**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(w*t, 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.