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The document is a homework assignment for the EE 7003 Principles of Digital Communication course at IIT Bombay, dated August 16, 2025. It includes various questions related to digital communication principles, such as frequency domain representations, signal processing, and probability theory. The assignment covers topics like upconversion and downconversion of signals, waveform generation using sinusoidal sources, and properties of random variables.

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1 views2 pages

Homework - 1 - 16 08 25

The document is a homework assignment for the EE 7003 Principles of Digital Communication course at IIT Bombay, dated August 16, 2025. It includes various questions related to digital communication principles, such as frequency domain representations, signal processing, and probability theory. The assignment covers topics like upconversion and downconversion of signals, waveform generation using sinusoidal sources, and properties of random variables.

Uploaded by

manas210282
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Indian Institute of Technology Bombay

Department of Electrical Engineering

Handout 2 EE 7003 Principles of Digital Communication


Homework-1 Aug 16, 2025

Question 1) The baseband representation of x(t) = xR (t) + jxI (t) is shown below in the
frequency domain.

X(f )

f (kHz)
−2 −1 0 1 2

This was upconverted to a sufficiently high frequency at the transmitter. While per-
forming downconversion at the receiver kept at a distance, there was a frequency difference
of 500Hz between the receiver and the transmitter. Draw the frequency domain represen-
tation of the baseband received waveform. Assume an ideal low pass filter with sufficient
cutoff to remove the carrier component.

Question 2) A real signal xb (t) was upconverted by a carrier frequency fc = 100MHz. The
frequency domain representation of xb (t) is given below.

Xb (f )

1.5

0.5
f (×106 Hz)
−2 −1 0 1 2

The upconverted signal was passed through a medium which passes all frequencies till
101MHz, but suppresses anything beyond this cut-off. Assuming no phase offset between
the transmitter and receiver oscillators, express the downconverted received baseband wave-
form y(t), as a function of xb (t). Comment whether the receive waveform is real or not?

Question 3) GNURADIO: Suppose you have to generate an approximate square waveform


of 1 + γ kHz frequency, but only using no more than 6 sinusoidal sources, and further linear
operations on them.
(a) Let γ be the last two digits of your roll number. Sketch the generated square waveform
and write the value of gamma near the plot.
(b) Observe the waveform in GNURADIO QT Time Sink. Does the square wave plot stay
the same over time, or are there variations as the simulations run? If you find variations,

Page 1-of-2
comment whether the time variations are related to Gibbs phenomenon or not (with an
one line justification).

Question 4) Consider an odd signal g(x) which is non-increasing in interval x ∈ [−γ, γ],
with γ > 0. Consider the pulse-shaping filter p(t), specified in the frequency domain as

1 1
P (f ) = 1I{∣f ∣≤a} + + g(f − a − γ)I{a≤f ≤a+2γ} .
2 2g(−γ)

If P (f ) = 0 for ∣f ∣ > a + 2γ, find the sampling rate which satisfies ISI free transmission in
the absence of other impairments.

Question 5) For a real non-negative random variable X, show that

X
P (X ≥ a) ≤ E[ ], ∀a > 0,
a

where E denotes the expectation.

Question 6) For a sequence of events An , n ≥ 1 belonging to some sigma algebra F, show


that
(a)
P ( ⋃ An ) ≤ ∑ P (An ).
n≥1 n≥1

(b)
N
P ( ⋃ An ) ≥ ∑ P (An ) − ∑ P (An ⋂ Am ) .
1≤n≤N n=1 1≤n<m≤N

Question 7) With β a positive constant, consider the family of functions

k
xk (t) = sinc (β(t − )) , k ∈ Z,
β

where Z stands for the set of all integers. For all k, l ∈ Z, find

∫R xk (t)xl (t)dt,

and express the answer in terms of appropriate Kronecker delta measure.

Question 8) Given a real random vector X̄ having pdf fX̄ (x), find the pdf of Ȳ = AX̄,
where A is an invertible matrix. You have to show the derivation.

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