Detection and EstimationChapter 2. Signal Detection

Husheng Li

Min Kao Department of Electrical Engineering and Computer ScienceUniversity of Tennessee, Knoxville

Spring, 2015

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Signal Detection Model

Consider the above hypothesis testing, where Y is the observation, N isthe noise, and S0 and S1 are samples from the two possible signals(could be random).

We need to calculate the following likelihood ratio

L(y) =E [pN(y − S1)]

E [pN(y − S0)].

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Deterministic Signals inIndependent Noise

If the signals are deterministic and the noises are independent in thetime, the testing is the comparison of the sum of log likelihood ratio.

For the coherent detection in i.i.d. Gaussian noise, the optimal detectionis the correlator (or matched filter).

For the coherent detection in i.i.d. Laplace noise, the optimal detector isa soft limiter.

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Locally Optimal Detection

What if the form of the signal is known but the amplitude is unknown?

When s0 = 0, the optimal detector is a nonlinear correlator, in which thestatistic for detection is given by

n∑k=1

sk glo(yk ), glo(x) =1

pN1(x)

dpN1(x)

dx.

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Detection with ColoredGaussian Noise

With colored Gaussian noise, the optimum detection is to use the testingstatistic (s1 − s0)T Σ−1y , which is compared with a threshold.

The performance is given by

PD = 1−Ψ(Ψ−1(1− α)− d),

where d = (s1 − s0)T Σ−1(s1 − s0). Here d can be interpreted as thesignal-to-noise ratio (SNR).

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Prewhitening Filters

A covariance matrix can be decomposed to

Σ =n∑

k=1

λk vk vTk ,

where λk and vk are the eigenvalues and eigenvectors. This is calledthe spectral decomposition, which can be used to whiten colored noise.

Another approach for prewhitening is the Cholesky decomposition:

Σ = CCT ,

where C is a lower triangular matrix.

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Signal Selection

What if we have the freedom to select s0 and s1?

The optimal selection is to let s0 − s1 equal the eigenvector of ΣN

corresponding to the minimum eigenvalue.

At this time, the SNR d is given by

d2 =1λmin‖s1 − s0‖2.

Is this water-filling?

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Detections of Signals withRandom Parameters

When the prior distribution ofthe parameter is known, wecan compute

L(y) =E1[pN (y − s1(Θ))]

E0[pN (y − s1(Θ))].

Example: Noncoherentdetection of a modulatedsinusoidal carrier:

s0 = 0

sk = ak sin((k − 1)wcTs + θ),

for k = 1, ..., n.

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Detection of StochasticSignals

Consider the Gaussian case:

H0 : Y ∼ N(µ0,Σ0) H1 : Y ∼ N(µ1,Σ1).

Special Case:H0 : Y = N H1 : Y = N + S,

whose detector could be

Quadratic detectorEnergy detectorLinear detector

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Performance Analysis:Direct

We can analyze the performance directly. If the testing statistic is T , wehave

PF = (1− FT ,0(τ)) + γ

(FT ,0(τ)− lim

σ→τ−FT ,0(σ)

),

and

PM = FT ,1(τ)− γ(

FT ,1(τ)− limσ→τ−

FT ,1(σ)

).

For the case in which we have T (Y ) =∑n

k=1 gk (yk ), then we have

PF =1

2π

∫ ∞τ

∫ ∞−∞

e−iutn∏

k=1

φgk,0 (u)dudt ,

where φ is the generating function.

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Performance Analysis:Chernoff Bounds

The Chernoff bound was proposed by Herman Chernoff, a professor inMIT (now in Harvard). Still alive! (92 years old!)

The goal is to evaluate P(X ≥ t), where X is the random variable understudy (e.g., the testing statistic). We want to bound it using momentgenerating function of X :

ΨX (λ) = log EeλX (λ ≥ 0), Ψ∗X (t) = supλ≥0

λt −ΨX (λ).

It is an elementary concentration inequality.

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Performance Analysis:Chernoff Inequality

We begin from the Markov inequality: If P(X ≥ 0) = 1, then

P(X ≥ a) ≤ E [X ]/a, ∀a > 0.

Then, we obtainPF ≤ e−Ψ∗

T ,0(τ),

where Ψ∗T ,0(τ) is for the testing statistic T under H0 andΨ∗T ,0(τ) = λ0τ −ΨT ,0(λ0) where λ0 satisfies Ψ′T ,0(λ0) = τ .

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Asymptotic PerformanceAnalysis: Neyman-Pearson

When the number of samples tends to infinity, we may have elegantasymptotic expressions for the performance of signal detection.

Chernoff-Stein Lemma: Let X1, ..., Xn be i.i.d. and of distribution Q.Consider the hypothesis test between two alternatives, Q = P0 andQ = P1, where D(P0||P1) ≤ ∞ (the Kullback-Leibler distance). An is theacceptance region of H1. The error probabilities are

PnF = P0(Ac

n) PnM = P1(An).

Define Pmin,n,εM = minAn,Pn

F≤αPn

M . Then, we have

limn→∞

1n

log Pmin,n,εM = −D(P0||P1).

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Asymptotic PerformanceAnalysis: Bayesian

Consider the Bayesian case, in which the total error probability is givenby Pn

e = π0PnF + π1Pn

M . Define the error exponent as

D∗ = − limn→∞

1n

log Pne .

Chernoff Theorem: The optimal achievable exponent D∗ is given by

D∗ = D(Pλ∗ ||P0) = D(Pλ∗ ||P1),

where

Pλ =Pλ0 (x)P1−λ

1 (x)∑a Pλ0 (a)P1−λ

1 (a),

and λ∗ makes D(Pλ∗ ||P0) = D(Pλ∗ ||P1).

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