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0981_cn541_fnl

# 0981_cn541_fnl - N 1 = 4 N 2 = 1 N 3 = 2 and N 4 = 6 Assume...

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CN 541 – Information Thoery – Spring 2010 Final Examination 1. (30%) Channel capacity. Calculate the channel capacity and provide the capacity-achieving distri- bution of the input alphabet (i.e. p ( x )) for each of the following channels with probability transition matrices: (a) (10%) X = Y = { 0 , 1 } p ( y | x ) = " 1 0 0 1 # (1) (b) (10%) X = Y = { 0 , 1 } p ( y | x ) = " 3 4 1 4 1 4 3 4 # (2) (c) (10%) X = Y = { 0 , 1 , 2 } p ( y | x ) = 1 0 0 0 3 4 1 4 0 1 4 3 4 (3) 2. (25%) Diﬀerential entropy. Let X 1 and X 2 be two zero-mean Gaussian random variables with variances σ 2 1 and σ 2 2 , respectively. X 1 and X 2 are independent. (a) (15%) Find h ( X 1 ) ,h ( X 2 | X 1 ) , and h ( X 1 ,X 2 ) . (b) (10%) Find h (2 X 1 + X 2 ,X 1 + 3 X 2 ) . 3. (25%) Parallel Gaussian channels. Consider a set of 4 parallel Gaussian channels: Y i = X i + Z i , i = 1 , 2 , 3 , 4 , (4) where Z i N (0 ,N i ) , i = 1 , 2 , 3 , 4 are independent, and
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Unformatted text preview: N 1 = 4 , N 2 = 1 , N 3 = 2 , and N 4 = 6 . Assume that there is a transmit power constraint: E " 4 X i =1 X 2 i # ≤ 8 . (5) Find the channel capacity. What is the capacity-achieving allocation of transmit power? 4. (20%) Fading channel. Consider an additive noise fading channel: Y = V X + Z, (6) where Z is additive noise, and V is a random variable representing fading. Z, V, and the input X are independent. Argue that knowledge of the fading factor V improves the capacity by showing that I ( X ; Y | V ) ≥ I ( X ; Y ) . (7) 1...
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