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notes11 - ECE 562 Fall 2010 Dierential Phase Modulation and...

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ECE 562 Fall 2010 Differential Phase Modulation and Detection Consider MPSK signaling on an ideal AWGN channel with phase offset φ ( t ) that may change with time. Over N symbol intervals r ( t ) = N n =1 E e n g ( t - ( n - 1) T s ) e ( t ) + w ( t ) . Suppose φ ( t ) changes slowly with time so that we can assume that it is constant over two consec- utive symbol intervals. For standard MPSK θ n = θ m n = 2 π ( m n - 1) M , m n ∈ { 1 , 2 , . . . , M } . We know that this scheme performs poorly if we cannot estimate φ at the receiver. But if φ ( t ) changes slowly, a differential modulation approach can be taken where the sequence { θ n } is generated from { m n } as θ n - θ n - 1 = δ m n = 2 π ( m n - 1) M (with θ 0 = 0) . Sufficient statistics for demodulation over the N -symbol interval are given by R n = r ( t ) , g ( t - ( n - 1) T s ) , n = 1 , 2 , . . . , N. Note that under the slowly changing phase assumption R n = E e n e n + W n where φ n φ n - 1 for all n . Symbol m n affects not just R n , but R for all n . Thus, for optimum detection, we would need
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