1. 1
Semester - IV Academic Year 2012-13
The LNM Institute of Information Technology,
Jaipur, Rajasthan
Mid Semester Exam
ECE219, Signals, Systems and Controls
Time : 1:30 Hours Maximum Marks : 50
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Instructions and information for students
• It is an open book exam. (Book: Signals and Systems by A. V. Oppenheim, A. S. Willsky and S. H.
Nawab)
• Do not write anything on the question paper and also inside the book.
• All questions are compulsory.
• Make suitable assumptions if necessary; write them with your answer.
1) a) Consider a continuous-time linear time-invariant (LTI) system whose response to the signal
x1 (t) in Figure 1(a) is the signal y1 (t) illustrated in Figure 1(b). Determine and sketch the
response of the system to the input x2 (t) depicted in Figure 1(c).
b) Determine the energy for each of the following continuous-time signals:
i) x1 (t) = u(t + 3) − u(t − 3), where u(·) denotes the continuous-time unit step function.
ii) x2 (t) = d x1t(t) .
d
[3+1.5+1.5 = 6 marks]

2. 2
2) Consider a continuous-time LTI system with input x(t) and output y(t) related through the equation
t
y(t) = e−(t−θ) x(θ − 2)dθ.
−∞
a) Determine the impulse response h(t) for this system.
b) Determine the step response s(t) for this system.
c) Determine the response of the system when the input x(t) = u(t + 1) − u(t − 2), where u(·)
denotes the continuous-time unit step function.
[3+3+4=10 marks]
3) a) A discrete-time periodic signal x[n] is real valued and has a fundamental period N = 5. The
nonzero Fourier series (FS) coefficients for x[n] are a0 = 2, a2 = a∗ = 2ejπ/6 , a4 = a∗ =
−2 −4
ejπ/3 . Express x[n] in the form x[n] = A0 + k=1 Ak sin(ωk n + φk ).
∞
b) Consider a discrete-time LTI system with impulse response h[n] = −δ[n + 2] − δ[n + 1] +
δ[n] + δ[n − 1] + δ[n − 2], where δ(·) denotes the discrete-time unit impulse function. Given
that the periodic input signal x[n] to this system is
+∞
x[n] = δ[n − 4k],
k=−∞
determine the FS coefficients of the output signal y[n].
[5+5=10 marks]
4) Let x(t) be a continuous-time periodic signal with fundamental period T and FS coefficients ak .
Derive the FS coefficients of each of the following signals in terms of ak
a) x(2 − t) + x(t − 2)
b) Od{x(t)}, where Od{·} denotes the odd part of the signal x(t)
c) Im{x(t)}, where Im{·} denotes the imaginary part of the signal x(t)
d) x(2 t − 2). [1.25 × 4 = 5 marks]
5) Let X(jω) be the Fourier transform (FT) of the continuous-time signal x(t)
a) Show that
1
x(t) sin(ω0 t) ←→ [X(j(ω − ω0 )) − X(j(ω + ω0 ))]
2j
b) Show that
x(t + T ) − x(t − T ) ←→ 2jX(jω) sin(T ω);
where T is a constant.
c) i) Sketch the signal g(t) = u(t + 4) − u(t + 2) − u(t − 2) + u(t − 4), where u(·) denotes the
continuous-time unit step function.
ii) Use the result of the part 5(b) to determine the FT of the aforementioned signal g(t).
[3.5+3.5+5=12 marks]
6) Consider a continuous-time LTI system whose response to the input x(t) = [e−t + e−3t ]u(t) is
y(t) = [2e−t − 2e−4t ]u(t), where u(·) denotes the continuous-time unit step function.
a) Determine the frequency response H(jω) of the system.
b) Determine the impulse response h(t) of the system. [3.5 × 2 = 7 marks]