by Ljiljana Milic
Supplemental material for Chapter VI:
6. Sampling Rate Conversion by a Fractional Factor
Table of Contents
6.1 Sampling Rate Conversion by a Rational Factor
Example 6.1
In this example, the MATLAB function resample is used for sampling-rate conversion by a rational factor.
Sampling-rate conversion by L/M = 3/2
close all, clear all
Fx = 20; Tx=1/Fx; % Original sampling frequency in Hz
tx = 0:Tx:1; % Time vector tx
x = 0.9*sin(2*pi*tx); % Input sequence
y = resample(x,3,2); % Re-sampling
ty = (0:(length(y)-1))*2*Tx/3; % New time vector ty
figure
subplot(2,1,1), stem(tx,x,'*k')
hold on
stem(ty,y,'-.k')
legend('original','resampled')
xlabel('Time'), ylabel('Amplitude')
axis([0,1,-1,1])
disp('END OF EXAMPLE 6.1)')
6.2 Efficient Implementation of Polynomial Interpolation Filters Using Farrow Structure
Remainder
Using the polynomial-based approach and the Farrow structure, the sampling rate conversion by an arbitrary factor can be implemented efficiently. The above mentioned structure is indicated in the Figure.
Farrow structure for polynomial-based interpolation filters:
(a) The overall structure. (b) FIR filter details.
In the following example, we demonstrate the application of the Farrow structure for converting the sampling rate of the input signal by the factor 16/15.
Example 6.2
This example demonstrates sampling rate alteration by the fractional factor R = 16/15.
clear all, close all
Generating the input signal x[n]
n = -2:32;
f1 = 0.1; f2 = 0.16;
x = n+2*sin(pi*f1*n)+5*cos(pi*f2*n)+1;
n=0:length(x)-5;
figure (1)
subplot(2,1,1),stem(n,x(3:length(x)-2),'k');
xlabel('Time index [n]'), ylabel('x[n]')
title('Original signal')
R = 16/15; % Re-sampling factor
s = 1:15;
mu = [0,1-(1-1/R)*s];
mu = repmat(mu,1,3); % Vector of fractional intervals
Farrow filter, coefficients:
C0 = [1/6,-1/2,1/2,-1/6]; % Vector of Lagrange coefficients for C_0(k)
C1 = [0,1/2,-1,1/2]; % Vector of Lagrange coefficients for C_1(k)
C2 = ([-1/6,1,-1/2,-1/3]); % Vector of Lagrange coefficients for C_2(k)
C3 = [0,0,1]; % Vector of Lagrange coefficients for C_3(k)
Farrow filter, initial states:
xs0 = [0,0,0]; xs1 = [0,0,0]; xs2 = [0,0,0]; xs3 = [0,0];
Farrow filtering:
l=1;
for n=1:1:31
xnl = x(n+3);
[v0,xs0] = filter(C0,1,xnl,xs0);
[v1,xs1] = filter(C1,1,xnl,xs1);
[v2,xs2] = filter(C2,1,xnl,xs2);
[v3,xs3] = filter(C3,1,xnl,xs3);
if mu(l)==0
y(l) = ((v0*mu(l)+v1)*mu(l)+v2)*mu(l)+x(n+2);
y(l+1) = ((v0*mu(l+1)+v1)*mu(l+1)+v2)*mu(l+1)+x(n+2);
l = l + 2;
else
y(l) = ((v0*mu(l)+v1)*mu(l)+v2)*mu(l)+x(n+2);
l = l + 1;
end
end
Displaying the results
figure
subplot(2,1,2)
stem(0:length(y) -1,y,'sk'), xlabel ('Time index [l]'),ylabel('y[l]')
title('Re-sampled signal')
axis([0,33,0,40])
disp('END OF EXAMPLE 6.2')
6.3 Fractional Delay Filters
Example 6.3
We generate impulse responses of a family of fractional delay filters using the MATLAB object dfilt.farrowlinearfd.
clear all, close all
Designing family of fractional delay filters
for k = 0:9
mu = k/10;
h(k+1)= dfilt.farrowlinearfd(mu);
end
Compute and plot phase delay characteristics
[phi,f] = phasedelay(h,512,2); % Computing the phase delay
figure
plot(f,phi/pi,'k')
xlabel('\omega/\pi'), ylabel('Phase delay (samples)'), grid
Compute and plot magnitude responses
[H,f] = freqz(h,512,2); % Computing the frequency response
figure
plot(f,abs(H),'k')
xlabel('\omega/\pi'), ylabel('Magnitude'), axis([0,1,0,1.1]),grid
Aplying the fractional delay filter
t=0:9;
x=0.5*sin(pi*0.2*t)+(sin(pi*0.1*t)); % Original sequence
y1=filter(h(3),x); % 0.2-sample delay
y2=filter(h(6),x); % 0.5-sample delay
figure
stem(t,x,'k'); hold on; stem(t-0.2,y1,'kv--','filled')
hold on; stem(t-0.5,y2,'ks-.','filled')
xlabel('n'),ylabel('Amplitude')
legend('\mu=0','\mu=0.2','\mu=0.5'), grid
disp('END OF EXAMPLE 6.3')
Example 6.4
This example illustrates processing of a sinusoidal sequence when the fractional delay factor is changed from 
to 
. Program generates the 1st order Lagrange fractional-delay filter, creates 10 samples of the sinusoidal sequence, and performs filtering with the sudden change of the delay factor from to at the time-index n = 5. The Matlab object dfilt.farrowlinearfd has been used.
to . Program generates the 1st order Lagrange fractional-delay filter, creates 10 samples of the sinusoidal sequence, and performs filtering with the sudden change of the delay factor from to at the time-index n = 5. The Matlab object dfilt.farrowlinearfd has been used.clear all, close all
t = 0:9;
x = sin(2*pi*0.1*t);
for i=1:10
if i <= 5, mu(i)=0.4;
else
mu(i)=0.2;
end
end
h = dfilt.farrowlinearfd(mu);
h.PersistentMemory = true;
for i=1:10
h.Fracdelay = mu(i);
ty(i) = t(i)-mu(i);
y(i) = filter(h,x(i));
end
figure
stem(t,x); hold on; stem(ty,y,'rs')
xlabel('n'),ylabel('Amplitude')
legend('Original Signal','Delayed Signal')
grid
disp('END OF EXAMPLE 6.4')
disp(' END OF CHAPRER VI')