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soln_6_229Aspr07

soln_6_229Aspr07 - EECS 229A Spring 2007 Solutions for...

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Unformatted text preview: EECS 229A Spring 2007 * * Solutions for Homework 6 1. We are given a set of k parallel independent additive Gaussian noise channels with noise variances N 1 ,...,N k respectively. A single transmitter is permitted to communicate to a single receiver over this set of channels. The transmitter is power constrained to P . Find the capacity of the system (in bits per use) in each of the following scenarios : (a) The transmitter can distribute its available power among the k channels in any way it likes and can choose the inputs to each channel in any way it likes (as a function of the message it wants to send) subject to the power constraints determined by the way it distributes power over the channels. The receiver receives information from each of the k channels separately. (b) The transmitter is constrained to use exactly the same input in each of the k channels (as a function of the message it wants to send). The receiver receives information from each of the k channels separately. (c) The transmitter can distribute its available power among the k channels in any way it likes. The inputs to each channel have to be scaled versions of a single input (as a function of the message the transmitter wishes to send) and subject to the individual power constraints determined by the way the transmitter distributes power over the channels. The receiver receives information from each of the k channels separately. (d) The transmitter is constrained to use exactly the same input in each of the k channels (as a function of the message it wants to send). The receiver, however, only sees the sum of the outputs of the k channels. Solution : (a) The capacity is given by water pouring the available power over the profile of the noise, as in Section 9.4 of the text. Thus, we choose a level ν = ν ( P ) such that k X l =1 ( ν- N l ) + = P , and the capacity is then given by C ( P ) = k X l =1 max(0 , 1 2 log( ν N l )) . (b) With the given constraints on the communication scheme the channel is in effect a scalar input vector output additive Gaussian noise channel described at each channel use by an equation of the form Y = 1 X + Z 1 where Z is Gaussian with mean zero and covariance matrix diag ( N 1 ,...,N k ), 1 denotes a column vector of ones, and X is power constrained to P k . The capacity of this channel is C ( P ) = max p X I ( X ; Y ) where the maximization is over all distributions satisfying E [ X 2 ] ≤ P k . We have I ( X ; Y ) = h ( Y )- h ( Y | X ) = h ( Y )- k X l =1 1 2 log(2 πeN l ) . Now, the covariance matrix of Y is E [ YY T ] = E [ X 2 ] 11 T + diag ( N 1 ,...,N k ) Hence I ( X ; Y ) ≤ 1 2 log det ( I + E [ X 2 ] N- 1 1 ... N- 1 k ....
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soln_6_229Aspr07 - EECS 229A Spring 2007 Solutions for...

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