COMMUNICATION SYSTEMS
6CCS3COS

COURSEWORK 1
M. R. Nakhai, Department of Engineering, October
2021
There are 3 Questions, answer ALL 3 questions.
Detailed answers are required for each one of the questions.
Graphs for the Q-function values are included.

Submit clearly scanned copies of your written answers by the
deadline of Tuesday 9 November 2021, 5 pm.
1
Graphs for the Q-Function Values

Labels of the vertical axes and the horizontal axes of the graphs are:

The vertical axis: ( ) duexQ
x
u


−
= 2
2
2
1



The horizontal axis: ( )x10log20













2







3
Question 1 [25 Marks]:

The baseband transfer function of an AWGN channel is given as

( )


 −
=
otherwise. 0
kHz500kHz500 1 f
fH

The channel signal-to-noise ratio is SNR=22 dB and the average number of nearest
neighbours of 4=eN is assumed for this system.

a) Find the largest integer number of bits per message, i.e., b , for Square -
Quadrature Amplitude Modulation (SQ-QAM) and Cross - Quadrature
Amplitude Modulation (CR-QAM) transmissions, such that the average
probability of symbol error of 610−eP is maintained.
[10 marks]

b) Find the largest possible symbol rate, i.e.,
T
1
, for this transmission and
explain your reason. [5 marks]


c) Identify the signal constellation in part (a), i.e., SQ-QAM or CR-QAM, which
gives you a higher data rate and compute this larger data rate when the
symbol rate is KHz 800
1
=
T
.
Hint: use the integer value of b that you have computed in part (a).
[3 marks]


d) Find the largest possible data rate for a Pulse Amplitude Modulation (PAM)
transmission on this channel with the same SNR = 22 dB to achieve the same
610−eP at a symbol rate of MHz 1
1
=
T
. Assume 2=eN for PAM.
[7 marks]













4
Question 2 [35 Marks]:

Five equally likely symbols of a constellation are given by the following vectors








+
+
=




−
=





−
=





−
−
=
1
1
,
1
1
,
1
1
,
1
1
3210 xxxx

,






=
3
0
4x

,

where the first dimension indicates the x-coordinate and the second dimension
indicates the y-coordinate. This constellation is used on an Additive White
Gaussian Noise (AWGN) channel with signal to noise ratio SNR=14 dB.

a) Draw the constellation and indicate the decision regions and their boundaries
on the constellation. [5 marks]

b) Find the noise variance 2 and the union bound on the probability of symbol
error, , for this transmission system. [5 marks]

c) Find the Nearest Neighbour Union Bound (NNUB) on the probability of symbol
error, , for this transmission system. [5 marks]

d) Assign binary equivalent of the given symbol indices for the corresponding
points on the constellation and find the upper bound on the probability of bit
error, ̅, for this transmission system. [5 marks]


e) Add two additional points 5 = [
3
0
] and 6 = [
−3
0
] to the given constellation
and assume that the resulting seven constellation points remain equally
likely.

I. Draw the constellation and indicate the decision regions and their
boundaries on the constellation. [7 marks]


II. Find the NNUB on the probability of symbol error for this new
transmission system. Assume SNR=14 dB.
[8 marks)












5
Question 3 [40 Marks]:

The frequency response of a passband channel in baseband, i.e., (), is given in
Figure 1. This channel is used for passband transmission with QAM.

Figure 1: The baseband frequency response of the Channel

The input signal to this channel is
() = √2[()] cos(2) − √2[()] sin(2)

where is the carrier frequency, is the symbol period.

a) Draw the passband channel frequency response ().
[5 marks]

b) Find the baseband equivalent of the channel input signal ().
[5 marks]

c) Assume () = √
2

sinc (
2

) and find the expression for the baseband
equivalent of the channel output (), i.e., in the absence of noise.
[15 marks]


To answer the following remaining parts of this question, use the expression
() = √( + ) sinc
2 (


) :

d) Find ℎ(), i.e., the analytic equivalent of the channel impulse response
ℎ(), in terms of .
[5 marks]

e) Write a simple expression for ℎ̌(), i.e., the Hilbert transform of the channel
impulse response ℎ(), in terms of . [5 marks]

f) Find the expression for the passband output of the channel () in terms of
. [5 marks]


















































































































































































































































































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