ECE 3300MATLAB Assignment 3This assignment explains how to use MATLAB to perform the convolution of two signals indiscrete time and in continuous time.Convolution in MATLABGiven two vectors...

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ECE 3300 MATLAB Assignment 3 This assignment explains how to use MATLAB to perform the convolution of two signals in discrete time and in continuous time. Convolution in MATLAB Given two vectors x and y, the command z=conv(x,y) assigns to z a vector consisting of the convolution of x and y. For example if, x[n] = δ[n]+δ[n−1] and y[n] = δ[n]+2δ[n−1], the convo- lution is (x∗h)[n] = y[n]+y[n−1] = δ[n]+2δ[n−1]+δ[n−1]+2δ[n−2] = δ[n]+3δ[n−1]+2δ[n−2]. Using MATLAB, If x=[1 1] and y=[1 2], then z=conv(x,y) gives output [1 3 2]. Convolution of Discrete-Time Signals The command conv does not keep track of time values, so we need to do this separately. We have learned that if x[n] is “on” from n = a1 to n = c1 and y[n] is “on” from n = a2 to n = c2, then (x ∗ h)[n] is “on” from n = a1 + a2 to n = c1 + c2. Given that x[n] is represented in MATLAB by n 1=a 1:c 1 and x and that y[n] is represented by n 2=a 2:c 2 and y, z[n] = (x ∗ h)[n] is obtained via n=n 1(1)+n 2(1):n 1(end)+n 2(end) and z=conv(x,y). PROBLEM STATEMENT: PART ONE In this problem we explore the claim that many signals. when convolved with themselves over and over, start to resemble a bell-shaped curve. 1. Let x[n] = u[n]− u[n− 50]. Let y[n] = (x ∗ x)[n], let z[n] = (y ∗ y)[n], let w[n] = (z ∗ z)[n], and let v[n] = (w ∗ w)[n]. Plot each of these signals on separate graphs using stemplot as per the instructinos for discrete-time plots in the first Matlab assignment. Note that each graph should use appropriate (different) horizontal and vertical ranges so that the resulting shapes are clearly seen, each with height taking up most of the vertical plot space. 2. Repeat this process with the signal x[n] = n(u[n] − u[n− 50]). Convolution of Continuous-Time Signals For continuous time signals x(t) and y(t) we can approximate the convolution integral as a sum of rectangles with width b, where b is the time between samples. Let x[m] = x(mb) and y[m] = y(mb), and set t = nb. Then [x ∗ y](t) = ∫ ∞ −∞ x(τ)y(t− τ)dτ = ∫ ∞ −∞ x(τ)y(nb− τ)dτ ≈ ∞∑ m=−∞ x(mb)y(nb−mb)b = ∞∑ m=−∞ x[m]y[n−m]b Apart from the final multiplication by b, the sum is readily implemented with the command conv. Thus, if x(t) is implemented in MATLAB via t 1=a 1:b:c 1 and x and y(t) is imple- mented via t 2=a 2:b:c 2 and y, then z(t) = [x ∗ y](t) can be implemented (approximately) via the following commands: t=t 1(1)+t 2(1):b:t 1(end)+t 2(end) and z=conv(x,y).*b. 1 PROBLEM STATEMENT: PART TWO In all parts of this problem use a sampling time of 0.01. 1. Use Matlab to determine the convolution of x(t) = (u(t − 7) − u(t − 10)) with h(t) = (u(t− 4) − u(t− 9)) and plot the result. 2. Compute this convolution by hand, and give the resulting expression. Plot the resulting expression and the previous result on a single graph, verifying that the answers are ap- proximately the same. (Scan your hand-computation and attach to the PUBLISH pdf file(s).) 2