2/6/01 Types of Communication Analog: continuous variables with noise ⇒ P{error = 0} = 0 (imperfect) Digital: decisions, discrete choices, quantized, noise ⇒ P{error} → 0 (usually perfect) The channel can be radio, optical, acoustic, a memory device (recorder), or other objects of interest as in radar, sonar, lidar, or other scientific observations. A1 Message S1, S2,…or SM Modulator ⇒ v(t) channel; adds noise and distortion M-ary messages, where M can be infinite Demodulator; hypothesize H1…HN (chooses one) message S1, S2,…or SM Lec14.10-1 Received2/6/01 Optimum Demodulator for Binary Messages P2OKERRORS2 P1ERROROKS1 H2H1 Probability a prioriHypothesis: Message: A2 How to define V1,V2? Demodulator design V1 V2 vb va 2-D case E.G. "Vv "Vv 22 11 ⇒∈ ⇒∈ vb va vc v t Lec14.10-2 measured v H " H "2/6/01 Optimum Demodulator for Binary Messages va How to define V1,V2? V1 V2 vb va 2-D case E.G. "Vv "Vv 22 11 ⇒∈ ⇒∈ vb vc v t Minimize { } { }2 1e 1 1 2 2V VP P P S P S∆ = = +∫ ∫ { } { }11 2 2 1 1VP S S= + −⎡ ⎤⎣ ⎦∫ { } { } 1 21 V 1V 1 =+ ∫∫Note: replace with ∫ 1V A3 Lec14.10-3 measured v H " H " error p v dv p v dv P pv P pv dv vd S v p vd S v p2/6/01 Optimum Demodulator for Binary Messages { } { }[ ]∫ −+= 1V 11221e dvPP A4 { } { }1 1 1 2 2i i , S S∋ > Very general solution ↑ [i.e., choose maximum a posteriori Lec14.10-4 S v p P S v p P error To m nim ze P choose V P p v P p v P (“MAP” estimate)]2/6/01 Example: Binary Scalar Signal Case A5 S 2 n21 ∆∆∆ =σ== { } 2)Av(1 e 1 −− π =∴ { }2 e 1 − π = :P21 = { }2p { }1p 0 AA/2 Decision threshold if 21 PP = 1P2P v 1P 2P 0 AA/2 v 2H 21 PP >Threshold if { }p v S(bias choise toward H2 and a priori information) Lec14.10-5 noise Gaussian , N , volts O S , volts A N2 N 2 S v p N2 v N 2 S v p P If S v S v2/6/01 Rule For Defining V1 : (Binary Scalar Case) A6 { } { } " P P 1 1 2 2 1 ⇒>= ∆A ( ) "Pn 112 ⇒> AAA (binary case) “Likelihood ratio” or (equivalently) For additive Gaussian noise, ( )[ ] ( ) ( )12 ?222 PAv-n AAA >−=+= ( ) ( ) 2 21 1 2 1 A P ANV i , P2A 2 A +∴ > > +A A   { } { }1 1 1 2 2S S∋ > { } 22 1 pv S e 2N −= π Lec14.10-6 " V S v p S v p " V P n P n N2 vA 2 N2 N2 A -v bias 2N n P choose f v or v n P   Choose V P p v P p v v2N2/6/01 Binary Vector Signal Case For better performance, use multiple independent samples: { } { } 1 2? 2 1 P P p p >= ∆A 2S { } { } m 12 m 1 i 1i1 S S = =π (independent noise samples) { } ( )Sv ii 2 i1ie 1 −− π =Where { } ( ) ( )m 2i 1ii1 vSi m 1p v S e 2N = − −∑ = π B1 v(t) 0 t 1S 12 m Lec14.10-7 S v S v Here P v ,v ,...,v p v N2 N 2 S v p 2N2/6/01 Binary Vector Signal Case B2 { } ( ) ( )Sv mi m 1i 2 i1i e 1 p ∑ −− − π = Thus the test becomes: ( ) ( )i i m m ?2 2 2i 2 i 1 1i1 i1 P1n v S v S n2N P = = ⎡ ⎤ = − − − >⎢ ⎥ ⎣ ⎦ ∑ ∑AA A 2 2Sv − 2 1Sv − 2 2 2 1 2 1 2 2 1 1But v S v S 2v S S S S S S− − − • + • + • − • ( ) 1 1 2 2 21 21 1 PS S S Sv V i v S S N n2 P ∗ ∗ • − • ⎛ ⎞∈ • − > + ⎜ ⎟⎝ ⎠ ATherefore if energy E1 = E2 if P2 = P1 { } { } ?1 2 12 pv S P = Ppv S ∆ >A Lec14.10-8 N2 N 2 S v 2v =− ff Bias = 0 Bias = 02/6/01 Binary Vector Signal Case B3 Multiple hypothesis generalization: This “matched filter” receiver minimizes Perror ?i iii i i j i SSH if f v S N n P all f2 ∆ ≠ • = • − + >A ∑τ m ∑τ m operator H1 H2 + -× × 1S 2S v i=1 i=1 ( ) 1 1 2 2 21 1 21 1 PSS S SS S N n2 P • − • ⎛ ⎞• − > + ⎜ ⎟⎝ ⎠ A Lec14.10-9 Choose V iff v2/6/01 Graphical Representation of Received Signals B4 2 2 i i i1 AS P = =∑ )t(1S 11S 12S 13S t 3-D Case: n n 1S 2S 0 1V 2V Lec14.10-10 verage energy2/6/01 Design of Signals iS 1ESenergyAverage 2 == B5 E.G. consider +1 -1 vs. 21 SS −= 2S 2S1 += 0S2 = ( ) 21 PE == 0 1S 2S 3S 4S boundaries b n E.G. 2-D space ( ) 12111 4321 SS : = ( ) 1312111 SS = 3-D space 1S 2S 3S4S a b ratio a bBetter Lec14.10-11 P 2 decision S , S , S , S , S for S , S ,2/6/01 Design of Signals iS B6 2-D space: 16-ary signals 161 or magnitude/phase vs 2iS 1iS n-Dimensional sphere packing optimization unsolved equilateral triangle slightly lower average p{error} 19 Lec14.10-12 S ,..., S signal energy for same2/6/01 :eP∆ = Binary case: For additive Gaussian noise, optimum is ( ) 1 2 2 2 2 1 211 P P nN2 SSSSvif A+ − >−• nSvWhere += )t(nN so 2 === ∆ [ ]( )No × ( ) ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ + − <−•= 1 2 2 2 2 1 21 P P nN2 SS SSP 1 A ( ) { } 2 1 2 21 2 1 S S P pn S S N n b2 P ⎧ ⎫− −⎪ ⎪ = • − < + =⎨ ⎬ ⎪ ⎪⎩ ⎭ A D1 Lec14.10-13 p{error} of n Calculatio " H " B kT B N sideband double ,B2 Hz W 2 1-S e v p p y <− y • 2B[GRVZM] -b • 2BLec14.10-14 2/6/01 Duality of Continuous and Sampled Signals ( ) [ ] { } bypP P nN2 SSSSnpP B2b 1 2 2 21 GRVZMB2y 21Se 1 − <= ⎪ ⎪ ⎭ ⎪ ⎪⎬ ⎫ ⎪ ⎪ ⎩ ⎪ …