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ECE101_2008FALL_EXAM2__[0]

# ECE101_2008FALL_EXAM2__[0] - Midterm 2 ECElOl Fall 2008...

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Unformatted text preview: Midterm 2 ECElOl Fall 2008 Problem 1 (25 points) The signals sl(t) and sz(t) shown are proposed for use in a digital transmission system. s1(t) 526) 2A 2A A A o T o T a) Find a non—orthonormal basis for the signal space. Write the basis functions, and draw a representation of each, that is fully labeled. (Hint: You may want to represent one as part of the other.) b) Find an orthonormal basis for the signal space. Write the basis functions. c) Draw and label the signal constellation. d) Sketch a receiver block diagram. How many correlators are used by the receiver? c) Find an expression in terms of the Q function for the probability of bit error, assuming the signals are received in white noise with a PSD of No/2 w/Hz. Problem 2 (25 points) Consider an 8—ary QAM system with the following signal coordinates relative to an orthonormal basis {wl , w2}: (iA, 0), (i3A, 0), (iA, i2A). The signals are sent with equal probability, and are corrupted by AWGN with two-sided power spectral density No/2 w/Hz. a) Sketch and label a constellation diagram with the maximum likelihood decision regions. b) Find, in terms of A2: 1. The average symbol energy ii. The average energy per bit c) Find a union bound in terms of the error (Q) function on the probability of symbol error, given that the symbol with coordinates (A, 2A) is transmitted. (Hint: Use a minimum number of terms.) (1) Find the maximum symbol error probability in terms of A, N0 and the Q function, for the maximum likelihood receiver. (Hint: The maximum symbol error probability is the maximum, over all eight possible transmitted symbols, of the symbol error probability.) Problem 3 (12 points) a) Name one advantage and one disadvantage of non—coherent detection over coherent detection. b) What sort of modulation would you suggest for low rate communication to a satellite in deep space, and why? c) What sort of modulation would you suggest for high-speed data communication over a one-kilometer long pair of twisted wires (such as a central ofﬁce to a home telephone line), and why? d) Can QPSK be identical to 4-QAM? Why or why not? Problem 4 (19 points) a) Find a set of orthonormal basis functions to represent the three signals shown. b) What is an expression of each of these three signals, in terms of the found basis functions? c) Draw and label a constellation diagram of these functions. 51(1) :30) 4 4 3 3 2 2 1 1 j 1 2 3 ' .3 1 2 3 t -2 —2 -3 '3 Problem 5 (19 points) Given the two signal constellations in the ﬁgure, each being a form of 8QAM, with minimum distance between nearest adjacent points of 2A. If each point is equiprobable, a) Find the average transmitted power for each constellation. b) Which one is more power efﬁcient? #1. (0"77- 0)) TABLE OF Q(x) VALUES _____—_—_______————————-———-—————— x Q(X) X Q(X) X Q(x) ——_____________________.—._..______ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 23 5.000000e-01 4.6017226—01 4.2074036—01 3.8208860—01 3.445783e—01 3.085375e—01 2.7425316—01 2.419637e—01 2.1185545-01 1.8406016—01 1.5865539-01 1.356661e—01 1.1506976—01 9.680049e-02 8.0756669roz 6.680720e—02 5479929602 4456546902 3593032503 23716566412 2.275013e—02 1.7864425—02 1.3903450-02 1.0724116-02 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 8.197534e-03 6.2096656—03 4.661 1899—03 3.466973e—03 2.555131e—03 1.8658125—03 1.349898e—03 9.676035e—04 6.8713786—04 4.834242o-O4 3.3692919—04 2.3262910—04 1.591086e—04 1.07 7997e—04 7.2348069—05 4.809633M5 3.1671240—05 2.0657525—05 1.3345760—05 8.5398985-06 5.4125426—06 3.3976730—06 2.112456e—06 1.3008096—06 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.933274e-O7 4.7918305—07 2.866516é—07 1 .6982686—‘07 9.9644379—06 5 .7901286—08 3.3320439—08 1.8989566—08 1.071760e—08 5.990378e—09 3.3157429—09 1.817507609 9.8658760—10 5.3034269-10 2.8231618—10 1.4882260—10 7 .768843e—11 4.0]60016—1 1 2.0557900-1 1 1.0420999—11 5.230951c—12 2.600125e-12 1.279813e-12 QC!) 10'1 10‘5 Bounds on Q-function. ...
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• Spring '08
• Faculty
• Orthonormal basis, Quadrature amplitude modulation, Phase-shift keying, basis functions, symbol error probability

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ECE101_2008FALL_EXAM2__[0] - Midterm 2 ECElOl Fall 2008...

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