## Description

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

XiQ

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.