1 Copyright ? 2001,S,K,Mitra

Comb Filters

? The simple filters discussed so far are

characterized either by a single passband

and/or a single stopband

? There are applications where filters with

multiple passbands and stopbands are

required

? The comb filter is an example of such filters

2 Copyright ? 2001,S,K,Mitra

Comb Filters

? In its most general form,a comb filter has a

frequency response that is a periodic

function of w with a period 2p/L,where L is

a positive integer

? If H(z) is a filter with a single passband

and/or a single stopband,a comb filter can

be easily generated from it by replacing

each delay in its realization with L delays

resulting in a structure with a transfer

function given by )()( LzHzG ?

3 Copyright ? 2001,S,K,Mitra

Comb Filters

? If exhibits a peak at,then

will exhibit L peaks at,

in the frequency range

? Likewise,if has a notch at,

then will have L notches at,

in the frequency range

? A comb filter can be generated from either

an FIR or an IIR prototype filter

|)(| wjeH

|)(| wjeH

|)(| wjeG

|)(| wjeGpw

ow

Lkp /w

Lko /w

10 ??? Lk

10 ??? Lk

p?w? 20

p?w? 20

4 Copyright ? 2001,S,K,Mitra

Comb Filters

? For example,the comb filter generated from

the prototype lowpass FIR filter

has a transfer function

? has L notches

at w = (2k+1)p/L and L

peaks at w = 2p k/L,

)( 121 1 ?? z

?)( zH 0

)()()( LL zzHzG ???? 12100

10 ??? Lk,in the

frequency range

p?w? 20

|)(| 0 wjeG

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

C om b f i l t e r f r om l ow pa s s pr ot ot ype

5 Copyright ? 2001,S,K,Mitra

Comb Filters

? For example,the comb filter generated from

the prototype highpass FIR filter

has a transfer function

? has L peaks

at w = (2k+1)p/L and L

notches at w = 2p k/L,

|)(| 1 wjeG

)( 121 1 ?? z

?)( zH 1

)()()( LL zzHzG ???? 12111

10 ??? Lk,in the

frequency range

p?w? 20

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

C om b f i l t e r f r om hi ghpa s s pr ot ot ype

6 Copyright ? 2001,S,K,Mitra

Comb Filters

? Depending on applications,comb filters

with other types of periodic magnitude

responses can be easily generated by

appropriately choosing the prototype filter

? For example,the M-point moving average

filter

has been used as a prototype

)()( 11

1

?

?

?

??

zM

z MzH

7 Copyright ? 2001,S,K,Mitra

Comb Filters

? This filter has a peak magnitude at w = 0,

and notches at,

? The corresponding comb filter has a transfer

function

whose magnitude has L peaks at,

and notches at

,

1?M M/2 ?p?w 11 ??? M?

)()( L

ML

zM

zzG

?

?

?

??

1

1

Lk /2 p?w

10 ??? Lk )( 1?ML LMk /2 p?w

)( 11 ??? MLk

8 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

Definition

? An IIR transfer function A(z) with unity

magnitude response for all frequencies,i.e.,

is called an allpass transfer function

? An M-th order causal real-coefficient

allpass transfer function is of the form

w?w a l lf o r,1|)(| 2jeA

M

M

M

M

MM

MM

M zdzdzd

zzdzddzA

???

?

?

????

?

????

??????

1

1

1

1

1

1

1

1

1,..

...

)(

9 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? If we denote the denominator polynomial of

as,

then it follows that can be written as,

? Note from the above that if is a pole

of a real coefficient allpass transfer function,

then it has a zero at

)(zDM)(zAM

MMMMM zdzdzdzD ????? ????? 11111,..)(

)(zAM

)(

)()(

zD

zDz

M M M

MzA 1????

?? jrez

??? jr ez 1

10 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The numerator of a real-coefficient allpass

transfer function is said to be the mirror-

image polynomial of the denominator,and

vice versa

? We shall use the notation to denote

the mirror-image polynomial of a degree-M

polynomial,i.e.,

)( zDM~

)( zDM

)()( zDzzD MMM ??~

11 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The expression

implies that the poles and zeros of a real-

coefficient allpass function exhibit mirror-

image symmetry in the z-plane

)(

)()(

zD

zDz

M M M

MzA 1????

321

321

3 2.018.04.01

4.018.02.0)(

???

???

???

?????

zzz

zzzzA

-1 0 1 2 3

- 1.5

-1

- 0.5

0

0.5

1

1.5

R e a l P a r t

I

m

a

gi

na

r

y P

a

r

t

12 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? To show that we observe that

? Therefore

? Hence

)(

)(1

1)( ???

?

zD

zDz

M M M

MzA

)(

)(

)(

)(1

1

1)()(

?

????

zD

zDz

zD

zDz

MM M M

M

M

MMzAzA

1|)(| ?wjM eA

1)()(|)(| 12 ?? w??w jezMMjM zAzAeA

13 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? Now,the poles of a causal stable transfer

function must lie inside the unit circle in the

z-plane

? Hence,all zeros of a causal stable allpass

transfer function must lie outside the unit

circle in a mirror-image symmetry with its

poles situated inside the unit circle

14 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? Figure below shows the principal value of

the phase of the 3rd-order allpass function

? Note the discontinuity by the amount of 2p

in the phase q(w)

321

321

3 2.018.04.01

4.018.02.0)(

???

???

???

?????

zzz

zzzzA

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

w / p

P

ha

s

e

,de

gr

e

e

s

P r i nc i pa l va l ue of pha s e

15 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? If we unwrap the phase by removing the

discontinuity,we arrive at the unwrapped

phase function indicated below

? Note,The unwrapped phase function is a

continuous function of w

)(wqc

0 0.2 0.4 0.6 0.8 1

- 10

-8

-6

-4

-2

0

w / p

P

ha

s

e

,de

gr

e

e

s

U nw r a ppe d ph a s e

16 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The unwrapped phase function of any

arbitrary causal stable allpass function is a

continuous function of w

Properties

? (1) A causal stable real-coefficient allpass

transfer function is a lossless bounded real

(LBR) function or,equivalently,a causal

stable allpass filter is a lossless structure

17 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? (2) The magnitude function of a stable

allpass function A(z) satisfies,

? (3) Let t(w) denote the group delay function

of an allpass filter A(z),i.e.,

??

?

?

?

??

??

??

1f o r1

1f o r1

1f o r1

z

z

z

zA

,

,

,

)(

)]([)( wq??wt w cdd

18 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The unwrapped phase function of a

stable allpass function is a monotonically

decreasing function of w so that t(w) is

everywhere positive in the range 0 < w < p

? The group delay of an M-th order stable

real-coefficient allpass transfer function

satisfies,

)(wqc

p?w? wt

p

Md

0

)(

19 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

A Simple Application

? A simple but often used application of an

allpass filter is as a delay equalizer

? Let G(z) be the transfer function of a digital

filter designed to meet a prescribed

magnitude response

? The nonlinear phase response of G(z) can be

corrected by cascading it with an allpass

filter A(z) so that the overall cascade has a

constant group delay in the band of interest

20 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? Since,we have

? Overall group delay is the given by the sum

of the group delays of G(z) and A(z)

1|)(| ?wjeA

|)(||)()(| www ? jjj eGeAeG

G(z) A(z)

21 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? Consider the two 1st-order transfer functions,

? Both transfer functions have a pole inside the

unit circle at the same location and are

stable

? But the zero of is inside the unit circle

at,whereas,the zero of is at

situated in a mirror-image symmetry

11121 ???? ???? bazHzH azbzaz bz,,)(,)(

az ??

bz ??

bz 1??

)(zH1

)(zH 2

22 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? Figure below shows the pole-zero plots of

the two transfer functions

)(

1 zH )(2 zH

23 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? However,both transfer functions have an

identical magnitude function as

? The corresponding phase functions are

1122111 ?? ?? )()()()( zHzHzHzH

w? w?w? w?w ?? c o ss i n1c o ss i n11 t a nt a n)](a r g [ abjeH

w? w?w? w?w ?? c o ss i n1c o s1 s i n12 t a nt a n)](a r g [ abbjeH

24 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? Figure below shows the unwrapped phase

responses of the two transfer functions for

a = 0.8 and b = 5.0?

0 0.2 0.4 0.6 0.8 1

-4

-3

-2

-1

0

1

2

w / p

P

ha

s

e

,de

gr

e

e

s

H

1

( z )

H

2

( z )

25 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? From this figure it follows that has

an excess phase lag with respect to

? Generalizing the above result,we can show

that a causal stable transfer function with all

zeros outside the unit circle has an excess

phase compared to a causal transfer

function with identical magnitude but

having all zeros inside the unit circle

)(2 zH

)(1 zH

26 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? A causal stable transfer function with all

zeros inside the unit circle is called a

minimum-phase transfer function

? A causal stable transfer function with all

zeros outside the unit circle is called a

maximum-phase transfer function

? Any nonminimum-phase transfer function

can be expressed as the product of a

minimum-phase transfer function and a

stable allpass transfer function

27 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? A set of digital transfer functions with

complementary characteristics often finds

useful applications in practice

? Four useful complementary relations are

described next along with some applications

28 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Delay-Complementary Transfer Functions

? A set of L transfer functions,,

,is defined to be delay-

complementary of each other if the sum of

their transfer functions is equal to some

integer multiple of unit delays,i.e.,

where is a nonnegative integer

)}({ zH i

10 ??? Li

0,)(

1

0

???? ?

?

?

? on

L

i

i zzH

on

29 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? A delay-complementary pair

can be readily designed if one of the pairs is

a known Type 1 FIR transfer function of

odd length

? Let be a Type 1 FIR transfer function

of length M = 2K+1

? Then its delay-complementary transfer

function is given by

)}(),({ 10 zHzH

)()( 01 zHzzH K ?? ?

)(0 zH

30 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Let the magnitude response of be

equal to in the passband and less than

or equal to in the stopband where and

are very small numbers

? Now the frequency response of can be

expressed as

where is the amplitude response

)(0 zH

p??1

s? p?

s?

)(0 zH

)()( 00 w? w?w HeeH jKj~

)(0 wH~

31 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Its delay-complementary transfer function

has a frequency response given by

? Now,in the passband,

and in the stopband,

? It follows from the above equation that in the

stopband,and in the

passband,

)(1 zH

)](1[)()( 011 w??w? w?w?w HeHeeH jKjKj~ ~

,1)(1 0 pp H ???w??? ~

ss H ??w??? )(0

~

pp H ??w??? )(1

~

ss H ???w??? 1)(1 1

~

32 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? As a result,has a complementary

magnitude response characteristic to that of

with a stopband exactly identical to

the passband of,and a passband that

is exactly identical to the stopband of

? Thus,if is a lowpass filter,will

be a highpass filter,and vice versa

)(1 zH

)(0 zH

)(0 zH

)(1 zH)(0 zH

)(0 zH

33 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? The frequency at which

the gain responses of both filters are 6 dB

below their maximum values

? The frequency is thus called the 6-dB

crossover frequency

ow

ow

5.0)()( 10 ?w?w oo HH ~ ~

34 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - Consider the Type 1 bandstop

transfer function

? Its delay-complementary Type 1 bandpass

transfer function is given by

)45541()1()( 12108424264 1 ?????? ??????? zzzzzzzH BS

)45541()1( 121084242641 ?????? ??????? zzzzzz

)()( 10 zHzzH BSBP ?? ?

35 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Plots of the magnitude responses of

and are shown below

)( zH BS

)( zH BP

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

H

BS

( z ) H

BP

( z )

36 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Allpass Complementary Filters

? A set of M digital transfer functions,,

,is defined to be allpass-

complementary of each other,if the sum of

their transfer functions is equal to an allpass

function,i.e.,

)}({ zH i

10 ??? Mi

)()(

1

0

zAzH

M

i

i ??

?

?

37 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Power-Complementary Transfer Functions

? A set of M digital transfer functions,,

,is defined to be power-

complementary of each other,if the sum of

their square-magnitude responses is equal to

a constant K for all values of w,i.e.,

)}({ zH i

10 ??? Mi

w??

?

?

w a llf o r,)(

1

0

2 KeHM

i

j

i

38 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? By analytic continuation,the above

property is equal to

for real coefficient

? Usually,by scaling the transfer functions,

the power-complementary property is

defined for K = 1

)(zH i

w??

?

?

? a llf o r,)()(

1

0

1 KzHzH

M

i

ii

39 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? For a pair of power-complementary transfer

functions,and,the frequency

where,is

called the cross-over frequency

? At this frequency the gain responses of both

filters are 3-dB below their maximum

values

? As a result,is called the 3-dB cross-over

frequency

ow

ow

)(0 zH )(1 zH

5.0|)(||)(| 2120 ?? ww oo jj eHeH

40 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - Consider the two transfer functions

and given by

where and are stable allpass

transfer functions

? Note that

? Hence,and are allpass

complementary

)(0 zH )(1 zH

)]()([)( 10210 zAzAzH ??

)(0 zA )(1 zA

)]()([)( 10211 zAzAzH ??

)()()( 010 zAzHzH ??

)(0 zH )(1 zH

41 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? It can be shown that and are

also power-complementary

? Moreover,and are bounded-

real transfer functions

)(0 zH )(1 zH

)(0 zH )(1 zH

42 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Doubly-Complementary Transfer Functions

? A set of M transfer functions satisfying both

the allpass complementary and the power-

complementary properties is known as a

doubly-complementary set

43 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? A pair of doubly-complementary IIR

transfer functions,and,with a

sum of allpass decomposition can be simply

realized as indicated below

)(0 zH )(1 zH

?

?)(1 zA

)(0 zA)(zX )(0 zY

)(1 zY1?

2/1

)( )(00 )( zX zYzH ? )( )(11 )( zX zYzH ?

44 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - The first-order lowpass transfer

function

can be expressed as

where

??????? ?

?

??

???

1

1

1

1

2

1)(

z

z

LP zH

)]()([)( 1021

1

121 1

1 zAzAzH

z

z

LP ???

??

?

?? ?

??

????

?

?

1

1

1 1)( ?

?

??

????

z

zzA

,)( 10 ?zA

45 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Its power-complementary highpass transfer

function is thus given by

? The above expression is precisely the first-

order highpass transfer function described

earlier

???????? ?

?

??

?????

1

1

1

1211021 )]()([)(

z

z

HP zAzAzH

??????? ?

?

??

???

1

1

1

1

2

1

z

z

46 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Figure below demonstrates the allpass

complementary property and the power

complementary property of and )( zH LP

)( zH HP

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

|H

HP

(e

j

w

)|

|H

LP

(e

j

w

)|

|H

LP

(e

j

w

) + H

HP

(e

j

w

)|

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

|H

HP

(e

j

w

)|

2

|H

LP

(e

j

w

)|

2

|H

LP

(e

j

w

)|

2

+ | H

HP

(e

j

w

)|

2

47 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Power-Symmetric Filters

? A real-coefficient causal digital filter with a

transfer function H(z) is said to be a power-

symmetric filter if it satisfies the condition

where K > 0 is a constant

KzHzHzHzH ???? ?? )()()()( 11

48 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? It can be shown that the gain function G(w) of a

power-symmetric transfer function at w = p is

given by

? If we define,then it follows from

the definition of the power-symmetric filter that

H(z) and G(z) are power-complementary as

c o n s t a n ta)()()()( 11 ?? ?? zGzGzHzH

)()( zHzG ??

dBK 3lo g10 10 ?

49 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Conjugate Quadratic Filter

? If a power-symmetric filter has an FIR

transfer function H(z) of order N,then the

FIR digital filter with a transfer function

is called a conjugate quadratic filter of

H(z) and vice-versa

)()( 11 ??? zHzzG

50 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? It follows from the definition that G(z) is

also a power-symmetric causal filter

? It also can be seen that a pair of conjugate

quadratic filters H(z) and G(z) are also

power-complementary

51 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - Let

? We form

? Thus,H(z) is a power-symmetric transfer function

)3621)(3621( 32321 zzzzzz ??????? ???

)()()()( 11 ?? ??? zHzHzHzH

321 3621)( ??? ???? zzzzH

)3621)(3621( 32321 zzzzzz ??????? ??? )345043( 313 ?? ????? zzzz

1 0 0)345043( 313 ??????? ?? zzzz

52 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? We have shown that the BIBO stability of a

causal rational transfer function requires

that all its poles be inside the unit circle

? For very high-order transfer functions,it is

very difficult to determine the pole

locations analytically

? Root locations can of course be determined

on a computer by some type of root finding

algorithms

53 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

The Stability Triangle

? For a 2nd-order transfer function the

stability can be easily checked by

examining its denominator coefficients

54 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? Let

denote the denominator of the transfer

function

? In terms of its poles,D(z) can be expressed

as

? Comparing the last two equations we get

22111)( ?? ??? zdzdzD

2211211211 )(1)1)(1()( ???? ????????????? zzzzzD

212211 ),( ???????? dd

55 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? The poles are inside the unit circle if

? Now the coefficient is given by the

product of the poles

? Hence we must have

? It can be shown that the second coefficient

condition is given by

1||,1|| 21 ????

2d

1|| 2 ?d

21 1|| dd ??

56 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? The region in the ( )-plane where the

two coefficient condition are satisfied,

called the stability triangle,is shown below

21,dd

Stability region

57 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? Example - Consider the two 2nd-order

bandpass transfer functions designed

earlier,

21

2

3 7 6 3 817 3 4 3 4 2 401

11 8 8 1 90

??

?

??

???

zz

zzH

BP,..)('

21

2

7 2 6 5 4 2 5 305 3 3 5 3 101

11 3 6 7 30

??

?

??

??

zz

zzH

BP,..)("

58 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? In the case of,we observe that

? Since here,is unstable

? On the other hand,in the case of,we

observe that

? Here,and,and hence

is BIBO stable

)(' zH BP

3 7 6 3 8 1 9.1,7 3 4 3 4 2 4.0 21 ??? dd

7 2 6 5 4 2 5 2 8.0,5 3 3 5 3 0 9 8.0 21 ??? dd

)(' zH BP1|| 2 ?d

)(" zH BP

21 1|| dd ??1|| 2 ?d

)(" zH BP

Comb Filters

? The simple filters discussed so far are

characterized either by a single passband

and/or a single stopband

? There are applications where filters with

multiple passbands and stopbands are

required

? The comb filter is an example of such filters

2 Copyright ? 2001,S,K,Mitra

Comb Filters

? In its most general form,a comb filter has a

frequency response that is a periodic

function of w with a period 2p/L,where L is

a positive integer

? If H(z) is a filter with a single passband

and/or a single stopband,a comb filter can

be easily generated from it by replacing

each delay in its realization with L delays

resulting in a structure with a transfer

function given by )()( LzHzG ?

3 Copyright ? 2001,S,K,Mitra

Comb Filters

? If exhibits a peak at,then

will exhibit L peaks at,

in the frequency range

? Likewise,if has a notch at,

then will have L notches at,

in the frequency range

? A comb filter can be generated from either

an FIR or an IIR prototype filter

|)(| wjeH

|)(| wjeH

|)(| wjeG

|)(| wjeGpw

ow

Lkp /w

Lko /w

10 ??? Lk

10 ??? Lk

p?w? 20

p?w? 20

4 Copyright ? 2001,S,K,Mitra

Comb Filters

? For example,the comb filter generated from

the prototype lowpass FIR filter

has a transfer function

? has L notches

at w = (2k+1)p/L and L

peaks at w = 2p k/L,

)( 121 1 ?? z

?)( zH 0

)()()( LL zzHzG ???? 12100

10 ??? Lk,in the

frequency range

p?w? 20

|)(| 0 wjeG

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

C om b f i l t e r f r om l ow pa s s pr ot ot ype

5 Copyright ? 2001,S,K,Mitra

Comb Filters

? For example,the comb filter generated from

the prototype highpass FIR filter

has a transfer function

? has L peaks

at w = (2k+1)p/L and L

notches at w = 2p k/L,

|)(| 1 wjeG

)( 121 1 ?? z

?)( zH 1

)()()( LL zzHzG ???? 12111

10 ??? Lk,in the

frequency range

p?w? 20

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

C om b f i l t e r f r om hi ghpa s s pr ot ot ype

6 Copyright ? 2001,S,K,Mitra

Comb Filters

? Depending on applications,comb filters

with other types of periodic magnitude

responses can be easily generated by

appropriately choosing the prototype filter

? For example,the M-point moving average

filter

has been used as a prototype

)()( 11

1

?

?

?

??

zM

z MzH

7 Copyright ? 2001,S,K,Mitra

Comb Filters

? This filter has a peak magnitude at w = 0,

and notches at,

? The corresponding comb filter has a transfer

function

whose magnitude has L peaks at,

and notches at

,

1?M M/2 ?p?w 11 ??? M?

)()( L

ML

zM

zzG

?

?

?

??

1

1

Lk /2 p?w

10 ??? Lk )( 1?ML LMk /2 p?w

)( 11 ??? MLk

8 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

Definition

? An IIR transfer function A(z) with unity

magnitude response for all frequencies,i.e.,

is called an allpass transfer function

? An M-th order causal real-coefficient

allpass transfer function is of the form

w?w a l lf o r,1|)(| 2jeA

M

M

M

M

MM

MM

M zdzdzd

zzdzddzA

???

?

?

????

?

????

??????

1

1

1

1

1

1

1

1

1,..

...

)(

9 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? If we denote the denominator polynomial of

as,

then it follows that can be written as,

? Note from the above that if is a pole

of a real coefficient allpass transfer function,

then it has a zero at

)(zDM)(zAM

MMMMM zdzdzdzD ????? ????? 11111,..)(

)(zAM

)(

)()(

zD

zDz

M M M

MzA 1????

?? jrez

??? jr ez 1

10 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The numerator of a real-coefficient allpass

transfer function is said to be the mirror-

image polynomial of the denominator,and

vice versa

? We shall use the notation to denote

the mirror-image polynomial of a degree-M

polynomial,i.e.,

)( zDM~

)( zDM

)()( zDzzD MMM ??~

11 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The expression

implies that the poles and zeros of a real-

coefficient allpass function exhibit mirror-

image symmetry in the z-plane

)(

)()(

zD

zDz

M M M

MzA 1????

321

321

3 2.018.04.01

4.018.02.0)(

???

???

???

?????

zzz

zzzzA

-1 0 1 2 3

- 1.5

-1

- 0.5

0

0.5

1

1.5

R e a l P a r t

I

m

a

gi

na

r

y P

a

r

t

12 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? To show that we observe that

? Therefore

? Hence

)(

)(1

1)( ???

?

zD

zDz

M M M

MzA

)(

)(

)(

)(1

1

1)()(

?

????

zD

zDz

zD

zDz

MM M M

M

M

MMzAzA

1|)(| ?wjM eA

1)()(|)(| 12 ?? w??w jezMMjM zAzAeA

13 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? Now,the poles of a causal stable transfer

function must lie inside the unit circle in the

z-plane

? Hence,all zeros of a causal stable allpass

transfer function must lie outside the unit

circle in a mirror-image symmetry with its

poles situated inside the unit circle

14 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? Figure below shows the principal value of

the phase of the 3rd-order allpass function

? Note the discontinuity by the amount of 2p

in the phase q(w)

321

321

3 2.018.04.01

4.018.02.0)(

???

???

???

?????

zzz

zzzzA

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

w / p

P

ha

s

e

,de

gr

e

e

s

P r i nc i pa l va l ue of pha s e

15 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? If we unwrap the phase by removing the

discontinuity,we arrive at the unwrapped

phase function indicated below

? Note,The unwrapped phase function is a

continuous function of w

)(wqc

0 0.2 0.4 0.6 0.8 1

- 10

-8

-6

-4

-2

0

w / p

P

ha

s

e

,de

gr

e

e

s

U nw r a ppe d ph a s e

16 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The unwrapped phase function of any

arbitrary causal stable allpass function is a

continuous function of w

Properties

? (1) A causal stable real-coefficient allpass

transfer function is a lossless bounded real

(LBR) function or,equivalently,a causal

stable allpass filter is a lossless structure

17 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? (2) The magnitude function of a stable

allpass function A(z) satisfies,

? (3) Let t(w) denote the group delay function

of an allpass filter A(z),i.e.,

??

?

?

?

??

??

??

1f o r1

1f o r1

1f o r1

z

z

z

zA

,

,

,

)(

)]([)( wq??wt w cdd

18 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? The unwrapped phase function of a

stable allpass function is a monotonically

decreasing function of w so that t(w) is

everywhere positive in the range 0 < w < p

? The group delay of an M-th order stable

real-coefficient allpass transfer function

satisfies,

)(wqc

p?w? wt

p

Md

0

)(

19 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

A Simple Application

? A simple but often used application of an

allpass filter is as a delay equalizer

? Let G(z) be the transfer function of a digital

filter designed to meet a prescribed

magnitude response

? The nonlinear phase response of G(z) can be

corrected by cascading it with an allpass

filter A(z) so that the overall cascade has a

constant group delay in the band of interest

20 Copyright ? 2001,S,K,Mitra

Allpass Transfer Function

? Since,we have

? Overall group delay is the given by the sum

of the group delays of G(z) and A(z)

1|)(| ?wjeA

|)(||)()(| www ? jjj eGeAeG

G(z) A(z)

21 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? Consider the two 1st-order transfer functions,

? Both transfer functions have a pole inside the

unit circle at the same location and are

stable

? But the zero of is inside the unit circle

at,whereas,the zero of is at

situated in a mirror-image symmetry

11121 ???? ???? bazHzH azbzaz bz,,)(,)(

az ??

bz ??

bz 1??

)(zH1

)(zH 2

22 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? Figure below shows the pole-zero plots of

the two transfer functions

)(

1 zH )(2 zH

23 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? However,both transfer functions have an

identical magnitude function as

? The corresponding phase functions are

1122111 ?? ?? )()()()( zHzHzHzH

w? w?w? w?w ?? c o ss i n1c o ss i n11 t a nt a n)](a r g [ abjeH

w? w?w? w?w ?? c o ss i n1c o s1 s i n12 t a nt a n)](a r g [ abbjeH

24 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? Figure below shows the unwrapped phase

responses of the two transfer functions for

a = 0.8 and b = 5.0?

0 0.2 0.4 0.6 0.8 1

-4

-3

-2

-1

0

1

2

w / p

P

ha

s

e

,de

gr

e

e

s

H

1

( z )

H

2

( z )

25 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? From this figure it follows that has

an excess phase lag with respect to

? Generalizing the above result,we can show

that a causal stable transfer function with all

zeros outside the unit circle has an excess

phase compared to a causal transfer

function with identical magnitude but

having all zeros inside the unit circle

)(2 zH

)(1 zH

26 Copyright ? 2001,S,K,Mitra

Minimum-Phase and Maximum-

Phase Transfer Functions

? A causal stable transfer function with all

zeros inside the unit circle is called a

minimum-phase transfer function

? A causal stable transfer function with all

zeros outside the unit circle is called a

maximum-phase transfer function

? Any nonminimum-phase transfer function

can be expressed as the product of a

minimum-phase transfer function and a

stable allpass transfer function

27 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? A set of digital transfer functions with

complementary characteristics often finds

useful applications in practice

? Four useful complementary relations are

described next along with some applications

28 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Delay-Complementary Transfer Functions

? A set of L transfer functions,,

,is defined to be delay-

complementary of each other if the sum of

their transfer functions is equal to some

integer multiple of unit delays,i.e.,

where is a nonnegative integer

)}({ zH i

10 ??? Li

0,)(

1

0

???? ?

?

?

? on

L

i

i zzH

on

29 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? A delay-complementary pair

can be readily designed if one of the pairs is

a known Type 1 FIR transfer function of

odd length

? Let be a Type 1 FIR transfer function

of length M = 2K+1

? Then its delay-complementary transfer

function is given by

)}(),({ 10 zHzH

)()( 01 zHzzH K ?? ?

)(0 zH

30 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Let the magnitude response of be

equal to in the passband and less than

or equal to in the stopband where and

are very small numbers

? Now the frequency response of can be

expressed as

where is the amplitude response

)(0 zH

p??1

s? p?

s?

)(0 zH

)()( 00 w? w?w HeeH jKj~

)(0 wH~

31 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Its delay-complementary transfer function

has a frequency response given by

? Now,in the passband,

and in the stopband,

? It follows from the above equation that in the

stopband,and in the

passband,

)(1 zH

)](1[)()( 011 w??w? w?w?w HeHeeH jKjKj~ ~

,1)(1 0 pp H ???w??? ~

ss H ??w??? )(0

~

pp H ??w??? )(1

~

ss H ???w??? 1)(1 1

~

32 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? As a result,has a complementary

magnitude response characteristic to that of

with a stopband exactly identical to

the passband of,and a passband that

is exactly identical to the stopband of

? Thus,if is a lowpass filter,will

be a highpass filter,and vice versa

)(1 zH

)(0 zH

)(0 zH

)(1 zH)(0 zH

)(0 zH

33 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? The frequency at which

the gain responses of both filters are 6 dB

below their maximum values

? The frequency is thus called the 6-dB

crossover frequency

ow

ow

5.0)()( 10 ?w?w oo HH ~ ~

34 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - Consider the Type 1 bandstop

transfer function

? Its delay-complementary Type 1 bandpass

transfer function is given by

)45541()1()( 12108424264 1 ?????? ??????? zzzzzzzH BS

)45541()1( 121084242641 ?????? ??????? zzzzzz

)()( 10 zHzzH BSBP ?? ?

35 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Plots of the magnitude responses of

and are shown below

)( zH BS

)( zH BP

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

H

BS

( z ) H

BP

( z )

36 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Allpass Complementary Filters

? A set of M digital transfer functions,,

,is defined to be allpass-

complementary of each other,if the sum of

their transfer functions is equal to an allpass

function,i.e.,

)}({ zH i

10 ??? Mi

)()(

1

0

zAzH

M

i

i ??

?

?

37 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Power-Complementary Transfer Functions

? A set of M digital transfer functions,,

,is defined to be power-

complementary of each other,if the sum of

their square-magnitude responses is equal to

a constant K for all values of w,i.e.,

)}({ zH i

10 ??? Mi

w??

?

?

w a llf o r,)(

1

0

2 KeHM

i

j

i

38 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? By analytic continuation,the above

property is equal to

for real coefficient

? Usually,by scaling the transfer functions,

the power-complementary property is

defined for K = 1

)(zH i

w??

?

?

? a llf o r,)()(

1

0

1 KzHzH

M

i

ii

39 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? For a pair of power-complementary transfer

functions,and,the frequency

where,is

called the cross-over frequency

? At this frequency the gain responses of both

filters are 3-dB below their maximum

values

? As a result,is called the 3-dB cross-over

frequency

ow

ow

)(0 zH )(1 zH

5.0|)(||)(| 2120 ?? ww oo jj eHeH

40 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - Consider the two transfer functions

and given by

where and are stable allpass

transfer functions

? Note that

? Hence,and are allpass

complementary

)(0 zH )(1 zH

)]()([)( 10210 zAzAzH ??

)(0 zA )(1 zA

)]()([)( 10211 zAzAzH ??

)()()( 010 zAzHzH ??

)(0 zH )(1 zH

41 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? It can be shown that and are

also power-complementary

? Moreover,and are bounded-

real transfer functions

)(0 zH )(1 zH

)(0 zH )(1 zH

42 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Doubly-Complementary Transfer Functions

? A set of M transfer functions satisfying both

the allpass complementary and the power-

complementary properties is known as a

doubly-complementary set

43 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? A pair of doubly-complementary IIR

transfer functions,and,with a

sum of allpass decomposition can be simply

realized as indicated below

)(0 zH )(1 zH

?

?)(1 zA

)(0 zA)(zX )(0 zY

)(1 zY1?

2/1

)( )(00 )( zX zYzH ? )( )(11 )( zX zYzH ?

44 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - The first-order lowpass transfer

function

can be expressed as

where

??????? ?

?

??

???

1

1

1

1

2

1)(

z

z

LP zH

)]()([)( 1021

1

121 1

1 zAzAzH

z

z

LP ???

??

?

?? ?

??

????

?

?

1

1

1 1)( ?

?

??

????

z

zzA

,)( 10 ?zA

45 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Its power-complementary highpass transfer

function is thus given by

? The above expression is precisely the first-

order highpass transfer function described

earlier

???????? ?

?

??

?????

1

1

1

1211021 )]()([)(

z

z

HP zAzAzH

??????? ?

?

??

???

1

1

1

1

2

1

z

z

46 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Figure below demonstrates the allpass

complementary property and the power

complementary property of and )( zH LP

)( zH HP

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

|H

HP

(e

j

w

)|

|H

LP

(e

j

w

)|

|H

LP

(e

j

w

) + H

HP

(e

j

w

)|

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

w / p

M

a

gni

t

ude

|H

HP

(e

j

w

)|

2

|H

LP

(e

j

w

)|

2

|H

LP

(e

j

w

)|

2

+ | H

HP

(e

j

w

)|

2

47 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Power-Symmetric Filters

? A real-coefficient causal digital filter with a

transfer function H(z) is said to be a power-

symmetric filter if it satisfies the condition

where K > 0 is a constant

KzHzHzHzH ???? ?? )()()()( 11

48 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? It can be shown that the gain function G(w) of a

power-symmetric transfer function at w = p is

given by

? If we define,then it follows from

the definition of the power-symmetric filter that

H(z) and G(z) are power-complementary as

c o n s t a n ta)()()()( 11 ?? ?? zGzGzHzH

)()( zHzG ??

dBK 3lo g10 10 ?

49 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

Conjugate Quadratic Filter

? If a power-symmetric filter has an FIR

transfer function H(z) of order N,then the

FIR digital filter with a transfer function

is called a conjugate quadratic filter of

H(z) and vice-versa

)()( 11 ??? zHzzG

50 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? It follows from the definition that G(z) is

also a power-symmetric causal filter

? It also can be seen that a pair of conjugate

quadratic filters H(z) and G(z) are also

power-complementary

51 Copyright ? 2001,S,K,Mitra

Complementary Transfer

Functions

? Example - Let

? We form

? Thus,H(z) is a power-symmetric transfer function

)3621)(3621( 32321 zzzzzz ??????? ???

)()()()( 11 ?? ??? zHzHzHzH

321 3621)( ??? ???? zzzzH

)3621)(3621( 32321 zzzzzz ??????? ??? )345043( 313 ?? ????? zzzz

1 0 0)345043( 313 ??????? ?? zzzz

52 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? We have shown that the BIBO stability of a

causal rational transfer function requires

that all its poles be inside the unit circle

? For very high-order transfer functions,it is

very difficult to determine the pole

locations analytically

? Root locations can of course be determined

on a computer by some type of root finding

algorithms

53 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

The Stability Triangle

? For a 2nd-order transfer function the

stability can be easily checked by

examining its denominator coefficients

54 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? Let

denote the denominator of the transfer

function

? In terms of its poles,D(z) can be expressed

as

? Comparing the last two equations we get

22111)( ?? ??? zdzdzD

2211211211 )(1)1)(1()( ???? ????????????? zzzzzD

212211 ),( ???????? dd

55 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? The poles are inside the unit circle if

? Now the coefficient is given by the

product of the poles

? Hence we must have

? It can be shown that the second coefficient

condition is given by

1||,1|| 21 ????

2d

1|| 2 ?d

21 1|| dd ??

56 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? The region in the ( )-plane where the

two coefficient condition are satisfied,

called the stability triangle,is shown below

21,dd

Stability region

57 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? Example - Consider the two 2nd-order

bandpass transfer functions designed

earlier,

21

2

3 7 6 3 817 3 4 3 4 2 401

11 8 8 1 90

??

?

??

???

zz

zzH

BP,..)('

21

2

7 2 6 5 4 2 5 305 3 3 5 3 101

11 3 6 7 30

??

?

??

??

zz

zzH

BP,..)("

58 Copyright ? 2001,S,K,Mitra

Algebraic Stability Test

? In the case of,we observe that

? Since here,is unstable

? On the other hand,in the case of,we

observe that

? Here,and,and hence

is BIBO stable

)(' zH BP

3 7 6 3 8 1 9.1,7 3 4 3 4 2 4.0 21 ??? dd

7 2 6 5 4 2 5 2 8.0,5 3 3 5 3 0 9 8.0 21 ??? dd

)(' zH BP1|| 2 ?d

)(" zH BP

21 1|| dd ??1|| 2 ?d

)(" zH BP