digital filter banks

7/30/2019 Digital Filter Banks

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1Copyright 2001, S. K. Mitra

Digital Filter Banks

The digital filter bank is set of bandpass

filters with either a common input or a

summed output AnM-band analysis filter bankis shown

below


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The subfilters in the analysis filter

bank are known as analysis filters

The analysis filter bank is used to

decompose the input signalx[n] into a set of

subband signals with each subband

signal occupying a portion of the originalfrequency band

)(zHk

][nvk


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AnL-band synthesis filter bankis shown

below

It performs the dual operation to that of the

analysis filter bank


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The subfilters in the synthesis filter

bank are known assynthesis filters

The synthesis filter bank is used to combine

a set ofsubband signals (typically

belonging to contiguous frequency bands)

into one signaly[n] at its output

)(zFk

][nvk^


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Uniform Digital Filter Banks

A simple technique to design a class of

filter banks with equal passband widths is

outlined next

Let represent a causal lowpass digital

filter with a real impulse response :

The filter is assumed to be an IIR

filter without any loss of generality

)(0 zH

n nznhzH ][)( 00

][0 nh

)(0 zH


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Assume that has its passband edge

and stopband edge aroundp/M,whereM

is some arbitrary integer, as indicated below

)(0 zH

wp0 2p

pw sw

M

p

pw

sw


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i.e.,

The corresponding frequency response is

given by

Thus, the frequency response of is

obtained by shifting the response of

to the right by an amount2pk/M

),()( 0kMk zWHzH 10 Mk

),()( )/2(0Mkjj

k eHeHpww 10 Mk

)(zHk)(0 zH


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The responses of , , . . . ,

are shown below

)(zHk )(zHk )(zHk


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Note:The impulse responses are, in

general complex, and hence does

not necessarily exhibit symmetry withrespect tow = 0

The responses shown in the figure of the

previous slide can be seen to be uniformlyshifted version of the response of the basic

prototype filter

][nhk|)(| wjk eH

)(0 zH


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11 Copyright 2001, S. K. Mitra


TheMfilters defined by

could be used as the analysis filters in the

analysis filter bank or as the synthesis filters

in the synthesis filter bank

Since the magnitude responses of allM

filters are uniformly shifted version of that

of the prototype filter, the filter bank

obtained is called a uniform filter bank

),()( 0kMk zWHzH 10 Mk


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Uniform DFT Filter Banks

Polyphase Implementation

Let the prototype lowpass transfer function

be represented in itsM-band polyphase

form:

where is the -th polyphase

component of :

1

00M MzEzzH

)()(

)(zH0

,][][)(

0 00 nn

nn znMhznezE

)(zE

10

M


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Substitutingzwith in the expression

for we arrive at theM-band polyphase

decomposition of :

In deriving the last expression we have used

the identity

)(zH0

kMzW

1

0M kM

MMk

Mk WzEWzzH )()(

)(zHk

101

0

MkzEWzM Mk

M

,)(

1kMMW


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The equation on the previous slide can be

written in matrix form as

].... )([)( kMMk

Mk

Mk WWWzH12

1

)(

)(

)(

)(

)(

MM

M

M

M

M

zEz

zEz

zEz

zE

11

22

110

10 Mk

..

.


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AllMequations on the previous slide can

be combined into one matrix equation as

In the aboveDis the DFT matrixMD

1

M

)(

)(

)(

)(

)(

M

M

M

M

M

M

zEz

zEz

zEz

zE

1

1

22

11

0

...

21121

1242

121

1

2

1

0

1

1

1

1111

)()()(

)(

)(

)(

)(

)()(

MM

MM

MM

MMMM

MMMM

M WWW

WWW

WWW

zH

zH

zHzH

.

.

.

.

.

.

.

.

.

.

.

.

...

...

...

...

.

.

.

.

.

.


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An efficient implementation of theM-band

uniform analysis filter bank, more

commonly known as the uniform DFT

analysis filter bank, is then as shown below


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The computational complexity of anM-band

uniform DFT filter bank is much smaller than

that of a direct implementation as shownbelow


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For example, anM-band uniform DFT

analysis filter bank based on anN-tap

prototype lowpass filter requires a total ofmultipliers

On the other hand, a direct implementation

requiresNMmultipliers

NMM 22log


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Following a similar development, we can

derive the structure for a uniform DFT

synthesis filter bankas shown below

Type I uniform DFT Type II uniform DFTsynthesis filter bank synthesis filter bank


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Now can be expressed in terms of

The above equation can be used to

determine the polyphase components of an

IIR transfer function

)(

)(

)(

)(

)(

MM

M

M

M

M

zEz

zEzzEz

zE

11

22

1

10

M

1

)(

)(

)(

)(

zH

zH zH

zH

M 1

21

0

D.

.

.

.

.

.

)( Mi zE

)(zH0


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Nyquist Filters

Under certain conditions, a lowpass filtercan be designed to have a number of zero-valued coefficients

When used as interpolation filters thesefilters preserve the nonzero samples of theup-sampler output at the interpolator output

Moreover, due to the presence of thesezero-valued coefficients, these filters arecomputationally more efficient than otherlowpass filters of same order


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Lth-Band Filters

These filters, called theNyquist filtersorLth-band filters, are often used in single-rate and multi-rate signal processing

Consider the factor-of-L interpolator shownbelow

The input-output relation of the interpolatorin thez-domain is given by

L][nx ][ny)(zH][nxu

)()()( LzXzHzY


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Lth-Band Filters

IfH(z) is realized in theL-band polyphase

form, then we have

Assume that thek-th polyphase component

ofH(z) is a constant, i.e., :

10 )()( Li Lii zEzzH

)(zEk

)(...)()()( 1)1(

11

0L

kkLL zEzzEzzEzH

)(...)( 1)1(

1)1( L

LLL

kk zEzzEz


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Lth-Band Filters

Then we can expressY(z) as

As a result,

Thus, the input samples appear at the outputwithout any distortion for all values ofn,

whereas, in-between output samples

are determined by interpolation

1

0

)()()()(L

LLLk zXzEzzXzzY

k

][][ nxkLny

)1( L


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Lth-Band Filters

A filter with the above property is called aNyquist filteror anLth-band filter

Its impulse response has many zero-valued

samples, making it computationallyattractive

For example, the impulse response of an

Lth-bandfilter fork= 0 satisfies thefollowing condition

][Lnh

otherwise,0

0, n


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Lth-Band Filters

Figure below shows a typical impulse

response of athird-band filter(L = 3)

Lth-band filterscan be either FIR or IIR

filters


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Lth-Band Filters

If the 0-thpolyphase component ofH(z) is aconstant, i.e., then it can be shown

that

(assuming = 1/L) Since the frequency response of is

the shifted version of ,

the sum of all of theseLuniformly shiftedversions of add up to a constant

)(0 zE

10 1)(

Lk kL LzWH

)( kLzWH

)( )/2( LkjeH pw )( wjeH

)( wjeH


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Half-Band Filters

AnLth-bandfilter forL = 2 is called ahalf-

band filter

The transfer function of a half-band filter isthus given by

with its impulse response satisfying

)()( 211 zEzzH

]2[ nh

otherwise,00, n


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Half-Band Filters

The condition

reduces to

(assuming = 0.5)

IfH(z) has real coefficients, then

Hence

)()( 211 zEzzH

1)()( zHzH

)()( )( wpw jj eHeH

1)()( )( wpw jj eHeH


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Half-Band Filters

and add up

to 1 for allq

Or, in other words, exhibits asymmetry with respect to the half-band

frequencyp/2, hence the name half-band

filter

)( )2/( qpjeH )( )2/( qpjeH

)( wjeH


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Half-Band Filters

Figure below illustrates this symmetry for ahalf-band lowpass filter for which passband

and stopband ripples are equal, i.e.,

and passband and stopband edges aresymmetric with respect top/2, i.e.,

sp

pww sp


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Half-Band Filters

Attractive property:About 50% of the

coefficients ofh[n] are zero

This reduces the number of multiplicationsrequired in its implementation significantly

For example, ifN= 101,an arbitrary Type 1

FIR transfer function requires about 50multipliers, whereas, aType 1half-band

filter requires only about 25 multipliers


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Half-Band Filters

An FIR half-band filter can be designed

with linear phase

However, there is a constraint on its length

Consider a zero-phase half-band FIR filter

for which , with

Let the highest nonzero coefficient beh[R]

][*][ nhnh 1||


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34

Half-Band Filters

ThenRis odd as a result of the condition

ThereforeR = 2K+1 for some integerK

Thus the length ofh[n] is restricted to be of

the form 2R+1 = 4K+3 [unlessH(z) is a

constant]

]2[ nhotherwise,0

0, n

digital filter banks

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