Copyright ? 2001,S,K,Mitra

1

Stability Condition in Terms of

the Pole Locations

? A causal LTI digital filter is BIBO stable if

and only if its impulse response h[n] is

absolutely summable,i.e.,

? We now develop a stability condition in

terms of the pole locations of the transfer

function H(z)

??? ?

?

???n

nh ][S

Copyright ? 2001,S,K,Mitra

2

Stability Condition in Terms of

the Pole Locations

? The ROC of the z-transform H(z) of the

impulse response sequence h[n] is defined

by values of |z| = r for which is

absolutely summable

? Thus,if the ROC includes the unit circle |z|

= 1,then the digital filter is stable,and vice

versa

nrnh ?][

Copyright ? 2001,S,K,Mitra

3

Stability Condition in Terms of

the Pole Locations

? In addition,for a stable and causal digital

filter for which h[n] is a right-sided

sequence,the ROC will include the unit

circle and entire z-plane including the point

? An FIR digital filter with bounded impulse

response is always stable

??z

Copyright ? 2001,S,K,Mitra

4

Stability Condition in Terms of

the Pole Locations

? On the other hand,an IIR filter may be

unstable if not designed properly

? In addition,an originally stable IIR filter

characterized by infinite precision

coefficients may become unstable when

coefficients get quantized due to

implementation

Copyright ? 2001,S,K,Mitra

5

Stability Condition in Terms of

the Pole Locations

? Example - Consider the causal IIR transfer

function

? The plot of the impulse response coefficients

is shown on the next slide

21 8 5 0 5 8 608 4 511

1

?? ??? zzzH,.)(

Copyright ? 2001,S,K,Mitra

6

Stability Condition in Terms of

the Pole Locations

? As can be seen from the above plot,the

impulse response coefficient h[n] decays

rapidly to zero value as n increases

0 10 20 30 40 50 60 70

0

2

4

6

T i m e i nde x n

A

m

pl

i

t

ude

h[n]

Copyright ? 2001,S,K,Mitra

7

Stability Condition in Terms of

the Pole Locations

? The absolute summability condition of h[n]

is satisfied

? Hence,H(z) is a stable transfer function

? Now,consider the case when the transfer

function coefficients are rounded to values

with 2 digits after the decimal point,

21 8508511

1

?? ??? zzzH,.)(

^

Copyright ? 2001,S,K,Mitra

8

Stability Condition in Terms of

the Pole Locations

? A plot of the impulse response of is

shown below

][nh^

0 10 20 30 40 50 60 70

0

2

4

6

T i m e i nde x n

A

m

pl

i

t

ude

][nh^

Copyright ? 2001,S,K,Mitra

9

Stability Condition in Terms of

the Pole Locations

? In this case,the impulse response coefficient

increases rapidly to a constant value as

n increases

? Hence,the absolute summability condition of

is violated

? Thus,is an unstable transfer function

][nh^

)(zH^

Copyright ? 2001,S,K,Mitra

10

Stability Condition in Terms of

the Pole Locations

? The stability testing of a IIR transfer

function is therefore an important problem

? In most cases it is difficult to compute the

infinite sum

? For a causal IIR transfer function,the sum S

can be computed approximately as

??? ? ? ???n nh ][S

? ??? 10Kn nh ][S K

Copyright ? 2001,S,K,Mitra

11

Stability Condition in Terms of

the Pole Locations

? The partial sum is computed for increasing

values of K until the difference between a

series of consecutive values of is

smaller than some arbitrarily chosen small

number,which is typically

? For a transfer function of very high order

this approach may not be satisfactory

? An alternate,easy-to-test,stability condition

is developed next

S K

610?

Copyright ? 2001,S,K,Mitra

12

Stability Condition in Terms of

the Pole Locations

? Consider the causal IIR digital filter with a

rational transfer function H(z) given by

? Its impulse response {h[n]} is a right-sided

sequence

? The ROC of H(z) is exterior to a circle

going through the pole farthest from z = 0

?

?

?

?

?

?

? N

k

k

k

M

k

k

k

zd

zp

zH

0

0)(

Copyright ? 2001,S,K,Mitra

13

Stability Condition in Terms of

the Pole Locations

? But stability requires that {h[n]} be

absolutely summable

? This in turn implies that the DTFT

of {h[n]} exists

? Now,if the ROC of the z-transform H(z)

includes the unit circle,then

??? ? jezj zHeH )()(

)( ?jeH

Copyright ? 2001,S,K,Mitra

14

Stability Condition in Terms of

the Pole Locations

? Conclusion,All poles of a causal stable

transfer function H(z) must be strictly inside

the unit circle

? The stability region (shown shaded) in the

z-plane is shown below

1 Re z

j Im z

1?

j?

j

unit circle

stability region

Copyright ? 2001,S,K,Mitra

15

Stability Condition in Terms of

the Pole Locations

? Example - The factored form of

is

which has a real pole at z = 0.902 and a real

pole at z = 0.943

? Since both poles are inside the unit circle,

H(z) is BIBO stable

21 8 5 0 5 8 6.0845.01

1)(

?? ??? zzzH

)94 3.01)(90 2.01(

1)(

11 ?? ??? zzzH

Copyright ? 2001,S,K,Mitra

16

Stability Condition in Terms of

the Pole Locations

? Example - The factored form of

is

which has a real pole on the unit circle at z =

1 and the other pole inside the unit circle

? Since one pole is not inside but on the unit

circle,H(z) is unstable

21 85.085.11

1)(

?? ??? zzzH

^

)85.01)(1(

1)(

11 ?? ??? zzzH

^

Copyright ? 2001,S,K,Mitra

17

Types of Transfer Functions

? The time-domain classification of an LTI

digital transfer function sequence is based

on the length of its impulse response,

- Finite impulse response (FIR) transfer

function

- Infinite impulse response (IIR) transfer

function

Copyright ? 2001,S,K,Mitra

18

Types of Transfer Functions

? Several other classifications are also used

? In the case of digital transfer functions with

frequency-selective frequency responses,

one classification is based on the shape of

the magnitude function or the form

of the phase function q(?)

? Based on this four types of ideal filters are

usually defined

|)(| ?jeH

Copyright ? 2001,S,K,Mitra

19

Ideal Filters

? A digital filter designed to pass signal

components of certain frequencies without

distortion should have a frequency response

equal to one at these frequencies,and

should have a frequency response equal to

zero at all other frequencies

Copyright ? 2001,S,K,Mitra

20

Ideal Filters

? The range of frequencies where the

frequency response takes the value of one is

called the passband

? The range of frequencies where the

frequency response takes the value of zero

is called the stopband

Copyright ? 2001,S,K,Mitra

21

Ideal Filters

? Frequency responses of the four popular types

of ideal digital filters with real impulse

response coefficients are shown below,

Lowpass Highpass

Bandpass Bandstop

Copyright ? 2001,S,K,Mitra

22

Ideal Filters

? Lowpass filter,Passband -

Stopband -

? Highpass filter,Passband -

Stopband -

? Bandpass filter,Passband -

Stopband -

? Bandstop filter,Stopband -

Passband -

c????0

????? c

????? c

c????0

21 cc ?????

10 c???? ????2cand

21 cc ?????

10 c???? ????? 2cand

Copyright ? 2001,S,K,Mitra

23

Ideal Filters

? The frequencies,,and are called

the cutoff frequencies

? An ideal filter has a magnitude response

equal to one in the passband and zero in the

stopband,and has a zero phase everywhere

c? 1c? 2c?

Copyright ? 2001,S,K,Mitra

24

Ideal Filters

? Earlier in the course we derived the inverse

DTFT of the frequency response

of the ideal lowpass filter,

? We have also shown that the above impulse

response is not absolutely summable,and

hence,the corresponding transfer function

is not BIBO stable

)( ?jLP eH

sin[ ] sin c,c c c

LP

nnh n n

n

? ? ?

? ? ?

??? ? ? ? ? ? ?

????

Copyright ? 2001,S,K,Mitra

25

Ideal Filters

? Also,is not causal and is of doubly

infinite length

? The remaining three ideal filters are also

characterized by doubly infinite,noncausal

impulse responses and are not absolutely

summable

? Thus,the ideal filters with the ideal,brick

wall” frequency responses cannot be

realized with finite dimensional LTI filter

][nhLP

Copyright ? 2001,S,K,Mitra

26

Ideal Filters

? To develop stable and realizable transfer

functions,the ideal frequency response

specifications are relaxed by including a

transition band between the passband and

the stopband

? This permits the magnitude response to

decay slowly from its maximum value in

the passband to the zero value in the

stopband

Copyright ? 2001,S,K,Mitra

27

Ideal Filters

? Moreover,the magnitude response is allowed

to vary by a small amount (ripple) both in the

passband and the stopband

? Typical magnitude response specifications of

a lowpass filter are shown below

Copyright ? 2001,S,K,Mitra

28

Zero-Phase and Linear-Phase

Transfer Functions

? A second classification of a transfer

function is with respect to its phase

characteristics

? In many applications,it is necessary that the

digital filter designed does not distort the

phase of the input signal components with

frequencies in the passband

Copyright ? 2001,S,K,Mitra

29

Zero-Phase and Linear-Phase

Transfer Functions

? One way to avoid any phase distortion is to

make the frequency response of the filter

real and nonnegative,i.e.,to design the

filter with a zero phase characteristic

? However,it is possible to design a causal

digital filter with a zero phase

Copyright ? 2001,S,K,Mitra

30

Zero-Phase and Linear-Phase

Transfer Functions

? For non-real-time processing of real-valued

input signals of finite length,zero-phase

filtering can be very simply implemented

by relaxing the causality requirement

? One zero-phase filtering scheme is sketched

below

x[n] v[n] u[n] w[n] H(z) H(z)

][][],[][ nwnynvnu ????

Copyright ? 2001,S,K,Mitra

31

Zero-Phase and Linear-Phase

Transfer Functions

? It is easy to verify the above scheme in the

frequency domain

? Let,,,,and

denote the DTFTs of x[n],v[n],

u[n],w[n],and y[n],respectively

? From the figure shown earlier and making

use of the symmetry relations we arrive at

the relations between various DTFTs as

given on the next slide

)( ?jeX )( ?jeV )( ?jeU )( ?jeW

)( ?jeY

Copyright ? 2001,S,K,Mitra

32

Zero-Phase and Linear-Phase

Transfer Functions

? Combining the above equations we get

x[n] v[n] u[n] w[n] H(z) H(z)

][][],[][ nwnynvnu ????

( ) ( ) ( ),j j jV e H e X e? ? ?? )()()( ??? ? jjj eUeHeW

,)(*)( ?? ? jj eVeU )(*)( ?? ? jj eWeY

)(*)(*)(*)( ???? ?? jjjj eUeHeWeY

)()()(*)()(* ????? ?? jjjjj eXeHeHeVeH )()( 2 ??? jj eXeH

Copyright ? 2001,S,K,Mitra

33

Zero-Phase and Linear-Phase

Transfer Functions

? The function filtfilt implements the above

zero-phase filtering scheme

? In the case of a causal transfer function with

a nonzero phase response,the phase

distortion can be avoided by ensuring that

the transfer function has a unity magnitude

and a linear-phase characteristic in the

frequency band of interest

Copyright ? 2001,S,K,Mitra

34

Zero-Phase and Linear-Phase

Transfer Functions

? The most general type of a filter with a

linear phase has a frequency response given

by

which has a linear phase from ? = 0 to ? =

2?

? Note also

Djj eeH ??? ?)(

1)( ??jeH

D??? )(

Copyright ? 2001,S,K,Mitra

35

Zero-Phase and Linear-Phase

Transfer Functions

? The output y[n] of this filter to an input

is then given by

? If and represent the continuous-

time signals whose sampled versions,

sampled at t = nT,are x[n] and y[n] given

above,then the delay between and

is precisely the group delay of amount D

njAenx ??][

)(][ DnjnjDj AeeAeny ????? ??

)(txa

)(txa

)(tya

)(tya

Copyright ? 2001,S,K,Mitra

36

Zero-Phase and Linear-Phase

Transfer Functions

? If D is an integer,then y[n] is identical to

x[n],but delayed by D samples

? If D is not an integer,y[n],being delayed by

a fractional part,is not identical to x[n]

? In the latter case,the waveform of the

underlying continuous-time output is

identical to the waveform of the underlying

continuous-time input and delayed D units

of time

Copyright ? 2001,S,K,Mitra

37

Zero-Phase and Linear-Phase

Transfer Functions

? If it is desired to pass input signal

components in a certain frequency range

undistorted in both magnitude and phase,

then the transfer function should exhibit a

unity magnitude response and a linear-phase

response in the band of interest

Copyright ? 2001,S,K,Mitra

38

Zero-Phase and Linear-Phase

Transfer Functions

? Figure below shows the frequency response

if a lowpass filter with a linear-phase

characteristic in the passband

Copyright ? 2001,S,K,Mitra

39

Zero-Phase and Linear-Phase

Transfer Functions

? Since the signal components in the stopband

are blocked,the phase response in the

stopband can be of any shape

? Example - Determine the impulse response

of an ideal lowpass filter with a linear phase

response,

??

??? )( j

LP eH ?????

??????

c

c

nj oe

,0

0,

Copyright ? 2001,S,K,Mitra

40

Zero-Phase and Linear-Phase

Transfer Functions

? Applying the frequency-shifting property of

the DTFT to the impulse response of an ideal

zero-phase lowpass filter we arrive at

? As before,the above filter is noncausal and

of doubly infinite length,and hence,

unrealizable

()[ ] sin c,c c o

LP

nnh n n??

??

???? ? ? ? ? ?

????

Copyright ? 2001,S,K,Mitra

41

Zero-Phase and Linear-Phase

Transfer Functions

? By truncating the impulse response to a

finite number of terms,a realizable FIR

approximation to the ideal lowpass filter

can be developed

? The truncated approximation may or may

not exhibit linear phase,depending on the

value of chosen on

Copyright ? 2001,S,K,Mitra

42

Zero-Phase and Linear-Phase

Transfer Functions

? If we choose = N/2 with N a positive

integer,the truncated and shifted

approximation

will be a length N+1 causal linear-phase

FIR filter

on

NnNn Nnnh cLP ???? ??? 0,)2/( )2/(si n][^

Copyright ? 2001,S,K,Mitra

43

Zero-Phase and Linear-Phase

Transfer Functions

? Figure below shows the filter coefficients

obtained using the function sinc for two

different values of N

0 2 4 6 8 10 12

- 0.2

0

0.2

0.4

0.6

T i m e i nde x n

A

m

pl

i

t

ude

N = 12

0 2 4 6 8 10 12

- 0.2

0

0.2

0.4

0.6

T i m e i nde x n

A

m

pl

i

t

ude

N = 13

Copyright ? 2001,S,K,Mitra

44

Zero-Phase and Linear-Phase

Transfer Functions

? Because of the symmetry of the impulse

response coefficients as indicated in the two

figures,the frequency response of the

truncated approximation can be expressed as,

where,called the zero-phase

response or amplitude response,is a real

function of ?

)(][)( 2/

0

??? ??

?

??? ?

LP

Nj

N

n

nj

LP

j

LP HeenheH ^ ^

~

)(?LPH~

1

Stability Condition in Terms of

the Pole Locations

? A causal LTI digital filter is BIBO stable if

and only if its impulse response h[n] is

absolutely summable,i.e.,

? We now develop a stability condition in

terms of the pole locations of the transfer

function H(z)

??? ?

?

???n

nh ][S

Copyright ? 2001,S,K,Mitra

2

Stability Condition in Terms of

the Pole Locations

? The ROC of the z-transform H(z) of the

impulse response sequence h[n] is defined

by values of |z| = r for which is

absolutely summable

? Thus,if the ROC includes the unit circle |z|

= 1,then the digital filter is stable,and vice

versa

nrnh ?][

Copyright ? 2001,S,K,Mitra

3

Stability Condition in Terms of

the Pole Locations

? In addition,for a stable and causal digital

filter for which h[n] is a right-sided

sequence,the ROC will include the unit

circle and entire z-plane including the point

? An FIR digital filter with bounded impulse

response is always stable

??z

Copyright ? 2001,S,K,Mitra

4

Stability Condition in Terms of

the Pole Locations

? On the other hand,an IIR filter may be

unstable if not designed properly

? In addition,an originally stable IIR filter

characterized by infinite precision

coefficients may become unstable when

coefficients get quantized due to

implementation

Copyright ? 2001,S,K,Mitra

5

Stability Condition in Terms of

the Pole Locations

? Example - Consider the causal IIR transfer

function

? The plot of the impulse response coefficients

is shown on the next slide

21 8 5 0 5 8 608 4 511

1

?? ??? zzzH,.)(

Copyright ? 2001,S,K,Mitra

6

Stability Condition in Terms of

the Pole Locations

? As can be seen from the above plot,the

impulse response coefficient h[n] decays

rapidly to zero value as n increases

0 10 20 30 40 50 60 70

0

2

4

6

T i m e i nde x n

A

m

pl

i

t

ude

h[n]

Copyright ? 2001,S,K,Mitra

7

Stability Condition in Terms of

the Pole Locations

? The absolute summability condition of h[n]

is satisfied

? Hence,H(z) is a stable transfer function

? Now,consider the case when the transfer

function coefficients are rounded to values

with 2 digits after the decimal point,

21 8508511

1

?? ??? zzzH,.)(

^

Copyright ? 2001,S,K,Mitra

8

Stability Condition in Terms of

the Pole Locations

? A plot of the impulse response of is

shown below

][nh^

0 10 20 30 40 50 60 70

0

2

4

6

T i m e i nde x n

A

m

pl

i

t

ude

][nh^

Copyright ? 2001,S,K,Mitra

9

Stability Condition in Terms of

the Pole Locations

? In this case,the impulse response coefficient

increases rapidly to a constant value as

n increases

? Hence,the absolute summability condition of

is violated

? Thus,is an unstable transfer function

][nh^

)(zH^

Copyright ? 2001,S,K,Mitra

10

Stability Condition in Terms of

the Pole Locations

? The stability testing of a IIR transfer

function is therefore an important problem

? In most cases it is difficult to compute the

infinite sum

? For a causal IIR transfer function,the sum S

can be computed approximately as

??? ? ? ???n nh ][S

? ??? 10Kn nh ][S K

Copyright ? 2001,S,K,Mitra

11

Stability Condition in Terms of

the Pole Locations

? The partial sum is computed for increasing

values of K until the difference between a

series of consecutive values of is

smaller than some arbitrarily chosen small

number,which is typically

? For a transfer function of very high order

this approach may not be satisfactory

? An alternate,easy-to-test,stability condition

is developed next

S K

610?

Copyright ? 2001,S,K,Mitra

12

Stability Condition in Terms of

the Pole Locations

? Consider the causal IIR digital filter with a

rational transfer function H(z) given by

? Its impulse response {h[n]} is a right-sided

sequence

? The ROC of H(z) is exterior to a circle

going through the pole farthest from z = 0

?

?

?

?

?

?

? N

k

k

k

M

k

k

k

zd

zp

zH

0

0)(

Copyright ? 2001,S,K,Mitra

13

Stability Condition in Terms of

the Pole Locations

? But stability requires that {h[n]} be

absolutely summable

? This in turn implies that the DTFT

of {h[n]} exists

? Now,if the ROC of the z-transform H(z)

includes the unit circle,then

??? ? jezj zHeH )()(

)( ?jeH

Copyright ? 2001,S,K,Mitra

14

Stability Condition in Terms of

the Pole Locations

? Conclusion,All poles of a causal stable

transfer function H(z) must be strictly inside

the unit circle

? The stability region (shown shaded) in the

z-plane is shown below

1 Re z

j Im z

1?

j?

j

unit circle

stability region

Copyright ? 2001,S,K,Mitra

15

Stability Condition in Terms of

the Pole Locations

? Example - The factored form of

is

which has a real pole at z = 0.902 and a real

pole at z = 0.943

? Since both poles are inside the unit circle,

H(z) is BIBO stable

21 8 5 0 5 8 6.0845.01

1)(

?? ??? zzzH

)94 3.01)(90 2.01(

1)(

11 ?? ??? zzzH

Copyright ? 2001,S,K,Mitra

16

Stability Condition in Terms of

the Pole Locations

? Example - The factored form of

is

which has a real pole on the unit circle at z =

1 and the other pole inside the unit circle

? Since one pole is not inside but on the unit

circle,H(z) is unstable

21 85.085.11

1)(

?? ??? zzzH

^

)85.01)(1(

1)(

11 ?? ??? zzzH

^

Copyright ? 2001,S,K,Mitra

17

Types of Transfer Functions

? The time-domain classification of an LTI

digital transfer function sequence is based

on the length of its impulse response,

- Finite impulse response (FIR) transfer

function

- Infinite impulse response (IIR) transfer

function

Copyright ? 2001,S,K,Mitra

18

Types of Transfer Functions

? Several other classifications are also used

? In the case of digital transfer functions with

frequency-selective frequency responses,

one classification is based on the shape of

the magnitude function or the form

of the phase function q(?)

? Based on this four types of ideal filters are

usually defined

|)(| ?jeH

Copyright ? 2001,S,K,Mitra

19

Ideal Filters

? A digital filter designed to pass signal

components of certain frequencies without

distortion should have a frequency response

equal to one at these frequencies,and

should have a frequency response equal to

zero at all other frequencies

Copyright ? 2001,S,K,Mitra

20

Ideal Filters

? The range of frequencies where the

frequency response takes the value of one is

called the passband

? The range of frequencies where the

frequency response takes the value of zero

is called the stopband

Copyright ? 2001,S,K,Mitra

21

Ideal Filters

? Frequency responses of the four popular types

of ideal digital filters with real impulse

response coefficients are shown below,

Lowpass Highpass

Bandpass Bandstop

Copyright ? 2001,S,K,Mitra

22

Ideal Filters

? Lowpass filter,Passband -

Stopband -

? Highpass filter,Passband -

Stopband -

? Bandpass filter,Passband -

Stopband -

? Bandstop filter,Stopband -

Passband -

c????0

????? c

????? c

c????0

21 cc ?????

10 c???? ????2cand

21 cc ?????

10 c???? ????? 2cand

Copyright ? 2001,S,K,Mitra

23

Ideal Filters

? The frequencies,,and are called

the cutoff frequencies

? An ideal filter has a magnitude response

equal to one in the passband and zero in the

stopband,and has a zero phase everywhere

c? 1c? 2c?

Copyright ? 2001,S,K,Mitra

24

Ideal Filters

? Earlier in the course we derived the inverse

DTFT of the frequency response

of the ideal lowpass filter,

? We have also shown that the above impulse

response is not absolutely summable,and

hence,the corresponding transfer function

is not BIBO stable

)( ?jLP eH

sin[ ] sin c,c c c

LP

nnh n n

n

? ? ?

? ? ?

??? ? ? ? ? ? ?

????

Copyright ? 2001,S,K,Mitra

25

Ideal Filters

? Also,is not causal and is of doubly

infinite length

? The remaining three ideal filters are also

characterized by doubly infinite,noncausal

impulse responses and are not absolutely

summable

? Thus,the ideal filters with the ideal,brick

wall” frequency responses cannot be

realized with finite dimensional LTI filter

][nhLP

Copyright ? 2001,S,K,Mitra

26

Ideal Filters

? To develop stable and realizable transfer

functions,the ideal frequency response

specifications are relaxed by including a

transition band between the passband and

the stopband

? This permits the magnitude response to

decay slowly from its maximum value in

the passband to the zero value in the

stopband

Copyright ? 2001,S,K,Mitra

27

Ideal Filters

? Moreover,the magnitude response is allowed

to vary by a small amount (ripple) both in the

passband and the stopband

? Typical magnitude response specifications of

a lowpass filter are shown below

Copyright ? 2001,S,K,Mitra

28

Zero-Phase and Linear-Phase

Transfer Functions

? A second classification of a transfer

function is with respect to its phase

characteristics

? In many applications,it is necessary that the

digital filter designed does not distort the

phase of the input signal components with

frequencies in the passband

Copyright ? 2001,S,K,Mitra

29

Zero-Phase and Linear-Phase

Transfer Functions

? One way to avoid any phase distortion is to

make the frequency response of the filter

real and nonnegative,i.e.,to design the

filter with a zero phase characteristic

? However,it is possible to design a causal

digital filter with a zero phase

Copyright ? 2001,S,K,Mitra

30

Zero-Phase and Linear-Phase

Transfer Functions

? For non-real-time processing of real-valued

input signals of finite length,zero-phase

filtering can be very simply implemented

by relaxing the causality requirement

? One zero-phase filtering scheme is sketched

below

x[n] v[n] u[n] w[n] H(z) H(z)

][][],[][ nwnynvnu ????

Copyright ? 2001,S,K,Mitra

31

Zero-Phase and Linear-Phase

Transfer Functions

? It is easy to verify the above scheme in the

frequency domain

? Let,,,,and

denote the DTFTs of x[n],v[n],

u[n],w[n],and y[n],respectively

? From the figure shown earlier and making

use of the symmetry relations we arrive at

the relations between various DTFTs as

given on the next slide

)( ?jeX )( ?jeV )( ?jeU )( ?jeW

)( ?jeY

Copyright ? 2001,S,K,Mitra

32

Zero-Phase and Linear-Phase

Transfer Functions

? Combining the above equations we get

x[n] v[n] u[n] w[n] H(z) H(z)

][][],[][ nwnynvnu ????

( ) ( ) ( ),j j jV e H e X e? ? ?? )()()( ??? ? jjj eUeHeW

,)(*)( ?? ? jj eVeU )(*)( ?? ? jj eWeY

)(*)(*)(*)( ???? ?? jjjj eUeHeWeY

)()()(*)()(* ????? ?? jjjjj eXeHeHeVeH )()( 2 ??? jj eXeH

Copyright ? 2001,S,K,Mitra

33

Zero-Phase and Linear-Phase

Transfer Functions

? The function filtfilt implements the above

zero-phase filtering scheme

? In the case of a causal transfer function with

a nonzero phase response,the phase

distortion can be avoided by ensuring that

the transfer function has a unity magnitude

and a linear-phase characteristic in the

frequency band of interest

Copyright ? 2001,S,K,Mitra

34

Zero-Phase and Linear-Phase

Transfer Functions

? The most general type of a filter with a

linear phase has a frequency response given

by

which has a linear phase from ? = 0 to ? =

2?

? Note also

Djj eeH ??? ?)(

1)( ??jeH

D??? )(

Copyright ? 2001,S,K,Mitra

35

Zero-Phase and Linear-Phase

Transfer Functions

? The output y[n] of this filter to an input

is then given by

? If and represent the continuous-

time signals whose sampled versions,

sampled at t = nT,are x[n] and y[n] given

above,then the delay between and

is precisely the group delay of amount D

njAenx ??][

)(][ DnjnjDj AeeAeny ????? ??

)(txa

)(txa

)(tya

)(tya

Copyright ? 2001,S,K,Mitra

36

Zero-Phase and Linear-Phase

Transfer Functions

? If D is an integer,then y[n] is identical to

x[n],but delayed by D samples

? If D is not an integer,y[n],being delayed by

a fractional part,is not identical to x[n]

? In the latter case,the waveform of the

underlying continuous-time output is

identical to the waveform of the underlying

continuous-time input and delayed D units

of time

Copyright ? 2001,S,K,Mitra

37

Zero-Phase and Linear-Phase

Transfer Functions

? If it is desired to pass input signal

components in a certain frequency range

undistorted in both magnitude and phase,

then the transfer function should exhibit a

unity magnitude response and a linear-phase

response in the band of interest

Copyright ? 2001,S,K,Mitra

38

Zero-Phase and Linear-Phase

Transfer Functions

? Figure below shows the frequency response

if a lowpass filter with a linear-phase

characteristic in the passband

Copyright ? 2001,S,K,Mitra

39

Zero-Phase and Linear-Phase

Transfer Functions

? Since the signal components in the stopband

are blocked,the phase response in the

stopband can be of any shape

? Example - Determine the impulse response

of an ideal lowpass filter with a linear phase

response,

??

??? )( j

LP eH ?????

??????

c

c

nj oe

,0

0,

Copyright ? 2001,S,K,Mitra

40

Zero-Phase and Linear-Phase

Transfer Functions

? Applying the frequency-shifting property of

the DTFT to the impulse response of an ideal

zero-phase lowpass filter we arrive at

? As before,the above filter is noncausal and

of doubly infinite length,and hence,

unrealizable

()[ ] sin c,c c o

LP

nnh n n??

??

???? ? ? ? ? ?

????

Copyright ? 2001,S,K,Mitra

41

Zero-Phase and Linear-Phase

Transfer Functions

? By truncating the impulse response to a

finite number of terms,a realizable FIR

approximation to the ideal lowpass filter

can be developed

? The truncated approximation may or may

not exhibit linear phase,depending on the

value of chosen on

Copyright ? 2001,S,K,Mitra

42

Zero-Phase and Linear-Phase

Transfer Functions

? If we choose = N/2 with N a positive

integer,the truncated and shifted

approximation

will be a length N+1 causal linear-phase

FIR filter

on

NnNn Nnnh cLP ???? ??? 0,)2/( )2/(si n][^

Copyright ? 2001,S,K,Mitra

43

Zero-Phase and Linear-Phase

Transfer Functions

? Figure below shows the filter coefficients

obtained using the function sinc for two

different values of N

0 2 4 6 8 10 12

- 0.2

0

0.2

0.4

0.6

T i m e i nde x n

A

m

pl

i

t

ude

N = 12

0 2 4 6 8 10 12

- 0.2

0

0.2

0.4

0.6

T i m e i nde x n

A

m

pl

i

t

ude

N = 13

Copyright ? 2001,S,K,Mitra

44

Zero-Phase and Linear-Phase

Transfer Functions

? Because of the symmetry of the impulse

response coefficients as indicated in the two

figures,the frequency response of the

truncated approximation can be expressed as,

where,called the zero-phase

response or amplitude response,is a real

function of ?

)(][)( 2/

0

??? ??

?

??? ?

LP

Nj

N

n

nj

LP

j

LP HeenheH ^ ^

~

)(?LPH~