D- Filter Bank

7/24/2019 D- Filter Bank

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Copyright 2001, S. K. Mitra1

Digital Filter BanksDigital 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 shownbelow


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Digital Filter BanksDigital Filter Banks

The subfilters in the analysis filterbank 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

)(zk

][nvk


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AnL-band synthesis filter bankis shownbelow

It performs the dual operation to that of the

analysis filter bank


4/40Copyright 2001, S. K. Mitra4


The subfilters in the synthesis filterbank are known as synthesis filters

The synthesis filter bank is used to combine

a set of subband signals (typically

belonging to contiguous frequency bands)

into one signaly[n]at its output

)(zFk

][nvk^


5/40Copyright 2001, S. K. Mitra

5

Uniform Digital Filter BanksUniform Digital Filter Banks

A simple technique to design a class offilter 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

][0nh

)(0 z



6


Assume that has its passband edgeand stopband edge around%/M,whereMis some arbitrary integer, as indicated below

)(0 z

&%0 2%

p&

s&

%

p&s&



7


Now, consider the transfer functionwhose impulse response is given by

where we have used the notation

Thus,

)(zk][nhk

,][][][0

/2

0

knnkj

k

Wnhenhnh #% $$

jeW /2%#$

10 #''k

( ) ,][][)( 0!! " #"$#"

#"$ # $$

n

nkn

nkk zWnhznhzH

10 #''k


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i.e.,

The corresponding frequency response isgiven by

Thus, the frequency response of is

obtained by shifting the response of

to the right by an amount2%k/M

),()( 0k

k zWHzH $ 10 #''k

),()( )/2(0kjj

k eHeH %#&& $ 10 #''k

)(zk)(0 zH


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Uniform Digital Filter BanksUniform Digital Filter Banks The responses of , , . . . ,

are shown below

)(zHk )(zHk )(zHk


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Note:The impulse responses are, ingeneral complex, and hence does

not necessarily exhibit symmetry with

respect to&= 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|)(| &jk eH

)(0 zH


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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

),()( 0k

k zWHzH $ 10 #''k


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Uniform DFT Filter BanksUniform DFT Filter BanksPolyphase Implementation

Let the prototype lowpass transfer functionbe represented in itsM-band polyphase

form:

where is the -th polyphase

component of :

! #$ #$ 100 MzEzzH ! !! )()(

)(z0

!

,][][)( !! "$#"

$# *$$

0 00 nn

nn znMhznezE !!!

)(zE!

10 #'' !


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Uniform DFT Filter BanksUniform DFT Filter Banks

Substitutingzwith in the expressionfor we arrive at theM-band polyphase

decomposition of :

In deriving the last expression we have used

the identity

)(zH0

k

zW

! #$ ##$1

0kMM

MkMk WzEWzzH ! !

!! )()(

)(zk

101

0 #''$! #

$

##

MkzEWz Mk

M! !

!!

,)(

1$kMW


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The equation on the previous slide can bewritten in matrix form as

].... )(

[)( kkk

k WWWzH12

1 ####

$

++

+++

,

-

..

.

..

/

0

###

#

#

)(

)(

)(

)(

)(

MM

M

M

M

M

zEz

zEz

zEz

zE

11

22

11

0

10 #''k

..

.


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Uniform DFT Filter BanksUniform DFT Filter Banks AllMequations on the previous slide can

be combined into one matrix equation as

In the aboveDis the DFT matrix

M D1#

1

++++

+

,

-

.

.

..

.

/

0

#

##

#

#

)(

)(

)(

)(

)(

M

M

M

M

M

M

zEz

zEz

zEz

zE

1

1

22

11

0

...++++++

,

-

.

.

.

.

.

.

/

0

$

++++

+

,

-

.

.

.

.

.

/

0

######

####

####

# 21121

1242

121

1

21

0

1

1

1

1111

)()()(

)(

)(

)(

)( )(

)(

MMM

MMMM

MMMM

M WWW

WWW

WWW

zH

zH zH

zH

......

......

...

...

...

...

......


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Uniform DFT Filter BanksUniform DFT Filter Banks An efficient implementation of theM-band

uniform analysis filter bank, morecommonly known as the uniform DFT

analysis filter bank, is then as shown below


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The computational complexity of anM-banduniform DFT filter bank is much smaller than

that of a direct implementation as shown

below


18/40



For example, anM-band uniform DFTanalysis filter bank based on anN-tap

prototype lowpass filter requires a total of

multipliers

On the other hand, a direct implementation

requiresNMmultipliers

NMM *22 log


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Uniform DFT Filter BanksUniform DFT Filter Banks Following a similar development, we can

derive the structure for a uniform DFTsynthesis filter bankas shown below

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


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Uniform DFT Filter BanksUniform DFT Filter Banks Now can be expressed in terms of

The above equation can be used todetermine the polyphase components of an

IIR transfer function

+++

++

,

-

.

..

.

.

/

0

###

#

#

)(

)(

)(

)(

)(

MM

M

M

M

M

zEz

zEz

zEz

zE

11

2

21

10

1$

++

+

,

-

..

.

/

0

# )(

)(

)(

)(

zH

zH

zH

zH

1

2

1

0

D

......

)(i zE

)(zH0


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Nyquist FiltrsNyquist Filtrs Under certain conditions, a lowpass filter

can be designed to have a number of zero-valued coefficients

When used as interpolation filters these

filters 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 other

lowpass filters of same order


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LLthth-Band Filters-Band Filters

These filters, called theNyquist filtersor

Lth-band filters, are often used in single-rate

and multi-rate signal processing

Consider the factor-of-Linterpolator shown

below

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

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

)()()( L

zXzHzY $


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IfH(z) is realized in theL-band polyphaseform, then we have

Assume that thek-th polyphase component

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

! #$ #$ 1

0

)()( L

i

Li

i zEzzH

2$)(zEk

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

11

0L

kkLL zEzzEzzEzH #### ***$

)(...)( 1)1(

1)1( L

LLL

kk zEzzEz #

##*

*# ***


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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

!

#

$## *2$

1

0)()()()(

L

LLLk zXzEzzXzzY!

!!

k3!

][][ nxkLny 2$*

)1( #L


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A filter with the above property is called a

Nyquist filteror anLth-band filter

Its impulse response has many zero-valued

samples, making it computationally

attractive

For example, the impulse response of an

Lth-bandfilter fork= 0 satisfies thefollowing condition

456$][Lnh otherwise,0 0, $2 n


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LLthth-Band Filters-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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If the0-thpolyphase component ofH(z) is a

constant, i.e., then it can be shown

that

(assuming2= 1/L)

Since the frequency response of isthe shifted version of ,

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

2$)(0 zE

! #$ $2$10

1)(Lk

kL LzWH

)( kLzWH)( )/2( LkjeH %#& )( &jeH

)( &jeH


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Half-Band FiltersHalf-Band Filters

AnLth-bandfilter forL= 2 is called ahalf-band filter

The transfer function of a half-band filter is

thus given by

with its impulse response satisfying

)()( 211 zEzzH #*2$

456$]2[ nh

otherwise,0

0, $2 n


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The condition

reduces to

(assuming2= 0.5)

IfH(z)has real coefficients, then

Hence

)()( 211 zEzzH #*2$

1)()( $#* zzH

)()( )( %& $# jj eHeH

1)()( )( $* %& jj eHeH


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and add upto1 for all7

Or, in other words, exhibits a

symmetry with respect to the half-band

frequency%/2, hence the name half-band

filter

)(

)2/( 7#%j

eH )(

)2/( 7*%j

eH

)( &jeH


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Half-Band FiltersHalf-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 are

symmetric with respect to%/2, i.e.,

sp 8$8

%$&*& sp


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Attractive property:About50% of thecoefficients ofh[n] are zero

This reduces the number of multiplications

required in its implementation significantly

For example, ifN= 101,an arbitrary Type 1

FIR transfer function requires about50multipliers, whereas, aType 1half-band

filter requires only about25 multipliers


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An FIR half-band filter can be designedwith 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 #2$ 1|| $2


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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 form2R+1 = 4K+3 [unlessH(z) is a

constant]

456$]2[ nh

otherwise,00, $2 n


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Design of Linear-PhaseDesign of Linear-Phase


A lowpass linear-phaseLth-band FIR filter

can be readily designed via the windowed

Fourier series approach

In this approach, the impulse response

coefficients of the lowpass filter are chosen

as where is theimpulse response of an ideal lowpass filter

with a cutoff at %/Land w[n]is a suitablewindow function

][][][ nwnhnh LP 9$ ][nhLP


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Now, the impulse response of an idealLth-

band lowpass filterwith a cutoff at

is given by

It can be seen from the above that

"''"#%%$ nn

LnnhLP ,)/sin(][

Lc /%$&

...,2,for0][ LLnnhLP ::$$


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Hence, the coefficient condition of theLth-

band filter

is indeed satisfied

Hence, anLth-band FIR filtercan be

designed by applying a suitable window

w[n]to

45

6

$][Lnh otherwise,00, $2 n

][nhLP


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There are many other candidates forLth-

band FIR filters

Program 10_8can be used to design an Lth-

band FIR filter using the windowed Fourierseries approach

The program employs the Hammingwindow

However, other windows can also be used

D i f Li Ph


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Figure below shows the gain response of a

half-band filter of length-23 designed using

Program 10_8

0 0.2 0.4 0.6 0.8 1

-80

-60

-40

-20

0

&/%

Gain,

dB

fD i f Li Ph


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The filter coefficients are given by

As expected,h[n] = 0for

;0]10[]10[;002315.0]11[]11[ $$##$$# hhhh

;0]8[]8[;005412.0]9[]9[ $$#$$# hhhh

;0]6[]6[;001586.0]7[]7[ $$##$$# hhhh

;0]4[]4[;003584.0]5[]5[ $$#$$# hhhh

;0]2[]2[;089258.0]3[]3[ $$##$$# hhhh;5.0]0[;3122379.0]1[]1[ $$$# hhh

10,8,6,4,2 :::::$n

D- Filter Bank

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