20121ee132A_1_EE132A_Problems_HW2

# 20121ee132A_1_EE132A_Problems_HW2 - Communication Systems...

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Unformatted text preview: Communication Systems EE 132A UCLA Winter quarter 2011/2012 Prof. Suhas Diggavi Handout # 8, Due: Thursday, January 26, 2012 Homework Set # 2 Hand in either in class on the due date or before 5:00 PM on the due date at Boelter Hall 6731J. Problem 1 (QPSK Decision Regions) Let H ∈ { , 1 , 2 , 3 } and assume that when H = i you transmit the signal s i shown in the figure. Under H = i , the receiver observes Y = s i + Z .- 6 s s s s s 2 s s 1 s 3 y 1 y 2 (a) Draw the decoding regions assuming that Z ∼ N (0 ,σ 2 I 2 ) and that P H ( i ) = 1 / 4, i ∈ { , 1 , 2 , 3 } . (b) Draw the decoding regions (qualitatively) assuming Z ∼ N (0 ,σ 2 I ) and P H (0) = P H (2) > P H (1) = P H (3). Justify your answer. (c) Assume again that P H ( i ) = 1 / 4, i ∈ { , 1 , 2 , 3 } and that Z ∼ N (0 ,K ), where K = σ 2 4 σ 2 . How do you decode now? Justify your answer. Problem 2 (QAM with Erasure) Consider a QAM receiver that outputs a special symbol called “erasure” and denoted by δ whenever the observation falls in the shaded area shown in the figure. Assume that s is transmitted and that Y = s + N is received where N ∼ N (0 ,σ 2 I 2 ). Let P i , i = 0 , 1 , 2 , 3 be the probability that the receiver outputs ˆ H = i and let P δ be the probability that it outputs δ . Determine P 00 , P 01 , P 02 , P 03 and P δ . 1 b- a y 1 y 2 b s s 1 s 2 s 3 Comment: If we choose b small enough we can make sure that the probability of the error is very small. When ˆ H = δ , the receiver can ask for a retransmission of...
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20121ee132A_1_EE132A_Problems_HW2 - Communication Systems...

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