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Systems and Signals Homework 3 Solved

Systems and Signals Homework 3 Solved

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Compute the impulse response functions h(t) for each of the LTI systems given below. Comment on the causality and BIBO stability of the systems. (a) S1 : y(t) = −t−∞1 et cos(2τ + 2 − 2t)x(τ)e−τ+2dτ (b) S2 : y(t) = e−t −t∞ eτ [cos(t) cos(τ) − sin(t) sin(τ)]x(τ)dτ S3 : y(t) = −t−∞1 e−(t−τ)x(τ…

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  1. Compute the impulse response functions h(t) for each of the LTI systems given below. Comment on the causality and BIBO stability of the systems.

(a)

S1 :

y(t) = −t−∞1 et cos(2τ + 2 − 2t)x(τ)e−τ+2dτ

(b)

S2 :

y(t) = e−t

−t∞ eτ [cos(t) cos(τ) − sin(t) sin(τ)]x(τ)dτ

    1. S3 : y(t) = −t−∞1 e−(t−τ)x(τ − 2)dτ

  1. Given below are two cascaded LTV systems S1 and S2. The input-output relation for system S1 is given by eqn. (1), where x(t) and y(t) are inputs and output of system S1, respectively. The input-output relation for system S2 is given by eqn. (2), where y(t) and w(t) are inputs and output of system S2 respectively.

y(t) = x(t)u(t)

−

t−2

e−(t−τ)x(τ)u(τ)dτ, t

∈

( ,

)

(1)

−∞

−∞ ∞

t

(2)

w(t) =

y(σ)u(σ)dσ, t ∈ (−∞, ∞)

−∞

  1. Compute IRF of S1 and S2 : h1(t, τ) and h2(t, τ) respectively.

  1. Compute IRF h12(t, τ) of cascaded system S1S2.

  1. Is the cascaded system (with input x(t) and output w(t) ) a stable system? Explain.

  1. Obtain the IRF h21(t, τ)

  1. Matlab assignment 2: For the following problems, include screenshots/images of your MATLAB figures. You should have 5 total figures for this problem. Also, copy and paste your MATLAB code into your homework.

    1. Using MATLAB, plot x(t) and h(t) individually over the range t ∈ [−3, 3].

x(t) = sin(2πt)u(t + 1)u(−t + 1) ; h(t) = u(t) − u(t − 1)

  1. Compute y1(t) = x(t) ∗ h(t) (∗ : convolution operator) analytically as a piecewise function.

(c) Solve for y2(t) = x(t) · h(t) (where · is the multiplication operator) analytically.

  1. Using MATLAB, plot y1(t) and y2(t) individually over the range t ∈ [−3, 3]. Are y1(t) and y2(t) the same?

  1. Using MATLAB, numerically convolve x(t) and h(t) and plot the result over the same range. Confirm that the result is the same as y1(t).

Hint for Matlab assignments: Here is a sample code snippet to help you get started. The following code plots the signal z(t) = sin (2πt) − 12 cos 3 π5 − π3 t in the time interval [−3, 3].

1

clear; clc; close all;

2

t=linspace(-3,3);

% Define time axis : 100 equally spaced points in [-3,3]

3

z = sin(2*pi*t) – 0.5*cos(3*pi/5-pi/3*t);

% Define signal z(t) as a function of

t

4

5

figure(1); % Create figure with label (1)

6

plot(t,z); % Plot signal z(t) against t

7

xlabel(’t’,’Interpreter’,’latex’,’fontsize’,16); ylabel(’z(t)’,’Interpreter’,’latex

’,’fontsize’,16); % Label the x and y axes

8

title(’z(t) vs t’,’Interpreter’,’latex’,’fontsize’,18); % Set the graph title

9

grid on; % Apply a mesh grid to the graph

Some useful Matlab functions:

  1. heaviside(x) : generates unit step function. eg. u=heaviside(t); generates function u(t)

  1. conv(x,y): returns convolution of input signals/vectors x and y.

  1. Elementwise multiplication of vectors in matlab: x.*y

Figure 1: Output of sample code snippet

2

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