End Sem EE530

Department of Electronics and Electrical Engineering
Indian Institute of Technology Guwahati

EE530: Communication Systems Theory
End-Semester Examination
Jul.-Nov. Sem., 2021
Time: 2.5 hr. Marks: 50
Instruction: Use of scientific calculator is allowed. Answer all questions, unless otherwise stated.
1) Consider a binary PAM scheme transmitting symbols xn = ±1 at rate 1/T . The received
signal r(t) is given by
√ X
∞
r(t) = E xn p(t − ǫ − nT ) + z(t) ,
n=−∞
where z(t) is the additive white Gaussian noise process with power spectral density N20 ,
and ǫ > 0 is the constant that arises from the propagation delay. The pulse waveform
p(t) has nonzero value only in the duration [0, T ], and its energy spectrum is given by
Sp (f ) = T sinc2 (f T ). Assume that we employ a matched filter receiver, whose impulse
response is matched to p(t). We also assume that the binary source is biased, and therefore
has a probability p > 0.5 for generating the symbol +1.
(i) Determine the decision region for the optimal detection for the above transmission
scenario. Also, compute the corresponding error performance. What happens to the
error performance if p = 0.5?
(ii) What is the minimum possible received signal-to-interference and noise power ratio
(SINR) in the above system? (assume p = 0.5).
(iii) For this above scenario, compute the outage probability, if the minimum received
SINR required for detecting any transmission is set to 0 dB, and if ǫ is assumed to be
uniformly distributed in the interval [0, T /100]. (Hint: here, the event outage refers to
the case where the received SINR falls below the preset threshold value )
[12 + 4 + 5]
2) Let us consider a modulation scheme, which first encodes the information bit sequence
{bk } to the encoded bit sequence {dk }, before applying BPSK modulation on the encoded
data stream, and transmitting it onto a pulse waveform, whose Fourier transform is constant
1 1
with amplitude 1 in [− 2T , 2T ], where T = 0.002 second. The relation between {bk } and
1
{dk } is given by dk = bk ⊕dk−1, where ‘⊕’ operator represents the logical XOR operation.
Note that we have a discrete memoryless source generating the independent bit sequence
{bk }, with following probabilities: P[bk = 1] = p, P[bk = 0] = 1 − p, ∀k.
(a) Assuming that the initial state of the encoder is 0 (i.e., d−1 = 0), find the marginal
probability mass function for the encoder output, i.e., qk , P[dk = 1], ∀k > 0.
(b) Comment on the independence of the encoded sequence {dk : k ≥ 0} for any general
value of p. Find the value of p, that ensures that both {bk } and {dk } are independent
sequences. Does such p exist? If not, then justify your answer.
[5 + (3 + 2)]
3) The non-coherent communication receiver as shown in Fig. 1 (see next page) is used for
a packet radio transmission system. Each packet consists of N binary digits, which are
transmitted using BFSK modulation. We assume a discrete memoryless unbiased source
that generates the binary digits, {bk }. In the absence of fading in the channel, the received
signal at the receiver for the k-th binary digit is given by
( √
2A cos(ω0 t + φ), bk = 0
s(t) = √
2A cos(ω1 t + φ), bk = 1
where kT ≤ t ≤ (k + 1)T . Note that ω0 6= ω1 , and ωi (i = 0, 1) are multiples of 2π T

. The
front-end of the receiver also receives the signal mixed with an additive white Gaussian
noise X(t), with spectral density N20 . So, the input to the receiver is Y (t) = s(t) + X(t).
(i) Assume that A is a constant.What would be the packet error rate for the above
transmission scenario? Comment on the relation between packet error rate and bit
error rate at very high SNR. (Hint: Use the accessory information)
(ii) Now, assume that A is a random variable, and is independently distributed as N (0, β 2).
Compute the bit error rate for this scenario.
(iii) For the scenario in (ii), we assume that a packet is unacceptable, if the bit error rate
exceeds a fixed threshold value p, and we call that event as outage in the system.
Derive an expression for the outage probability.
[(5 + 2) + 3 + 5]
Answer either Q. 4 or Q. 5.
Eb
4) Compare the required N 0
for the coherent reception of BPSK, non-coherent detection of
BFSK, and differentially coherent detection of DBPSK, to the tenth of a dB, for the bit
p Eb
error rate of 10−6 in each case. Assume that Q( 2Eb /N0 ) = 10−6 for N 0
= 10.5 dB.
Eb
Comment on the values of N0 obtained. (Hint: Use the accessory information) [3]
5) Suppose a digital communication system employs a Gaussian-shaped pulse of the form
2
2
x(t) = e−πkt (k > 0). To reduce the level of inter-symbol interference (ISI), we impose the
condition that x(T ) is 1% of x(0), where T is the symbol duration. The pulse bandwidth
is defined as W , such that x
b(W ) is 1% of xb(0), where xb(f ) is the Fourier Transform of
x(t). Determine the value of W , and compare it with a raised cosine spectrum with 100%
roll-off. [3]
Fig. 1. Optimum Receiver for Non-coherent BFSK.
ACCESSORY I NFORMATION
BER for Some well-known Modulation Schemes (Per-bit SNR is γ):
1) BPSK:
p
Pe = Q( 2γ)
2) Differential BPSK:
1
Pe = e−γ
2
3) NC-BFSK:
1
Pe = e−γ/2
2

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Department of Electronics and Electrical Engineering

Indian Institute of Technology Guwahati

where kT ≤ t ≤ (k + 1)T . Note that ω0 6= ω1 , and ωi (i = 0, 1) are multiples of 2π T

Fig. 1. Optimum Receiver for Non-coherent BFSK.

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