DSP Lab 1 Handout - Lab Manual PDF

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International Islamic University, Islamabad Digital Signal Processing LAB

EXPERIMENT # 01: Introduction to Discrete Signals and Signal Operations in MATLAB Name of Student: ………………………………….. Roll No.: …………………………………………… Date of Experiment: ……………………………….. Report submitted on: ………………………………..

Marks obtained: …………………………………… Remarks: …………………………………………… Instructor’s Signature: ……………………………... Digital Signal Processing Lab

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Introduction to Discrete Signals and Signal Operations in MATLAB 1. Objective This lab focuses on the creation of discrete time signals in MATLAB and explains the required MATLAB commands. After performing the lab, the students will be able to • Create discrete time signals in MATLAB • Use MATLAB commands to perform simple operations on discrete time signals 2. Resources Required • A computer • MATLAB 7.0 or higher 3. Introduction Digital signal processing involves the processing of a discrete time signal (input signal) to produce another discrete time signal (output signal) which has some desired properties. A strong background of discrete time signals is essential for analyzing discrete time systems. Following are some of the commonly used discrete time signals generated using MATLAB: 3.1 Unit Impulse and Unit Step sequences A unit impulse sequence of length N can be generated in MATLAB by the following command: N=10; u = [1

zeros(1, N-1)]

A unit impulse sequence delayed by M samples where M < N can be generated as follows: M=5; ud = [zeros(1,M)

1

zeros(1,N-M- 1)];

Likewise, a unit step sequence s[n] of length N can be generated using the MATLAB command: s = [ones(1,N)];

A delayed unit step sequence can be generated in a manner similar to that used in the generation of a delayed unit sample sequence. % Program P1_1 % Generation of a Unit Sample Sequence clear all close all clc % Generate a vector from -10 to 20 n = -10:20; % Generate the unit sample sequence u = [zeros(1,10) 1 zeros(1,20)]; % Plot the unit sample sequence stem(n,u); xlabel('Time index n'); ylabel('Amplitude'); title('Unit Sample Sequence'); axis([-10 20 0 1.2]); grid on;

3.2 Sinusoidal sequences Sinusoidal sequences can be generated in MATLAB using “sin” and “cos” functions. Program P1_2 is a MATLAB code that generates a sinusoidal signal.

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% Program P1_2 % Generation of a sinusoidal sequence clear all; close all; clc; n = 0:40; f = 0.1; phase = 0; A = 1.5; arg = 2*pi*f*n - phase; x = A*cos(arg); stem(n,x,'r'); % Plot the generated sequence axis([0 40 -2 2]); grid; title('Sinusoidal Sequence'); xlabel('Time index n'); ylabel('Amplitude');

3.3 Linear and Nonlinear Systems We now investigate the linearity property of a causal system. Consider the system given by 𝑦[𝑛] − 0.4𝑦[𝑛 − 1] + 0.75𝑦[𝑛 − 2] = 2.2403𝑥[𝑛] + 2.4908𝑥[𝑛 − 1] + 2.2403𝑥[𝑛 − 2] MATLAB Program P1_3 is used to simulate the system, to generate three different input sequences x1[n], x2[n], and x[n] = a · x1[n] + b · x2[n], and to compute and plot the corresponding output sequences y1[n], y2[n], and y[n]. % Program P1_3 % Generate the input sequences close all; clear all; clc n = 0:40; a = 2; b = -3; x1 = cos(2*pi*0.1*n); x2 = cos(2*pi*0.4*n); x = a*x1 + b*x2; num = [2.2403 2.4908 2.2403]; den = [1 -0.4 0.75]; ic = [0 0]; % Set zero initial conditions y1 = filter(num,den,x1,ic); % Compute the output y1[n] y2 = filter(num,den,x2,ic); % Compute the output y2[n] y = filter(num,den,x,ic); % Compute the output y[n] yt = a*y1 + b*y2; d=y-yt;% Compute the difference output d[n] % Plot the outputs and the difference signal subplot(3,1,1); stem(n,y); ylabel('Amplitude'); title('Output Due to Weighted Input: a \cdot x_{1}[n] + b \cdot x_{2}[n]'); subplot(3,1,2); stem(n,yt); ylabel('Amplitude'); title('Weighted Output: a \cdot y_{1}[n] + b \cdot y_{2}[n]'); subplot(3,1,3); stem(n,d); ylabel('Amplitude');

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xlabel('Time index n'); title('Difference Signal')

3.4 Time-Invariant and Time-Varying Systems We next investigate the time-invariance property of a causal system. MATLAB Program P1_4 is used to simulate the system, to generate two different input sequences x[n] and x[n - D], and to compute and plot the corresponding output sequences y1[n], y2[n], and the difference y1[n] - y2[n + D]. % Program P1_4 % Generate the input sequences close all; clear all; clc n = 0:40; D = 10; a = 3.0; b = -2; x = a*cos(2*pi*0.1*n) + b*cos(2*pi*0.4*n); xd = [zeros(1,D) x]; num = [2.2403 2.4908 2.2403]; den = [1 -0.4 0.75]; ic = [0 0];% Set initial conditions % Compute the output y[n] y = filter(num,den,x,ic); % Compute the output yd[n] yd = filter(num,den,xd,ic); % Compute the difference output d[n] d=y- yd(1+D:41+D); % Plot the outputs subplot(3,1,1); stem(n,y); ylabel('Amplitude'); title('Output y[n]'); grid; subplot(3,1,2); stem(n,yd(1:41)); ylabel('Amplitude'); title(['Output Due to Delayed Input x[n - ', num2str(D),']']); grid; subplot(3,1,3); stem(n,d); xlabel('Time index n'); ylabel('Amplitude'); title('Difference Signal'); grid;

3.5 Impulse Response Computation The MATLAB command y = impz(num,den,N) can be used to compute the first N samples of the impulse response of the causal LTI discrete-time system. MATLAB Program P1_5 given below computes and plots the impulse response of the system. % Program P1_5 % Compute the impulse response y clf; N = 40; num = [2.2403 2.4908 2.2403]; den = [1 -0.4 0.75]; y = impz(num,den,N);

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% Plot the impulse response stem(y); xlabel('Time index n'); ylabel('Amplitude'); title('Impulse Response'); grid;

3.6 Convolution The convolution operation is implemented in MATLAB by the command conv, provided the two sequences to be convolved are of finite length. For example, the output sequence of an FIR system can be computed by convolving its impulse response with a given finite-length input sequence. ∞

𝑦[𝑛] = ∑ ℎ[𝑘]𝑥[𝑛 − 𝑘] 𝑘=−∞ ∞

𝑦[𝑛] = ∑ ℎ[𝑛 − 𝑘]𝑥[𝑘] 𝑘=−∞

Or can be represented as

𝑦[𝑛] = ℎ[𝑛] ⊛ 𝑥[𝑛] The following MATLAB program illustrates this approach.

% Program P1_6 close all; clear all; clc h= [3 2 1 -2 1 0 -4 0 3]; % impulse response x = [1 -2 3 -4 3 2 1]; % input sequence y = conv(h,x); n = 0:14; subplot(2,1,1); stem(n,y); xlabel('Time index n'); ylabel('Amplitude'); title('Output Obtained by Convolution');grid; x1 = [x zeros(1,8)]; y1 = filter(h,1,x1); subplot(2,1,2); stem(n,y1); xlabel('Time index n'); ylabel('Amplitude'); title('Output Generated by Filtering');grid;

3.7 Stability of LTI Systems As indicated, an LTI discrete-time system is BIBO stable if its impulse response is absolutely sum able. It therefore follows that a necessary condition for an IIR LTI system to be stable is that its impulse response decays to zero as the sample index gets larger. Program P1_7 is a simple MATLAB program used to compute the sum of the absolute values of the impulse response samples of a causal IIR LTI system. It computes N samples of the impulse response sequence, evaluates 𝐾

𝑆(𝐾) = ∑|ℎ[𝑛]| 𝑛=0

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for increasing values of K, and checks the value of |h[K]| at each iteration step. If the value of |h[K]| is smaller than 10−6, then it is assumed that the sum S(K) of equation has converged and is very close to S(∞). % Program P1_7 % Stability test based on the sum of the absolute % values of the impulse response samples clear all; close all; clc num = [1 -0.8]; den = [1 1.5 0.9]; N = 200; h = impz(num,den,N+1); parsum = 0; for k = 1:N+1; parsum = parsum + abs(h(k)); if abs(h(k)) < 10^(-6) break; end end % Plot the impulse response n = 0:N; stem(n,h); xlabel('Time index n'); ylabel('Amplitude'); % Print the value of abs(h(k)) disp('Value ='); disp(abs(h(k)));

3.8 The Moving Average System 1 𝑦[𝑛] = (𝑥 [𝑛 + 1] + 𝑥 [𝑛] + 𝑥 [𝑛 − 1]) 3 Examination of above eq. reveals that the three-point smoothing filter considered here is an LTI FIR system. Moreover, as y[n] depends on a future input sample x[n + 1], the system is non-causal. A causal version of the three-point smoothing filter is obtained by simply delaying the output by one sample period, resulting in the FIR filter described by 1 (1) 𝑦[𝑛] = (𝑥 [𝑛] + 𝑥 [𝑛 − 1] + 𝑥 [𝑛 − 2]) 3 𝑀−1 1 𝑦[𝑛] = ∑ 𝑥[𝑛 − 𝑘] (2) 𝑀 𝑘

This defines a causal M-point smoothing FIR filter. The system of Eq. 2 is also known as a moving average filter. We illustrate its use in filtering high-frequency components from a signal composed of a sum of several sinusoidal signals. % Program P1_8 % Generate the input signal clear all; close all; clc n = 0:100; s1 = cos(2*pi*0.05*n); % A low frequency sinusoid s2 = cos(2*pi*0.47*n); % A high frequency sinusoid x = s1+s2; % Implementation of the moving average filter

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M = input('Desired length of the filter = '); num = ones(1,M); y = filter(num,1,x)/M; % Display the input and output signals subplot(2,2,1); plot(n,s1); axis([0, 100, -2, 2]); xlabel('Time index n'); ylabel('Amplitude'); title('Signal # 1'); subplot(2,2,2); plot(n,s2); axis([0, 100, -2, 2]); xlabel('Time index n'); ylabel('Amplitude'); title('Signal # 2'); subplot(2,2,3); plot(n,x); axis([0, 100, -2, 2]); xlabel('Time index n'); ylabel('Amplitude'); title('Input Signal'); subplot(2,2,4); plot(n,y); index n'); axis([0, 100, -2, 2]); xlabel('Time ylabel('Amplitude'); title('Output Signal');

LAB WORKSHEET (Lab # 1) 1

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Run the Program P1_8 for M = 2 to generate the output signal with x[n] = s1[n] + s2[n] as the input. Which component of the input x[n] is suppressed by the discrete-time system simulated by this program? If the LTI system is changed from y[n] = 0.5(x[n] + x[n - 1]) to y[n] = 0.5(x[n] - x[n - 1]), what would be its effect on the input x[n] = s1[n] + s2[n]? Run Program P1_8 for other values of filter length M, and various values of the frequencies of the sinusoidal signals s1[n] and s2[n]. Comment on your results. Use sinusoidal signals with different frequencies as the input signals and compute the output signal for each input. How do the output signals depend on the frequencies of the input signal? Run Program P1_3 and compare y[n] obtained with weighted input with yt[n] obtained by combining the two outputs y1[n] and y2[n] with the same weights. Are these two sequences equal? Is this system linear? Consider another system described by: y[n] = x[n] x[n − 1]. Modify Program P1_3 to compute the output sequences y1[n], y2[n], and y[n] of the above system. Compare y[n] with yt[n]. Are these two sequences equal? Is this system linear? Run Program P1_4 and compare the output sequences y[n] and yd[n - 10]. What is the relation between these two sequences? Is this system time-invariant? Consider another system described by Modify Program P1_4 to simulate the above system and determine whether this system is timeinvariant or not.

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