ECE300 Communication Theory Problem Set VI: Digital Communications

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1. For binary transmission over the AWGN channel of two symbols with correlation coe¢ cient , the exact probability of error for coherent detection is:
Pe = Q
p
(1 Re ()) b

For binary orthogonal transmission with noncoherent detection, the exact probability
of error is:
Pe =
1
2
e
b=2
Note that in MATLAB, the inverse of Q () is the function qfuncinv.
For binary FSK, the minimum value of  (which achieves minimum probability of error
for coherent detection) is:
 = 0:217
Compute b
(in dB) necessary to achieve Pe = 105
for various situations:
(a) Coherent demodulation of binary FSK with the optimal  as above.
(b) Coherent demodulation of binary orthogonal FSK.
(c) Noncoherent demodulation of binary orthogonal FSK.
Remark: You should observe that the SNR requirements in the order given are increasing!
2. Assuming an initial phase of 0

, specify the sequence of phases for encoding the following binary sequence in =4 DQP SK, using the diagram provided in the notes.
01 11 11 00 10
3. Reference the constellation shown. Note that:
d
2
min = 4E0
Also be careful: E0 is a reference value but it does not equal the average energy per
symbol.
(a) Assign binary codes to the symbols using Gray coding rules.
(b) Express the energy per bit Eb in terms of E0.
(c) Determine all distinct distances between pairs of points, sorted as d1 < d2 <    ,
with d1 = dmin, and the number of pairs k1; k2; etc. As a check, Pki = 8 7
(remember going from symbol i to symbol j, and symbol j to symbol i is counted
twice!). [You can do this manually, or code this in MATLA
(d) Note that, with Gray coding, the probability of bit error Pb is related to the
probability of symbol error Pe as Pb =
1
3
Pe (as there are 8 symbols and we used
Gray coding). Set up the union bound to obtain an expression of the form:
Pb 
X iQ
p
i b

for constants i
; i
. SpeciÖcally, use MATLAB to compute vectors containing the
list of i
; i values.
(e) Obtain an approximation for the upper bound keeping only the term with the
(smallest/largest) i
(which should you keep, if you keep only one?).
(f) Use MATLAB to superimpose graphs of the bounds, with vertical axis probability
on a log scale, and horizontal axis b
in dB, with 2dB  b  8dB. [The di§erence
between the curves may not be visible!] It turns out the size of the error decreases
with increasing b
. Also, the probability approximations get very small, so the
best way to compare the two would be to examine their ratio (rather than their
di§erence):
1 Perr;only one term=Perr;all terms
Compute this value at 2dB and at 8dB.