# Convolution for discrete-time signals Consider two sequences {x(n} and {h(n)} of lengths N_x, and N_h, respectively…

###### Home/Education/Expert Q&A/Convolution for discrete-time signals Consider two sequences {x(n} and {h(n)} of lengths N_x, and N_h, respectively…

Convolution for discrete-time signals Consider two sequences {x(n} and {h(n)} of lengths N_x, and N_h, respectively. Let N_h lessthanorequalto N_x. The convolution of {x(n)[and {h(n)} is defined as follows: y(n) = sigma^N_ -1_k = 0 h(k) x (n – k) Here {y(n)} is the convolution of the sequences {x(n)} and {h(n)}. You can see it yourself that the length of sequence {y(n)} is N_y = N_x + N_h – 1. Convolution for continuous-time signals Convolution can be used to compute the output of a linear time-invariant (LTI) system when the input signal and the impulse response of the LTI system are given. Let the input signal be x(t) and the impulse response of the LTI system be h(t). The output y(t) of the LTI system is obtained by convolving x(t) with h(t), i.e., y(t) = integral^infinity_-infintiy h(tau) x (t – tau) d tau = integral^infinity_-infinity x (tau) h (t – tau) d tau Periminary Consider two sequences defined as: {x(n)} = {4, 3, 2, 1} and {h(n)} = {5, 6, 7} Compute their convolution. Compute analytically convolution of x(t) and h(l) defined as follows: x(t) = {1, if -1 lessthanorequalto t lessthanorequalto 1 0, otherwise and h(t) = {1, if -1 lessthanorequalto t lessthanorequalto 1 0, otherwise Experiment 1 Consider two sequences |x(n)} and {h(n)} defined by equations (3) and (4). respectively. Write a program in MATLAB to carry out convolution defined by equation I. List the sequence {y(n)}. Are you getting the same result as obtained in Preliminary 1? Experiment 2 Consider two sequences {x(n)} and {h(n)} as follows: x(n) = n^5 – 5n + 1, n = 0,1, …, N_x – 1 and h(n) = 1 – n, n = 0, 1, …, N_h – 1 Take N_x = 9 and N_h = 3. Use the MATLAB program (from Experiment 1) to compute the convolution {y(n)} equation 1. List the sequence {y(n)}. Experiment 3 Consider signals x(t) and h(t) defined by equations (5) and (6), respectively, over the time interval [-4,4). Write a program in MATLAB to carry out convolution defined in equation 2. Use this MATLAB program to compute convolution y(t) of x(t) and h(t). Plot the signals x(t), h(t) and y(t). Are you getting the same result as obtained in Preliminary 2? Experiment 4 Use the MATLAB program (from Experiment 3) to compute convolution y(t) of x(t) and h(t) defined by equations (9) and (10), respectively. Plot the signals x(t), lift) and y(t). x(t) = {1, if 0 lessthanorequalto t lessthanorequalto 2 0, otherwise and h(t) = {1, if 0 lessthanorequalto t lessthanorequalto 1 0, otherwise Note the durations of x(t), h(t) and y(t). What is the relationship between the three durations.

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