ece4305_L11 - Optimal Detection ECE4305 Software-Dened...

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Optimal Detection ECE4305: Software-Defined Radio Systems and Analysis Professor Alexander M. Wyglinski Department of Electrical and Computer Engineering Worcester Polytechnic Institute Lecture 11 Professor Alexander M. Wyglinski ECE4305: Software-Defined Radio Systems and Analysis
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Optimal Detection Signal Vector Framework Probability Density Function of n Decision Rules Recall Simple Digital Transceiver Model I Receiver only observes the corrupted version of s ( t ) by n ( t ), namely r ( t ) I Usually n ( t ) represents the culmination of all noise sources into a single variable I Detection problem: Given r ( t ) for 0 t T , determine which s i ( t ), i = 1 , 2 , . . . , M , is present in r ( t ) Transmitting Device Receiving Device Additive Noise Channel s i (t) r(t) n(t) Figure: Simple Digital Transceiver Model. Professor Alexander M. Wyglinski ECE4305: Software-Defined Radio Systems and Analysis
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Optimal Detection Signal Vector Framework Probability Density Function of n Decision Rules Mathematical Formulation I Decompose waveforms s i ( t ), n ( t ), and r ( t ) into a collection of weights applied to a set of orthonormal basis functions: s i ( t ) = N X k =1 s ik φ k ( t ) , r ( t ) = N X k =1 r k φ k ( t ) , n ( t ) = N X k =1 n k φ k ( t ) I Thus, waveform model r ( t ) = s i ( t ) + n ( t ) now becomes r ( t ) = s i ( t ) + n ( t ) N X k =1 r k φ k ( t ) = N X k =1 s ik φ k ( t ) + N X k =1 n k φ k ( t ) r = s i + n Vector Model Professor Alexander M. Wyglinski ECE4305: Software-Defined Radio Systems and Analysis
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Optimal Detection Signal Vector Framework Probability Density Function of n Decision Rules n ( t ) is Gaussian I We know that the noise vector element n k is equal to: n k = T Z 0 n ( t ) φ k ( t ) dt (1) I Since n ( t
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