Maximally Decimated Filter Banks

A filter bank decomposes the input signal into M non-overlapping

band-pass channels Bk, with k = 0, 1, , (M 1) such that(M1)

k=0

Bk =

[, ].

2MM

0

H (Z)0

H (Z)1 M1

H (Z)

w

H0(z) in /M to /M, H1(z) in /2M to /M and /M to /2M and

so on

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Assuming brick wall response shown above, input signal can be

recovered by just summing these M channels.

Exact recovery of the original signal (just by summing) may not

be possible, if each channel is decimated by M, because of the

expansion of each channel spectra by M.

Decimation is essential for data compression.

A recovery is possible by the proper design of synthesis (recon-

struction) section of the filter bank.

A filter bank is said to be maximally (critically) decimated if the

input signal x[n], spectrum is divided into M non-overlapping bands

and each band signal is decimated by a factor M.

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Because this is the maximum factor by which we can decimate

without losing information.

x(n) sampled at 8 kHz, passed thro two filters (0-2 and 2-4).

Decimation can be max of 2 to recover without losing information.

If decimation by 4, then we will lose information from 1 to 3 kHz

in the reconstructed signal.

Effects of Using Filter Banks:

Introduces several aspects into reconstructed signal: aliasing, am-

plitude distortion, phase distortion and quantization effect

First three can be eliminated by proposer design but not the fourth

one. Hence we assume its effect is negligible.

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The analysis filter bank will have M parallel filters.

Uniform Filter Bank of Order M:

M

M

2M

2M

0

H (Z)0

H (Z)1 M1

H (Z)

w

Ideal Response

Practical Response

w

0

:

:

Since, we cannot use ideal filters in practice, aliasing is in-

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evitable in the frequency decomposition.

Aliasing must therefore be canceled by proper design of the

reconstruction filters in the synthesis filter bank.

Also, the M filters will introduce magnitude and phase distor-

tions into the decomposed signals.

The reconstruction filters must take care of eliminating these

distortions.

Quadrature Mirror Filter (QMF) Bank:

Filters in the filter bank must be carefully designed to avoid dis-

tortion and also cancel aliasing.

This will lead to perfect reconstruction after the synthesis filter

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bank.

In fact, filters in analysis and synthesis blocks are designed simul-

taneously for aliasing cancelation and distortion free performance.

Such filters are called Quadrature Mirror Filters (QMFs).

Two Channel QMF Bank:

H (Z)0

H (Z)1

x [n]0

x[n]

x [n]1

Q2 2

Q 22

F (Z)

F (Z)

0

1

x[n]

Input signal x[n] is filtered in parallel by H0(Z) and H1(Z), downsam-

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pled, coded and transmitted.

Coder is represented by Q, where the signal is coded into a speci-

fied number of bits. This is where quantization of the signal takes

place.

2

0H (Z) 1H (Z)

2

0 w

Quadrature

Mirror version with respect to quadrature ( )

Hence QMF

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In general, H0(Z) is a LPF and H1(Z) is a HPF, having a bandwidth

of .

Choices of H0(Z) and H1(Z):

These filters will have non-zero transmission bandwidths.

Non-Overlapping Case:

20 w

Merit: If stop band attenuation of the filters are sufficiently

large, then aliasing due to downsampling is negligible.

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Demerit: Input signal energy is drastically reduced around 2.

Remedy: This can be compensated by designing the synthesis

filters F0(Z) and F1(Z) to have high gain around 2. However, this

will cause amplification of channel noise. Hence this is not a

good remedy.

Overlapping Case:

20 w

Merit: No attenuation of signal energy around 2.

Demerit: Cause significant aliasing due to downsampling.

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Remedy: Aliasing can be eliminated by the proper design of

reconstruction filters F0(Z) and F1(Z).

Hence what is done in practice to use always overlapping cases.

This will also easy the strict specifications needed on the filters.

Analysis of 2-Channel QMF Filter Bank:

H (Z)0

H (Z)1

2 2

2 2

F (Z)0

F (Z)1

x[n]

y [n]0 x [n]

0

x [n]1y [n]1v [n]1x [n]1

x [n]0

v [n]0

x[n]

Analysis Bank Decimator Interpolator Synthesis Bank

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Expression for reconstructed signal:

Note:

1. V0(ejw) = 1M .(M1)

l=0

X0(ej(w2lM )) = 12.

1

l=0

X0(ej(w2l

2))

2. X(ejw) = X(z)

X(ejw2 ) = X(z

1

2)

X(ej(w)) = X(z)

Upper Channel:

X0(z) = H0(z)X(z)

V0(z) = ( 2)X0(z)

V0(z) =12[X0(z

1

2) + X0(z1

2)]

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= 12[H0(z1

2)X(z1

2) + H0(z1

2)X(z1

2)]

Y0(z) = ( 2)V0(z) = V0(z2)

= 12[H0(z)X(z) + H0(z)X(z)]

X0(z) = Y0(z)F0(z)

X0(z) =F0(z)

2[H0(z)X(z) + H0(z)X(z)]

In a similar fashion, the output of lower channel is:

X1(z) =F1(z)

2[H1(z)X(z) + H1(z)X(z)]

The output of the synthesis filter is:

X(z) = X0(z) + X1(z)

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= F0(z)2 [H0(z)X(z) + H0(z)X(z)] +F1(z)

2 [H1(z)X(z) + H1(z)X(z)]

X(z) =1

2[F0(z)H0(z) + F1(z)H1(z)]X(z) +

1

2[F0(z)H0(z) + F1(z)H1(z)]X(z) (1)

For perfect reconstruction we should have

X(z) = X(z) (2)

The term corresponding to X(z) represents aliasing, and the term

corresponding to X(z) represents distortion.

Alias Cancelation:

From equation (1) we cancel aliasing by choosing the filter such

that term corresponding to X(z) = 0. That is,

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H0(z)F0(z) + H1(z)F1(z) = 0

Thus the following choice cancels aliasing:

F0(z) = H1(z) & F1(z) = H0(z)

H0(z)H1(z) H1(z)H0(z) = 0

Given H0(z) and H1(z), it is thus possible to completely cancel alias-

ing by this choice of synthesis filters.

If analysis filters have large transition bandwidths and low stop

band attenuation, then large aliasing errors, but are canceled by

above choice.

Summary: In QMF bank aliasing is permitted in analysis bank

and synthesis filters are chosen such that the alias component is

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canceled.

Distortion Cancelation

H (Z)0

H (Z)1

2 2

2 2

F (Z)0

F (Z)1

x[n]

y [n]0 x [n]

0

x [n]1y [n]1v [n]1x [n]1

x [n]0

v [n]0

x[n]

X0(z) = F0(z)Y0(z)

= F0(z)V0(z2)

= F0(z)(12(X0(z) + X0(z)))

= F0(z)[12[H0(z)X(z) + H0(z)X(z)]]

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X0(z) =12[H0(z)F0(z)X(z) + H0(z)F0(z)X(z)]

X1(z) =12[H1(z)F1(z)X(z) + H1(z)F1(z)X(z)]

X(z) = X0(z) + X1(z)

= 12[H0(z)F0(z) + H1(z)F1(z)]X(z) +12[H0(z)F0(z) + H1(z)F1(z)]X(z)

If the QMF is free of aliasing then,

H0(z)F0(z) + H1(z)F1(z) = 0

= F0(z) = H1(z) and F1(z) = H0(z)

For alias free QMF filter bank we have

X(z) = 12[H0(z)F0(z) + H1(z)F1(z)]X(z)

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Substituting the condition for aliasing we have

X(z) = 12[H0(z)H1(z) + H1(z)(H0(z))]X(z)

X(z) = 12[H0(z)H1(z) H0(z)H1(z)]X(z)

Let T (z) = 12[H0(z)H1(z) H0(z)H1(z)]

X(z) = T (z)X(z)

where, T (z) is called the Distortion Transfer Function.

In the frequency domain let

T (ejw) = |T (ejw)|.ej6 T (ejw)

X(z) = T (z)X(z)

X(ejw) = T (ejw)X(ejw)

X(ejw) = (|T (ejw)|ej6 T (ejw))X(ejw)

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If |T (ejw)| = constant 6= 0 (i.e all pass) for all w, then we do not have

any amplitude distortion.

Similarly, if T (ejw) has linear phase i.e 6 T (ejw) = a+ b.w, for constants

a and b, then we have no phase distortion.

Summary:

For the QMF filter bank to be free from distortion we need T (ejw)

to be a linear phase all-pass filter.

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