sol4

# sol4 - Boston University Department of Electrical and...

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Boston University Department of Electrical and Computer Engineering ENG EC515 Digital Communication ( Fall 2009 ) Solution to Problem set 4 Posted: Mon 12 Oct Suggested reading: Class-notes, handouts, and Ch. 4 textbook Notation: Q ( x ) denotes the Q-function deﬁned as Q ( x ) = 1 2 π R x exp( - t 2 / 2)d t and SNR in deci- bels (dB) = 10 log 10 (SNR). Problem 4.1 ( 16 -ary QAM) [10pts] One of 16 equally likely messages is to be transmitted over a vector (zero-mean) AWGN channel with noise variance σ 2 . The transmitter uses the 16-ary QAM signal constellation where the points have coordinates { ( d (2 i - 1) 2 , d (2 j - 1) 2 ) } , i , j = - 1 , 0 , 1 , 2. (a) [4pts] Find the exact expression for the minimum probability of error in terms of the Q function using the method which involves enumerating all types of decoding regions. Com- pare your answer with the method discussed in the class which is based on the idea of two independent 4-ary PAM decoders. (b) [3pts] Is it possible to modify this signal constellation to reduce the probability of error without changing E max ? If yes, explain how. If no, explain why. (c) [3pts] Plot the graph of log 10 ( P 16 - PAM e ) versus SNR 16 - PAM in dB. Also plot the graph of log 10 ( P 16 - QAM e ) versus SNR 16 - QAM in dB. For small values of error probability, roughly how much more SNR in decibels is needed to get the same probability of error? What insight does this o er? Solution: [10pts] E max d d φ 1 φ 2 (0 , 0) decision regions boundary 4 corner decision regions 8 non - corner 1

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(a) [4pts] There are 3 types of decoding regions: I ) 4 corners, II ) 4 squares, and III ) 8 non- corner boundary rectangular strips. Let P I c | m denote the probability of correct decoding given that message M = m belonging to region I was transmitted. Similarly, let P II c | m and P III c | m denote
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• Fall '09
• VenkateshSaligrama
• Quadrature amplitude modulation, Phase-shift keying, Emax, constellation diagram, signal constellation

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sol4 - Boston University Department of Electrical and...

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