Untitled

by Ljiljana Milic

Supplemental material for Chapter VIII:

8. Complementary Filter Pairs

Table of Contents

8. Complementary Filter Pairs 8.1 FIR Complementary Filter Pairs 8.1.1 Delay-Complementary and Magnitude-Complementary FIR Filter Pair Example 8.1 8.1.2 Power-Complementary FIR Filter Pairs Example 8.2 8.2 IIR complementary filter pairs 8.2.1 Class I: Power-Complementary and All-Pass complementary Filter Pairs Implemented as a Parallel Connection of Two All-Pass Subfilters Example 8.3 8.2.2 Class II: Power-Complementary Filter Pairs Implemented as a Tapped Cascaded Interconnection of Two Identical All-Pass Subfilters Example 8.4

8.1 FIR Complementary Filter Pairs

8.1.1 Delay-Complementary and Magnitude-Complementary FIR Filter Pair

Remainder

Two filters

and

make the delay-complementary filter pair when satisfy

where N is the filter length, an odd number.

The magnitude-complementary property is satisfied when

and

are designed to meet the following criteria:

and

for

and

for

Here,

and

are passband/stopband edge frequencies,

is the crossover frequency and

denotes minimal attenuation in the stopband.

Example 8.1

Design magnitude-complementary and delay-complementary halfband FIR filter pair to meet the criteria defined above.

Filter order

=18, the passband/stopband ripple

, the crossover frequency

Solution:

close all, clear all

Setting the input parameters

Nord = 18; delta = 0.04; fc = 0.5;

Step 1: Adjusting the ripple value

delp = delta/(2*(1 - delta));

Step 2: Design of the intermediate filter Elp(z)

elp = firceqrip(Nord,fc,[delp,delp]);

Step 3: Modifying the intermediate filter Elp(z)

hlp = (1 - delta)*elp;

Step 3: Modifying the intermediate filter Elp(z)

hlp(Nord/2 + 1) = hlp(Nord/2+1)+delta/2;%

Zero settings

hhp=zeros(size(1:length(hlp)));

Step 4: Complementary highpass filter

hhp(10) = 1; hhp = hhp - hlp;

Display impulse responses

figure

stem((0:18),hlp), xlabel('n'),ylabel('h_L_P[n]')

title('Lowpass filter impulse response')

axis([0,18,-0.4,0.6])

figure

stem((0:18),hhp), xlabel('n'),ylabel('h_L_P[n]')

title('Highpass filter impulse response')

axis([0,18,-0.4,0.6])

Display the pole-zero locations of lowpass filter

figure

zplane(hlp,1)

title('Zero-pole locations of lowpass filter')

Computing the magnitude responses

f(1)=0;
omega(1)=pi*f(1);
K=(length(hlp)+1)/2;
n=1:K-1;
h_red=hlp(1:K-1);
h_red=fliplr(h_red);
hh_red=hhp(1:K-1);
hh_red=fliplr(hh_red);
for p=1:101
Alp(p)=1/2+2*sum(h_red.*cos(omega(p)*n));
Ahp(p)=1/2+2*sum(hh_red.*cos(omega(p)*n));
f(p+1)=f(p)+0.01;
omega(p+1)=pi*f(p+1);
end

Displaying magnitude responses of the filter pair

figure

plot(f(1:length(Alp)),Alp,f(1:length(Ahp)),Ahp,'b')

xlabel('Normalized frequency \omega/\pi'),ylabel('Magnitude'),grid

title('Complementary filter pair')

axis([0,1,-0.1,1.1])

disp('END OF EXAMPLE 8.1')

END OF EXAMPLE 8.1

8.1.2 Power-Complementary FIR Filter Pairs

The power-complementary property is satisfied when the sum of squared magnitude responses of the lowpass and highpass filters of the pair

is equal to unity for all frequencies,

The power-complementary property is achieved when the filter magnitude responses meet the following criteria:

and

for

and

for

Example 8.2

Design the power-complementary halfband FIR filter pair to meet the specifications given bellow:

Filter order

, the passband/stopband ripple

, the crossover frequency

Solution:

close all, clear all

Setting the input papameters

Nord = 15; Nord = 2*Nord; delta = 0.1;

Adjusting the ripple value, Substep 1.1

dfir =(delta^2)/(2*(1 - (delta^2)));

Designing the intermediate filter Elp(z), Substep 1.2

elp = firceqrip(Nord,0.5,[dfir,dfir]);

Modifying the intermediate filter Elp(z), Substep 1.3

eh = (1-(delta^2))* elp;

Modifying the intermediate filter Elp(z), Substep 1.3

eh(Nord/2+1) = eh(Nord/2 + 1) + (delta^2)/2;%

Generating the minimum-phase lowpass filter Hlp(z), Step 2

[hlp,ssp,iter] = minphase(eh(Nord/2 + 1:Nord+1));

Indexing the filter coefficients

n = 0:length(hlp) - 1;

Complementary highpass filter, Step 3

hhp = fliplr(hlp).*((-1).^n);

Displaying the results

figure

stem((0:15),hlp), xlabel('n'),ylabel('h_L_P[n]')

title('Lowpass filter impulse response')

axis([0,15,-0.5,0.6])

figure

stem((0:15),hhp),xlabel('n'),ylabel('h_H_P[n]')

title('Highpass filter impulse response')

axis([0,15,-0.5,0.6])

figure

zplane(hlp,1)

title('Lowpass filter, pole-zaro locations')

figure

zplane(hhp,1)

title('Highpass filter, pole-zaro locations')

figure

[Hlp,f]=freqz(hlp,1,512,2);

[Hhp,f]=freqz(hhp,1,512,2);

plot(f,abs(Hlp),f,abs(Hhp),'b')

xlabel('Normalized frequency \omega/\pi'),ylabel('Magnitude'),grid

title('Magnitude response of the complementary filter pair')

axis([0,1,-0.1,1.1])

g(1)=axes('Position',[0.20 0.70 0.25 0.1]);

plot(f(1:222),(abs(Hlp(1:222)))), axis([0,0.45,0.99,1.002]),grid

g(2)=axes('Position',[0.62 0.70 0.25 0.1]);

plot(f(285:512),(abs(Hhp(285:512)))), axis([0.55,1,0.99,1.002]),grid

disp('END OF EXAMPLE 8.2')

END OF EXAMPLE 8.2

8.2 IIR complementary filter pairs

8.2.1 Class I: Power-Complementary and All-Pass complementary Filter Pairs Implemented as a Parallel Connection of Two All-Pass Subfilters

Remainder

The lowpass (highpass) filter transfer function

is an odd order minimum-phase IIR transfer function whose magnitude response meets the following criteria

and

for

and

for

The implementation structure of the filter pair

is based on the parallel connection of two all-pass subfilters

and

Example 8.3

close all, clear all

Setting the input parameters for lowpass filter

N = 5; fp = 0.4;

Lowpass halfband filter design

[b,a,z,p,k] = halfbandiir(N,fp);

Sorting the filter poles

pabs = sort(abs(p));

Coefficients of allpass branches

beta0 = (abs(pabs(2)))^2; beta1=(abs(pabs(4)))^2;

Frequency response of

[A0,f] = freqz([beta0,0,1],[1,0,beta0],512,2);

Frequency response of

[A1,f] = freqz([0,beta1,0,1],[1,0,beta1],512,2);

Lowpass filter frequency response

GLp = (A0 + A1)/2;

Highpass filter frequency response

GHp = (A0 - A1)/2;

Displaying the magnitude response of the complementary filter pair

figure

plot(f,(abs(GLp)),f,abs(GHp),'b')

xlabel ('Normalized frequency \omega/\pi'),ylabel('Magnitude'), grid,

axis([0,1,0,1.05])

title('Complementary IIR filter pair [G_0(z),G_1(z)]')

g(1)=axes('Position',[0.24 0.70 0.25 0.1]);

plot(f(1:205),(abs(GLp(1:205)))), axis([0,0.4,0.9998,1.00002])

g(2)=axes('Position',[0.62 0.70 0.25 0.1]);

plot(f(307:512),(abs(GHp(307:512)))), axis([0.6,1,0.9998,1.00002])

disp('END OF EXAMPLE 8.3')

END OF EXAMPLE 8.3

8.2.2 Class II: Power-Complementary Filter Pairs Implemented as a Tapped Cascaded Interconnection of Two Identical All-Pass Subfilters

Remainder

The filter pair

exhibits power-complementary property as defined for example 8.2.

The solution for Class II filter pair

is achieved with the aid of the following transfer functions:

a) FIR prototype filter pair

designed to provide passbans/stopband ripple and power-complementary property of the overall pair

b) IIR prototype filter pair

designed to provide edge frequencies of the pair

Example 8.4

Design the Class II power-complementary IIR filter pair

to meet the following specifications:

Minimal attenuation in the stopband

. The edge frequencies of the filter pair are:

and

Implementation structure of the Class II power-complementary IIR filter pair:

Solution:

Functions firceqrip, minphase were used.

close all, clear all

Design of the FIR prototype filter pair

Nord = 5; Nord = 2*Nord;

delta = 0.001;

dfir =(delta^2)/(2*(1-(delta^2)));

hl = firceqrip(Nord,0.5,[dfir,dfir]);

eh = (1-(delta^2))*hl;

eh(Nord/2+1) = eh(Nord/2+1)+(delta^2)/2;

[alp,ssp,iter] = minphase(eh(Nord/2+1:Nord+1));

n = 0:length(alp)-1;

ahp = fliplr(alp).*((-1).^n);

len = Nord/2;

figure

stem((0:len),alp), xlabel('n'),ylabel('a_L[n]')

title('Impulse response of lowpass FIR filter')

axis([0,len,-0.5,0.6])

figure

stem((0:len),ahp),xlabel('n'),ylabel('a_H[n]')

title('Impulse response of highpass FIR filter')

axis([0,len,-0.5,0.6])

figure

[Hlp,f]=freqz(alp,1,512,2);

[Hhp,f]=freqz(ahp,1,512,2);

plot(f,20*log10(abs(Hlp)),f,20*log10(abs(Hhp)),'b')

xlabel('\omega/\pi'),xlabel('\theta/\pi'),ylabel('Gain [dB]'),grid

title('Gain responses of prototype FIR filter pair')

axis([0,1,-80,2])

Design of the IIR prototype filter pair

beta = 0.545577589; % For explanation see pages 266-267 in the book.

Aa0 = [1,0,beta];

Ab0 = fliplr(Aa0);

Ab1 = [0,1];

Aa1 = 1;

[H0,f] = freqz(Ab0,Aa0,512,2);

[H1,f] = freqz(Ab1,Aa1,512,2);

Hiil=(H0+H1);

Hiih=(H0-H1);

figure

subplot(5,1,1:2)

plot(f,20*log10(abs(Hiil/2)),f,20*log10(abs(Hiih/2)),'b'),grid

xlabel('\omega/\pi'),ylabel('Gain [dB]')

title('Gain responses of prototype IIR filter pair')

axis([0,1,-30,2])

a = alp;

Lattice coefficients of the 5th order minimum phase halfband FIR filter

a5=a/a(1);
b5=a5;
b5(1:2:6)=-a5(1:2:6);
b5=fliplr(b5);
k5=a5(6);
a3=(a5+k5*b5)/(1+k5^2);
a3=a3(1:4);
b3=(b5-k5*a5)/(1+k5^2);
b3=b3(3:6);
k3=a3(4);
a1=(a3+k3*b3)/(1+k3^2);
a1=a1(1:2);
b1=(b3-k3*a3)/(1+k3^2);
b1=b1(3:4);
k1=a1(2);
disp('=================================================')
=================================================
disp([sprintf(' lattice coefficient k1 = %15.9f', k1)])
 lattice coefficient k1 =     2.382963175
disp([sprintf(' lattice coefficient k3 = %15.9f', k3)])
 lattice coefficient k3 =    -0.541364466
disp([sprintf(' lattice coefficient k5 = %15.9f', k5)])
 lattice coefficient k5 =     0.108528870
disp('=================================================')
=================================================

Coefficients of allpass branches

Q=[beta,0,1];
P=fliplr(Q);
QQ=conv(Q,Q);
PP=conv(P,P);

Computing and plotting the gain response of the overall filter pair

u=zeros(size(1:1024));

u(1)=1;

v1=[0,1];

v2=[0,0,1];

x1=filter(Q,P,u)-k1*filter(v1,1,u);

y1=k1*filter(Q,P,u)+filter(v1,1,u);

x3=filter(QQ,PP,x1)-k3*filter(v2,1,y1);

y3=k3*filter(QQ,PP,x1)+filter(v2,1,y1);

x5=filter(QQ,PP,x3)-k5*filter(v2,1,y3);

y5=k5*filter(QQ,PP,x3)+filter(v2,1,y3);

yLP = x5;

yHP = y5;

[H0,w]=freqz(yLP,1,4096);

dop=20*log10(max(abs(H0)));

figure

plot(w/(pi),20*log10(abs(H0))-dop)

hold on

[H1,w]=freqz(yHP,1,4096);

plot(w/(pi),20*log10(abs(H1))-dop,'b'), grid

xlabel('\omega/\pi'),ylabel('Gain [dB]')

title('Complementary filter pair [H_0(z),H_1(z)], gain response')

axis([0,1,-80,2])

disp('Specifications are met with two very low order filters.')

Specifications are met with two very low order filters.

disp('END OF EXAMPLE 8.4')

END OF EXAMPLE 8.4

disp(' END OF CHAPTER VIII')

                     END OF CHAPTER VIII

function [y,ssp,iter]=minphase(h);
% Reproduced with permission of IEEE from: IEEE Trans. Circuits and Systems-I: Fundamental Theory
% and Applications-I: 50(3), 365-375.
y=[1 zeros(1,length(h)-1)];
ssp=realmax;
ss=ssp/2;
iter=0;
d=0;
while (ss<ssp)
    y=y+d';
    ssp=ss;
    iter=iter+1;
    Ar=toeplitz([y(1),zeros(1,length(h)-1)],y);
    Al=fliplr(toeplitz([y(length(h)),zeros(1,length(h)-1)],fliplr(y)));
    A=Al+Ar;
    b=h'-Al*y';
    d=A\b;
    ss=norm(d);
end;
end