 1 X n W M - n k
(1)
so
Yz=n=(1Mk=0M1xn W M - n k )znM=1Mk=0M1xn W M + k z1Mn=1Mk=0M1Xz1M W M k Y z n 1 M k 0 M 1 x n W M - n k z n M 1 M k 0 M 1 x n W M + k z 1 M n 1 M k 0 M 1 X z 1 M W M k
(2)
Armed with these results, let's determine X ^ z x ^ n X ^ z x ^ n . (Figure 6) Note U 1 z=Xz H 0 z U 1 z X z H 0 z U 2 z=12k=01Xz12ej2πk2 H 0 z12e(jπk)=12Xz12 H 0 z12+12Xz12 H 0 z12 U 2 z 1 2 k 0 1 X z 1 2 2 k 2 H 0 z 1 2 k 1 2 X z 1 2 H 0 z 1 2 1 2 X z 1 2 H 0 z 1 2 U 3 z=12Xz H 0 z+12Xz H 0 z U 3 z 1 2 X z H 0 z 1 2 X z H 0 z U 4 z=12 F 0 z H 0 zXz+12 F 0 z H 0 zXz U 4 z 1 2 F 0 z H 0 z X z 1 2 F 0 z H 0 z X z and L 4 z=12 F 1 z H 1 zXz+12 F 1 z H 1 zXz=12 F 1 z H 1 zXz+12 F 1 z H 1 zXz L 4 z 1 2 F 1 z H 1 z X z 1 2 F 1 z H 1 z X z 1 2 F 1 z H 1 z X z 1 2 F 1 z H 1 z X z Finally then,
X ^ z= U 4 z+ L 4 z=12( H 0 z F 0 zXz+ H 0 z F 0 zXz+ H 1 z F 1 zXz+ H 1 z F 1 zXz)=12( H 0 z F 0 z+ H 1 z F 1 z)Xz+12( H 0 z F 0 z+ H 1 z F 1 z)Xz X ^ z U 4 z L 4 z 1 2 H 0 z F 0 z X z H 0 z F 0 z X z H 1 z F 1 z X z H 1 z F 1 z X z 1 2 H 0 z F 0 z H 1 z F 1 z X z 1 2 H 0 z F 0 z H 1 z F 1 z X z
(3)
Note that the XzXω+π X z X ω corresponds to the aliasing terms!

There are four things we would like to have:

1. No aliasing distortion
2. No phase distortion (overall linear phase → simple time delay)
3. No amplitude distortion
4. FIR filters

## No aliasing distortion

By insisting that H 0 z F 0 z+ H 1 z F 1 z=0 H 0 z F 0 z H 1 z F 1 z 0 , the Xz X z component of X ^ z X ^ z can be removed, and all aliasing will be eliminated! There may be many choices for H 0 H 0 , H 1 H 1 , F 0 F 0 , F 1 F 1 that eliminate aliasing, but most research has focused on the choice F 0 z= H 1 z : F 1 z= H 0 z F 0 z H 1 z : F 1 z H 0 z We will consider only this choice in the following discussion.

## Phase distortion

The transfer function of the filter bank, with aliasing cancelled, becomes Tz=12( H 0 z F 0 z+ H 1 z F 1 z) T z 1 2 H 0 z F 0 z H 1 z F 1 z , which with the above choice becomes Tz=12( H 0 z H 1 z H 1 z H 0 z) T z 1 2 H 0 z H 1 z H 1 z H 0 z . We would like Tz T z to correspond to a linear-phase filter to eliminate phase distortion: Call Pz= H 0 z H 1 z P z H 0 z H 1 z Note that Tz=12(PzPz) T z 1 2 P z P z Note that Pz1npn P z 1 n p n , and that if pn p n is a linear-phase filter, 1npn 1 n p n is also (perhaps of the opposite symmetry). Also note that the sum of two linear-phase filters of the same symmetry (i.e., length of pn p n must be odd) is also linear phase, so if pn p n is an odd-length linear-phase filter, there will be no phase distortion. Also note that Z-1pzpz=pn1npn={2pn  if  n is odd0  if  n is even Z p z p z p n 1 n p n 2 p n n is odd 0 n is even means pn=0 p n 0 , when nn is even. If we choose h 0 n h 0 n and h 1 n h 1 n to be linear phase, pn p n will also be linear phase. Thus by choosing h 0 n h 0 n and h 1 n h 1 n to be FIR linear phase, we eliminate phase distortion and get FIR filters as well (condition 4).

## Amplitude distortion

Assuming aliasing cancellation and elimination of phase distortion, we might also desire no amplitude distortion ( |Tω|=1 T ω 1 ). All of these conditions require Tz=12( H 0 z H 1 z H 1 z H 0 z)=czD T z 1 2 H 0 z H 1 z H 1 z H 0 z c z D where cc is some constant and DD is a linear phase delay. c=1 c 1 for |Tω|=1 T ω 1 . It can be shown by considering that the following can be satisfied! Tz=PzPz=2czD{2pz=2cδnD  if  n is oddpn=anything  if  n is even T z P z P z 2 c z D 2 p z 2 c δ n D n is odd p n anything n is even Thus we require Pz=n=0 N p2nz(2n)+zD P z n 0 N p 2 n z 2 n z D Any factorization of a Pz P z of this form, Pz=AzBz P z A z B z can lead to a Perfect Reconstruction filter bank of the form H 0 z=Az H 0 z A z H 1 z=Bz H 1 z B z [This result is attributed to Vetterli.] A well-known special case (Smith and Barnwell) H 1 z=(z(2D)+1 H 0 z-1) H 1 z z 2 D 1 H 0 z Design techniques exist for optimally choosing the coefficients of these filters, under all of these constraints.

H 1 z= H 0 z H 1 ω= H 0 π+ω= H 0 * πω H 1 z H 0 z H 1 ω H 0 ω H 0 * ω
(4)
for real-valued filters. The frequency response is "mirrored" around ω=π2 ω 2 . This choice leads to Tz= H 0 2z H 0 2z T z H 0 z 2 H 0 z 2 : it can be shown that this can be a perfect reconstruction system only if H 0 z= c 0 z(2 n 0 )+ c 1 z(2 n 1 ) H 0 z c 0 z 2 n 0 c 1 z 2 n 1 which isn't a very flexible choice of filters, and not a very good lowpass! The Smith and Barnwell approach is more commonly used today.

## 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