# 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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