# Hw4 - Boston University Department of Electrical and Computer Engineering ENG EC515 Digital Communication(Fall 2009 Problem set 4 Assigned Mon 12

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Unformatted text preview: Boston University Department of Electrical and Computer Engineering ENG EC515 Digital Communication (Fall 2009) Problem set 4 Assigned: Mon 12 Oct Suggested reading: Class-notes, handouts, and Ch. 4 textbook Oﬃce hours: M, W 4:00–6:00pm PHO438 Notation: Q( x) denotes the Q-function deﬁned as Q( x) = = 10 log10 (SNR). ∞ √1 exp(−t2 /2)dt 2π x Due: Mon 19 Oct and SNR in decibels (dB) Problem 4.1 (16-ary QAM) One of 16 equally likely messages is to be transmitted over a vector (zeromean) AWGN channel with noise variance σ2 . The transmitter uses the 16-ary QAM signal constellation j i where the points have coordinates {( d(22−1) , d(22−1) )}, i, j = −1, 0, 1, 2. (a) 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. Compare your answer with the method discussed in the class which is based on the idea of two independent 4-ary PAM decoders. (b) Is it possible to modify this signal constellation to reduce the probability of error without changing Emax ? If yes, explain how. If no, explain why. (c) Plot the graph of log10 (P16−PAM ) versus SNR16−PAM in dB. Also plot the graph of log10 (P16−QAM ) e e versus SNR16−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 oﬀer? Problem 4.2 The two-dimensional signal constellation {(d/2, d/2), (−d/2, d/2), (−d/2, −d/2), (d/2, −d/2)}, with d > 0, is used for transmission over a vector (zero-mean) AWGN channel with noise variance σ2 . All four constellation points are equally likely. The optimum receiver for this signal constellation is modiﬁed to put out a special symbol, called an erasure, when the received signal coordinates are “close” to the optimum decision boundaries to indicate that the receiver is “unsure”. Speciﬁcally, the optimum decision regions are modiﬁed as shown in the following ﬁgure. When the received signal coordinates fall in the cross-hatched 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 1 0 11111111111111 1 00000000000000 0 11 00 1 0 111111111 000000000 11111111111111 1 1 00000000000000 0 0 111 000 1 0 11111111111111 1 1 00000000000000 0 0 d 11 00 11111111111111 1 1 00000000000000 0 0 111 000 11 00 11111111111111 1 1 00000000000000 0 0 1 0 1 0 1 0 111111111 000000000 11111111111111 1 00000000000000 0 1 0 1 0 11111111111111 1 00000000000000 0 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 d 2 region, the receiver outputs a special symbol to indicate an erasure. (a) Compute the exact expression for the probability of erasure in terms of the Q function. 1 (b) Compute the exact expression for the probability of wrongly decoding a message. Note: an erasure is not regarded as a wrong decision. Problem 4.3 The following two-dimensional signal constellation is used for transmission over a vector (zero-mean) AWGN channel with noise variance σ2 . All constellation points are equally likely. 3 −3 3 −3 (a) What is the average energy of this signal constellation? (b) Sketch the ML decoding regions. (c) What is the minimum distance dmin of this signal constellation? (d) What is the union-bound on the probability of error of the ML decoder? (e) What is the nearest-neighbor union-bound on the probability of error of the ML decoder? 2 ...
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## This note was uploaded on 11/09/2010 for the course ECE 515 taught by Professor Venkateshsaligrama during the Fall '09 term at BU.

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