Spectral Transformations of IIR Digital Filters

(1)

1

Spectral Transformations of Spectral Transformations of

IIR Digital Filters IIR Digital Filters

• Objective-Transform a given lowpass digital transfer function to another digital transfer function that could be a lowpass, highpass, bandpass or bandstop filter

• has been used to denote the unit delay in the prototype lowpass filter and to denote the unit delay in the transformed filter to avoid confusion

) (z G_L

ˆ) (z G_D

ˆ⁻1

z

−1

z

) (z G_L

ˆ) (z G_D

2

Spectral Transformations of Spectral Transformations of

IIR Digital Filters IIR Digital Filters

• Unit circles in z- and -planes defined by ,

• Transformation from z-domain to -domain given by

• Then

zˆ

=e^jω

z zˆ=e^j^ω^ˆ

ˆ) (z F z=

)}

ˆ ( { ) ˆ

(z G F z G_D = _L

3

Spectral Transformations of Spectral Transformations of

IIR Digital Filters IIR Digital Filters

• From , thus , hence

• Recall that a stable allpass functionA(z) satisfies the condition

ˆ) (z F

z= z= F(zˆ)

⎪⎩

⎪⎨

⎧

<

=

>

1 if , 1

1 if , 1 ˆ) (

z z z z

F

4

Spectral Transformations of Spectral Transformations of

IIR Digital Filters IIR Digital Filters

• Therefore must be a stable allpass function whose general form is

⎪⎩

⎪⎨

⎧

<

>

=

>

<

1 if , 1

1 if , 1 ) (

z z z z

A

ˆ) ( / 1 F z

1 ˆ ,

1 ˆ ˆ)

( 1

1

* ⎟⎟⎠ α <

⎜⎜ ⎞

⎝

⎛ α

− α

± −

= ∏

= l

l l

L l

z z z

F

5

Lowpass

Lowpass- -to to- -Lowpass Lowpass Spectral Transformation Spectral Transformation

• To transform a lowpass filter with a cutoff frequency to another lowpass filter

with a cutoff frequency , the transformation is

where αis a function of the two specified cutoff frequencies

) (z G_L

ˆ) (z G_D

ωc

ωˆc

α

−α

= −

− =

z z z

z F

ˆ 1 ˆ ˆ) (

1 1

6

Lowpass

Lowpass- -to to- -Lowpass Lowpass Spectral Transformation Spectral Transformation

• On the unit circle we have

• From the above we get

• Taking the ratios of the above two expressions

ω

− ω ω −

−

α

− −α

= ^ˆ _ˆ

1 ^j

j j

e e e

) 2 / ˆ 1 tan(

) 1 2 /

tan( ⎟ ω

⎠

⎜ ⎞

⎝

⎛ −+αα

= ω

ω

− ω

− ω ω −

−

α

⋅ − α

± α =

− α

= ^ˆ− _ˆ ^ˆ _ˆ

1 ) 1 1 ( 1 1

1 _j

j j

e e e

e m e m m

(2)

7

Lowpass

Lowpass- -to to- -Lowpass Lowpass Spectral Transformation Spectral Transformation

• Solving we get

• Example-Consider the lowpass digital filter

which has a passband from dcto with a 0.5dB ripple

• Redesign the above filter to move the passband edge to

( )

(

( ˆ )/2

)

sin

2 / ˆ ) ( sin

c c

c c+ω ω

ω

−

= ω α

) 3917 . 0 6763 . 0 1 )(

2593 . 0 1 (

) 1 ( 0662 . ) 0

( ₁ ₁ ₂

3 1

−

+

−

= +

z z

z z z

G_L

π 25 . 0

π 35 .

0 8

Lowpass

Lowpass- -to to- -Lowpass Lowpass Spectral Transformation Spectral Transformation

• Here

• Hence, the desired lowpass transfer function is 1934

. ) 0 3 . 0 sin(

) 05 . 0

sin( =−

π

− π

= α

1 1 1

ˆ 1934 . 0 1

1934 . 0

) ˆ

( ) ˆ (

−

− −

+

= +

=

z z z

L

D z G z

G

0 0.2 0.4 0.6 0.8 1

-40 -30 -20 -10 0

ω/π

Gain, dB G

L(z) G D(z)

9

Lowpass

Lowpass- -to to- -Lowpass Lowpass Spectral Transformation Spectral Transformation

• The lowpass-to-lowpass transformation

can also be used as highpass-to-highpass, bandpass-to-bandpassand bandstop-to- bandstoptransformations

α

−α

= −

− =

z z z

z F

ˆ ˆ 1 ˆ) (

1 1

10

Lowpass

Lowpass- -to to- -Highpass Highpass Spectral Transformation Spectral Transformation

• Desired transformation

• The transformation parameter is given by

where is the cutoff frequency of the lowpass filter and is the cutoff frequency of the desired highpass filter

1 1 1

1 ˆ ˆ

−

− −

α +

α

− +

= z z z

( )

(

( ˆ ⁾^/²

)

cos

2 / ) ˆ ( cos

c c

ω

− ω

ω +

− ω

= α

α

ωc

ωˆc

11

Lowpass

Lowpass -to - to- -Highpass Highpass Spectral Transformation Spectral Transformation

• Example-Transform the lowpass filter

• with a passband edge at to a highpass filter with a passband edge at

• Here

• The desired transformation is

) 3917 . 0 6763 . 0 1 )(

2593 . 0 1 (

) 1 ( 0662 . ) 0

( ₁ ₁ ₂

3 1

−

+

−

= +

z z

z z z

G_L

π 25 .

0 0.55π 3468 . 0 ) 15 . 0 cos(

/ ) 4 . 0

cos( π π =−

−

= α

1 1

1

3468ˆ . 0 1

3468 . ˆ 0

−

− −

−

− −

= z

z z

12

Lowpass

Lowpass- -to to- -Highpass Highpass Spectral Transformation Spectral Transformation

• The desired highpass filter is

1 1 1

ˆ 3468 . 0 1

3468 . ˆ 0

) ( ) ˆ (

−

− −

−

− −

= =

z z z

D z G z

G

0 0.2π 0.4π 0.6π 0.8π π

−80

−60

−40

−20 0

Normalized frequency

Gain, dB

(3)

13

Lowpass

Lowpass- -to to- -Highpass Highpass Spectral Transformation Spectral Transformation

• The lowpass-to-highpasstransformation can also be used to transform a highpassfilter with a cutoff at to a lowpassfilter with a cutoff at

and transform a bandpassfilter with a center frequency at to a bandstopfilter with a center frequency at

ωc

ωˆc

ωˆo

ωo

14

Lowpass

Lowpass- -to to- -Bandpass Bandpass Spectral Transformation Spectral Transformation

ˆ 1 1 ˆ 2

1 1

1 ˆ 1 1 ˆ 2

1 2

1

+ + β

− αβ + β

−

β β+

− +β + β

− αβ

−

= − −

−

z z

z

15

Lowpass

Lowpass- -to to- -Bandpass Bandpass Spectral Transformation Spectral Transformation

• The parameters and are given by

where is the cutoff frequency of the lowpass filter, and and are the desired upper and lower cutoff frequencies of the bandpass filter

α β

(

(ˆ ˆ ⁾^/²

)

^tan( ^/²⁾

cot ω_c₂−ω_c₁ ω_c

= β

( )

(

(ˆ ˆ ⁾^/²

)

cos

2 / ) ˆ ˆ ( cos

1 2

c c

ω

− ω

ω +

= ω α

ωc

ˆ_c1

ω ωˆ_c₂

16

Lowpass

Lowpass- -to to- -Bandpass Bandpass Spectral Transformation Spectral Transformation

• Special Case-The transformation can be simplified if

• Then the transformation reduces to

where with denoting the desired center frequency of the bandpass filter

1

2 ˆ

ˆ_c _c

c=ω −ω ω

ωo

=

α cosˆ ωˆ_o

1 1 1 1

1 ˆ

ˆ ˆ ₋

− −

−

α

− α

− −

= z

z z z

17

Lowpass

Lowpass- -to to- -Bandstop Bandstop Spectral Transformation Spectral Transformation

ˆ 1 1 ˆ 2 1 1

1 ˆ 1 1 ˆ 2

1 2

1

+ + + −

− +

+ −

− +

= − −

−

z z

z

β αβ β

β β

αβ

18

Lowpass

Lowpass- -to to- -Bandstop Bandstop Spectral Transformation Spectral Transformation

• The parameters and are given by

where is the cutoff frequency of the lowpass filter, and and are the desired upper and lower cutoff frequencies of the bandstop filter

ωc

α β

ˆ_c1

ω ωˆ_c₂

( )

(

(ˆ ˆ )/2

)

cos

2 / ) ˆ ˆ ( cos

1 2

c c

c c −ω ω

ω +

= ω α

(

(ˆ ˆ )/2

)

tan( /2) tan ω_c₂−ω_c₁ ω_c

= β

(4)

19

Generation of

Generation of Allpass Allpass Function Using MATLAB Function Using MATLAB

• The allpass function needed for the spectral transformation from a specified lowpass transfer function to a desired highpassor bandpassor bandstoptransfer function can be generated using MATLAB

20

Generation of

Generation of Allpass Allpass Function Using MATLAB Function Using MATLAB

• Lowpass-to-Highpass Transformation

• Basic form:

[AllpassNum,AllpassDen] = allpasslp2hp(wold,wnew) where woldis the specified angular bandedge frequency of the original lowpass filter, and wnewis the desired angular bandedge frequency of the highpass filter

21

Generation of

Generation of Allpass Allpass Function Using MATLAB Function Using MATLAB

• Lowpass-to-Bandpass Transformation

• Basic form:

[AllpassNum,AllpassDen] = allpasslp2bp(wold,wnew) where woldis the specified angular bandedge frequency of the original lowpass filter, and wnewis the desired angular bandedge frequency of thebandpassfilter

22

Generation of

Generation of Allpass Allpass Function Using MATLAB Function Using MATLAB

• Lowpass-to-Bandstop Transformation

• Basic form:

[AllpassNum,AllpassDen] = allpasslp2bs(wold,wnew) where woldis the specified angular bandedge frequency of the original lowpass filter, and wnewis the desired angular bandedge frequency of the bandstopfilter

23

Generation of

Generation of Allpass Allpass Function Using MATLAB Function Using MATLAB

• Lowpass-to-Highpass Example – wold= 0.25π, wnew= 0.55π

• The MATLAB statement [APnum, APden]

= allpasslp2hp(0.25, 0.55) yields the mapping

1 3468 0

3468 0

1 1 1

+

− +

→ − ⁻ ₋

−

z z z

. .

24

Spectral Transformation Spectral Transformation

Using MATLAB Using MATLAB

• The pertinent M-files areiirlp2lp, iirlp2hp,iirlp2bp, andiirlp2bs

• Lowpass-to-Highpass Example –

Passband edge wold=0.25π

Desired passband edge of highpass filter wnew=0.55π

3 2

1

3 1

1015 0 5669 0 9353 0 1

1 066 0

−

− +

−

= +

z z

z z z

G_LP

. .

.

) ( ) .

(

(5)

25

Spectral Transformation Spectral Transformation

Using MATLAB Using MATLAB

• The MATLAB code fragments used are

b = 0.066*[1 3 3 1];

a = [1.00 -0.9353 0.5669 -0.1015];

[num,den,APnum,APden]

= iirlp2hp(b,a,0.25,0.55);

• The desired highpassfilter obtained is

3 2

1

3 1

0329 0 3661 0 3521 0 1

1 218 0

−

− +

−

= −

z z

z z z

G_HP

. .

.

) ( ) .

(

26

IIR Digital Filter Design Using IIR Digital Filter Design Using

MATLAB MATLAB

• Order Estimation-

• For IIR filter design using bilinear transformation, MATLAB statements to determine the order and bandedge are:

[N, Wn] = buttord(Wp, Ws, Rp, Rs);

[N, Wn] = cheb1ord(Wp, Ws, Rp, Rs);

[N, Wn] = cheb2ord(Wp, Ws, Rp, Rs);

[N, Wn] = ellipord(Wp, Ws, Rp, Rs);

27

IIR Digital Filter Design Using IIR Digital Filter Design Using

MATLAB MATLAB

• Example-Determine the minimum order of a Type 2 Chebyshev digital highpass filter with the following specifications:

kHz, kHz, kHz, dB, dB

• Here, ,

• Using the statement

[N, Wn] = cheb2ord(0.5, 0.3, 1, 40);

we get N = 5 and Wn = 0.3224

=1

Fp F_s=0.6 F_T =4

=1

α_p α_s=40 5 . 0 4 / 1 2

Wp= × = Ws=2×0.6/4=0.3

28

IIR Digital Filter Design Using IIR Digital Filter Design Using

MATLAB MATLAB

• Filter Design-

• For IIR filter design using bilinear

transformation, MATLAB statements to use are:

[b, a] = butter(N, Wn) [b, a] = cheby1(N, Rp, Wn) [b, a] = cheby2(N, Rs, Wn) [b, a] = ellip(N, Rp, Rs, Wn)

29

IIR Digital Filter Design Using IIR Digital Filter Design Using

MATLAB MATLAB

• The form of transfer function obtained is

• The frequency response can be computed using the M-filefreqz(b, a, w)wherewis a set of specified angular frequencies

• It generates a set of complex frequency response samples from which magnitude and/or phase response samples cn be computed

N N

z N a z

a

z N b z

b b z A

z z B

G ₋ ₋

−

+ + + +

+ + +

= +

= 1 (2) ( 1)

) 1 ( )

2 ( ) 1 ( ) (

) ) (

( ₁

1

L L

30

IIR Digital Filter Design Using IIR Digital Filter Design Using

MATLAB MATLAB

• Example-Design an elliptic IIR lowpass filter with the specifications: kHz,

kHz, kHz, dB, dB

• Here, ,

• Code fragments used are:

[N,Wn] = ellipord(0.4, 0.5, 0.5, 40);

[b, a] = ellip(N, 0.5, 40, Wn);

8 .

=0 Fp

=1

Fs F_T =4 α_p=0.5

=40 α_s

π

= π

=

ω_p 2 F_p/F_T 0.4 ω_s =2πF_s/F_T=0.5π

(6)

31

IIR Digital Filter Design Using IIR Digital Filter Design Using

MATLAB MATLAB

• Gain response plot is shown below:

0 0.2 0.4 0.6 0.8 1

-60 -40 -20 0

ω/π

Gain, dB

Elliptic IIR Lowpass Filter

0 0.1 0.2 0.3 0.4

-0.5 -0.4 -0.3 -0.2 -0.1 0

ω/π

Gain, dB

Passband Details

32

Computer

Computer -Aided Design of - Aided Design of IIR Digital Filter IIR Digital Filter

• The IIR filter design algorithms discussed so far are used in applications requiring filters with a frequency-selective magnitude response having either a lowpass or a highpass or a bandpass or a bandstop characteristics

• Designing IIR filters with other types of frequency responses usually involve the use of some type of iterative optimization techniques that are used to minimize the error between the desired response and that of the computer-generated filter

33

Computer

Computer- -Aided Design of Aided Design of IIR Digital Filter IIR Digital Filter

• Basic Idea -

• Let denote the frequency response of the computer generated transfer function H(z)

• Let denote the desired frequency response

• Objective is to design H(z)so that approximates in some sense

) (e^j^ω H

) (e^j^ω D

) (e^j^ω H )

(e^j^ω D

34

Computer

Computer -Aided Design of - Aided Design of IIR Digital Filter IIR Digital Filter

• Usually the difference between and specified as a weighted error

function E(ω)

is minimized for all values of ωover closed subintervals of

where is some user-specified positive weighting function

) (e^j^ω H )

(e^j^ω D

)]

( ) ( )[

( )

(ω =W e^j^ω H e^j^ω −D e^j^ω E

π

≤ ω

≤ 0 ) (e^j^ω W

35

Computer

Computer- -Aided Design of Aided Design of IIR Digital Filter IIR Digital Filter

• A commonly used approximation measure, called Chebyshevor minimax criterion, is to minimize the peak absolute valueof E(ω) given by

where Ris the set of disjoint frequency bands in the range , on which the desired frequency response is defined

π

≤ ω 0≤

) (

max ω

= ε ω∈ E

R

36

Computer

Computer -Aided Design of - Aided Design of IIR Digital Filter IIR Digital Filter

• In filtering applications, Ris composed of the desired passbands and stopbands of the filter to be designed

• For example, for a lowpass filterdesign, R is the disjoint union of the frequency ranges

and , where and are, respectively, the passbandand stopband edges

] ,

[0 ω_p [0,ω_s] ω_p ω_s

(7)

37

Computer

Computer- -Aided Design of Aided Design of IIR Digital Filter IIR Digital Filter

• Another approximation measure, called the least-p criterion, is to minimize the integral of the p-th power of E(ω):

over the specified frequency range Rwith p a positive integer

ω

−

= ε ∫

∈ ω

ω ω

ω H e D e d

e

W ^p

R

j j

j )[ ( ) ( )]

(

38

Computer

Computer -Aided Design of - Aided Design of IIR Digital Filter IIR Digital Filter

• The least-squares criterionobtained withp= 2 is often used for simplicity

• It can be shown that as , theleast p-th solutionapproaches theminimax solution

• In practice, the integral error measure is approximated by a finite sum given by

where , , is a suitably chosen dense grid of digital angular frequencies

K p i

j j

j _i _i _i

e D e H e

∑W

=

ω ω

ω −

= ε

1

)]

( ) ( )[

(

∞

→ p

ωi 1≤i≤K

39

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

• For a distortion-free transmission of an input signal in a prescribed frequency range through a digital filter, the transfer function of the filter should exhibit a unity

magnitude responseand a linear-phase response, i.e., a constant group delay, in the frequency band of interest

40

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

• The IIR digital filter design methods discussed so far lead to transfer functions with nonlinear phase responses

• Thus, to arrive at a frequency selective IIR digital filter with a constant group delay, a practical approach is to cascade the IIR digital filter meeting the magnitude response specifications with an allpass filter so that the overall group delay is a constant group delay in the band of interest

41

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

• The allpass delay equalizer is usually designed using a computer-aided optimization method

• We outline one such method next

• LetH(z) be the transfer function of the IIR digital filter with a group delay given by

(ω) τ_H

42

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

• Objective is to design a stable allpass section with a transfer function

with a group delay and cascade it with the IIR digital filterH(z) such that the overall group delay is a constant

H(z) A(z)

∏= − −

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+ +

+

= ^M +

z d z d

z z d z d

A

1 2

1 2 1

2 1 1

2

l 1 l l

l l

, ,

,

) ,

(

(ω) τ_A

) ( ) ( )

(ω =τ ω +τ ω

τ _H _A

(8)

43

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

• Moreover, to guarantee stability we need to ensure that

• The allpass delay equalizer design problem can be formulated as a minimax optimization problem in which we minimize the peak absolute value of the error

τo

− ω τ

= ω) ( ) E(

l l

l , ,

, , ₁ ₂

2 1 d 1 d

d ≤ < +

44

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

• The adjustable parameters are the desired delay and the coefficients , of the allpass filter

• The M-file iirgrpdelaycan be used to design the allpass delay equalizer

l 1,

d d₂_,_l τo

45

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

• Example–We design an 8-th order allpass equalizer to equalize the group delay of a 4-th order elliptic lowpass filter with a passband edgeat 0.3π, passband rippleof 1 dBand a minimum stopband attenuationof 30 dBusingProgram 9_4.m

• The group delays of the lowpass filter and the overall cascade are shown on the next slide

46

Group Delay Equalization of Group Delay Equalization of

IIR Digital Filter IIR Digital Filter

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

0 0.2 0.4 0.6 0.8 1

0 5 10 15

ω/π

Group delay, samples

Original Filter

It can be shown that the designed allpass filter is stable