EGR324L: Linear Systems and Signals LaLab 4: Convolution SumObjectiveTo learn how to compute...

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EGR324L: Linear Systems and Signals LaLab 4: Convolution  SumObjectiveTo learn how to compute convolution sum in Matlab.BackgroundConvolution Sum in MatlaLet x[n] be a discrete-time signal such that x[n] = 0 for n y a pair , where vx = [ x[ox] x[ox+1 ] ... ] is an vector and x[ox] ≠ 0. In Fig. 1, x[n] =, where ox = -2 and vx = [ 1 -2]. And h[n] = , where oh = 3 and vh = [ XXXXXXXXXX].Fig. l Representing general discrete time signals in MatlaIt is easy to see that if x[ n] = , then x[n-n0] = .A built-in function vy = conv( vx, vh) is provided by Matlab to compute the convolution sum of two vectors vx and vh. With our notation, this function actually computes =  *.In order to use function conv to calculate convolution sum for general discrete-time signals, a signal x[n] =  must be shifted to the left for ox steps (if ox > 0), Let x1[n]  =0, vx> = x[n+ox] = x[n] * δ[n+ox]. For h[n] = , we may have h1[n] = = h[n+oh] = h[n] * δ[n+oh].By invoking Matlab function vy = conv( vx, vh), we actually obtainy 1 [n]    =  = x1[n] * h1[n]= (x[n] * δ[n+ox]) * (h[n] * δ[n+oh])= (x[n] * h[n]) * δ[n+ox+oh]Thus3x[n] * h[n] = y1[n] * δ[n-(ox+oh)]= y1[n -(ox+oh)] = = Example:For signals x[n] and h[n] shown in Fig. 1, Matlab instructions for representing  these two signals, and that for calculate the convolution sum of the two are given below.Fig. 2 Computing convolution sum of general discrete-time signals in MatlaPreparationYou need to read the Matlab book for the following topics:· more functions and commands for graphical plotting· functions conv and deconv, rand and sound· for statement and if statement in Matla· relational operators >, = · logic operators &, | and ~.Problems(1) Write  a function  [oy , vy ]  = gconv(ox , vx, oh, vh)  that  computes  generalized convolution sum  =  * , as defined in the background section of this handout.(2) Using the function gconv defined in (2), and function trainplot defined in Lab 3, calculate and plot y[n] = x[n]  * h[n], where x[n] and h[n] are shown as in Fig. 4.Fig. 4 Compute convolution sum of two signals(3) Using Matlab functions vy = conv( vx, vh) and [vh,  vr]  = deconv( vy, vx),  solve Problem 9.17 in the textbook.(4) A discrete-time system generating echoes of a sampled acoustic source is modeled as a causal and LTI system with unit impulse responsewhere 0 Write a function y = gecho( x, a, N0, d), where column vector y is obtained by convolving input vector x with the system impulse response h, which depends on attenuation factor a, time delay N0  and dead band ratio d defined as above.(5) Test function gecho with the following parameters.Plot the vector generated by function gecho.(6) Apply  function gecho to a signal hi.mat (sampled at 8,192 Hz) using the following parameters.attenuation factor: 0.7 echo delay time: 1 second dead band ratio: 0.05Play the processed signal using Matlab function sound(x).(7) Repeat Step (6) for the following four sets of parameteters:attenuation factor: 0.3    attenuation factor: 0.7echo delay time: 1 second    echo delay time: 2 seconddead band ratio: 0.05    dead band ratio: 0.05attenuation factor: 0.7    attenuation factor: 0.5echo delay time: 1 second    echo delay time: 0.1 seconddead band ratio: 0.2    dead band ratio: 0.05How do they compare with each other ? Do they sound like what you predicted?You may wish to experiment this function with some other signals you sampled in the lab.ReportAs usual, you should turn in all the Matlab work. This includes all Matlab programs (.m files) as well as all graphical plots. Again, the audio signals should be plotted against time in seconds -- not in aay indices. Explain in detail how Matlab was used to solve the problems.Describe the experiment you did in (6) and (7).How do they compare with each other ? Do they sound like what you predicted?(8) (9) You may wish to experiment this function with some other signals you sampled in the lab.(10) (11) Report(12) (13) As usual, you should turn in all the Matlab work. This includes all Matlab programs (.m files) as well as all graphical plots. Again, the audio signals should be plotted against time in seconds -- not in aay indices. Explain in detail how Matlab was used to solve the problems.(14) (15) Describe the experiment you did in (7) and (8).(16) ReportAs usual, you should turn in all the Matlab work. This includes all Matlab programs (.m files) as well as all graphical plots. Explain in detail how Matlab was used to solve the problems. ((5))

Kshitij · Accepted Answer

nw gray/ans2.m
ox=-2
vx=[1 -2];
oh=3;
vh=[-1 0 3 2];
[oy,vy]=gconv(ox,vx,oh,vh)
nw gray/ans5.m
x=rand(10,1)
a=0.5;
N0=100;
d=0.01
[y] = gecho(x, a, N0, d)
plot(y)
xlabel('Time')
ylabel('amplitude')
nw gray/ans6.m
x=rand(100,1)
sound(x)
a=0.7;
N0=100;
d=0.05
[y] = gecho(x, a, N0, d)
plot(y)
xlabel('Time')
ylabel('amplitude')
nw gray/ans7.m
x=rand(100,1)
a=0.7;
N0=10;
d=0.05
[y] = gecho(x, a, N0, d)
plot(y)
xlabel('Time')
ylabel('amplitude')
nw gray/answer3.m
%a) if N=4
clc;
clear all;
close all;
n=0:5;%%creating time vector
N=4;%%taking N=4
x=(n>=0)-(n>=(N+1));%%generating x[n] sequence
h=(n>=0)-(n>=4);%%generating h[n] sequence
y=conv(x,h);%%convolution of x[n] and h[n]
disp(y);%%displaying y[n]
stem(y);%%plootting y[n]
xlabel('n--->');
ylabel('y[n]--->');
title('convoltion of x[n) and h[n]')
%%b)calculting the value of N=0)-(n>=(N+1));%%calculting x[n] for each value of N
y1=conv(x,h);%% calculating new convolution sequence
if (y1(3+1)==3 & y1(6+1)==0)%%comparing x[3]=3 and y[6]=0
fprintf('the N values for y[3]=3 and y[6]=0 is %d 
 ',N)
end
end
nw gray/gconv.m
function [oy,vy]=gconv(ox,vx,oh,vh)
nx=ox:1:length(vx)+ox; % Defining Axis for Vx vector
nh=oh:1:length(vh)+oh; % Defining Axis for Vh vector
vy=conv(vx,vh);

EGR324L: Linear Systems and Signals Lab Lab 4: Convolution Sum Objective To learn how to compute convolution sum in Matlab. Background Convolution Sum in Matlab Let x[n] be a discrete-time signal such...

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