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Licentiates Presentation Fundamental Estimation and Detection Limits in Linear Non-Gaussian Systems Gustaf Hendeby Automatic Control Department of Electrical Engineering Linkpings universitet AUTOMATIC CONTROL Gustaf Hendeby

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SLIDE 1

Licentiate’s Presentation

Fundamental Estimation and Detection Limits in Linear Non-Gaussian Systems

Gustaf Hendeby Automatic Control Department of Electrical Engineering Linköpings universitet

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 2

Motivation

Estimation and detection are used everywhere Vital functions rely on it Information is expensive

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 3

Motivation

System description System

wt ut

Measure

et yt xt

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 4

Motivation

Noise approximation

True distribution Gaussian approximation

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 5

Motivation

Noise approximation

True distribution

Gaussian approximation Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 6

Motivation

Noise approximation

True distribution

Gaussian approximation

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 7

Motivation

Noise approximation

True distribution Gaussian approximation

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 8

Outline

  • 1. Introduction
  • 2. Noise and Information
  • 3. Estimation Limits
  • 4. Detection Limits
  • 5. Conclusions

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 9

Outline

  • 1. Introduction
  • 2. Noise and Information
  • 3. Estimation Limits
  • 4. Detection Limits
  • 5. Conclusions

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 10

Noise

Random effects

  • Measurement noise
  • Process noise

More or less informative Description:

  • PDF p(x)
  • Expected value: E(x) = µ
  • Variance: var(x) = Σ

−4 −3 −2 −1 1 2 3 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 x p(x)

p(x) = N (x;0,1)

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 11

Information and Accuracy

yt = θ +et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e p(e) True Gauss Approx.

et

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 12

Information and Accuracy

yt = θ +et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e p(e) True Gauss Approx. Meas.

et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 θ p(θ|Yt) True

  • Approx. Based

Meas.

θ|Yt

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 13

Information and Accuracy

yt = θ +et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e p(e) True Gauss Approx. Meas.

et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 θ p(θ|Yt) True

  • Approx. Based

Meas.

θ|Yt

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 14

Information and Accuracy

yt = θ +et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e p(e) True Gauss Approx. Meas.

et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 θ p(θ|Yt) True

  • Approx. Based

Meas.

θ|Yt

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 15

Information and Accuracy

yt = θ +et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e p(e) True Gauss Approx. Meas.

et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 θ p(θ|Yt) True

  • Approx. Based

Meas.

θ|Yt

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 16

Information and Accuracy

yt = θ +et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e p(e) True Gauss Approx. Meas.

et

−4 −3 −2 −1 1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 θ p(θ|Yt) True

  • Approx. Based

Meas.

θ|Yt

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 17

Definitions: information measures

Fisher information (true parameter θ 0):

I x(θ) = −Ex

  • ∆θ

θlogp(x|θ)

  • θ=θ 0
  • Intrinsic accuracy (true mean µ0):

I x = −Ex

  • ∆x

xlogp(x|µ0)

  • Relative accuracy:

Ψx = var(x)I x

Kullback-Leibler information:

I KL

p(·),q(·)

  • =
  • p(x)log

p(x)

q(x)

  • dx

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 18

Definitions: information measures

Fisher information (true parameter θ 0):

I x(θ) = −Ex

  • ∆θ

θlogp(x|θ)

  • θ=θ 0
  • Intrinsic accuracy (true mean µ0):

I x = −Ex

  • ∆x

xlogp(x|µ0)

  • Relative accuracy:

Ψx = var(x)I x

Kullback-Leibler information:

I KL

p(·),q(·)

  • =
  • p(x)log

p(x)

q(x)

  • dx

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 19

Example: intrinsic accuracy

Assume yi = θ +ei, ei ∼ N (µ = 0,Σ), then the intrinsic accuracy is

I e = −Ee

  • ∆e

elogN (e;µ,Σ)

  • = −Ee
  • ∆e

elog

1

√

2πΣ e− (e−µ)2

2Σ

  • = Ee
  • ∆e

e

  • log

√

2πΣ+ (e − µ)2 2Σ

  • = Ee
  • ∇e

(e − µ) Σ

  • = Ee

1 Σ

  • = 1

Σ =

1 var(e).

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 20

Example: intrinsic accuracy

Assume yi = θ +ei, ei ∼ N (µ = 0,Σ), then the intrinsic accuracy is

I e = −Ee

  • ∆e

elogN (e;µ,Σ)

  • = −Ee
  • ∆e

elog

1

√

2πΣ e− (e−µ)2

2Σ

  • = Ee
  • ∆e

e

  • log

√

2πΣ+ (e − µ)2 2Σ

  • = Ee
  • ∇e

(e − µ) Σ

  • = Ee

1 Σ

  • = 1

Σ =

1 var(e).

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 21

Example: intrinsic accuracy

Assume yi = θ +ei, ei ∼ N (µ = 0,Σ), then the intrinsic accuracy is

I e = −Ee

  • ∆e

elogN (e;µ,Σ)

  • = −Ee
  • ∆e

elog

1

√

2πΣ e− (e−µ)2

2Σ

  • = Ee
  • ∆e

e

  • log

√

2πΣ+ (e − µ)2 2Σ

  • = Ee
  • ∇e

(e − µ) Σ

  • = Ee

1 Σ

  • = 1

Σ =

1 var(e).

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-22
SLIDE 22

Example: intrinsic accuracy

Assume yi = θ +ei, ei ∼ N (µ = 0,Σ), then the intrinsic accuracy is

I e = −Ee

  • ∆e

elogN (e;µ,Σ)

  • = −Ee
  • ∆e

elog

1

√

2πΣ e− (e−µ)2

2Σ

  • = Ee
  • ∆e

e

  • log

√

2πΣ+ (e − µ)2 2Σ

  • = Ee
  • ∇e

(e − µ) Σ

  • = Ee

1 Σ

  • = 1

Σ =

1 var(e).

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-23
SLIDE 23

Example: intrinsic accuracy

Assume yi = θ +ei, ei ∼ N (µ = 0,Σ), then the intrinsic accuracy is

I e = −Ee

  • ∆e

elogN (e;µ,Σ)

  • = −Ee
  • ∆e

elog

1

√

2πΣ e− (e−µ)2

2Σ

  • = Ee
  • ∆e

e

  • log

√

2πΣ+ (e − µ)2 2Σ

  • = Ee
  • ∇e

(e − µ) Σ

  • = Ee

1 Σ

  • = 1

Σ =

1 var(e).

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-24
SLIDE 24

Example: intrinsic accuracy

Assume yi = θ +ei, ei ∼ N (µ = 0,Σ), then the intrinsic accuracy is

I e = −Ee

  • ∆e

elogN (e;µ,Σ)

  • = −Ee
  • ∆e

elog

1

√

2πΣ e− (e−µ)2

2Σ

  • = Ee
  • ∆e

e

  • log

√

2πΣ+ (e − µ)2 2Σ

  • = Ee
  • ∇e

(e − µ) Σ

  • = Ee

1 Σ

  • = 1

Σ =

1 var(e).

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 25

Intrinsic Accuracy: outliers

p1(x;ω,k) = (1−ω)N (x;0,Σ)+ωN (x;0,kΣ) Inverse relative accuracy:

Ψ−1

x

= (cov(x)I x)−1 Σ−1 := 1+(k −1)ω,

to get cov(x) = 1 Red is informative, blue is not k = 1 yields Gaussian distribution

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 26

Outline

  • 1. Introduction
  • 2. Noise and Information
  • 3. Estimation Limits
  • 4. Detection Limits
  • 5. Conclusions

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 27

Estimation

Extract hidden information Find p(xt|Yτ):

  • Measurement update phase:

p(xt|Yτ) = p(yτ|xt)p(xt|Yτ−1) p(yτ|Yτ−1)

  • Time update phase:

p(xt+1|Yτ) =

  • p(xt+1|xt)p(xt|Yτ)dxt,

Approximation of p(xt|Yτ) often needed

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 28

State-Space Model

General State-Space Model: xt+1 = f(xt,wt) yt = h(xt,et) Linear State-Space Model: xt+1 = Ftxt +Gtwt yt = Htxt +et Qt = cov(wt) and Rt = cov(et)

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 29

Estimation: algoritms

Kalman Filter

The best linear unbiased estimator (BLUE)

  • 1. Initiate: ˆ

x0|−1, P0|−1

  • 2. Measurement update phase:

Kt = Pt|t−1HT

t

  • HtPt|t−1HT

t +Rt

−1 ˆ

xt|t = ˆ xt|t−1 +Kt

  • yt −Ht ˆ

xt|t−1

  • Pt|t =
  • I −KtHt
  • Pt|t−1
  • 3. Time update phase:

ˆ

xt+1|t = Ft ˆ xt|t Pt+1|t = FtPt|tF T

t +Qt EKF, IEKF, UKF, filter banks

Particle Filter

Asymptotically (in N) correct PDF

  • 1. Initiate: {x(i)

0 }N i=1 ∼ p(x0),

{ω(i)

0|−1}N i=1 = 1 N

  • 2. Measurement update phase:

ω(i)

t|t = p(yt|x(i)

t )ω(i) t

∑j p(yt|x(j)

t )ω(j) t

  • 3. Resample!
  • 4. Time update phase:

{x(i)

t+1}N i=1 ∼ q(xt+1|x(i) t ,Yt),

ω(i)

t+1|t =

ω(i)

t|t p(x(i) t+1|x(i) t )

q(x(i)

t+1|x(i) t ,Yt)

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 30

Parametric Cramér-Rao Lower Bound (CRLB)

Estimation performance, assuming correct trajectory exists. Bound given by: Pt|t = Pt|t−1 −Pt|t−1HT

t (HtPt|t−1HT t +Rt)−1HtPt|t−1

Pt+1|t = FtPt|tF T

t +GtQtGT t ,

initialized with P−1

0|−1 = I x0 and with

F T

t = ∇xt f(xt,w0 t )

  • xt=x0

t ,

GT

t = ∇wt f(x0 t ,wt)

  • wt=w0

t ,

HT

t R−1 t

Ht = −Eyt

  • ∆xt

xtp(yt|xt)

  • ,

Q−1

t

= −Ext

  • ∆wt

wtp(xt|w0 t )

  • .

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 31

Posterior Cramér-Rao Lower Bound (CRLB)

Estimation performance, assuming only trajectory distribution. Bound given by: P−1

t+1|t = Q−1 t

−ST

t (P−1 t|t−1 +R−1 t

+Vt)−1St

P−1

t+1|t+1 = Q−1 t

+R−1

t+1 −ST t (P−1 t|t +Vt)−1St

initiated with P−1

0|−1 = I −1 x0 , P−1 0|0 = (P−1 0|−1 +R−1 0 )−1, with:

Vt = −Ext,wt

  • ∆xt

xtlogp(xt+1|xt)

  • ,

R−1

t

= −Ext,yt

  • ∆xt

xtlogp(yt|xt)

  • ,

St = −Ext,wt

  • ∆xt+1

xt

logp(xt+1|xt)

  • ,

Q−1

t

= −Ext,wt

  • ∆xt+1

xt+1logp(xt+1|xt)

  • .

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

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SLIDE 32

Linear Systems CRLB

Parametric and posterior CRLB are identical: Pt|t =

  • P−1

t|t−1 +HT t I et Ht

−1,

Pt+1|t = FtPt|tF T

t +Gt I −1 wt GT t ,

initiated with P0|−1 = I −1

x0

Effects of I wt and I et examined Compare to BLUE performance given by Kalman filter (same expression)

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-33
SLIDE 33

Example: DC motor setup

State-space model: xt+1 =

  • 1

1−e−1 e−1

  • xt +
  • e−1

1−e−1

  • wt

yt =

  • 1
  • xt +et

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-34
SLIDE 34

Example: DC motor setup

State-space model: xt+1 =

  • 1

1−e−1 e−1

  • xt +
  • e−1

1−e−1

  • wt

yt =

  • 1
  • xt +et

with the noise wt ∼ N

  • 0,

π

180

2

et ∼ 0.9N

  • 0,

π

180

2 +0.1N

  • 0,

10π

180

2

wt

−0.1 −0.05 0.05 0.1 0.15 5 10 15 20 25 w p(w)

var(wt) = 3.0·10−4

I wt = 3.3·103 Ψwt = 1

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-35
SLIDE 35

Example: DC motor setup

State-space model: xt+1 =

  • 1

1−e−1 e−1

  • xt +
  • e−1

1−e−1

  • wt

yt =

  • 1
  • xt +et

with the noise wt ∼ N

  • 0,

π

180

2

et ∼ 0.9N

  • 0,

π

180

2 +0.1N

  • 0,

10π

180

2

et

−0.1 −0.05 0.05 0.1 0.15 5 10 15 20 25 30 35 40 45 e p(e)

var(et) = 8.3·10−4

I et = 1.1·104 Ψet = 9.0

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-36
SLIDE 36

Example: filtering analysis

Normalized filtering performance Red is improvement, blue is not Note axis

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-37
SLIDE 37

Example: filtering result

10 20 30 40 50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10

−3

Time MSE CRLB PF KF BLUE

Monte Carlo simulations Dashed lines indicate asymptotic limits Note improvement

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-38
SLIDE 38

Observations

Improved performance with nonlinear filter on non-Gaussian noise. Comparing CRLB and BLUE performance indicates the gain. System properties and used methods affect actual gain.

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-39
SLIDE 39

Outline

  • 1. Introduction
  • 2. Noise and Information
  • 3. Estimation Limits
  • 4. Detection Limits
  • 5. Conclusions

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-40
SLIDE 40

Detection

Determine if a change/fault has occurred Decide between hypotheses; H0 and H1 Common design criteria:

  • Minimize probability of false alarm

PFA = Pr(decide H1|H0)

  • Maximize probability of detection

PD = Pr(decide H1|H1)

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-41
SLIDE 41

Detection: methods

Decide using L(Y)

H1

≷

H0

γ

Generalized likelihood ratio (GLR) test statistic L(Y) = supf|H1 p(Y|H1) supf|H0 p(Y|H0) Composite hypotheses are more difficult and more common

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-42
SLIDE 42

Detection Models

Residuals: rt = yt −h(xt,f 0

t )

Fault models: ft = ϕT

t θ

Stacked linear residuals Rt =

  • r T

t−L+1

r T

t−L+2

...

r T

t

T : Rt = Yt −Otxt−L+1 = ¯

Hθ

t θ + ¯

Hv

t Vt

Known initial state and detection in parity-space can be described this way

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-43
SLIDE 43

GLR Performance

Asymptotically (in information) uniformly most powerful (UMP) Test statistics L′(Y) := 2logL(Y)

a

∼

  • χ2

nθ ,

under H0

χ′2

nθ (λ),

under H1

,

with

λ = θ 1T ¯

HθT

t

I ¯

Hv

t V ¯

Hθθ 1

= θ 1T ¯

HθT

t

( ¯

Hv

t I −1

V ¯

HvT

t )−1 ¯

Hθθ 1

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-44
SLIDE 44

Example: DC motor setup

State-space model: xt+1 =

  • 1

1−e−1 e−1

  • xt +
  • e−1

1−e−1

  • (wt +ft)

yt =

  • 1
  • xt +et

with the noise wt ∼ N

  • 0,

π

180

2

et ∼ 0.9N

  • 0,

π

180

2 +0.1N

  • 0,

10π

180

2

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-45
SLIDE 45

Example: DC motor setup

State-space model: xt+1 =

  • 1

1−e−1 e−1

  • xt +
  • e−1

1−e−1

  • (wt +ft)

yt =

  • 1
  • xt +et

with the noise wt ∼ N

  • 0,

π

180

2

et ∼ 0.9N

  • 0,

π

180

2 +0.1N

  • 0,

10π

180

2

10 20 30 40 50 −0.01 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time ft

ft =

  • 0,

t ≤ 25

2π 180,

t > 25

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-46
SLIDE 46

Example: detection analysis

Normalized detection performance, PFA = 1% Red is improvement, blue is not Note axis Level 1 (blue) equals PD = 37%

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-47
SLIDE 47

Example: detection result

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PFA PD GLR limit GLR Gaussian GLR Gaussian GLR limit

Monte Carlo simulations Dashed lines indicate asymptotic limits Note improvement

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-48
SLIDE 48

Observations

Utilizing non-Gaussian effects may substantially improve detection performance. It is sometimes difficult to predict the effect of Gaussian approximations. The system determines how difficult it is to improve performance.

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-49
SLIDE 49

Outline

  • 1. Introduction
  • 2. Noise and Information
  • 3. Estimation Limits
  • 4. Detection Limits
  • 5. Conclusions

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-50
SLIDE 50

Conclusions

Consider non-Gaussian effects Estimation: compare CRLB to

BLUE performance

Detection: compare asymptotic GLR performance under different assumptions Improvement shown in simulations Further work

  • When are asymptotic

results reached?

  • Other performance

measures?

  • Robustness?
  • Treat nonlinear systems

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS

slide-51
SLIDE 51

Gustaf Hendeby Licentiate’s Presentation

AUTOMATIC CONTROL LINKÖPINGS UNIVERSITET COMMUNICATION SYSTEMS