Question

use matlab

y(t)=10*(cos(2*pi*500*t)+cos(2*pi*1000*t)+ cos(2*pi*1500*t)).

e) Down sample y(t) by a factor of 6. Sketch the Fourier transform with appropriate frequency axis. Check if all frequency components are correct?

Up-sample the time-domain signal obtained in e) by a factor of 6. Use appropriate filter for interpolation. Sketch the Fourier transform of the up-sampled and filtered signal. Does the resulting signal show all frequency components of the original signal y(t)?

Answer 1

Sorry for the inconvenience caused to you. Pls. find below the Matlab program as per the given question:

clc
clear all
close all
%%
fs= 4000; %%(assumed)
Ts= 1/fs;
t=0:Ts:5e-2;
f1=500;
f2=1000;
f3=1500;
y= 5*sin(2*pi*f1*t)+ 5*sin(2*pi*f2*t)+ sin(2*pi*f3*t);
figure(1);
plot(t,y);
title('Input signal');
%% Fourier Transform of original signal
N=length(y);
N1=2.^nextpow2(N);
fy=fft(y,N1);
fy=fy(1:N1/2);
K=(0:N1/2-1)/(N1/fs);
figure(2);
plot(K,abs(fy/max(fy)));
title('Magnitude spectrum of original signal');

%% Downsample by 6
y1= 5*sin(2*pi*f1*t*6)+ 5*sin(2*pi*f2*t*6)+ sin(2*pi*f3*t*6);
figure(3);
plot(1:6:6*length(y1),y1);
title('Downsampled signal by 6');
%% Fourier Transform of Downsampled signal
N=length(y1);
N1=2.^nextpow2(N);
fy=fft(y1,N1);
fy=fy(1:N1/2);
K=(0:N1/2-1)/(N1/fs);
figure(4);
plot(K,abs(fy/max(fy)));
title('Magnitude spectrum of Downsampled signal by 6');
%%
%%Upsampled by 6
y2= 5*sin(2*pi*f1*t/6)+ 5*sin(2*pi*f2*t/6)+ sin(2*pi*f3*t/6);
figure(5);
plot(1:6:6*length(y2),y2);
title('Upsampled signal by 6');
%% Fourier Transform of Upsampled signal
N=length(y2);
N1=2.^nextpow2(N);
fy=fft(y2,N1);
fy=fy(1:N1/2);
K=(0:N1/2-1)/(N1/fs);
figure(6);
plot(K,abs(fy/max(fy)));
title('Magnitude spectrum of Upsampled signal by 6');
%%
%%Interpolation Filter
upsampling = 6;
Ripple = 0.5;
h = intfilt(upsampling,2,Ripple);
b = fir1(40,Ripple);
%%rng('default')
x1 = filter(b,1,y2);
x2 = upsample(x1,upsampling);
y = filter(h,1,x2);
delay = mean(grpdelay(h));
y(1:delay) = [];
stem(1:upsampling:upsampling*length(x1),x1)
hold on
plot(y)
title(' Interpolated signal');