Introduction to Detection Theory

Transcription

1 Introduction to Detection Theory Detection Theory (a.k.a. decision theory or hypothesis testing) is concerned with situations where we need to make a decision on whether an event (out of M possible events) of interest occurs or not.

2 Detection Eamples Eampleincludes :. Radar (target detection) : presence or absence of approaching aircraft.. Communications (BSK) : whether + or -transmitted. 3.Speech Recognition : determine which word was spoken from a group of possible words, e.g.{,, 9}. 4.Sonar : detect enemy submarine. 5. Biomedical: detect presence of abnormal heartbeat. 6.Seismology : detect presence of underground oil deposit.

3 Estimation Follows Detection! In all of these tasks, after detecting the presence of an event, we are interested in etracting more information about it, e.g. bearing, range and speed of aircraft or carrier frequency offset, timing offset, etc. of a signal is done using the tools of estimation theory. which

4 Detection roblem Statement Detection problems can be classified into binary or multiplehypothesis testing problems roblem Statement : Given data set {(), (),, (N -)}, find T () (or detection rule) and based on it make a Test decision

5 Binary ypothesis Testing Eample: DC levelin AWGN, N () A w() ; + w(); (noise only) (signal + noise) Two hypotheses: Η noise only how to choose? Η signal + noise

6 Binary ypothesis Testing (Cont.) DF's : () ~ Ν(, ) : Η ; () Ν(A, ): Η p( (); A) ep () A π Detection : is "A"present or not? estimation :estimate"a" ((;)) ~ Error II (we decide but is correct) A/ A R Error I (we decide but is correct) X()

7 Numerical Eample: Assume A Test Type Type : Decide I II if () error probability : decided error probability : the other one decreases ; threshold at { } ; r () ; true { } ; ; r () < decided true Remark : As threshold changes, one error type increases while

8 False Alarm & Missed Detection Radar Application : : no aircraft, : presence of aircraft Type I error false alarm (FA) Type II error missed detection (MD) Neymen - earson Criterion : Given a certain probability of FA, min imize correct detction MD - or maimize MD D

9 Neyman - earson Test R FA D { : Decide } ( ; ) d R R choose R threshold ( ; ) N test tells us how to o d ;i.e.set the : R Threshold R Decide Decide A (; ) (; ) R R FA D

10 N detection rule : To maimize D for a given FA α : Decide Likelihood ratio, L() R iff { : L() γ} (; ) (; ) γ threshold Sketch of proof : form constrained Lagrangian cost function in terms of Integrals on previous slide. To maimize it, integrand should be positive

11 ) ( - ep ) ( - ep ; ; : N - test N -,,, n ; ) ( ) ( : N -,,, n ; w(n) ) ( : N, w ~, A : DC levelin WGN γ γ + n n n A n n w A n n Eample

12 Eample (Cont.) by taking log of sample mean under NA erformance T() N : n ( n) T() both sides ln( γ ) + Evaluation : ~ A ' γ threshold N, X(n) N A n ( n) ln ( γ ) + NA Too high r, FA r n Too low r, FA

13 FA ' γ r Eample (Cont.) { } ' ' T() γ ; Q γ N where, Q() NQ ( ) FA ep π t dt Q() Q - () / /

14 Eample (Cont.) Under Q ' γ : T() A N ~ QQ ( ) { ' N & r T() γ ; } N A, - ( ) FA D NA SNR What happens for the special cases of very low and very high SNR? Why? SNR3 SNR D SNR D FA - FA - FA -3 SNR SNR FA -4 FA Receiver Operating Characteristics (ROC) FA -5 log NA

15 Minimum Consider To minimize ( detection rule) e e and ( ) ( ) + ( ) ( ) e, prior probabilities (while N - test doesn' t assign prior probabilities). test :(suitable for digital comm.) L() as random events and assign we decide if ( ) ( ) γ ( ) ( )

16 Alternatively, But i.e. ( ) ( ) ( ) ( ) ( ) decide we decide on ( ) i i if if ( Baye's Rule) ( ) ( ) ( MA detection rule) i ence, MA test and Min. e test are equivalent.

17 Special case : For MA test decide ( or Min. e test) ( ) ( ) if with ( ) ( ) ( ML detection rule)

18 Bayes Test Generalizes each type of decide Eample: we want C i : but : part defective, min error. Define C j C is e test by assigning costs true. ij as the cost if to we : part functional better to decide part defective when it's not than deciding it's functional when it's not!

19 Minimize Assume C Decide Bayes detection rule Remark :Typically,C If C C i j if, C C ij ( ) ( ) and C ( ) ( ) C we get the min i j C γ e j test. ( C ) C ( C C ) ( ) & C C

20 { } ep ep and :, : ; y y :, ; z y N, ~ z,, (Binary MA detector) : y y MA test Eample + π π

21 Eample (Cont.) { } { } ln ep ep ep y y y y y y

22 Eample (Cont.) ln ln y if decide y X() r + ln

23 Eample (Cont.) Special cases :. ML y. BSK + ( +, ) 3. ML and BSK y y ln

24 Eample (Cont.) ' ' ' ' ep ep ep ep calculation for BSK : d d d d r r r r e e π π π π - + ln r

25 Eample (Cont.).59 r ML detector ; case : Special. for vs. lot : : e e + < < + + Q Q Q Eercise r Q r Q Q Repeat this eercise for the Laplacian noise DF Is it better to have Gaussian or Laplacian noise? Why?

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