Types of Transfer Functions

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1

Types of Transfer Functions 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

2

Types of Transfer Functions Types of Transfer Functions

• In the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications

• (1) Classification based on the shape of the magnitude function

• (2) Classification based on the the form of the phase functionθ(ω)

| ) (

|H e^j^ω

3

Classification Based on Classification Based on Magnitude Characteristics Magnitude Characteristics

• One common classification is based on an idealmagnitude response

• A digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to oneat these frequencies, and should have a frequency response equal to zeroat all other frequencies

4

Ideal Filters Ideal Filters

• The range of frequencies where the

frequency response takes the value of oneis called thepassband

• The range of frequencies where the frequency response takes the value of zero is called thestopband

5

Ideal Filters 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 6

• Lowpass filter: Passband - Stopband -

• Highpass filter: Passband - Stopband -

• Bandpass filter: Passband - Stopband -

• Bandstop filter: Stopband - Passband -

ωc

≤ ω

≤ 0

π

≤ ω

<

ω_c π

≤ ω

≤ ω_c

ωc

<

ω

≤ 0

2

1 c

c ≤ω≤ω ω

0≤ω<ω_c1andω_c₂<ω≤π

2

1 c

c <ω<ω ω

0≤ω≤ω_c1and ω_c₂≤ω≤π

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7

Ideal Filters Ideal Filters

• The frequencies , , and are called thecutoff 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

ωc ω_c₁ ω_c₂

8

• 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

) ( ^j^ω

LP e H

∞

<

∞ π −

= ω n

n n n

h_LP sin ^c , ]

[

9

Ideal Filters 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

] [n h_LP

10

Ideal Filters Ideal Filters

• To develop stable and realizable transfer functions, the ideal frequency response specifications are relaxed by including a transition bandbetween 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

11

Ideal Filters Ideal Filters

• Moreover, the magnitude response is allowed to vary by a small amount both in the passband and the stopband

• Typical magnitude response specifications of a lowpass filter are shown below

12

Bounded Real Transfer Bounded Real Transfer

Functions Functions

• A causal stable real-coefficient transfer functionH(z) is defined as a bounded real (BR) transfer functionif

• Let x[n]and y[n]denote, respectively, the input and output of a digital filter

characterized by a BR transfer function H(z) with and denoting their DTFTs

) (e^j^ω

X Y(e^j^ω) 1

| ) (

|H e^j^ω ≤ for all values ofω

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13

Bounded Real Transfer Bounded Real Transfer

Functions Functions

• Then the condition implies that

• Integrating the above from toπ, and applying Parseval’s relationwe get

1

| ) (

|H e^j^ω ≤

2

2 ( )

)

(e^j^ω ≤ X e^j^ω Y

∑

^∞

−∞

=

∞

−∞

= ≤

n n

n x n

y[ ]² [ ]² π

−

14

Functions Functions

• Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy implying that a digital filter characterized by a BR transfer function can be viewed as apassive structure

• If , then the output energy is equal to the input energy, and such a digital filter is therefore alossless system

1

| ) (

|H e^j^ω =

15

Bounded Real Transfer Bounded Real Transfer

Functions Functions

• A causal stable real-coefficient transfer functionH(z) with is thus called alossless bounded real(LBR) transfer function

• The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity

1

| ) (

|H e^j^ω =

16

Functions Functions

• Example –Consider the causal stable IIR transfer function

where Kis a real constant

• Its square-magnitude function is given by 1 0 1 ,

)

( ₁ <α<

α

= − ₋ z z K H

ω α

− α

= +

= _ω

=

− ω

cos 2 ) 1 ) (

( ) ( )

( ₂

1 2

2 K

z H z H e

H _j

e z j

17

Functions Functions

• The maximum value of is obtained when in the denominator is a maximumand the minimum value is obtained when is a minimum

• For α> 0, maximumvalue of is equal to 2αat ω= 0, and minimumvalue is

at ω= π

ω αcos 2

α

−2

|2

) (

|H e^j^ω

18

Bounded Real Transfer Bounded Real Transfer

Functions Functions

• Thus, for α> 0, the maximum value of is equal to at ω= 0 and the minimum value is equal to

at ω= π

• On the other hand, for α< 0, the maximum value of is equal to at ω= π and the minimum value is equal to 2αat ω

= 0

2 2/(1−α)

2 K

| ) (

|H e^j^ω

ω

2αcos −2α

2 2/(1+α) K

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19

Functions Functions

• Here, the maximum value of is equal to at ω= πand the minimum value is equal to at ω= 0

• Hence, the maximum value can be made equal to 1by choosing , in which case the minimum value becomes

|2

) (

|He^j^ω

2 2/(1−α)

K 2 2

1 )

/( −α K

) 1 ( −α

±

= K

2 2/(1 ) )

1

( −α +α

20

Bounded Real Transfer Bounded Real Transfer

Functions Functions

• Hence,

is a BR function for

• Plots of the magnitude function for with values of Kchosen to make H(z)a BR functionare shown on the next slide

) 1 ( −α

±

= K

1 0 1 ,

)

( ₁ <α<

α

= − ₋ z z K H

5 .

±0

= α

21

Functions Functions

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

ω/π

Magnitude

K = ± 0.5, α = 0.5

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

ω/π

Magnitude

K = ± 0.5, α = -0.5

Lowpass Filter Highpass Filter

22

Allpass

Allpass Transfer Function Transfer Function

Definition

• An IIR transfer functionA(z) with unity magnitude response for all frequencies, i.e., is called anallpass transfer function

• AnM-th order causal real-coefficient allpass transfer function is of the form

ω

ω)| =1, forall (

|Ae^j ²

M M M M

M M

M M M

z d z

d z

d

z z

d z

d z d

A ₋ ₊ ₋

− −

− +

−

− −

+ +

+

± +

= ₁

1 1

1

1 1 1 1

1 ...

. . ) .

(

23

Allpass

Allpass Transfer Function Transfer Function

• If we denote the denominator polynomials 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

) (z D_M ) (z A_M

M M M M

M z d z d z d z

D ( )=1+ ₁ ⁻¹+...+ ₋₁ ⁻ ⁺¹+ ⁻ )

(z A_M

) (

)

) (

( D z

z D z

M M

z A

−1

± −

=

=re^jφ

z

φ

=_re−^j

z ¹

24

Allpass

• The numerator of a real-coefficient allpass transfer function is said to be themirror- image polynomialof the denominator, and vice versa

• We shall use the notation to denote the mirror-image polynomial of a degree-M polynomial , i.e.,

) (z D^~_M

) (z D_M

) ( )

(z =z⁻ D z⁻¹ D^~_M ^M _M

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25

Allpass

• The expression implies that the poles and zeros of a real- coefficient allpass function exhibitmirror- image symmetryin thez-plane

) (

)

) (

( D z

z D z

M M

z A

−1

± −

=

3 2 1

3 1 0.4 0.18 0.2 4 . 0 18 . 0 2 . ) 0

( ₋ ₋ ₋

−

− + +

+ + +

=−

z z z

z z z z

A

-1 0 1 2 3

-1.5 -1 -0.5 0 0.5 1 1.5

Real Part

Imaginary Part

26

Allpass

• To show that we observe that

• Therefore

• Hence

) (

) 1 (

) 1

( ⁻ =± ₋

z D

z D z

M M

z A

) (

) ( )

( ) 1 (

1 1

) ( )

( ⁻ = ⁻ ⁻ ₋

z D

z D z z D

z D z M

M

M M M M M M

z A z A

1

| ) (

|A_M e^j^ω =

1 )

( ) (

| ) (

| ²= ¹ _ω =

=

− ω

ej M z

j M

M e A z A z

A

27

Allpass

• 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 symmetrywith its poles situated inside the unit circle

28

Allpass

• Figure below shows the principal value of the phase of the 3rd-order allpass function

• Note the discontinuity by the amount of2π in the phaseθ(ω)

3 2 1

3 2 1 3 1 0.4 0.18 0.2

4 . 0 18 . 0 2 . ) 0

( ₋ ₋ ₋

−

− + +

+ + +

=−

z z z

z z z z

A

0 0.2 0.4 0.6 0.8 1

-4 -2 0 2 4

ω/π

Phase, degrees

Principal value of phase

29

Allpass

• 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 ofω

(ω) θ_c

0 0.2 0.4 0.6 0.8 1

-10 -8 -6 -4 -2 0

ω/π

Phase, degrees

Unwrapped phase

30

Allpass

• The unwrapped phase function of any arbitrary causal stable allpass function is a continuous function ofω

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

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31

Allpass

• (2)The magnitude function of a stable allpass functionA(z) satisfies:

• (3)Letτ(ω) denote the group delay function of an allpass filterA(z), i.e.,

⎪⎩

⎪⎨

⎧

<

>

=

>

<

1 for 1

z z z z

A , , , ) (

)]

( [ )

(ω =− θ ω

τ _ω _c

d d

32

Allpass

• The unwrapped phase function of a stable allpass function is a monotonically decreasing function ofωso thatτ(ω) is everywhere positive in the range0 < ω< π

• The group delay of an M-th order stable real-coefficient allpass transfer function satisfies:

(ω) θ_c

π

=

∫ ωτ ω

π

M d

0

) (

33

Allpass

A Simple Application

• A simple but often used application of an allpass filter is as adelay equalizer

• LetG(z) be the transfer function of a digital filter designed to meet a prescribed

magnitude response

• The nonlinear phase response ofG(z) can be corrected by cascading it with an allpass filterA(z) so that the overall cascade has a constant group delay in the band of interest

34

Allpass

• Since , we have

• Overall group delay is the given by the sum of the group delays ofG(z) andA(z)

1

| ) (

|Ae^j^ω =

| ) (

|

| ) ( ) (

|Ge^j^ω Ae^j^ω =G e^j^ω

G(z) A(z)

35

Allpass

• Example –Figure below shows the group delay of a 4^thorder elliptic filter with the following specifications: ,

dB, dB

π

= ω_p 0.3

=1

δ_p δ_s=35

0 0.2 0.4 0.6 0.8 1

0 5 10 15

ω/π

Group delay, samples

Original Filter

36

Allpass

• Figure below shows the group delay of the original elliptic filter cascaded with an 8^th order allpass section designed to equalize the group delay in the passband

0 0.2 0.4 0.6 0.8 1

0 5 10 15 20 25 30

ω/π

Group delay, samples

Group Delay Equalized Filter

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37

Classification Based on Phase Classification Based on Phase

Characteristics Characteristics

• 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

38

Zero Zero- -Phase Transfer Function Phase Transfer Function

• 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 not possible to design a causal digital filter with a zero phase

39

Zero- Zero -Phase Transfer Function Phase Transfer Function

• 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] H(z) v[n] u[n] H(z) w[n]

] [ ] [ ], [ ]

[n v n yn w n

u = − = −

40

• It is easy to verify the above scheme in the frequency domain

• Let , , , , and denote the DTFTs ofx[n],v[n], u[n],w[n], andy[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

) (e^j^ω

X V(e^j^ω) U(e^j^ω) W(e^j^ω) )

(e^j^ω Y

41

Zero- Zero -Phase Transfer Function Phase Transfer Function

• Combining the above equations we get

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

] [ ] [ ], [ ]

[n v n yn w n

u = − = −

), ( ) ( )

(e^j^ω =H e^j^ω X e^j^ω

V W(e^j^ω)=H(e^j^ω)U(e^j^ω) ,

) (

* )

(e^j^ω =V e^j^ω

U Y(e^j^ω)=W*(e^j^ω)

) (

* ) (

* )

(e^j^ω =W e^j^ω =H e^j^ωU e^j^ω Y

) ( ) ( ) (

* ) ( ) (

* ^ω ^ω = ^ω ^ω ^ω

=H e^j V e^j H e^j He^j X e^j ) ( )

( ^ω ² ^ω

=H e^j X e^j 42

• The functionfiltfiltimplements 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-phasecharacteristic in the frequency band of interest

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43

Zero- Zero -Phase Transfer Function Phase Transfer Function

• The most general type of a filter with a linear phase has a frequency response given by

which has a linear phase fromω= 0 toω= 2π

• Note also

D j

j e

e

H( ^ω)= ⁻ ^ω

1 ) (e^j^ω = H

=D ω τ( )

44

Linear

Linear- -Phase Transfer Phase Transfer Function

Function

• The outputy[n] of this filter to an input is then given by

• If and represent the continuous- time signals whose sampled versions, sampled att= nT, arex[n] andy[n] given above, then the delay between and is precisely the group delay of amountD

n

Aej

n x[ ]= ^ω

)

] (

[n Ae ^j ^De^j ⁿ Ae^j ⁿ ^D y = ⁻ ^ω ^ω = ^ω ⁻

) (t x_a

) (t x_a )

(t y_a

) (t y_a

45

Linear

Function

• If Dis an integer, then y[n]is identical to x[n], but delayed by Dsamples

• If Dis 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 Dunits

of time ⁴⁶

Linear

Function

• 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

47

Linear

Function

• Figure below shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband

48

Linear

Function

• 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:

⎩⎨⎧

ω)= ( ^j

LP e

H ⁻ ^ω ω<≤ωω<≤ωπ

c n c j _o

e , 0

0 ,

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49

Linear

Function

• 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

∞

<

∞

− − π

−

= ω n

n n

n n n

h

o o

LP c ,

) (

) ( ] sin [

50

Linear

Function

• 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 chosenn_o

51

Linear

Function

• If we choose = N/2 withNa positive integer, the truncated and shifted approximation

will be a lengthN+1 causal linear-phase FIR filter

no

N N n

n N n n

h_LP ^c ≤ ≤

− π

−

= ω , 0

) 2 / (

) 2 / ( ] sin

^ [

52

Linear

Function

• Figure below shows the filter coefficients obtained using the functionsincfor two different values ofN

0 2 4 6 8 10 12

-0.2 0 0.2 0.4 0.6

Time index n

Amplitude

N = 12

0 2 4 6 8 10 12

-0.2 0 0.2 0.4 0.6

Time index n

Amplitude

N = 13

53

Zero- Zero -Phase Response Phase Response

• 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 responseor amplitude response, is a real function of ω

) ( ]

[ )

( ^/²

0

ω

=

= ⁻ ^ω

=

ω

−

ω

∑

N j N LP

n

n LP j

LP ej h n e e H

H^{^} ^{^} ^∼

(ω) H^∼LP

54

Minimum

Minimum- -Phase and Maximum Phase and Maximum- - Phase Transfer Functions 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 1

1 1

2

1 z = ₊⁺ H z = ₊⁺ a< b<

H z a

bz a

z b

z , ( ) , ,

) (

a z=−

b z=− z=−b¹

) (z H₁

) (z H₂

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55

Minimum

• Figure below shows the pole-zero plots of the two transfer functions

)

1(z

H H₂(z)

56

Minimum

• However, both transfer functions have an identical magnitude function as

• The corresponding phase functions are ) ( ) ( ) ( )

( ₁ ¹ ₂ ₂ ¹

1 zH z− =H z H z−

H

ω + ω ω −

+ ω

−

ω = − ¹ ^sin_cos

cos 1 sin

1( )] tan tan

arg[ _a

b

ej

H

ω + ω ω −

+ ω

−

ω = _cos − ¹ ^sin_cos

1 1 sin

2( )] tan tan

arg[ b a

j b

e H

57

Minimum

• Figure below shows the unwrapped phase responses of the two transfer functions for a = 0.8 andb = −0.5

0 0.2 0.4 0.6 0.8 1

-4 -3 -2 -1 0 1 2

ω/π

Phase, degrees

H1(z)

H2(z)

58

Minimum

• From this figure it follows that has an excess phase lag with respect to

• The excess phase lag property of with respect to can also be explained by observing that we can write

)

2(z H

)

1(z H

)

2(z H )

1(z H

⎟⎠

⎜ ⎞

⎝

⎛ +

⎟ +

⎠

⎜ ⎞

⎝

⎛ +

= + +

= +

b z bz a z

b z a z z bz

H 1 1

2( )

) (z ) A (z H₁

59

Minimum

where is a stable allpass function

• The phase functions of and are thus related through

• As the unwrapped phase function of a stable first-order allpass function is a negative function of ω, it follows from the above that

has indeed an excess phase lag with respect to

) /(

) ( )

(z bz z b

A = +1 +

)

2(z H )

1(z H

)]

( arg[

)]

( arg[

)]

(

arg[H₂ e^j^ω = H₁e^j^ω + Ae^j^ω

)

2(z H

)

1(z

H ₆₀

Minimum

• Generalizing the above result, let be a causal stable transfer function with all zeros inside the unit circle and let H(z)be another causal stable transfer function satisfying

• These two transfer functions are then related through where A(z)is a causal stable allpass function

) (z H_m

) ( ) ( )

(z H z A z

H = _m

) ( )

(e^j^ω =H_m e^j^ω H

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61

Minimum

• The unwrapped phase functions of and H(z)are thus related through

• H(z)has an excess phase lag with respect to

• A causal stable transfer function with all zeros inside the unit circle is called a minimum-phase transfer function

) (z H_m

)]

( arg[

)]

( arg[

)]

(

arg[H e^j^ω = H_m e^j^ω + Ae^j^ω

62

Minimum

• A causal stable transfer function with all zeros outside the unit circle is called a maximum-phase transfer function

• A causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function

63

Minimum

• Example –Consider the mixed-phase transfer function

• We can rewrite H(z)as

) . )(

. (

) . )(

. ) (

( 1 1

1 1

5 0 1 2 0 1

4 0 3 0 1 2

−

+

−

= +

z z

z z z

H

⎟⎟⎠

⎞

⎜⎜⎝

⎛

−

⎥ −

⎦

⎢ ⎤

⎣

⎡

+

−

= + ₋⁻₁¹ ₋⁻₁¹ ⁻₋¹₁ 4 0 1

4 0 5

0 1 2 0 1

4 0 1 3 0 1 2

z z z

z

z z z

H

. . ) . )(

. (

) . )(

. ) (

(

Minimum-phase function Allpass function