by Ljiljana Milic
Supplemental material for Chapter VII:
7. Lth-Band Digital Filters
Table of Contents
Introduction
The common characteristic of Lth-band lowpass filters is that the 6 dB (or 3dB) cutoff angular frequency is located at 
, and the transition band is approximately symmetric around this frequency. In time domain, the impulse response of an Lth-band digital filter has zero valued samples at the multiples of L samples counted away from the central sample to the right and left directions. Actually, an Lth-band filter has the zero crossings at the regular distance of L samples thus satisfying the intersymbol interference property. The Lth-band filters are also called the Niquist filters.
, and the transition band is approximately symmetric around this frequency. In time domain, the impulse response of an Lth-band digital filter has zero valued samples at the multiples of L samples counted away from the central sample to the right and left directions. Actually, an Lth-band filter has the zero crossings at the regular distance of L samples thus satisfying the intersymbol interference property. The Lth-band filters are also called the Niquist filters.7.1 Lth-Band Linear-Phase FIR Filters
Example 7.1
This example presents a factor-of-3 interpolator constructed with the third-band filter (L = 3) of the length N = 21. The m-file firnyquist is used to compute impulse response coefficients. The example is aimed to demonstrate the following property of an Lth-band filter: the sample values of the original signal are preserved in the interpolated signal, and in-between samples are the interpolated values.
close all, clear all
3rd-band filter design
h = firnyquist(20,3,0.3);
[H,F] = freqz(h,1,1000,2);
Displayng the 3rd-band filter impulse response
figure
stem((0:20),h)
xlabel('Time index n')
ylabel('Amplitude')
title('Impulse response')
axis([0,20,-0.1,0.4])
Displaying the 3rd-band filter gain response
figure
plot(F,20*log10(abs(H))), grid
axis([0,1,-60,5])
xlabel('\omega/\pi')
ylabel('Gain, dB')
title('Gain Response')
Generating the test signal x
n=0:16;
x=0.6*cos(2*pi*1.2*n/16)+0.8*sin(2*pi*0.7*n/16);
Displaying the test signal
figure
stem((0:16),x)
xlabel('Time index n')
ylabel('Amplitude')
title('Test signal')
axis([0,16,-1,1])
Upsampling and interpolating the test signal
xu=upsample(x,3); % Upsampling-by-3
y=3*filter(h,1,xu); % Interpolating
y= 3*conv(h,xu);
Displaying interpolated signal y
figure
stem((10:3:length(xu)+7),x,'r')
hold on
stem(0:length(y)-1,y,'-.*k')
xlabel('Time index, n'), ylabel('y[m]')
legend('Exact values','Interpolated values')
axis([0,60,-1,1]);
title('Interpolated signal')
disp('In the interpolated signal y[m], every 3rd amplitude preserves')
disp('the value of the original test signal x[n], see red lines.')
disp(' ')
disp('END OF EXAMPLE 7.1')
7.2 Linear-Phase FIR Halfband Filters
Example 7.2
In this example, we design a minimum order equaripple halfband FIR filter to satisfy the following specifications:
The minimum stopband attenuation 
(maximum stopband gain, 
(maximum stopband gain, The stopband edge frequency 
.
.The MATLAB function firhalfband is used for the FIR filter design.
clear all, close all
Specifying the minimal stopband attenuation
as = 40;
Computation of the peak ripple
delta = 10^(-as/20);
Specifying the stopband edge frequency
fs = 0.60;
Computation of the passband edge frequency
fp = 1-fs;
Halfband filter design
h = firhalfband('minorder',fp,delta);
figure
stem(0:length(h)-1,h)
xlabel('n'), ylabel('h[n]')
title('Impulse response')
axis([0,22,-0.2,0.6])
Halfband filter frequency response
[H,f] = freqz(h,1,512,2);
figure
plot(f,20*log10(abs(H))),grid
xlabel('\omega/\pi'), ylabel('Gain [dB]')
title('Gain response')
axis([0,1,-60,5])
disp('END OF EXAMPLE 7.2')
Example 7.3
In this example , we design a halfband filter using the truncated impulse response window method. We take for the filter length N = 23, and the hamming window,
clear all,close all
Specifying the filter order
Nord = 22;
Specifying the window
win = hamming(23);
Halfband filter design
h = firhalfband(Nord,win);
figure
stem(0:length(h)-1,h)
xlabel('n'), ylabel('h[n]')
title('Impulse response')
axis([0,22,-0.2,0.6])
Halfband filter frequency response
[H,f] = freqz(h,1,512,2);
figure
plot(f,20*log10(abs(H))),grid
xlabel('\omega/\pi'), ylabel('Gain [dB]')
title('Gain response')
axis([0,1,-60,5])
disp('END OF EXAMPLE 7.3')
Example 7.4
This example demonstrates in MATLAB the process of decimation-by-2 based on the efficient structure shown bellow. The Figure shows polyphase memory saving implementation on the example of the linear-phase 10th order FIR halfband filter. The coefficient symmetry property is exploited.
Efficient implementation of the linear-phase FIR halfbf and filter.
(a) Polyphase configuration. (b) Implementation scheme for N=11.
This program performs:
- Design of linear-phase FIR halfband filter of Nord = 22.
- Generates 257 samples of the original signal x
- Implements the decimation-by-2 according to the efficient scheme presented above. The implementation structure is extended since the filter ordeer in this example is 22 (instead of 11).The delay chain has 11 elements, and the number of multiplication constants is 6.
- Computes and plots the spectrum of the original signal and that of the decimated signal.
clear all, close all
Designing 22nd order linear-phase FIR filter
Nord = 22; % Hafband filter order
h = firhalfband(22,0.4); % FIR filter design
Generation of the input signal
F = [0,0.05,0.45,1]; A = [0,1,0.1,0.05];
x1 = fir2(256,F,A); x2 = sin(2*pi*(0:256)*0.4)/150;
x = x1 + x2;
[X,f] = freqz(x,1,512,2); % Spectrum of the original signal
Polyphase decomposition
e0 = h(1:2:(length(h)-1)/2);
Setting the initial states
xk = zeros(size(1:(length(h)+1)/2));
Decimation
rot = 0; % Initial swithch position
y = [];
for n=1:length(x)
xn = x(n);
if rot==0 ss=1;
pro = xn*e0;
xk = [pro+xk(1:length(xk)/2),fliplr(pro) + xk((length(xk)/2+1):length(xk))];
yn = xk(length(xk));
y = [yn,y];
xk = [0,xk(1:length(xk)-1)];
else
end
if rot==1 ss=0;
xk(length(xk)/2+1) = 0.5*xn + xk(length(xk)/2+1);
else
end
rot=ss;
end
Displaying the results
figure
subplot(2,1,1)
plot(f,abs(X(1:512))), xlabel('\omega/\pi'),ylabel('|X(e^{j\omega})|')
title('Spectrum of the input signal')
axis([0,1,0,1.1])
Y = freqz(y,1,512,2); % Spectrum of the decimated signal
subplot(2,1,2)
plot(f,abs(Y(1:512))), xlabel('\omega/\pi'),ylabel('|Y(e^{j\omega})|')
title('Spectrum of the decimated signal')
axis([0,1,0,0.6])
disp('END OF EXAMPLE 7.4')
7.3 IIR Halfband Filter Design in MATLAB
Example 7.5
Design an elliptic halfband filter that is specified with the stopband edge frequency of 
and the minimal stopband attenuation 
.
and the minimal stopband attenuation .For IIR halfband filter design we use the function halfbandiir which is included in this folder.
The design procedure is performed in three steps:
clear all, close all
Step 1: Adjusting the design parameters
as = 45
as = 45
ap = 10*log10(1+1/(10^(as/10)-1))
ap = 1.3734e-04
fs = 0.57
fs = 0.5700
fp = 1-fs
fp = 0.4300
Step 2: Computation of the minimal filtr order
[N,fp] = ellipord(fp,fs,ap,as)
N = 7
fp = 0.4300
Step 3: Computation of the halfband filter parameters
[b,a,z,p,k] = halfbandiir(7,0.43);
Halfband filter frequency response
[H,f]=freqz(b,a,1024,2);
Setting the specifications
fspec=[0.57,0.57,1];
aspec=[2,-45,-45];
Displaying the results
figure
plot(f,20*log10(abs(H)))
hold on
plot(fspec,aspec,'r'), xlabel('\omega/\pi'), ylabel('Gain, dB')
axis([0,1,-80,2])
figure
zplane(z,p)
disp('Notice that filter poles are located on the imaginary axis.')
disp('END OF EXAMPLE 7.5')
7.4 IIR Halfband Filters with Approximately Linear Phase
Remainder
The main disadvantage of minimum phase IIR filters is their nonlinear phase response. A solution for the approximately linear-phase IIR halfband filter can be achieved when representing the transfer function H(z) as a sum of two subfilters, one of them a pure delay term,
where the subfilter 
iz an all-pass function expressible as a product of the second-order and the fourth-order sections.
iz an all-pass function expressible as a product of the second-order and the fourth-order sections.Example 7.6
Design the approximately linear-phase IIR halfband filter specified by the following parameters:
- Given the passband/stopband edge frequencies:
, . - The all-pass branch is of degree
, and the degree of the delay branch is 
.
Solution:
The design method is based on the paper: H.W. Schussler, P. Stefen (2001), Recursive Halfband Filters, AEU Int. J. Electron. Commun., 55(6), 377-388.
The paper was accompanied with the MATLAB program Hblipha, which were available at
http://www-nt.e-technik.uni-erlangen.de/~hws/programs/halfbandfilters/ . The program Hblipha is included into this folder.
Filter design
[b1,b2,a,aA,dP] = Hblipha(10,0.4,'Cheby');
Frequency response
[H,f]=freqz(b1,a,1024,2);
Group delay
[gd,f] = grpdelay(b1,aA,1024,2);
Displaying the results
figure
plot(f,20*log10(abs(H)))
title('Gain response')
axis([0,1,-70,2]),xlabel('\omega/\pi'), ylabel('Gain, dB'),grid
figure
zplane(b1,aA)
title('pole-zero locations')
figure
plot(f,unwrap(angle(H))),xlabel('\omega/\pi'), ylabel('Phase, rad'),grid
title('Phase response')
figure
plot(f,gd),,xlabel('\omega/\pi'), ylabel('Group delay, samples'),grid
title('Group delay')
disp('END OF EXAMPLE 7.6')
disp(' END OF CHAPTER VII')