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 s⁢l+n⁢l s l n l , where s⁢l s l denotes the l th l th signal value and n⁢l 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.

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 r⁢l r l , l∈0…L−1 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.

Figure 1: The two most common methods of converting continuous-time observations into discrete-time ones are shown. In the upper panel, the integrate-and-dump method is shown: the input is integrated over an interval of duration ΔΔ and the result sampled. In the lower panel, the sampling method merely samples the input every ΔΔ seconds.

(a) Integrate-and-Dump (b) Sampling

With either method, the continuous-time detection problem of selecting between models (a binary selection is used here for simplicity) ℳ 0 : r⁢t= s0 ⁢t+n⁢t    0≤t<T ℳ 0 : r t s0 t n t    0 t T ℳ 1 : r⁢t= s1 ⁢t+n⁢t    0≤t<T ℳ 1 : r t s1 t n t    0 t T where si ⁢t si t denotes the known signal set and n⁢t n t denotes additive noise modeled as a stationary stochastic process 1 is converted into the discrete-time detection problem ℳ 0 : r l = s l 0 + n l    0≤l<L ℳ 0 : r l s l 0 n l    0 l L ℳ 1 : r l = s l 1 + n l    0≤l<L ℳ 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=r⁢0…r⁢L−1T r r 0 … r L 1 . The binary detection problem is to distinguish between two possible signals present in the noisy output waveform. ℳ 0 : r= s 0 +n ℳ 0 : r s 0 n ℳ 0 : r= s 1 +n ℳ 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 | ℳ i r=p n r− s i p r ℳ i r p n r s i and the corresponding likelihood ratio by Λ⁢r=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).