Indian Institute of Technology Kharagpur
Department of Electrical Engineering
Subject: Signals and Networks (EE21101)
Part-I Signals and Systems
UG-II Autumn 2016-2017
Tutorial-III
Q1. Find the impulse response h[n] of the system described by the difference equation 8y[n] + 6y[n − 1] = x[n].
Is the system causal? Is the system having damped oscillation?
Q2. (a) Determine the impulse, step and pulse response of the first-order RC series circuit, shown in Fig. 1b.
Assume that the system is initially relaxed. The waveform of the pulse is shown in Fig. 1a.
R
+
x(t)
+
x(t)
−0.5
0.5
t (s)
C
y(t)
−
−
(a) Rectangular Pulse.
(b) RC series circuit.
Figure 1: Rectangular pulse and RC series circuit.
(b) In the RL series circuit, shown in Fig. 2b the input signal voltage is vi (t) and the output signal voltage
is v0 (t). Find the impulse response of the system. Sketch the step response.
x(t)
1
T
+
T →0
− T2
T
2
R
+
vi(t)
t (s)
L v0(t)
−
(a) Impulse function.
−
(b) RL series circuit.
Figure 2: Impulse function and RL series circuit.
Q3. (a) Find the impulse response of length-2, length-4 and length-N moving average filters.
(b) Find the impulse response of length-2 moving difference filter.
(c) if the input signal |x[n]|<Bx , discuss about the Bounded-Input-Bounded-Output (BIBO) stability of
the length-2 moving average filter. hM A = { 12 , 12 }
(d) if the input signal |x[n]|<Bx , discuss about the Bounded-Input-Bounded-Output (BIBO) stability of
the length-2 moving difference filter. hM D = { 12 , − 12 }
Q4. (a) If x(t) = u(t) − u(t − 1), sketch y(t) = x(t) ∗ x(t).
(b) If the impulse response h[n] = {4, 3, 2, 1} and the input signal x[n] = {−3, 7, 4}, n = 0, 1, ..., are given,
find the output sequence y[n] = h[n] ∗ x[n].
(c) Compute and sketch the impulse response sequences of the following systems, shown in Fig. 3. Infer
about the BIBO stability of these systems.
y[n]
x[n]
+
y[n]
x[n]
+
+
+
−
0.9
D
0.9
(a) LTI positive feedback system.
x[n]
D
(b) LTI negative feedback system.
y[n]
y[n]
x[n]
+
+
+
+
+
+
−
1.1
D
1.1
(c) LTI positive feedback system.
D
(d) LTI negative feedback system.
Figure 3: LTI feedback system.
Q5. (a) Determine Trigonometric Fourier series coefficients for the periodic pulse train, shown in Fig. 4a.
(b) If the periodic pulse train is modified, shown in Fig. 4b, what will be the revised coefficients of
Trigonometric Fourier series?
(c) If the periodic pulse train is modified, shown in Fig. 4c, what will be the revised coefficients of
Trigonometric Fourier series? Note that this is an odd function of t.
(d) If the periodic pulse train is modified, shown in Fig. 4d, what will be the revised coefficients of
Trigonometric Fourier series?
(e) Draw the maginitude spectrum (|ck | vs. Ωk ) in each of these cases, where ck is the complex coefficient
of Fourier series.
x(t)
x(t)
1
−1.5
− 21
0
1.5
1
2
2
3
− 32
t
0
− 21
T0
1
2
x(t)
1
t
0
1
t
(b)
x(t)
0.5
−1
− 31
T0
(a)
−0.5
3
2
t
0
−1
−1
T0
(c)
1
T0
(d)
Figure 4: Pulse train.
Q6. (a) Consider the periodic pulse train depicted in Fig. 5. The duty cycle of a pulse train is defined as
the ration of the pulse width to the period, d = Tτ0 . Assuming the value of d = 0.1 and d = 0.2,
draw the magnitude spectrum of Exponential Fourier Series (EFS) coefficients |ck | versus k. Check the
magnitude of c0 and compare the roll-off of these two spectra.
(b) Consider a sinusoidal signal (sin()) of unity amplitude and time period T0 . This sinusoidal signal is
converted to its full wave rectified version. Compute the EFS coefficients.
5π
Q7. (a) For the CT periodic signal x(t) = 2 + cos( 2π
3 t) + 4sin( 3 t), determine the fundamental frequency Ω0
∞
P
and the Fourier series coefficients ck such that x(t) =
ck ejΩ0 t .
k=−∞
x(t)
1
−T0
− τ2 0
τ
2
T0
t
Figure 5: Periodic pulse train.
(b) Use the Fourier series analysis equation to calculate the coefficients ck for the CT periodic signal
1.5, 0 ≤ t < 1
x(t) =
−1.5, 1 ≤ t < 2
with the fundamental frequency Ω0 = π.
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