ECE 614 – Principles of Digital Communications
Homework 6
Assigned on: 04/12/2016
Due by: 04/28/2016
6.1
A binary communication scheme uses two equiprobable messages m  1, 2 corresponding to
signals s1 t  and s2 t  , where
s1 t   xt 
s2 t   xt  1
and xt  is shown below
x(t)
1
1
3
4
t
The noise is white Gaussian with power spectral density of N 0 / 2 .
1. Design an optimal matched filter receiver for this system. Carefully label the diagram and
determine all the required parameters.
2. Determine the error probability for this communication system.
3. Show that the receiver can be implemented using only one matched filter.
4. Now assume that s1 t   xt  and
 xt  1
s 2 t   
 xt 
with probabilit y 0.5
with probabilit y 0.5
In other words, in this case for m  1 the transmitter always sends xt  , but for m  2 it is
equally likely to send either xt  or xt  1 . Determine the optimal detection rule for this
case, and find the corresponding error probability.
6.2
Compare the performances of the following systems. Assume binary PAM modulation and
message bits are equally probable. Assume the objective bit error probability is Pb  10 6 .
(1) No error correction coding.
(2) (7,4) Hamming code with a soft (ML) decoder.
(3) (7,4) Hamming code with a hard (two step) decoder.
6.3
Suppose 4 binary symbols x1 , x2 , x3 , x4 are transmitted through a channel. The binary
symbols are independently distributed, and each takes values 0,1 with equal probability.
Your received symbols are y1  0.6 , y 2  0.9 , y3  1.3 , and y4  0.3 .
(a) Suppose the channel model is y k  xk  nk , where n k , for all k, are independent
Gaussian noise with zero mean and unit variance. Find the ML detection output of the
message symbols.
(b) Suppose the channel model is y k  xk  0.5 xk 1  nk , where n k , for all k, are
independent Gaussian noise with zero mean and unit variance. Assume x0  0 . Find the
ML detection output of the message symbols.
6.4
A fading channel can be represented by the vector channel model r  as  n , where a is a
random variable denoting the fading, whose density function is given by the Rayleigh
distribution
2ae  a
p a   
0
2
a0
a0
1. Assuming that equiprobable signals, binary antipodal signaling with bit energy Eb ,
N0
, and coherent detection are
2
employed (meaning that a is known at the receiver), what is the structure of the optimal
receiver?
additive white Gaussian noise with spectral density
2. Show that the bit error probability in this case can be written as
Pb 
Eb / N 0
1 1

2 2 1  Eb / N 0
and for large SNR values we have
Pb 
1
4 Eb / N 0
3. Assuming an error probability of 10 5 is desirable, determine the required SNR per bit (in
dB) if (i) the channel is nonfading with a  1 and (ii) the channel is a fading channel. How
much more power is required by the fading channel to achieve the same bit error probability?
6.5
Assume a sequence of  1 -valued binary symbols bn are transmitted through an intersymbol interference channel with additive Gaussian noise. The channel is modeled by
rn  bn  0.5bn  1  vn
where vn is a sequence of independent Gaussian random variables each with zero mean
and variance  2 . Assume optimal sequence detection, i.e. the Viterbi algorithm, is
implemented. Use MATLAB simulations to plot the bit error probability versus  2 of the
communication system. You should use the log scale on both  2 and the obtained error
probability. The plot should at least cover error probability from 10 2 to 10 5 . Please
provide the MATLAB code and the plot in your answer.