ELE637_Chapter_9.pdf - Gaussian Channel Besma SMIDA ECE 595K Chapter 7 Fall 2011 B Smida(ECE 595K Gaussian Channel Fall 2011 1 36 Today\u2019s outline

ELE637_Chapter_9.pdf - Gaussian Channel Besma SMIDA ECE...

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Gaussian Channel Besma SMIDA ECE 595K: Chapter 7 Fall 2011 B. Smida (ECE 595K) Gaussian Channel Fall 2011 1 / 36
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Today’s outline Discrete-time Gaussian Channel Capacity Continuous Typical Set and AEP Gaussian Channel Coding Theorem Bandlimited Channel Parallel Gaussian Channels Channels with Colored Gaussian Noise Gaussian Channel with Feedback B. Smida (ECE 595K) Gaussian Channel Fall 2011 2 / 36
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Discrete-time Gaussian Channel Capacity Definition of Gaussian Channel X i Y i Z i Definition: (Gaussian channel) Discrete-time channel with input X i , noise Z i , and output Y i at time i . This is Y i = X i + Z i , where the noise Z i is drawn i.i.d. from N (0 , N ) and assumed to be independent of the signal X i . Average power constraint n i =1 x 2 i n P E [ X 2 ] P . E [ Y 2 ] = E [( X + Z ) 2 ] = E [ X 2 ] + 2 E [ X ] E [ Z ] + E [ Z 2 ] P + N B. Smida (ECE 595K) Gaussian Channel Fall 2011 3 / 36
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Discrete-time Gaussian Channel Capacity Information Capacity Definition: The information capacity with power constraint P is C = max E [ X 2 ] P I ( X ; Y ) . I ( X ; Y ) = h ( Y ) - h ( Y | X ) = h ( Y ) - h ( X + Z | X ) = h ( Y ) - h ( Z | X ) = h ( Y ) - h ( Z ) 1 2 log(2 π e ( P + N )) - 1 2 log(2 π eN ) = 1 2 log(1 + P N ) The optimum input is Gaussian and the worst noise is Gaussian B. Smida (ECE 595K) Gaussian Channel Fall 2011 4 / 36
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Discrete-time Gaussian Channel Capacity Gaussian Channel Code Encoder Decoder X i Y i Z i W 1 : M ˆ W 1 : M Definition: An ( M , n ) code for the Gaussian channel with power constraint P consists of the following: 1 An index set { 1 , 2 , ..., M } . 2 An encoding function x : { 1 , 2 , ..., M } X n , yielding codewords x n (1) , x n (2) , ..., x n ( M ), satisfying the power constraint P ; that is for every codeword n i =1 x 2 i ( w ) nP , w = 1 , 2 , ..., M . 3 A decoding function g : Y n { 1 , 2 , ..., M } . B. Smida (ECE 595K) Gaussian Channel Fall 2011 5 / 36
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Discrete-time Gaussian Channel Capacity Coding Theorem for the Gaussian Channel Definition: A rate R is said to be achievable with a power constraint P if there exists a sequence of (2 nR , n ) codes with codewords satisfying the power constraint such that the maximal probability of error λ ( n ) tends to zero. The capacity of the channel is the supremum of the achievable rates. Theorem: The capacity of a Gaussian channel with power constraint P and noise variance N is C = 1 2 log 1 + P N bits per transmission . Conversely, the rates R > C are not achievable. B. Smida (ECE 595K) Gaussian Channel Fall 2011 6 / 36
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Discrete-time Gaussian Channel Capacity Sphere Packing Each transmitted x i is received as a probabilistic cloud y i Cloud ’radius’ = Var( Y | X ) = nN Energy of y i constrained to n ( P + N ) so clouds must fit into a hypersphere of radius n ( P + N ) Volume of hypersphere r n Max number of non-overlapping clouds: ( nP + nN ) n 2 ( nN ) n 2 = 2 n 1 2 log(1+ P N ) Max rate is 1 2 log(1 + P N ) B. Smida (ECE 595K) Gaussian Channel Fall 2011 7 / 36
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Continuous AEP Jointly Typical Set Definition: The set J ( n ) of jointly typical sequences { ( x n , y n ) } with respect to the density f X , Y ( x , y ) is defined as follows: J ( n ) = ( x n , y n ) R n × R n : - 1 n log f X ( x n ) - h ( X ) < , - 1 n log f Y ( y n ) - h ( Y ) < , - 1 n log f X , Y ( x n , y n ) - h ( X , Y ) < , B. Smida (ECE 595K) Gaussian Channel Fall 2011 11 / 36
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Continuous AEP Properties of Jointly Typical Set 1 Individual pdf: ( x n , y n ) R n × R n , log f X , Y ( x n , y n ) = - nh ( X ) ± n 2 Total Prob.: Pr( J ( n ) ) > 1
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