ECE2111 review questions for topic 1–3
The following 10 questions cover concepts introduced in topics 1–3. They are indicative of the style of problems on the first mid-semester test for ECE2111. Any of the problems on tutorial sheets 1–4 are also useful for review, as are many problems at the end of chapter 1 and 2 of the recommended text A. V. Oppenheim and A. S. Willsky, ‘Signals and Systems’.
1. Let x be a discrete-time signal defined by
x[n] = 2 cos(4πn/5) + 1 for all n.
Which of the following is true?
(a) x is odd and periodic with fundamental period 5 samples. (b) x is even and periodic with fundamental period 5/2 samples.
(c) x is even and periodic with fundamental period 5 samples. (d) x is odd and periodic with fundamental period 5/2 samples.
2. Thegraphofxfor−1≤t≤4isshownbelow: x(t)
Which of the following is the graph of |x(3 − t)| restricted to −1 ≤ t ≤ 4? (a)
|x(3 − t)| 1
−1 1 2 4 |x(3 − t)|
|x(3 − t)|
|x(3 − t)| 1
3. Suppose x is a continuous-time bounded signal with period T. If S is a BIBO stable, time-invariant, continuous-time system then which of the following statements is true of the output y = S(x)?
(a) The output satisfies y(t + T ) = y(t) for all t but we cannot determine whether the output is bounded.
(b) The output is bounded and we cannot determine whether it is periodic.
(c) The output is bounded and y(t + T ) = y(t) for all t.
(d) We cannot determine whether the output is bounded or whether it is periodic.
4. Let S be a linear system. Let h0 be the output of the system when the input is the unit impulse. Let h1 be the output when the input is Delay1(δ), the unit impulse delayed by one sample. Which of the following is true?
(a) h1[n]=h0[n−1]foralln
(b) If the system input is x[n] = δ[n] + 2δ[n − 1] then the system output is h0[n] + 2h1[n]
(c) The output of the system is the convolution of the input and h0. (d) The output of the system is never the zero signal.
5. A discrete-time linear time invariant system with input x and output y is described by the following difference equation
y[n]=1.1y[n−1]+x[n] foralln
subject to the condition of initial rest. Which of the following statements are true?
(a) The impulse response of the system is h[n] = (1.1)n for all n and the system is not BIBO stable.
(b) The impulse response of the system is h[n] = (1.1)nu[n] for all n and the system is BIBO stable.
(c) The impulse response of the system is h[n] = (1.1)nu[n] for all n and the system is not BIBO stable.
(d) The impulse response of the system is h[n] = (0.1)nu[n] for all n and the system is BIBO stable.
6. The following block diagram describes the relationship between the input x and output y of a discrete-time system.
Which of the following difference equations describes the same input-output relationship?
(a) y[n]=x[n]+2x[n+1]+0.5x[n+2]foralln. (b) y[n]=x[n]+2x[n−1]+0.5x[n−2]foralln. (c) y[n]=x[n]+2y[n−1]+0.5x[n−2]foralln. (d) y[n]=x[n]+2y[n+1]+0.5x[n+2]foralln.
7. A discrete-time signal x is defined by
x[n] = 2δ[n + 2] + δ[n] + 3δ[n − 2] + δ[n − 3].
Which of the following is a plot of x[n] vs n for −3 ≤ n ≤ 4? (a)
−3 −2 −1 0 1 2 3 4
−3 −2 −1 0 1 2 3 4 x[n]
7 6 5 4 3 2 1
−3 −2 −1 0 1 2 3 4 x[n]
−3 −2 −1 0 1 2 3 4
8. Which of the following systems are causal and time-invariant? (a) The discrete-time system defined by the difference equation
y[n]=0.5y[n−2]+x[n] foralln 4
subject to y[n] = 0 for all n ≤ 0.
(b) A discrete-time LTI system with impulse response h[n] = cos(πn)u[n + 1] for all n
(c) The continuous-time system with input x and output y related by y(t) = (x(t − 1))2 for all t.
(d) The continuous-time system with input x and output y related by y(t) = tx(t) for all t.
9. Which of the following is true for a discrete-time infinite impulse response LTI system?
(a) The system is never BIBO stable. (b) The system is always causal.
(c) The system is always BIBO stable.
(d) A possible output of the system is the zero signal, i.e., y[n] = 0 for all n.
10. Let x be the discere-time signal defined by x[n] = (−1)n for all n. If x is the input to the discrete-time LTI system with impulse response defined by
h[n] = n(u[n] − u[n − 4]) for all n, which of the following describes the output y of the system.
(a) y[n] = 2(−1)n−1 for all n. (b) y[n] = 2(−1)n for all n.
(c) y[n] = 3(−1)n−1 for all n. (d) y[n] = 3(−1)n for all n.