MIT 6 661 - Types of Communication (23 pages)

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Types of Communication



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Types of Communication

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Pages:
23
School:
Massachusetts Institute of Technology
Course:
6 661 - Receivers, Antennas, and Signals
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Types of Communication Analog continuous variables with noise P error 0 0 imperfect Digital decisions discrete choices quantized noise P error 0 usually perfect Message S1 S2 or SM Modulator v t channel adds noise and distortion M ary messages where M can be infinite Received message S1 S2 or SM Demodulator hypothesize H1 HN chooses one 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 Lec14 10 1 2 6 01 A1 Optimum Demodulator for Binary Messages Hypothesis H1 Message S1 OK Probability H2 a priori ERROR P1 S2 ERROR vb va E G V1 P2 v Demodulator design 2 D case OK vb v measured t V2 va v V 1 H1 v V2 H2 Lec14 10 2 2 6 01 How to define V1 V2 vc A2 Optimum Demodulator for Binary Messages v vb E G V1 2 D case v measured va V2 vb t How to define V1 V2 va v V H 1 1 v V2 H2 vc Minimize Perror Pe P1 p v S1 dv P2 p v S2 dv V2 V1 replace with V 1 P1 P2p v S2 P1p v S1 dv V1 Note Lec14 10 3 2 6 01 V1 p v S1 dv V2 p v S1 dv 1 A3 Optimum Demodulator for Binary Messages Pe P1 P2p v S2 P1p v S1 dv V1 To minimize Perror choose V1 P1p v S1 P2p v S2 Very general solution i e choose maximum a posteriori P MAP estimate Lec14 10 4 2 6 01 A4 Example Binary Scalar Signal Case S1 A volts S2 O volts p v S1 p v S 2 2 2N 1 v A e 2 N p v S 2 If P1 P2 P2 bias choise toward H2 and a priori information noise 1 e v 2N 2 N p v S1 P1 0 Decision threshold if P1 P2 Lec14 10 5 2 6 01 2 n N Gaussian A 2 v A p v S P2 A 2 0 H2 P1 A Threshold if P1 P2 v A5 Rule For Defining V1 Binary Scalar Case Choose V1 P1p v S1 P2p v S2 1 v 2 2N p v S2 e 2 N binary case Likelihood ratio P A 2 V1 p v S2 P1 or equivalently An A An P2 P1 V1 p v S1 For additive Gaussian noise An A v A 2 2N v 2 2N 2vA A 2 2N An P2 P1 A 2 2NAn P2 P1 A N or v A n P2 P1 choose V1 if v 2A 2 A Lec14 10 6 2 6 01 bias A6 Binary Vector Signal Case For better performance use multiple independent samples v t S1 p v S1 P2 A p v S2 P1 t 0 1 2 m S 2 m Here P v1 v 2 vm S1 p vi S1 independent noise samples i 1 v i S1i 2 2N 1 e Where p v i Si 2 N i 1 m p v Si Lec14 10 7 2 6 01 1 2 N m e vi S1i 2 2N B1 Binary Vector Signal Case m p v Si 1 e v i S1i 2 2N i 1 2 N m Thus the test becomes p v S1 P2 A p v S2 P1 m m P2 2 2 1 An A vi S2i vi S1i An 2N i 1 P1 i 1 v S2 But v S2 2 2 v S1 2 2 v S1 2v S2 2v S1 S2 S2 S1 S1 S S S S 2 2 N An P2 Therefore v V1 iff v S1 S2 1 1 P 2 1 Lec14 10 8 2 6 01 Bias 0 if energy E1 E2 Bias 0 if P2 P1 B2 Binary Vector Signal Case P V1 iff v1 S1 S2 S1 S1 S2 S2 N An 2 2 P1 m S1 v i 1 operator H1 H2 m S2 i 1 Multiple hypothesis generalization Choose Hi if fi v Si Si Si N An Pi all f j i 2 This matched filter receiver minimizes Perror Lec14 10 9 2 6 01 B3 Graphical Representation of Received Signals 2 3 D Case Average energy Si Pi 2 i 1 S1 V1 n V2 0 S2 S1 t S13 S11 n t S12 Lec14 10 10 2 6 01 B4 Design of Signals Si 1 E G consider S1 S 2 S1 2 vs Average energy S 2 E G 2 D space for S1 S 2 S3 S 4 E 1 1 S2 S2 b a S4 n S4 S1 S2 S1 0 decision boundaries Lec14 10 11 2 6 01 E 2 P1 P2 3 D space S1 S11 S12 S13 S1 S11 S12 S3 S2 0 Better b S3 b ratio a B5 Design of Signals Si 2 D space S1 S16 Si2 19 Si1 16 ary signals or magnitude phase vs equilateral triangle slightly lower average signal energy for same p error n Dimensional sphere packing optimization unsolved Lec14 10 12 2 6 01 B6 Calculation of p error Pe Binary case For additive Gaussian noise optimum is 2 S1 S2 H1 if v S1 S2 2 Where v S n 2 2 P N An 2 P1 N n t NoB kTsB No 2 W Hz 1 2B double sideband 2 2 S S 1 2 P Pe S1 p v S1 S2 N An 2 2 P1 2 S1 S2 P2 p n S1 S2 N An p y b 2 P1 Lec14 10 13 2 6 01 y 2B GRVZM b 2B D1 Duality of Continuous and Sampled Signals 2 S1 S2 P2 Pe S1 p n S1 S2 N An p y b P1 y 2B GRVZM 2 b 2B Conversion to continuous signals assuming nyquist sampling is helpful here S1 t 0 t T S1 2BT samples sampling theorem T y n t S1 t S2 t dt No WHz 1 2 noise o T No 1 2 b S1 t S2 t dt An P2 P1 2o 2 B 0 B f 2 2BT 1 2 2 y E y E n j S1j S2 j 2B j 1 Lec14 10 14 2 6 01 D2 Calculation of Pe continued 2 2BT 1 2 2 y E y E n j S1j S2 j 2B j 1 2 2BT 2BT 1 E nin j S1i S2i S1j S2 j 2B i 1 j 1 where E nin j N ij 2 T 2 No 2 2 1 y N S1 S2 S t S t dt 1 2 2 o 2B N oB T 2B S1 t S 2 t dt 2 o Lec14 10 15 2 6 01 D3 Calculation of Pe continued 2 T 2 N 1 2y N S1 S2 o S1 t S2 t 2 dt 2 o 2B NoB T 2B S1 t S 2 t dt 2 o p y 1 2 2y e y 2 2 2y GRVZM Therefore P y Pe S1 …


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