ECE2111 review questions for topics 5–8
The following questions cover concepts and skills introduced in topics 5–8. They are indica- tive of the style of problems on the second mid-semester test for ECE2111. (The test will have fewer problems though!) Any of the problems on tutorial sheets 6–9 are also useful for review, as are problems at the end of chapters 3, 4, and 5 of the recommended text A. V. Oppenheim and A. S. Willsky, ‘Signals and Systems’.
1. Suppose the input x and the output y of a continuous-time LTI system are related by the differential equation
y(t) = −dy(t) + 0.5x(t) − dx(t) + 0.5dx2(t) for all t dt dt dt2
subject to the condition of initial rest.
(a) Find the frequency response H of the system.
(b) If the input to the system is x(t) = 2 + cos(t), find an expression for y(t) for all t. 2. Suppose the input x and the output y of a discrete-time LTI system are related by the
difference equation
subject to the condition of initial rest.
y[n]=ay[n−1]+x[n−1]−0.5x[n−2] foralln
(a) Assuming that |a| < 1, find the frequency response H of the system. Express your
answer in terms of a.
(b) Use the table of discere-time Fourier transforms, together with the properties of the DTFT, to find the impulse response of the system. Express your answer in terms of a.
(c) If the input to the system is x[n] = jn for all n then the output is y[n] = jn+3 for all n. Find the value of a.
x(t) = cos(2t) + sin(t) − 3 cos(3t) (a) Find the fundamental frequency ω0 of x.
for all t.
find X−3, X−2, X−1, X0, X1, X2, X3.
(c) Let the frequency response of a continuous-time LTI system be
0 Sketch H(ω) vs ω for −2 ≤ ω ≤ 2.
if ω ≥ 1.5
0 if ω ≤ −1.5 1.5+ω if−1.5≤ω≤0
1.5−ω if0≤ω≤1.5
x(t) = Xkejω0kt k=−∞
(d) If x is the input to an LTI system with frequency response H, find the output y of the system.
4. A discrete-time LTI system has impulse response
h1[n] = u[n] − u[n − 3].
(a) Find the frequency response H1 of the system.
(b) A second discrete-time LTI system has impulse response
h2[n] = δ[n] + δ[n − 2]. Find the frequency response H2 of the system.
(c) The system with impulse response h1 and the system with impulse response h2 are connected in series to form a single system with frequency response H. Find an expression for H(ω) for all ω.
(d) Find h1 ∗ h2 without using the convolution formula.
(e) Find a difference equation relating the input x and the output y of the system with
frequency response H.
5. A continuous-time LTI system has frequency response that satisfies
a for |ω| ≤ 5π/4
and its value for 5π/4 < |ω| < 7π/4 is left unspecified.
The input to the system is
x(t) = Xkejω0kt k=−∞
where ω0 = π and Xk = (0.5)|k| for all k.
(a) Find the fundamental period of x.
(b) Is x a real signal? Explain why or why not.
(c) If the output of the system is
y(t) = x(t) − (1 + cos(πt))
find the values of a and b.
b for |ω| > 7π/4