EN 2063 Signals and Systems
Semester 3
Assignment 1
September 2023
Submission Date: 17th October, 2023
Prathapasinghe Dharmawansa
1. From the viewpoint of an observer on a mobile unit, the signal received from a
continuous wave transmission as the mobile moves with constant velocity may be
represented as a carrier whose phase and amplitude are randomly varying. Many of
the statistical characteristics of this random process can be determined from its power
spectrum. In this respect, capitalizing on certain assumptions, the power spectra of
the individual filed components assume the form, for |f | < fm ,
1
SE (f ) = p
2 − f2
fm
f2
SH (f ) = p
2 − f2
fm
where f denotes the frequency. Determine the inverse Fourier transforms of the above
spectra.1
[10 marks]
2. The moving average of a process x(t) is denoted by
1
y(t) =
2T
Z
t+T
x(λ)dλ.
t−T
(a) Show that y(t) is the output of a linear time invariant ( LTI) system with input
x(t).
[5 marks]
(b) Determine the impulse response of the above LTI system.
1
The Bessel function of the first kind and order n ∈ Z can be expressed as
Z
j −n π jz cos θ
Jn (z) =
e
cos nθdθ
π 0
√
where j = −1.
1
[5 marks]
(c) Find the frequency response H(f ) of the system and identify the nature of
filtration that this system performs.
[5 marks]
(d) Alternatively, take the derivative of y(t) with respect to t and thereby determine
the frequency response of the above LTI system.
[5 marks]
3. A certain second-order system assumes the following frequency response
fn2
H(f ) =
(jf )2 + 2ξfn jf + fn2
where ξ > 0 and fn > 0. Equivalently, we can rewrite it as
H(ω) =
ωn2
(jω)2 + 2ξωn jω + ωn2
where we have used the relations ω = 2πf and ωn = 2πfn .
(a) Determine the values of α1 and α2 such that
H(ω) =
[5 marks]
ωn2
.
(jω − α1 ) (jω − α2 )
(b) Determine the impulse response for ξ > 1.
[5 marks]
(c) Determine the impulse response corresponding to the case ξ = 1.
[5 marks]
(d) Determine the impulse response for 0 < ξ < 1.
[5 marks]
(e) Plot the impulse responses corresponding to ξ = 0.1, 0.2, 0.4, 0.7, 1, 1.5 on the
same graph assuming ωn = 1.
[5 marks]
(f) Generate the exact Bode plots (i.e., 20 log10 |H(ω)| vs log10 ω plot) corresponding
to the above values of ξ and ωn .
[5 marks]
(g) Draw the asymptotic Bode plot corresponding to the above system. [5 marks]
4. (a) Use the first principles to determine the Fourier transform of the following radio
frequency (RF) pulse
[5 marks]
x(t) = cos πt, t ∈ (−1/2, 1/2).
(b) Choose a suitable function g(t) such that
x(t) = g(t) cos πt
and hence determine the Fourier transform of x(t).
[5 marks]
(c) Obtain the second derivative of x(t) with respect to t and hence determine its
Fourier transform.
[5 marks]
2
(d) A certain digital communication transmitter uses the above RF pulse to transmit
binary antipodal signals as follows
N
X
1
s(t) =
ak x t − − k
2
k=0
where ak = {−1, +1}. Plot all possible energy spectral densities (i.e., |S(f )|2 )
for N = 4.
[5 marks]
5. A certain analog filter is realized via the two electronic circuits shown in Fig. 1 and
Fig. 2 in which Vs , Vo denote the input and output voltages, respectively.
R3
C2
R1
R2
Vs
−
Vo
C1
+
Figure 1: Active filter: Multiple feedback topology.
(a) Find the system transfer function H(ω) corresponding to the circuit in Fig. 1.
[5 marks]
(b) Find the system transfer function G(ω) corresponding to the circuit in Fig. 2.
[5 marks]
(c) Transfer function corresponding to the normalized third-order Butterworth filter
is given by
S(ω) =
1/2
.
(1 + jω) (−ω 2 + jω + 1)
Propose a cascade connection of two active filter circuits to realize the above
filter.
[5 marks]
3
C2
R1
R2
Vs
+
Vo
−
C1
Figure 2: Active filter: Sallen-Key topology.
4