Task #1
Create a 10 second time vector in MATLAB. Use fs (sampling rate) = 1000 Hz.
Using the formula for a chirp from last lab create a chirp signal from this time vector.
Use the initial starting frequency f0= 10 Hz and a modulation index, µ=1.
Plot this signal in both the time and frequency domains. In the frequency domain you should see a continuous band of frequencies starting at 10 Hz.
L = 10;
fs = 1000;
Ts = 1/fs;
t = 0:Ts:L-Ts;
f0 = 10;
w = 2*pi*f0;
mu = 1;
x = cos(2*pi*(f0+t*mu).*t);
N = length(t);
F = fs/N;
f = f0+2*t*mu;
soundsc(x, fs);
plot(t,x);
xlabel('Time, t');
ylabel('Amplitude, x');
title('Chirp Signal in Time Domain');
X=fftshift(fft(x))/N;
figure
plot(f,abs(X));
xlabel('Frequency, f');
ylabel('Magnitude of FFT');
title('Chirp Signal in Frequency Domain');
Task #2
Use the square() function to generate a train of pulses at a frequency well above the Nyquist rate. Choose the duty cycle to be very small (0.1 for example), to simulate ideal sampling.
p=0.5(1+square(wt, 0.1)); %Only positive pulses
Plot the pulse train on a new figure in both the time and frequency domain.
L = 10;
fs = 1000;
Ts = 1/fs;
t = 0:Ts:L-Ts;
f0 = 10;
w = 2*pi*f0;
p = 0.5*(1+square(w*t, 0.1));
mu = 1;
f = f0+2*t*mu;
plot(t,p);
xlabel('Time, t');
ylabel('Amplitude, p');
title('Signal in Time Domain');
N = length(t);
X=fftshift(fft(p))/N;
figure
plot(f,abs(X));
xlabel('Frequency, f');
ylabel('Magnitude of FFT');
title('Signal in Frequency Domain');
Task #3
Multiply the square wave generated in task 2 with the chirp you generated in task 1. Use the following frequencies for the pulse rate (sampling rate): 50 Hz, 100Hz, 200Hz, 250 Hz, 500 Hz.
Again plot the time graph and frequency spectra of these sampled signals.
%Sampling Rate 50 Hz
L = 10;
fs = 50;
Ts = 1/fs;
t = 0:Ts:L-Ts;
f0 = 10;
w = 2*pi*f0;
mu = 1;
x = cos(2*pi*(f0+t*mu).*t);
p = 0.5*(1+square(w*t, 0.1));
xx = x.*p;
f = f0+2*t*mu;
plot(t,xx);
xlabel('Time, t');
ylabel('Amplitude, xx');
title('Signal in Time Domain');
axis([0 10 -2 2]);
N = length(t);
X=fftshift(fft(xx))/N;
figure
plot(f,abs(X));
xlabel('Frequency, f');
ylabel('Magnitude of FFT');
title('Signal in Frequency Domain');
%Sampling Rate 100 Hz
L = 10;
fs = 100;
Ts = 1/fs;
t = 0:Ts:L-Ts;
f0 = 10;
w = 2*pi*f0;
mu = 1;
x = cos(2*pi*(f0+t*mu).*t);
p = 0.5*(1+square(w*t, 0.1));
xx = x.*p;
f = f0+2*t*mu;
figure
plot(t,xx);
xlabel('Time, t');
ylabel('Amplitude, xx');
title('Signal in Time Domain');
axis([0 10 -2 2]);
N = length(t);
X=fftshift(fft(xx))/N;
figure
plot(f,abs(X));
xlabel('Frequency, f');
ylabel('Magnitude of FFT');
title('Signal in Frequency Domain');
Task #4 Down-sampling and Up-sampling
Down Sampling is used to decrease sampling rate and Up Sampling is used to increase the sampling rate.
x = [1 2 3 4 5 6 7 8 9 10]; y = downsample(x,3) %Decrease the sampling rate of a sequence by 3: y = downsample(x,3,2) %Decrease the sampling rate of the sequence by 3 and add a phase offset of 2: %Decrease the sampling rate of a matrix by 3: x = [1 2 3; 4 5 6; 7 8 9; 10 11 12]; y = downsample(x,3); x,y %Upsampling x = [1 2 3 4]; y = upsample(x,3); x,y %Increase the sampling rate of the sequence by 3 and add a phase offset of 2: x = [1 2 3 4]; y = upsample(x,3,2); x,y %Increase the sampling rate of a matrix by 3: x = [1 2; 3 4; 5 6;]; y = upsample(x,3); x,y
Task #5 Interpolation
Interpolation increases the original sampling rate for a sequence to a higher rate. interp performs lowpass interpolation by inserting zeros into the original sequence and then applying a special lowpass filter.
%Interpolate a signal by a factor of four:
t = 0:0.001:1; % Time vector
x = sin(2*pi*30*t) + sin(2*pi*60*t);
y = interp(x,4);
stem(x(1:30));
title('Original Signal');
figure
stem(y(1:120));
title('Interpolated Signal');
Task #6 Loading and showing image
Image files can be read using imgread() function.
The function imshow() can be used to display the image in a new figure window.
Use a small bitmap file for use.
For example
data=imread('shuvro.bmp');
figure
imshow(data)
title('Test Image');
Task #7 Down sampling, up sampling and interpolation of image
Use previously used down sampling, up sampling and interpolation techniques to resize image and then, create a larger image using upsampling and interpolation technique and see the difference.
%down sampling, up sampling and interpolation of image
close all
clear
clc
data=imread('shuvro.bmp');
figure
imshow(data)
title('Original image');
%DOwnsampling an image by a scale factor of 2. The same task can be done by image resize function
data1=downsample(data,2); %Downsample horizontal by a scale factor 0f 2
data1=downsample(data1.',2); %Downsample vertical by a scale factor 0f 2
data1=downsample(data1.',1); % Re orienting the image
figure
imshow(data1)
title('Down Sampled image');
%Upsampling example and its problem
data2=upsample(data1,2);
data2=upsample(data2.',2);
data2=upsample(data2.',1);
figure
imshow(data2)
title('Up Sampled image');
%Interpolation
%to be done by students
Attachments
Sample Image
Sample speech and music files
Try Yourself [May come in Lab exam]
- Down sampling of audio file.
- Try to repeat this downsampling process until the aliasing effect
becomes very noticeable.
- Try to repeat this downsampling process until the aliasing effect
- Downsampling of speech files
- Contrast the effect reducing the sampling rate had on
speech as opposed to the music. - Also contrast the spectra and time waveforms of the
two different types of signals.
- Contrast the effect reducing the sampling rate had on
- For a bit more of a challenge try the colour image, you will need to take into account the three layers (red, green, blue) when downsampling and restructuring this image.
- Try saving downsampled, upsampled, interpolated image and sound files using wavwrite and imwrite function.
N.B.: This lab exercise is heavily influenced by Laboratory Exercise from Mr. Refat Kibria, Department of CSE, SUST.