detection1.pdf

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Detection Theory

Example

1- Radar

2- Communications

3- Speech

4- Sonar

5- Control

6- . . .

Definition

Assume a set of data {x[0], x[1], . . . , x[N − 1]} is available. To arrive at a decision,

first we form a function of the data or T (x[0], x[1], . . . , x[N −

1]) and then make a de-

cision based on its value. Determining the function T and its mapping to a decision

is the central problem addressed in Detection Theory.

1



Introduction

Example: BPSK phase detection

2



Introduction

Detection Problem

The simplest detection problem is to determine if a signal is present or not. Note

that such a signal is always embedded in noise. This type of detection problem is

called binary hypothesis testing problem. Assuming the received data at time n to

be x[n], the signal s[n] and the noise w[n], the binary hypothesis testing problem is

defined as follows

H0 : x[n] = w[n]

H1 : x[n] = s[n] + w[n]

Note that if the number of hypotheses is more than two, then the problem becomes

a multiple hypothesis testing problem. One example is detection of different digits

in speech processing.

3



Detection Problem

Example continued

The probability density function of x[0] under each hypothesis is as follows

p(x[0];H0) = 1

√ 2πσ2 exp− 1

2σ2 x2

[0]

p(x[0];H1) = 1√

2πσ2exp

− 1

2σ2(x[0]−1)2

Deciding between H0 and H1, we are essentially asking weather x[0] has beengenerated according to the pdf p(x[0];H0) or the pdf p(x[0];H1).

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Chi-Squared (Central)

A chi-squared pdf arises as the pdf of x, where x =vi=1

x2i , if xi is a standard normally

distributed random variable. The chi-squared pdf with v degrees of freedom is

defined as

p(x) =

1

2v2 Γ(v

2)

xv

2−1 exp

−12x

, x > 0

0, x < 0

and is denoted by χ2v. v is assumed to be integer and v

≥1. The function Γ(u) is

the Gamma function and is defined as

Γ(u) =

∞

0tu−1 exp(−t)dt

7



Chi-Squared (Noncentral)

If x =vi=1

x2i , where xi’s are i.i.d. Gaussian random variables with mean µi and

variance σ2 = 1, then x has a noncentral chi-squared pdf with v degrees of freedom

and noncentrality parameter λ =v

i=1 µ

2

i . The pdf then becomes

p(x) =

12

x

λ

v−2

4 exp− 1

2(x + λ)

I v2−1

√ λx

, x > 0

0, x < 0

8



Neyman-Pearson Theorem

Detection performance measures

Detection performance of a system is measured mainly by two factors:

1. Probability of false alarm: P FA = p(H1;H0)

2. Probability of detection: P D = p(H1;H1)

Note that sometimes instead of probability of detection, probability of miss detec-

tion, P M = 1−P D is used.

9




Problem statement

Assume a data set x = [x[0], x[1],...,x[N −1]]T is available. The detection problem

is defined as follow

H0 = T (x) < λ

H1 = T (x) > λ

where T is the decision function and λ is the detection threshold. Our goal is to

design T so as to maximize P D subject to P FA < α.

10





To maximize P D for a given P FA = α decide H1 if

L(x) = p(x

;H1) p(x;H0)

> λ

where the threshold λ is found from

P FA = {x:L(x)>λ}

p(x;H0)dx = α

The function L(x) is called the likelihood ratio and the entire test is called the likeli-

hood ratio test (LRT).

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Example: DC level in WGN

Consider the following signal detection problem

H0 : x[n] = w[n] n = 0, 1, . . . , N −1

H1 : x[n] = s[n] + w[n] n = 0, 1, . . . , N

−1

where the signal is s[n] = A for A > 0 and w[n] is AWGN with variance σ2. Now the

NP detector decides H1 if

1

(2πσ2)N 2 exp

− 1

2σ2N −1

n=0 (x[n]−A)

21

(2πσ2)N

2

exp− 1

2σ2

N −1n=0 x2[n]

> λ

Taking the logarithm of both sides and simplification results in

Aσ2

N −1n=0

x[n] > lnλ + N A2

2σ2

Since A > 0, we have finally

1N

N −1n=0

x[n] > σ

2

N A ln λ + A2 = λ′

12




Example continued

The NP detector compares the sample mean x̄ = 1N

N −1n=0 x[n] to a threshold λ

′

. To

determine the detection performance, we first note that the test statistic T (x) = x̄ is

Gaussian under each hypothesis and its distribution is as follows

T (x) ∼ N (0, σ

2

N ) under H0

N (A, σ2

N ) under H1

We have then

P FA = P r(T (x) > λ′

;H0) = Q

λ′

σ2/N

and

P D = P r(T (x) > λ′

;H

1) = Q λ′−A σ2/N

P D and P FA are related to each other according to the following equation

P D = QQ−1(P FA)

− N A2

σ2

13



Receiver Operating Characteristics

The alternative way of summarizing the detection performance of a NP detector

is to plot P D versus P FA. This plot is called Receiver Operating Characteristics(ROC). For the former DC level detection example, the ROC is shown here. Note

that here NA2

σ2 = 1.

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Minimum Probability of Error

Assume the prior probabilities of H0 and H1 are known and represented by P (H0)

and P (H1), respectively. The probability of error, P e, is then defined as

P e = P (

H1)P (

H0

|H1) + P (

H0)P (

H1

|H0) = P (

H1)P M + P (

H0)P FA

Our goal is to design a detector that minimizes P e. It is shown that the following

detector is optimal in this case

p(x|H1) p(x|H0)

> P (H0)P (H1)

= λ

In case P (H0) = P (H1), the detector is called the maximum likelihood detector.

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