1. Evaluate and sketch the convolution integral (the output y(t)) for a system with input x(t) and impulse response h(t), where x(t) = u(1-2) and h(t)= "u(t) u(t) is the unit step function. Please show clearly all the necessary steps of convolution. Determine the values of the output y(t) at 1 = 0,1 = 3,1 = 00. (3 pts)
1.[10pt] Compute the convolution X(t)* v(t). x(t) = 2u(t) – 2u(t – 2), s 2-t, 0<t<2 v(t) = { ö otherwise
Compute the convolution of the following pairs of signals x(t) and hlt) by and inverse transformin z(t)u(t), h(t)-e'u(-t)
By first expressing the triangular signal x(t) in Figure P9.8 as the convolution of a rectangular pulse with itself, determine the Fourier transform of x(t) x(t) -2 Figure P9.8
4. Graphically determine the discrete convolution of h(t) and x(t) for the case shown below. NOTE: ht)*x(t)-x(t)*h(t), so it does not matter which you calculate.] h(t) x(t)
4. Convolution EX4. The input X(t) and impulse response h(t) for a system are given. Using convolution evaluating the system output y(t). X(t)=1 O<t1 h(t)=sin pi*t 0<<2 =0 else where =0 elsewhere Xit) ↑ hlt) E mer
A system has an input, x(t) and an impulse response, h(t). Using the convolution integral, find and plot the system output, y(t), for the combination given below. x(t) is P3.2(e) and h(t) is P3.2(f). 1/2 cycle of 2 cos at -2. (e)
Obtain y(t) from the convolution of x(t) with h(t) Write the resultant y(t) ecuation Draw y(t)
2. Using direct convolution (i.e., the integral), determine the convolution between r(t) and h(t), where h(t) and r(t) are defined as (note: please do NOT just plug in the formulas we derived in the class): h(t) exp(-2t) u (t) and x(t) = exp(-t)u(t), u(t) is the unit step function. h(t) exp(-t)u (t) and r(t)= exp(-t)u(t)
b) Find x(t)= x1(t) * x2(t) using the convolution integral. Write the result by region Show all regions and plots in your calculations. eros 3 x Answer: x(t)= bnien l vo s 1Swans ls AVSV meldoy C) Repeat part b) using Laplace. Write the result in terms of delayed unit steps and verify that it is an equivalent result pnwlle Answer: x(t)= ) 3et-1)u(t) 6(t-2) = c) 3e--1u(t)o()= d) 3e--1)u(t)-8(1) = Hint: Is not the same multiplication by a delayed...