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132A_1_Final-W07

# 132A_1_Final-W07 - UCLA Electrical Engineering Dept EE132A...

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UCLA — Electrical Engineering Dept. EE132A: Communication Systems Final Exam — Solutions 1. A source generates information bits to be digitally modulated at a bit rate of R b = 18000 bps. The bits are grouped in triplets, say b 1 b 2 b 3 . If b 1 = 0 , then the bit pair b 2 b 3 is sent to the modulator M1. If b 1 = 1 , then the bit pair b 2 b 3 is sent to the modulator M2. M1 is a QPSK modulator that generates the signals s (1) i ( t ) = 2 E 1 T cos 2 πf c t + ( i - 1) π 2 , i = 1 , 2 , 3 , 4 , 0 t T, and M2 is a QPSK modulator which generates the signals s (2) j ( t ) = 2 E 2 T cos 2 πf c t + ( j - 1) π 2 , j = 1 , 2 , 3 , 4 , 0 t T, with f c T 1 , E 1 = 2 E 0 and E 2 = (2+ 2) 2 E 0 , for a given E 0 . The combined modulator output is then one out of the eight signals S = { s (1) 1 ( t ) , s (1) 2 ( t ) , s (1) 3 ( t ) , s (1) 4 ( t ) , s (2) 1 ( t ) , s (2) 2 ( t ) , s (2) 3 ( t ) , s (2) 4 ( t ) } . (a) What is the symbol period, T ? R s = 1 /T = R b /b = 18000 / 3 = 6000. (b) Draw a geometric representation of the constellation of signals, S , and the corre- sponding decision regions. s (1) 1 = [ 2 , 0] E 0 , s (1) 2 = [0 , 2] E 0 , s (1) 3 = [ - 2 , 0] E 0 , s (1) 4 = [0 , - 2] E 0 , s (2) 1 = [2 + 2 , 0] E 0 , s (2) 2 = [0 , 2 + 2] E 0 , s (2) 3 = [ - 2 - 2 , 0] E 0 , s (2) 4 = [0 , - 2 - 2] E 0 . See Fig. 1. (c) Compute d min in terms of E 0 . From the drawing, d min = 2 E 0 . (d) Compute P ( e | s (1) 1 ) and P ( e | s (2) 1 ) in terms of E 0 /N 0 , and again in terms of E b /N 0 , where E b is the energy per bit. We approximate the error probabilities by considering only the constellation point closest to the point under consideration: P ( e | s (1) 1 ) 3 Q (( d min / 2) / N 0 / 2) and P ( e | s (2) 1 ) Q (( d min / 2) / N 0 / 2) where Q (( d min / 2) / N 0 / 2) = Q ( 2 E 0 /N 0 ). (e) Compute the (approximate) probability of symbol error P 8 in terms of of E b /N 0 . Because the symbols are equiprobable, and because of the above, P 8 2 Q ( 2 E 0 /N 0 ).

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132A_1_Final-W07 - UCLA Electrical Engineering Dept EE132A...

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