Lecture 6. Goals: Determine the optimal threshold, filter, signals for a binary communications problem VI-1

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1 Lecue 6 Goals: Deemine e opimal esold, file, signals fo a binay communicaions poblem VI-

2 Minimum Aveage Eo Pobabiliy Poblem: Find e opimum file, esold and signals o minimize e aveage eo pobabiliy. s s Z γ dec s γ dec s n P e P e π Peos ansmied Peos ansmied Pobabiliy s ansmied VI-

3 π π π Pobabiliy s ansmied e aveage pobabiliy of eo is P ep e π Pe π () Le ŝ ŝ s s d d oupu due o s alone oupu due o s alone Since we assume a e eceive will decide s if e oupu of e file is lage an a esold and s if i is smalle, we need o assume a. ŝ ŝ P e P Z γ s ansmied VI-3

4 If s is ansmied en Z akes e fom Z ŝ η wee η is a Gaussian andom vaiable wi mean and vaiance σ N ; σ N N d N H f d f us P e P P ŝ η γ Q ŝ η γ ŝ γ () P e P P Z ŝ γ s ansmied η γ s ansmied P η γ ŝ VI-4

5 s Q γ ŝ (3) Subsiuing () and (3) ino () yields P e γ s π Q ŝ γ π Q γ ŝ (4) e poblem is o minimize e eo pobabiliy ove all coices of γ. s s and VI-5

6 π π Sep : Minimize P e ove γ Facs used: Q Q x x x e x π π e u du Meod: Se e deivaive of P e wi espec o γ equal o. d P e dγ exp ŝ π γ exp γ ŝ π VI-6

7 π exp ŝ π exp ŝ γ γ ŝ exp γ ŝ ŝ γ σn π π γ γŝ ŝ ŝ γŝ γ σ N ln π π γ ŝ ŝ σ N ln π π ŝ ŝ γ σ N ln π π ŝ ŝ ŝ ŝ γ op ŝ σ N ln π π ŝ ŝ (5) Special Case: If π π en γ op ŝ ŝ VI-7

8 Wa is P e fo e opimal esold? ŝ γ op ŝ ŝ σ N ln π π ŝ ŝ ŝ ŝ ŝ ŝ σ N ŝ ln π π ŝ γ op ŝ ŝ ŝ ŝ ln π (6) π Definiion: ŝ f g s ŝ s f g s s d s d s s d VI-8

9 us fom (6) Similaly Remembe a σ N Le λ ŝ γ op ŝ γ op N N s s s s s s ln π (7) π ln π (8) π d(9) d () s s s s s d s s d VI-9

10 Ē s s s s d E E s s Ē Ē E E s Ē s Ē () Combining (7), (8), (9), (), and () ŝ γ op s N N s ln π π s s Ē N 4Ē N s s ln π π VI-

11 Le λ s s en ŝ γ op λ β λ Ē N β 4Ē N ln π π Similaly γ op ŝ λ β λ VI-

12 Summay of Sep : γ op ŝ σ N ln π π ŝ ŝ πq λ ŝ P e γ op s s πq λ β λ β λ λ s s Ē N β 4Ē N ln π π Ē E E s s Ē VI-

13 πq λ λ Find e opimal file Sep : o minimize e aveage pobabiliy of eo Meod: Fis sow a P e is an deceasing funcion of λ by sowing e deivaive is negaive. en find e a maximizes λ (us minimizing P e ). P e s s Pe γ op πqλ s s β λ β λ P e λ π e π λ π β λ β λ β λ π exp exp β λ λ λ π β β e β β λ λ β λ π π π β λ VI-3

14 exp λ β λ π β λ e β π β λ e β β Ē N N 4Ē ln π π ln π ln π π π e β π π e β π π π e β ππ π e βπ π d P e dλ e λ π e β λ λ ππ β λ ππ β λ β λ VI-4

15 op s Since d P e dλ so a Pe is minimized by maximizing λ. λ s s Fom Scwaz s inequaliy s s s us s s λ wi equaliy if s. Coose λ ). Fo opimal esold and opimal file s P e γ op s s πq β πq β s o e signals. s is called e maced file because i is maced γ op op s E E N ln π π VI-5

16 Fo e opimal file e oupu due o signal alone ae ŝ E Ē ŝ Ē E If π π en P eq Q Ē N VI-6

17 Find e opimal signals s eo. Sep 3: and s Meod: P e depends on e signal only oug Ē and. Ē E E o minimize e aveage pobabiliy of s I is obvious a we could jus incease e enegy o infiniy and ge eo pobabiliy. Insead we will fix Ē and vay e signals o vay. Again we sow a P e is an inceasing funcion of and en coose e signals o minimize. s Ē P e πq β πq β Ē N β 4Ē N ln π π VI-7

18 d P e d π e β π β π e β π β e β π e β β π e β β e β π π Ē N d P e d Fom Scwaz s inequaliy s s Ē s Ē s VI-8

19 wi equaliy if s Ks K. Fo s Ks E E E E wi equaliy if E E. (Aimeic mean Geomeic mean). wo signals s and s ae said o be anipodal if s s Opimal signals ae anipodal. E N β N 8E ln π π VI-9

20 If π π en P eq Q E N VI-

21 Aside: Scwaz s inequaliy: Fo any f g f f g g Since e polynomial in is neve negaive ee mus be eie no zeos o a double zeo. us e discimanan mus be no be posiive. 4 f g 4 f g f g f g f g Equaliy occus wen f x Kgx. If K is posiive e inequaliy on e ig side becomes equaliy and if K is negaive e inequaliy on e ig side becomes equaliy. is is Scwaz s inequaliy. VI-

22 Aside: Aimeic mean Geomeic mean: Le a and a be eal nonnegaive numbes. a a wi equaliy if a a a aa a a a a a a a a a a a 4aa 4a a a a a a wi equaliy if a a VI-

23 s s ŝ ŝ σ N n Summay γ dec s γ dec s N d s s d d N H f d f VI-3

24 πq λ op P e γ s s π Q ŝ γ π Q γ ŝ Sep : Opimize wi espec o γ. P e γ op s s πq λ β λ β λ Ē N β 4Ē N ln π π λ s s s s s γ op ŝ ŝ ŝ σ N ŝ ln π π Sep : Opimize wi espec o. P e γ op s s πq β πq β VI-4

25 op ˆβ Ē N β 4Ē N ln π π γ op op op s E E s N ln π π e maced file. Sep 3: Opimize wi espec o s and s. P e γ op s op s op πq ˆ ˆβ πq ˆ ˆβ ˆ Ē N N 8Ē ln π π s s op s s s s op op s VI-5

26 γ op op s op s op N ln π π VI-6

27 SPECIAL CASE π P e π Q ŝ γ Q γ ŝ Sep : Opimize wi espec o γ P e Q λ Ē N λ s S Ē E E E s d E s s Ē s d VI-7

28 ˆ γ op ŝ ŝ Sep : Opimize wi espec o P e Q Ē N op s s maced file γ op E E Sep 3: Opimize wi espec o s and s. P e Q ˆ Ē N s s VI-8

29 op s γop VI-9

30 Example: s s Ap s Ap Baseband signals op Ap Assume s ansmied s A A Ap p p Ap p d d d VI-3

31 p e oupu due o signal alone: VI-3

32 ŝ A e oupu due o noise is a Gaussian andom vaiable wi mean zeo and vaiance σ N N 4A Le be e sampling ime. Since e signal ou is a maximum wen and e noise vaiance does no depend on e sample ime e opimum AN VI-3

33 sampling ime is. Equivalen fom of opimal eceive s s γ dec s γ dec s Z d s s s s d VI-33

34 Z s s s d s d If s and s ae ime limied o en Z s s d γ dec s γ dec s s s is is called e Coelaion Receive. VI-34

35 Example: Binay Pase Sif Keying (RF signals) s Acos ω p s s s i i Acos ω p Acos ω iπ p VI-35

36 ω s s γ dec s γ dec s Acos VI-36

37 ω Assume ω oω nπ E i s i d A A A A cos ω d cos sin ω ω d P eq E N Q A N π π γ VI-37

38 ω ω ω Subopimal Receives s i i Acos p ω nπ Z γ dec s γ dec s n cos Claim: σ n N (powe equals /). P e Q ŝ Q ŝ 4. e faco of / is due o muliplying by cos VI-38

39 ωs Poof: σ n E N 4 N 4 N 4 E n n N δ n cos N cos ω cos s s d sin ω cos cos ω ω ω n d s ω ω cos cos cos ωs ωs dds dds dds VI-39

40 povide a ω ŝ i oω nπ. i A i A i Acos ω p low pass cos d ig feq. d w d P e Q ŝ i VI-4

41 Example: Single pole RC file RC e e u RC u d e RC e d VI-4

42 d u d σ N N 4 e u d e e d d e e VI-4

43 e e e ŝ We would like o maximize ŝ ŝ is maximized a andσ N N 4 is consan e opimal sampling ime is is σn. Since VI-43

44 esuls in a signal-o-noise aio of ŝ SNR 3 e A N 4 Maximize w... f e f e e e e Le x en xe x e x x x e x e x VI-44

45 x e x We can numeically solve is o ge x is yeilds a signal-o-noise aio of 56 and so 56 RC 8. SNR A 95 N Loss due o subopimal eceive =.89 db VI-45

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