ECE 614 – Principles of Digital Communications
Homework 5
Assigned on: 03/29/2016
Due by: 04/14/2016
5.1
Suppose a sequence of binary symbols a k , a k   1 is transmitted over a fading channel.
The received signal is given by
rk  hk ak  nk
 N 
where hk is uniformly distributed in the interval 1  hk  2 ; nk ~ N  0, 0  is the additive
 2 
Gaussian noise. Assume source bits take two binary values with equal probability. All
parameters are independent over time.
(a) Assume the receiver knows hk  precisely. Derive the bit error probability of a ML
detector.
(b) Now assume ak   1, 3. Assume the receiver only knows the distribution of hk , but
not their values. Derive the ML detection rule (It is fine to use a long equation to
describe what you want to compute.)
5.2
Consider a binary modulation scheme defined by s1 t  
sketched below:
sin 3t 
sin 4t 
and s 2 t  
, as
3t
4t
1
s1 t 
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-3
-2
-1
0
1
2
3
0
1
2
3
1
s2 t 
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-3
-2
-1
(a) Place a point in the power-vs-bandwidth plane for this modulation scheme, assuming
ML detection in AWGN and Pe  10 6
(b) At this particular spectral efficiency, what is the gap to capacity (in dB)?
5.3
Put a point in the normalized-power versus normalized-bandwidth plane for the following
so-called “cross QAM” constellations:
M=32
M=128
Both are subsets of larger “square” QAM alphabets. Assume AWGN and ML detection
Pe  10 6 .
5.4
Consider a binary linear block code C with parity-check matrix H and minimum Hamming
distance d H ,min between codewords.
(a) Show that d H ,min is equal to the minimum number of columns of H that can be added
to produce 0.
(b) Use part (a) to show that for all binary linear block codes
d H ,min  n  k  1