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Course by: Laurence Riddle. E-mail the author

# Discrete-Time Detection Theory

Module by: Don Johnson. E-mail the author

## Introduction

Detection theory applies optimal model evaluation to signals (Helstrom, Poor, van Trees). Usually, we measure a signal in the presence of additive noise over some finite number of samples. Each observed datum is of the form sl+nl s l n l , where sl s l denotes the l th l th signal value and nl n l the l th l th noise value. In this and in succeeding sections of this chapter, 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 the form of the observations - the hypothesis - do not differ because of noise characteristics. Rather, the signal component determines model variations and the noise is statistically independent of the signal; such is the specificity of detection problems in contrast to the generality of model evaluation. For example, we may want to determine whether a signal characteristic of a particular ship is present in a sonar array's output (the signal is known) or whether no ship is present (zero-valued signal).

To apply optimal hypothesis testing procedures previously derived, we first obtain a finite number LL of observations rl r l , l0L1 l 0 L 1 . These observations are usually obtained from continuous-time observations 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 Figure 1.

### Integrate-and-Dump

In this procedure, no attention is paid to the bandwidth of the noise in selecting the sampling rate. Instead, the sampling interval ΔΔ is selected according to the characteristics of the signal set. Because of the finite duration of the integrator, successive samples are statistically independent when the noise bandwidth exceeds 1Δ 1 Δ 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. As in the integrate-and-dump procedure, the sampling rate is chosen according to signal properties. Presumably, changes in sampling rate would force changes in the filter. As we shall see, this linkage has dramatic implications on performance.

With either method, the continuous-time detection problem of selecting between models (a binary selection is used here for simplicity) 0 : rt= s0 t+nt    0t<T 0 : r t s0 t n t    0 t T 1 : rt= s1 t+nt    0t<T 1 : r t s1 t n t    0 t T where si t si t denotes the known signal set and nt n t denotes additive noise modeled as a stationary stochastic process 1 is converted into the discrete-time detection problem 0 : r l = s l 0 + n l    0l<L 0 : r l s l 0 n l    0 l L 1 : r l = s l 1 + n l    0l<L 1 : r l s l 1 n l    0 l L where the sampling interval is always taken to divide the observation interval T : L=TΔ T : L T Δ . We form the discrete-time observations into a vector: r=r0rL1T r r 0 r L 1 . The binary detection problem is to distinguish between two possible signals present in the noisy output waveform. 0 : r= s 0 +n 0 : r s 0 n 0 : r= s 1 +n 0 : r s 1 n To apply our model evaluation results, we need the probability density of rr under each model. As the only probabilistic component of the observations is the noise, the required density for the detection problem is given by p r | i r=p n r s i p r i r p n r s i and the corresponding likelihood ratio by Λr=p n r s 1 p n r s 0 Λ r p n r s 1 p n r s 0 Much of detection theory revolves about interpreting this likelihood ratio and deriving the detection threshold (either thresholdthreshold or γγ).

## Footnotes

1. We are not assuming the amplitude distribution of the noise to be Gaussian.

## References

1. C. W. Helstrom. (1968). Statistical Theory of Signal Detection. (Second Edition). Oxford: Pergamon Press.
2. H. V. Poor. (1988). An Introduction to Signal Detection and Estimation. New York: Springer-Verlag.
3. H. L. van Trees. (1968). Detection, Estimation, and Modulation Theory, Part I. New York: John Wiley & Sons.

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