ENSC428_10LeeNyquist

# ENSC428_10LeeNyquist - Digital Transmission through...

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Digital Transmission through bandlimited AWGN channels ENSC 428 – Spring 2008 Reference: Lecture 11 of Gallager Chapter 8 of Proakis & Salehi

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Digital Communication System
Review of AWGN channels ± Use of complete orthonormal set of L 2 . ² Projection, correlation, matched filter. ± No channel distortion, no channel bandwdith limitation ² Separate symbol-by-symbol detection, assuming statistically independent source symbols

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Band-limited channel possibly with distortion () Consider base-band PAM k k st apt kT =− Consider base-band PAM , where ( ) is a stair-well function. k k pt channel Let us model the channel by an LTI system with knwon impulse response ( ). Then, . Received waveform in time duration [ ,( 1) ] contains information of many symbols, not just one. k k ct rt aht kT nt kT k T + + Given ( ), we can celverly design pulse ( ) and a reciever filter to cope with it.
Sufficient statistics for optimal detection of L symbols () {} 1 1 12 1 ( ) ( ), real signal. Each can be one of symbols. Signal part is one of possible signals. Use orthonormal functions , , . , () , L nn n L L n n K L kk n k n rt aht nT wt a M M ff f rr t f t a h t n T f t w t = = = =− + == + " 2 11 0 0 ( ) , 1,2,. .., are sufficient statitstics. ( ), ( ) ( ), ( ) , 1, 2,. .... ,( ) 1 exp k k KL kn k K ft k K t t w t f t k KK ra h t n T f t pra N N φ π = = + +  −−   ∑∑

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For simple implementation () {} 2 11 0 0 2 2 2 1 2 1 Maximum Likelihood decision ,( ) 1 arg max exp arg min , ( ) arg min , ( ) 0 argmin ( ) KL kn k a K ak n k n k k k K L an n ra h t n T f t N N h t n T f t h t n T f t r rt aht nT d t π == = + = −− =− + ∑∑
() ( ) ( ) 2 1 2 1 11 1 arg min ( ) 2 arg min 2( ) arg min , L an n L n n a LL nm L n n a rt aht nT d t rt d t a rtht nTd t aa ht nTht mTd t ar t h tn T d t t r t h t nT dt n = = == = −∞ −−   = +−  = ∫∫ ∑∑ 1,2,. .., are sufficient statsitics. L = ,2 ,. .., tTT L T = Matched filter ( ) ht Signal processing

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Sufficient statistics for optimal detection of L symbols ± We will later consider the signal processing for optimal detection. ± For now, we conclude that we can design an optimal receiver by using a filter followed by a sampler.
Sufficient statistics for optimal detection of L complex symbols For band-pass modulation, we can use complex base-band Representation of signals.

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