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EE 624-Assignment 1

Due Aug. 17th, 2012

1. Using the result y2 (y x)2 + x2, prove the following inequality forthe Gaussian Q function,

Q(x) < ex2/2, for x > 0. (1)

2. In this problem we prove the Cauchy-Schwarz inequality restated herefor convenience,

|< s, r >| =

s(t)r(t)dt

s r , (2)

for any complex valued signals s(t) and r(t), with equality if and onlyif one is a scalar multiple of the other. For simplicity we assume in theproof that s(t) and r(t) are real-valued.

(a) Suppose we try to approximate s(t) by ar(t), a scalar multiple ofr, where a is a real number.That is we are trying to approximates by an element spanned in the subspace by r. Then the error inthe approximation is the signal e(t) = s(t) ar(t). Show that theenergy of the error signal, as a scalar a, is given by,

J(a) = e2 = s2 + a2 r2 2as, r. (3)

(b) Note that J(a) is a quadratic function ofa with a global minimum.Find the minimizing argument amin and evaluate J(amin). TheCauchy-Schwarz now follows by noting that the minimum errorenergy is non-negative. That is it is a restatement of the fact thatJ(amin) 0

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(c) Interpret the condition for equality in the Cauchy-Schwarz in-

equality.(d) Interpret the minimising argument amin as follows: the signal

aminr(t) corresponds to the projection of s(t) along the unit vec-tor in the direction of r(t). The Cauchy-Schwarz inequality thenamounts to saying that the error incurred in the projection hasnon-negative energy with equality if s(t) lies in the sub-spacespanned by r(t).

3. Modified From Tse(Problem 3.5) Consider the use of differentialBPSK scheme proposed in Section 3.1.3 for the AWGN channel

(a) Find a natural non-coherent scheme to detect u[m] based on y[m1] and y[m] assuming the channel is constant across the two sym-bol times. Your scheme doesnt have to be the ML detector.

(b) Analyse the performance of your detector at high SNR. You mayneed to do some approximations.

(c) What is the ideal coherent detector and how does the high SNRperformance of your detector compare to that of the coherent de-tector?

4. MATLAB Assignment- Simulate a BPSK communication channelin MATLAB. Generate the BER curve for BPSK detection in AWGNchannel. Superpose the analytical curve over it. The following MAT-LAB commands are useful:rand,randn,plot.

5. Simulate the repetition coding scheme for n = 7 with BPSK modulationin MATLAB. Plot the probability of codeword error from simulationsfor both the symbol-wise and block detection schemes vs SNR. Also ploton the same graph the probability of error obtained from the theoreticalexpressions and demonstrate that they coincide with the respectivesimulation results.

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