Applied Research Laboratory. Decision Theory and Receiver Design

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1 Decson Theor and Recever Desgn

2 Sgnal Detecton and Performance Estmaton Sgnal Processor Decde Sgnal s resent or Sgnal s not resent Nose Nose Sgnal? Problem: How should receved sgnals be rocessed n order to detect sgnals n nose? What knd of detecton erformance can be exected? The aroach to soluton: Must be statstcal, snce nose s nvolved Imlement hothess testng

3 Sgnal Detecton Inut to detector s sgnal lus nose. Requrements exressed n terms of robablt of detecton robablt of false alarm Aled Research Laborator Threshold for declarng detecton s set based on models for sgnal and nose Nose background estmaton can be erformed on data to mrove model. Oututs of detector are threshold crossngs Performance defned b recever oeratng characterstc ROC curve robablt of detecton vs. robablt of false alarm for a artcular SNR.

4 Detecton In Nose 3 sgnal nose mean nose nose mean T T sgnal + nose Tme

5 Performance Crtera: Detecton Threshold Probablt of detecton P D Probablt of false alarm P FA These crtera are not ndeendent: a lower threshold ncreases P D, but also ncreases P FA. Theoretcal ROC s used to set thresholds. True test s erformance n water.

6 Recever Oeratng Curve ROC Probablt of Detecton P D T Decreasng threshold T Probablt of False Alarm P FA

7 Possble Hotheses: H : Onl nose s resent Hothess Testng H : Sgnal s resent n addton to nose Stes n formng hotheses: Process arra outut to obtan a detecton statstc x. Calculate the a osteror robabltes PH x and PH x. Pck the hothess whose robablt s the hghest: the maxmum a osteror, or MAP estmate. P H P H x x, <, Choose H Choose H

8 Hothess Testng Cont d Equvalentl, we can use Baes rule to wrte: P H x P x P x H P H P H x P x P x H P H PH and PH are called a ror robabltes Then the test can be wrtten: P H P H x x P x H P x H P H P H, <, Choose H Choose H

9 Hothess Testng Cont d Aled Research Laborator An equvalent test s P x P x H H > P H P H P H P H,, Choose Choose H H P x H λ x s called the lkelhood rato P x H

10 Asde: Baes Rule and Notaton Probablt denst functons are often used to descrbe contnuous random varables: P x Baes Rule as wrtten for robabltes also holds for robablt denst functons df. x A comact notaton s used n what follows: Lkelhood raton test wrtten n terms of r x dx x H x x H x x x > P H P H P H P H,, Choose H Choose H

11 A Frst Examle: Constant Sgnal The ossble nuts are: H H : : x t x t n t μ + n t Nose Sgnal onl lus nose If n t s Gaussan dstrbute d and μ x H x and x H x, then are as shown below : x πσ x πσ / / x ex σ ex x μ σ

12 At tme t, we receve a sgnal xt. Knowng x and x, we can calculate the lkelhood rato λ x x x and comare t to a threshold λ and decde accordngl: λ, λ x < λ, P H P H Choose H Choose H Note that γ s the value of x at whch λx λ n the fgure.

13 Errors and Correct Decsons Aled Research Laborator The ossble errors are: False Alarm: We choose H when H s the rght answer. False Dsmssal: We choose H when H s the rght answer. The ossble correct decsons are: Detecton: We choose H when t s the rght answer. Correct Dsmssal: We choose H when t s the rght answer.

14 Probabltes of Errors and Correct Decsons P FA P FD γ γ x dx x dx Errors P D P CD γ x dx γ x dx Correct Decsons Note : P + P P + P because x dx CD FA FD D -

15 Neman-Pearson Crteron Aled Research Laborator Usuall we don t know PH and PH and thus cannot calculate λ from ther rato. Instead, we can secf a desred P FA, or false alarm rate, and use t to obtan γ. P FA γ x dx secfed false alarm robablt Then we can calculate λ γ γ or just comare x to γ drectl.

16 Same Examle: Multle Samles, σ μ πσ σ πσ / / x ex x x ex x For each samle x xt, the robabltes are:

17 If we have a set of M multle, ndeendent samles, then ther jont robablt denst functons under H and H are and,... x, x x M / M x ex x σ πσ M / M x ex σ πσ M / M x ex x σ μ πσ Same Examle: Multle Samles Cont d

18 The lkelhood rato becomes: where s the mean value of the samles. Note that each x s Gaussan wth mean under H or μ under H. Also, each x has varance σ under both H and H. Then s also Gaussan, wth the same mean, but wth varance M x M σ μ σ μ σ μ λ M M ex x x ex x x x M M σ Same Examle: Multle Samles Cont d

19 Same Examle: Multle Samles Cont d s a detecton statstc.e. t s a suffcent statstc Usng the Neman-Pearson crteron, the robablt of a false alarm P FA γ can be used to obtan a threshold γ for. M Note that usng x satsfes our ntuton that the M d recever should counter the effects of nose b averagng the samles.

20 Second Examle: Arbtrar But Known Sgnal Possble recever nuts are: H : xt nt H : xt st + nt Nose onl Sgnal lus nose If the sgnal s resent, we know ts shae exactl. Assume we have M samles s st n the nterval,t. The robabltes are: Under H : x M M / x πσ ex σ Under H : x M / x s πσ ex σ M

21 The lkelhood rato s: The second term can be calculated before recevng the samles. As we samle more fnel n the nterval,t, the summaton becomes the ntegral: where E s the energ n the sgnal. M M M s s x ex x s x ex x x x σ σ σ λ T M dt t s s E

22 The test statstc n ths case s: x M x s T xt st dt Note that the receved sgnal xt s beng correlated wth the sgnal we are trng to detect st. Equvalentl, we can flter xt usng a flter wth mulse resonse functon htst-τ as can be seen from ths equaton: T h τ xt τ dτ T st τ xt τ dτ st xt dt A flter whose mulse resonse functon s matched to the sgnal n ths wa s called a matched flter. T

23 We can defne the SNR of to be: Aled Research Laborator Test Statstc SNR The exected values of the test statstc under H and H are E H SNR E H [ E E ] E E var T T xt st dt xt + nt st dt E The varance of under H s usng the shorthand : TT [ ] E st s τ nt n τ dtd E var τ H

24 Let nt be Gaussan whte nose wth sectral level,.e.: R nn τ N δ τ N Then var TT τ δ τ τ N NE st s t dtd And so: SNR E N As long as nt s whte Gaussan nose WGN, there s no other recever,.e. no other test statstc, whch has a hgher SNR. For man other tes of nose, the matched flter s otmal or near otmal as well. Ths s wh the matched flter s used.

25 Thrd Examle: Sgnal Known Excet Amltude and Start Tme Ths s the most common case, n whch we are - Lookng for a target echo - Lstenng for a radated sgnal Aled Research Laborator Exact arrval tme and sgnal amltude are unknown. The hotheses are: H : x t n t Nose onl H : x t a s t t + n t a, t unknown As before, T s the duraton of st

26 We al the sgnal to a matched flter. under H, the outut s t T h τ x t τ dτ T st τ [ a s t τ t + n t τ ] dτ a R s t T t + T st τ n t τ dτ The frst term s the autocorrelaton functon as s at a lag of t T -t. It s maxmum when t T+ t, the tme corresondng to the end of the ulse arrval The second term s random due to the nose.

27 Assume st s a tone burst: The autocorrelaton functon s:

28 Autocorrelaton functon s wrtten: R s A a T cos πf t,, -T T otherwse Can get the enveloe of Rs b squarng and low-ass flterng A a [ R ] T s lf 8 Ths s maxmum when t-t-t or tt+t. Thus the eak n [ t] lf occurs at t T+t, and snce we know T, can get t

29 Therefore, we defne a new test statstc Zt: The robablt denst functons of Zt under H and H are shown b Burdc to be: Where and s the zero-order modfed Bessel functon. [ ] lf t Zt S - - E - zs z I z ex z N, z ex z σ σ σ σ σ σ S SNR I

30 The robablt denst functons are lotted below Can use the Neman-Pearson crteron to get γ, then calculate P D P FA γ z dz

31 Fourth Examle: Possble Doler Shft Non-zero radal moton between a transmtter or reflector and recever causes the frequenc of the receved sgnal to be shfted relatve to the transmtted sgnal. Ths s called Doler Shft. Ths comlcaton s usuall met b mlementng a arallel bank of flters or FFT, each matched to a dfferent frequenc. l L

32 Passve Broadband Detecton Want to detect targets wth broadband sgnatures: Aled Research Laborator Assume we know the ambent nose ower sectrum

33 Passve Broadband Detecton Cont d Use the recever shown below, where h t and h t are flters whose mulse functons need to be determned.

34 Passve Broadband Detecton cont. It has been shown that the Eckart Flter s otmal for h t: H f ψ f ψ f s n Eckart Flter Note: when the nose s whte, H f looks lke Ψ s f. Otherwse, H f s mnmzed when Ψ n f s large The ower sectrum of under H and H s then: Ψ Ψ and f Ψ f H f SNR Ψ f + Ψ f n s [ Ψ f Ψ f ] Ψ f n f df Ψ s f Ψ f H n f df Ψ s f Ψ f n Ψ f df Ψ f s n Ψ s f df Ψ f n Ψ s f + Ψ f n

35 Passve Broadband Detecton cont. Burdc shows that the SNR of the outut of the enveloe detector s SNR SNR The commonl-used ost detecton flter s an averager whose duraton s as long as ossble, h, T t, T τ otherwse The roduct of Τβ ε s tcall large, where β ε s the effectve nose bandwdth at the outut of the re-detecton flter h τ,.e. β ε s the wdth of a rectangular flter whch admts the same nose ower. The frequenc doman exresson for β ε s derved b Burdc n secton 8-4 to be T β ε [ ] Ψ f H f df Ψ n n f H f 4 df

36 Passve Broadband Detecton cont. Usng the Eckert Flter β ε Ψ s f df Ψ f Ψ f df Ψ f n s n Gven large Tβ ε, Burdc shows that the SNR at the averager outut s SNR z Tβ εsnr Tβε SNR Usng the exressons for SNR and β ε SNR Ψ s f Ψs f df df Ψ n f Ψn f Ψs f T T df Ψs f f n f df Ψ Ψ s Ψ f df n Ψ n f z Note the effect on SNR z of ncreasng T.

37 Passve Narrowband Detecton Aled Research Laborator Want to detect targets that emt ure tone sgnatures: Recever s shown below essentall a sectrum analzer

38 Passve Narrowband Detecton Cont d Tcall mlemented b Fourner transformng the nut sgnal. Second flter s an ntegrator averager. Long averages are usuall emloed, so that Tβ >>. If : Sgnal Sectrum : Ψ f s a δ f f Flter : H f,, - β f f β otherwse Nose Sectrum : Ψ f n Constant around f

39 Passve Narrowband Detecton Cont d Then SNR a ψ f n β As before, the SNR of the test statstc Z s SNR z Tβ SNR Puttng these together SNR z T a β ψ n f

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