ECE642 Detection and Estimation Theory by mr8ball3

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									                                        ECE642: Detection and Estimation Theory




       ECE642: Detection and Estimation Theory



                               Dr. Sudharman K. Jayaweera
                                     Assistant Professor

                      Department of Electrical and Computer Engineering

                                  University of New Mexico

                           Lecture 10 - September 25th , Tuesday

                                          Fall 2007




Dr. S. K. Jayaweera, Fall 07                                                      1
                                         ECE642: Detection and Estimation Theory

   Detection of Coherent Signals with Unknown Amplitude in iid Noise


     • In many detection problems, we know the form of the received signal
       except its amplitude.

     • We can use following composite hypothesis testing problem to model
       this situation:

                    H0 : Yk      =      Nk   for k = 1, 2 . . . , n
                               versus                                               (1)
                    H1 : Yk      =      θsk + Nk    for k = 1, 2 . . . , n, θ > 0

     • Here θ > 0 is a signal strength parameter




Dr. S. K. Jayaweera, Fall 07                                                              2
                                            ECE642: Detection and Estimation Theory

  Detection of Coherent Signals with Unknown Amplitude in iid Noise:
                          Vector Formulation


     • In vector notation:

                                      H0 : Y = N
                                                 versus
                                      H1 : Y = θs + N,         θ>0              (2)
                                                            
                               s1                         N1
                                               
                    s                      N 
                    2                      2 
        where s =       is known and N =        is a continuous random
                    .                      . 
                                               
                     sn                       Nn
        noise vector with iid components and marginal probability density
        function pNk

Dr. S. K. Jayaweera, Fall 07                                                          3
                                       ECE642: Detection and Estimation Theory

            Coherent Signals with Unknown Amplitude in iid Noise:
                               UMP Detection?


     • Since noise components are iid, given θ we can write the likelihood
       ratio between H0 and H1 as,
                                         n
                                            pNk (yk − θsk )
                               Lθ (y) = ∏                                  (3)
                                        k=1    pNk (yk )

     • Recall that model (1) is in fact a composite hypothesis testing
       problem.
     • Then from previous discussions we know that, in general, the critical
       region Γθ = {y ∈ Rn | Lθ (y) > τ} will depend on θ (except for some
       special cases).
     • Hence, UMP tests for model (2) exist only for some special noise
       models (eg. Gaussian noise case considered in example A.1).

Dr. S. K. Jayaweera, Fall 07                                                     4
                                  ECE642: Detection and Estimation Theory

     Locally Optimum (Locally Most Powerful) Detection of Coherent
              Signals with Unknown Amplitude in iid Noise


     • But of course we can easily find LMP tests for model (2)

     • These LMP tests have a simple and intuitively reasonable structure
       that makes their investigation interesting.

     • Assuming sufficient regularity, the locally optimum test for H0 versus
       H1 is (from (9.19)):
                             
                              1
                                                         >
                             
                             
                                             ∂
                  δl0 (y) =
                  ˜             γ       if ∂θ Lθ (y)      = τ            (4)
                             
                                                     θ=0
                             
                              0                          <




Dr. S. K. Jayaweera, Fall 07                                                   5
                                              ECE642: Detection and Estimation Theory

  Likelihood Ratio for Locally Optimum Detection of Coherent Signals
                               in iid Noise


     • Differentiating (3),
                                                                                                 
                                                                            ∂
                  ∂                         pNk (yk − θsk )               ∂θ (pN j (y j − θs j )) 
                                   n             n
                    Lθ (y) =      ∑  ∏ pN (yk ) 
                                      
                 ∂θ                                                               pN j (y j )
                                                                                                    
                                  j=1   k=1       k
                                               k= j
                                                                                                
            ∂                          n              n
                                             pN (yk )                s j p′ j (y j − θs j )
                                                                           N
              Lθ (y)           = − ∑  ∏ k                                                     θ=0 
                                       
           ∂θ                                pNk (yk )                         pN j (y j )
                                                                                                  
                         θ=0       j=1   k=1
                                                     k= j

                                       n     p′ j (y j )s j
                               = −∑
                                              N
                                                                                                    (5)
                                       j=1     pN j (y j )

                                                              ∂
                                where        p′ j (x)       =    pN j (x) .                         (6)
                                              N
                                                              ∂x
Dr. S. K. Jayaweera, Fall 07                                                                              6
                                              ECE642: Detection and Estimation Theory

    Locally Optimum Detection of Coherent Signals in iid Noise (ctd...)


     • Define the function:
                                                      p′ 1 (x)
                                                       N
                                          glo (x) = −          .                      (7)
                                                      pN1 (x)

     • Since we have assumed that noise samples are iid,

                               pN j (y j ) = pN1 (y j )    for j = 1, 2, . . . , n.   (8)

     • Hence, from (7) and (8) we can write (5) as,
                                      ∂                     n

                                     ∂θ
                                        Lθ (y)         =   ∑ s j glo(y j )            (9)
                                                 θ=0       j=1




Dr. S. K. Jayaweera, Fall 07                                                                7
                                  ECE642: Detection and Estimation Theory

          Locally Optimal Detector for Coherent Signals in iid Noise


     • Hence, from (4) and (5), the LMP test for H0 versus H1 is,
                             
                              1
                                                          >
                             
                 δl0 (y) =
                 ˜               γ    if ∑n sk glo (yk ) = τ
                                            k=1
                                                                       (10)
                             
                             
                                 0
                             
                                                           <




Dr. S. K. Jayaweera, Fall 07                                                  8
                                         ECE642: Detection and Estimation Theory

     Locally Optimal Detector for Coherent Signals in iid Noise (ctd...)




                   Figure 1: LMP Detector for Coherent Signals in iid Noise

     • Since this has a memoryless nonlinearity followed by a correlator, it
       is called a nonlinear correlator.

     • Just like the likelihood ratio, the locally optimum non linearity glo (.)
       shapes the observations yk ’s to reduce the detrimental effects of the
       noise as much as possible.


Dr. S. K. Jayaweera, Fall 07                                                       9
                                         ECE642: Detection and Estimation Theory

   Example A.3: Locally Optimum Detection of Coherent Signals in iid
                           Gaussian Noise


     • Suppose noise samples are all N (0, σ2 ). Then, for k = 1, . . . , n
                                                                 y2
                                                           1   − k2
                                        pN1 (yk ) = √          e2σ
                                                        2πσ2
                                    ∂                          yk
                               ∴       [pN1 (yk )] = −pN1 (yk ) 2             (11)
                                   ∂yk                         σ

     • Hence from (7) and (11),
                                                      yk
                                        gl0 (yk ) =                           (12)
                                                      σ2
     • Hence, in this case gl0 (yk ) is in fact a linear function of the
       observation yk


Dr. S. K. Jayaweera, Fall 07                                                     10
                                   ECE642: Detection and Estimation Theory

   Example A.3: Locally Optimum Detection of Coherent Signals in iid
                        Gaussian Noise (ctd...)


     • Hence, in this special case Fig. 1 in fact is the same correlator that we
       obtained in example A.1 (Fig. 9.3)

     • But, of course from the discussion on UMP tests we know that for
       Gaussian noise case the detector in Fig. 9.3 is in fact UMP for H0
       versus H1

     • Since it is UMP, then of course it is also LMP!.




Dr. S. K. Jayaweera, Fall 07                                                   11
                                             ECE642: Detection and Estimation Theory

       Example A.4: Locally Optimum Detection of Coherent Signals
                          in iid Laplacian Noise


     • Suppose now that noise samples are iid Laplacian with
                                              α −α|yk |
                               pN1 (yk ) =      e          for yk ∈ R
                                              2
     • Then,
            ∂                ∂ α −α√y2         α −α√y2 1 2                   −1
               pN1 (yk )) =       e       k = − e       k α     y             2
                                                                                  2yk
           ∂yk              ∂yk 2              2             2 k
                              α −α|yk | αyk    α2 −α|yk | yk
                          = − e             = − e
                              2          y2    2         | yk |
                                                   k

                                  α2 −α|yk |
                               = − e         sgn(yk )                             (13)
                                   2
                               = −αpN1 (yk )sgn(yk )                              (14)

Dr. S. K. Jayaweera, Fall 07                                                            12
                                  ECE642: Detection and Estimation Theory

       Example A.4: Locally Optimum Detection of Coherent Signals
                      in iid Laplacian Noise (ctd...)


     • Hence from (7) and (14):

                               glo (yk ) = α sgn(yk )                 (15)

     • Thus, LMP test for H0 versus H1 in Laplacian noise (from (10)) is:
                           
                            1
                                                           >
                           
               δl0 (y) =
               ˜              γ     if ∑n α sk sgn(yk ) = τ
                                          k=1
                                                                       (16)
                           
                           
                              0
                           
                                                            <




Dr. S. K. Jayaweera, Fall 07                                              13
                                        ECE642: Detection and Estimation Theory

  Locally Optimum Detector for Coherent Signals in iid Laplacian Noise




             Figure 2: LMP Detector for Coherent Signals in iid Laplacian Noise

     • Thus, in this case the LMP test correlates a scaled version of the
       signal with the sequence of signs of the observations.
     • The function glo (yk ) as given by (15) is called a hard limiter
          – Compare this with the soft limiter we encountered in example
            A.2 as the optimal detector for the same Laplacian noise (when
            hypotheses are simple).

Dr. S. K. Jayaweera, Fall 07                                                      14
                                           ECE642: Detection and Estimation Theory

       Example A.5: Locally Optimum Detection of Coherent Signals
                           in iid Cauchy Noise


     • A noise model that has an even heavier tail than the Laplacian noise
       is the Cauchy Noise.
     • Cauchy Noise has the pdf:
                                               1
                               pN1 (yk ) =                    for yk ∈ R           (17)
                                           π(1 + y2 )
                                                  k

     • Then
                       ∂                     −π 2yk                        2yk
                          pN1 (yk ) =                      = −pN1 (yk )            (18)
                      ∂yk                   π(1 + y2 )
                                                       2                  1 + y2
                                                                               k
                                                   k

     • Hence, from (7) and (18):
                                                      2yk
                                         glo (yk ) =
                                                     1 + y2
                                                          k

Dr. S. K. Jayaweera, Fall 07                                                          15
                                       ECE642: Detection and Estimation Theory

       Example A.5: Locally Optimum Detector for Coherent Signals
                           in iid Cauchy Noise


     • Then, the LMP detector for Cauchy noise is (from (10))
                             
                              1
                                                        >
                             
                             
                  δl0 (y) =
                  ˜              γ    if ∑n 1+y2 sk = τ
                                           k=1
                                                 2yk
                                                                                (19)
                             
                                                    k
                             
                              0                         <




              Figure 3: LMP Detector for Coherent Signals in iid Cauchy Noise


Dr. S. K. Jayaweera, Fall 07                                                       16
                                    ECE642: Detection and Estimation Theory

       Example A.5: Locally Optimum Detection of Coherent Signals
                       in iid Cauchy Noise (ctd...)


     • Note that, function glo (yk ) in this case, is approximately linear near
       yk = 0.

     • Hence, for small values of observations, the detector approximately
       correlates the observation yk with the signal sk .

     • However, for the observations with large magnitudes, the function
       glo (yk ) asymptotically goes to zero.
          – Hence, the detector makes the contribution from large
            observations to the accumulated sum very small (because there is
            a high probability that those large values are due to (Cauchy
            distributed) noise.)



Dr. S. K. Jayaweera, Fall 07                                                      17
                                ECE642: Detection and Estimation Theory

                                Next Time


      Detection of Deterministic Signals in Correlated Gaussian Noise
                               (Section III.B)




Dr. S. K. Jayaweera, Fall 07                                            18

								
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