Question # 4 Let x(t) u(t) be a signal, let h(t) = e -5tu(t) be a...
4. A linear time invariant system has the following impulse response: h(t) =2e-at u(t) Use convolution to find the response y(t) to the following input: x(t) = u(t)-u(t-4) Sketch y(t) for the case when a = 1
QUESTION 2 (20 MARKS) (a) A continuous time signal x(t) = 3e2tu(-t) is an input to a Linear Time Invariant system of which the impulse response h(t) is shown as h(t) = { .. 12, -osts-2 elsewhere Compute the output y(t) of the system above using convolution in time domain for all values of time t. [8 marks) (b) The impulse response h[n] of an LTI system is given as a[n] = 4(0.6)”u[n] Determine if the system is stable. [3...
Q2 (a) Given the signal x(t) and system h(t) as presented in Figure Q2(a). Determine the output y(t) using the graphical representation of convolution integral. (7 marks) x(1) h(t) 1 e-'u(t) e-2 (1) 0 Figure Q2(a) Q2 (b) Consider a system as shown in Figure Q2(b). t2 - 1 x(t) y(t) Advance by 1 second Х Figure Q2(b) Find the input-output relation between x(t) and y(t). (i) (1 mark) Examine whether the system is time variant or time invariant. (5...
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit impulse response h[nu(t) for some input signal z(t). Find r(t) Hint: Use the Laplace transform of y(t) and h(t) to first find the Laplace transform of r(t), and then find r(t) using inverse Laplace transform. 25 points
A CT window signal is given as x(t) = u(t+4) – uſt-4) The frequency response of a CT LTI system is a given as H(jw) = {2e-jw 1w 5 31/16 HV) 0 otherwise if xz(t) = x(t)* 8(t - 16k) is applied to this CT LTI system, k=-00 a) Sketch the magnitude spectrum of the output. b) Sketch the phase spectrum of the output. c) Give the mathematical expression for the output y(t)
For a continuous time linear time-invariant system, the input-output relation is the following (x(t) the input, y(t) the output): , where h(t) is the impulse response function of the system. Please explain why a signal like e/“* is always an eigenvector of this linear map for any w. Also, if ¥(w),X(w),and H(w) are the Fourier transforms of y(t),x(t),and h(t), respectively. Please derive in detail the relation between Y(w),X(w),and H(w), which means to reproduce the proof of the basic convolution property...
QUESTION 4 a. Determine the frequency spectrum of the signal x(t) b. What is the Nyquist rate for this signal? cos(t) + 5sn3t. me-invariant system defined by, dt dt what will the system output y(t) be? QUESTION 5 The system function of a casual LTI system is given as, 2s a. Find the impulse response of the system. b. Find the step response of the system. A causal discrete-time LTI system is described by, y[n] - (3/4) yin-1+(1/8) yin-21 xn]...
Prob. 5 (a) Let x(t) = u(t) and h(t) = e-looor u(t) + e-lotu(t). -00 <t< oo using graphical convolution(s). Determine y(t) = h(t) * x(t) for Prob. 5 (cont.) (b) Let zln] = uln] and h[n]-G)nuln] + (-))' hnnDetermine vinl -h) rin) for -00n< oo using graphical convolution(s) Prob. 5 (a) Let x(t) = u(t) and h(t) = e-looor u(t) + e-lotu(t). -00
4. Let S be a linear, time-invariant, and causal system whose input x(t) and corresponding output y(t) are shown below: r(t) Page 1 of 2 Please go to next page... y(t) ? (a) Find the impulse response function h(t) of ? (b) Find the output of S when its input is e*, t<0, t2, t20
(a) Let x(t) be a continuous-time signal known to have a first derivative ct) that is a smooth, continuous function over all t in (-00,00). Then the integral [ [x(t) – e(t – 7)][8(t – 3) + 6(t – 10)] dt evaluates to which of the following expressions: 1. x(t)8(t – 3) 2. x(3) 3. x(3) – č(-4) + x(10) – č(3) 4. x(3) – 3(-4) (b) A continuous-time dynamic system is described by the differential equation dyſt) + 4y(t)...