ELEN90057 - Communication Systems - Engineering Assignment Help

Assignment Task

Questions 1. Signal Space, Modulation and Error Probability Consider a transmitter which uses a baseband 4-ary modulation scheme with transmitted signals s1(t), s2(t), s3(t) and s4(t) as shown in Fig. 1. Assume the mapping of bits to symbols is 10, 01, 11, and 00, respectively. The channel adds white Gaussian noise, n(t), with noise power spectral density N0 2 so that the received signal is r(t) = sm(t) + n(t).

(a) Sketch the transmitted signal if the input bit sequence is 01001011.

(b) Show that the waveforms φ1(t) and φ2(t) shown in Fig. 2 are orthonormal.

(c) Express s1(t), s2(t), s3(t) and s4(t) as linear combinations of φ1(t) and φ2(t).

(d) Calculate the energy of sm(t) for m ∈ {1, 2, 3, 4}, the average energy per symbol (Es) and the average energy per bit (Eb).

(e) Sketch the signal space or constellation diagram and show the optimal decision regions, assuming equiprobable signals.

(f) Let r = [r1, r2] T be the projection of the received signal onto the signal space so that r1 = hr(t), φ1(t)i and r2 = hr(t), φ2(t)

i. What is the joint conditional probability density function (pdf) of r given sm, p(r1, r2 | sm) for m ∈ {1, 2, 3, 4}?

(g) Derive exact expressions for the symbol error probability (Pe) and the bit error probability (Pb) given s1 is transmitted. Now give an expression for the overall symbol error probability. (You might only be able to write these in terms of twodimensional integrals.)

(h) Use the union bound to give an upper bound on the probability of symbol error given s1 is transmitted. Now give an upper bound on the overall symbol error probability (Pe).

(i) (Harder) Use Matlab to calculate and plot the symbol error probability versus Eb/N0 for 0 dB ≤ Eb/N0 ≤ 20 dB. Also plot the upper bound obtained in

(h). (It might be that your error probability expressions don’t appear to involve Eb, but you have determined Eb in (d) and so can set Eb/N0 by choosing N0 appropriately.)

(j) Please list all of the resources you used to answer this question including references to the subject notes, workshops and problem sets as well as any websites or textbooks etc.

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