ECE 614 – Principles of Digital Communications
Homework 4
Assigned on: 02/23/2016
Due by: 03/10/2016
4.1
Suppose a binary symbol a   1 is transmitted across three separate channels to a receiver.
The receiver observations are r1 , r2 , r3 , as sketched below:
n1
a   1
E
n2
n3
r1
r2
r3
 N 
The noise n1 , n 2 , n3 , are i.i.d. N  0, 0  .
 2 
(a) Sketch the block diagram of the ML detector. Simplify it as much as possible.
(b) Find the MAP decision when r1  0.1 , r2  0.2 , r3  0.3 , and a priori pmf of the
transmitted symbol is Pra  1  0.4 . Assume E  4 and N 0  0.2 .
4.2
Consider a one-dimensional model for PAM in AWGN, where the received sample is:
r an
where a is the transmitted symbol, and where n ~ N 0,1 is independent of a. Instead of the
usual pulse-amplitude modulation scheme, in which a is chosen from an alphabet of possible
amplitudes, consider a noise-variance-modulation scheme, in which the transmitted symbol is a
Gaussian random variable, and information is conveyed by modulating its variance
 2  1,2,3,4. Assume all four variances are equally likely.
(a) Find the ML decision regions.
(b) Find a numerical value of the probability of error of the ML detector.
4.3
Instead of an additive-Gaussian-noise channel, consider a multiplicative-Gaussian-noise
channel, where the received sample r is related to the transmitted symbol a  A by:
r  an
 N 
Assume N  0, 0  is independent of a. Consider a binary alphabet A  1,2 , and assume
 2 
that Pra  1  2 Pra  2.
Find a numerical value for r such that the ML decision is different from the MAP decision.
Assume that N 0  1 .
4.4
Consider the scalar channel r  a  n , where the symbol alphabet is a   1 , and where
the noise n is independent of a . If the noise is Gaussian, then the ML detector reduces to a
simple quantizer: aˆ ML  sgn r  . The Guassian pdf is said to be symmetric because it satisfies
f  n  f n .
Give an example of a symmetric noise pdf, satisfying f  n  f n , for which the ML
detector does not reduce to aˆ ML  sgn r  .
A precise answer is not needed, a rough sketch of the pdf is sufficient.