Estimation and Detection

Transcription

1 Estimation and Detection Lecture : Detection Theory Unknown Parameters Dr. ir. Richard C. Hendriks //05

2 Previous Lecture H 0 : T (x) < H : T (x) > Using detection theory, rules can be derived on how to chose and how to find T (x). Neyman-Pearson Theorem: L(x)= (x;h ) (x;h 0 ) >, where found from P FA = R {x:l(x)> } (x;h 0)dx = Minimum robability of error: (x H ) (x H 0 ) > P (H 0) P (H ) = Bayesian detector: (x H ) (x H 0 ) > C 0 C 00 C 0 C P (H 0 ) P (H ) =. Similar test statistic, different threshold.

3 Deterministic Signals - Interretation Interretation : The resulting T (x)= P N n=0 x[n]s[n] is a correlator. The received data is correlated with a relica of the signal. Interretation : The resulting T (x)= P N n=0 x[n]s[n] is a matched filter. Fig. 4. from Kay-II. 3

4 Deterministic Signals Summary Deterministic signals: T (x)=x T C s Notice that is C is ositive definite, C can be written as C = D T D, leading to T (x)=x T D T Ds Fig. 4.7 Kay-II. 4

5 Random Signals Estimator Correlator Hence, we decide for H if T (x)=x T ŝ > 00 with ŝ = ale C s + I C s x = C s (C s + I) x Fig. 5. Kay-II. 5

6 Hence, we decide for H if Random Signals Exercise with ŝ = T (x)=x T ŝ > 00 ale C s + I C s x = C s (C s + I) x Fig. 5. Kay-II. Problem Ch. 5 Kay-II. 6

7 Detection under NP: Today Deterministic signals: Random Signals: with ŝ = T (x)=x T C s T (x)=x T ŝ > 00 ale C s + I C s x = C s (C s + I) x NP requires erfect knowledge of (x;h 0 ) and (x;h ). (and thus s and/or C s and ) What if this information is unknown? 7

8 Comosite Hyothesis Testing Remember this examle H 0 : x[n]=w[n] n =0,,...,N H : x[n]=a + w[n] n =0,,...,N where A>0 and w[n] is WGN with variance. NP detector decides H if L(x)= (x;h ) (x;h 0 ) > 8

9 Comosite Hyothesis Testing We then have ( ) N ( ) N h ex h ex P N n=0 (x[n] A) i P N n=0 x [n] i > Taking the logarithm of both sides and simlification results in N NX n=0 x[n] > NA ln + A = 0 Where the threshold is found by P FA = Pr(T (x) > 0 ;H 0 )=Q 0 ) 0 = r N Q (P FA ) 9

10 Comosite Hyothesis Testing What if the value of A is unknown? H 0 : x[n]=w[n] n =0,,...,N H : x[n]=a + w[n] n =0,,...,N where A is unknown, but we know A>0. Further we know that w[n] is WGN with variance. NP detector decides H if L(x)= (x;a,h ) (x;a,h 0 ) > PDFs arameterized by A This is called: Comosite Hyothesis testing. 0

11 We now have Comosite Hyothesis Testing ( ) N ( ) N h ex h ex P N n=0 (x[n] A) i P N n=0 x [n] i > Taking the logarithm of both sides and simlification results in N NX n=0 x[n] > NA ln + A = 0 In this case, Both the test statistic T (x) and threshold deend on A. 0 do not Even though A is unknown, we can thus imlement this detector. (Although P D does deend on A) Where the threshold is found by 0 P FA = Pr(T (x) > 0 ;H 0 )=Q r NA P D = Q Q (P FA ) ) 0 = r N Q (P FA )

12 Comosite Hyothesis Testing - UMP The test N NX n=0 x[n] > NA ln + A = r N Q (P FA ), leads to the highest P D (remember NP maximises P D ) for any value A. (as long as A>0). Such a test is called a Uniformly Most Powerful (UMP) test. Any other test will have a oorer erformance. However...often an UMP does not exist.

13 Comosite Hyothesis Testing - UMP Consider the same examle: What if A is <A< instead of A>0? In that case we will have a different test for negative and ositive A: A>0: A<0: N N NX n=0 NX n=0 x[n] > x[n] < r N Q (P FA ) r N Q (P FA ) In this case, the UMP does not exist (as A can either be > 0 or < 0). Instead of en otimal test we will have to use a sub-otimal test. However, the unrealizable otimal test can be used for erformance comarison (such a test is called a clairvoyant detector). 3

14 Examle DC level in WGN unknown A Let us first calculate the clairvoyant detector for the case <A< For A>0 A>0: A<0: P FA = Pr{ x> N N NX n=0 NX n=0 0 +;H 0 } = Q x[n] > 0 + x[n] < For A<0 0 0 P FA = Pr{ x< 0 ;H 0 } = Q = Q 4

15 Examle DC level in WGN unknown A The detection erformance of the clairvoyant detector: 0 for A>0 P D = Pr{ x> +;H 0 + A } = Q = Q Q (P FA ) r NA for A<0 P D = Q 0 A = Q 0 + A = Q Q (P FA )+ A Fig. 6.3 Kay-II. 5

16 Examle DC level in WGN unknown A Instead of the clairvoyant detector, let s look at the realisable detector: N NX n=0 x[n] > 00, where A is now unknown and <A<. P FA = Pr{ x > 00 ;H 0 } =Pr{ x> 00 ;H 0 } =Q = Q (P FA /) P D = Pr{ x > 00 ;H } = Q Q (P FA /) r NA + Q Q (P FA /) + A 6

17 Examle DC level in WGN unknown A P D = Pr{ x > 00 ;H } = Q Q (P FA /) r NA + Q Q (P FA /) + A The erformance of this realisable detector is thus not otimal, but close to the otimal clairvoyant detector. Fig. 6.4 Kay-II. 7

18 Exercise Problems Ch. 6 Kay-II. 8

19 Aroaches for comosite Hy. testing Two aroaches: Bayesian aroach: Consider unknown arameters as realizations of random variables and assign a rior df. Generalized likelihood ratio: Estimate unknown arameters using MLEs. Bayesian aroach: Assign riors to unknown arameters 0 and under hyotheses H 0 and H, resectively: Z (x;h 0 )= (x 0;H 0 )( 0 )d 0 Z (x;h )= NP detector: (x;h ) (x;h 0 ) = R (x ;H )( )d R (x 0;H 0 )( 0 )d 0 > (x ;H )( )d Need to choose rior df. Integration can be difficult. 9

20 Exercise Bayesian Comosite NP Detector Examle Bayesian NP detector: We observe one samle x[0] having an exonential df NP detector: (x;h ) (x;h 0 ) = R (x ;H )( )d R (x 0;H 0 )( 0 )d 0 > (x[0]) = ex( x[0]), where is unknown. For the hyothesis testing roblem H 0 : = 0 H : 6= 0, Determine the Bayesian comosite NP detector if it is given that between 0 and C. is uniformly distributed 0

21 Generalized Likelihood Ratio Test GLRT: Relace unknown arameters by their MLEs. GLRT: L G (x)= (x; ˆ,H ) (x; ˆ0,H 0 ) > with (x; ˆi,H i ) given by (x; ˆi,H i ) = max(x; i,h i ) i

22 Examle: DC in WGN with Unknown Amlitude - GRLT Remember this examle H 0 : x[n]=w[n] n =0,,...,N H : x[n]=a + w[n] n =0,,...,N where <A< and w[n] is WGN with variance. NP detector decides H if the GLRT: L G (x)= (x; ˆ,H ) (x; ˆ0,H 0 ) > with (x; ˆi,H i ) given by (x; ˆi,H i ) = max(x; i,h i ) i

23 Examle: DC in WGN with Unknown Amlitude - GRLT MLE of A: This will lead to  = N = max A (x;â,h ) = max A (x;a,h ) ( ) N P x [n]= x. h ex X (x[n] A) i. N n=0 Thus, the GLRT L G (x)= ( ) N ( ) N h ex h ex P N n=0 (x[n] x) i P N n=0 x [n] i > Taking the logarithm of both sides we have lnl G (x)= ( N x + N x )= N x 3

24 Examle: DC in WGN with Unknown Amlitude - GRLT We decide thus for H if lnl G (x)= ( N x + N x )= N x x > 00. This detector is identical to realisable detector we looked at before. Remember: P FA = Pr{ x > 00 ;H 0 } =Pr{ x> 00 ;H 0 } = Q = Q (P FA /) P D = Pr{ x > 00 ;H } = Q Q (P FA /) r NA + Q Q (P FA /) + A 4