# 2.1 Discrete-time detection theory

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## Introduction

Detection theory applies optimal model evaluation to signals ( Helstrom , Poor , van Trees ). Usually, we measure a signal in the presence of additive noiseover some finite number of samples. Each observed datum is of the form $s(l)+n(l)$ , where $s(l)$ denotes the ${l}^{\mathrm{th}}$ signal value and $n(l)$ the ${l}^{\mathrm{th}}$ noise value. In this and in succeeding sections of thischapter, we focus the general methods of evaluating models.

## Detection of signals in gaussian noise

For the moment, we assume we know the joint distribution of the noise values. In most cases, the various models for theform of the observations - the hypothesis - do not differ because of noise characteristics. Rather, the signal componentdetermines model variations and the noise is statistically independent of the signal; such is the specificity ofdetection problems in contrast to the generality of model evaluation. For example, we may want to determine whether asignal characteristic of a particular ship is present in a sonar array's output (the signal is known) or whether no shipis present (zero-valued signal).

To apply optimal hypothesis testing procedures previously derived, we first obtain a finite number $L$ of observations $r(l)$ , $l\in \{0, , L-1\}$ . These observations are usually obtained from continuous-timeobservations in one of two ways. Two commonly used methods for passing from continuous-time to discrete-time are known: integrate-and-dump and sampling . These techniques are illustrated in .

## Integrate-and-dump

In this procedure, no attention is paid to the bandwidth of the noise in selecting the sampling rate. Instead, thesampling interval  is selected according to the characteristics of the signalset. Because of the finite duration of the integrator, successive samples are statistically independent when thenoise bandwidth exceeds $\frac{1}{}$ Consequently, the sampling rate can be varied to some extent while retaining this desirable analytic property.

## Sampling

Traditional engineering considerations governed the selection of the sampling filter and the sampling rate. Asin the integrate-and-dump procedure, the sampling rate is chosen according to signal properties. Presumably, changesin sampling rate would force changes in the filter. As we shall see, this linkage has dramatic implications onperformance.

With either method, the continuous-time detection problem of selecting between models (a binary selection is used here forsimplicity) ${}_{0}:r(t)={s}^{0}(t)+n(t)0\le t< T$ ${}_{1}:r(t)={s}^{1}(t)+n(t)0\le t< T$ where $\{{s}^{i}(t)\}$ denotes the known signal set and $n(t)$ denotes additive noise modeled as a stationary stochasticprocess

We are not assuming the amplitude distribution of the noise to be Gaussian.
is converted into the discrete-time detection problem ${}_{0}:{r}_{l}={s}_{l}^{0}+{n}_{l}0\le l< L$ ${}_{1}:{r}_{l}={s}_{l}^{1}+{n}_{l}0\le l< L$ where the sampling interval is always taken to divide the observation interval $T:L=\frac{T}{}$ . We form the discrete-time observations into a vector: $r=\left(\begin{array}{c}r(0)\\ \\ r(L-1)\end{array}\right)$ . The binary detection problem is to distinguish between two possible signals present in the noisy outputwaveform. ${}_{0}:r={s}_{0}+n$ ${}_{0}:r={s}_{1}+n$ To apply our model evaluation results, we need the probabilitydensity of $r$ under each model. As the only probabilistic component of theobservations is the noise, the required density for the detection problem is given by $p(r, {}_{i}, r)=p(n, r-{s}_{i})$ and the corresponding likelihood ratio by $(r)=\frac{p(n, r-{s}_{1})}{p(n, r-{s}_{0})}$ Much of detection theory revolves about interpreting thislikelihood ratio and deriving the detection threshold (either $\mathrm{threshold}$ or  ).

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Abhi
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20/(×-6^2)
Salomon
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Salomon
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Salomon
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Salomon
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Abhi
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