By far the easiest detection problem to solve occurs when the noise vector consists of statistically independent, identically distributed, Gaussian random variables, what is commonly termed white Gaussian noise. The mean of white noise is usually taken to be zero1 and each component's variance is σ2 σ 2 . The equal-variance assumption implies the noise characteristics are unchanging throughout the entire set of observations. The probability density of the noise vector evaluated at r− s i r s i equals that of a Gaussian random vector having independent components with mean s i s i . p N r− s i =12⁢π⁢σ2L2⁢e−(12⁢σ2⁢(r− s i )T⁢(r− s i )) p N r s i 1 2 σ 2 L 2 1 2 σ 2 r s i r s i The resulting detection problem is similar to the Gaussian example we previously examined, with the difference here being a non-zero mean---the signal---under both models. The logarithm of the likelihood ratio becomes (r− s 0 )T⁢(r− s 0 )−(r− s 1 )T⁢(r− s 1 ) ≷ ℳ 0 ℳ 1 2⁢σ2⁢lnη r s 0 r s 0 r s 1 r s 1 ≷ ℳ 0 ℳ 1 2 σ 2 η The usual simplifications yield in rT⁢ s 1 − s 1 T⁢ s 1 2−(rT⁢ s 0 − s 0 T⁢ s 0 2) ≷ ℳ 0 ℳ 1 σ2⁢lnη r s 1 s 1 s 1 2 r s 0 s 0 s 0 2 ≷ ℳ 0 ℳ 1 σ 2 η The model-specific components on the left side express the signal processing operations for each model.2

Each term in the computations for the optimum detector has a signal processing interpretation. When expanded, the term s i T⁢ s i s i s i equals ∑ l =0L−1 s i 2l l 0 L 1 s i l 2 , the signal energy E i E i . The remaining term, rT⁢ s i r s i , is the only one involving the observations and hence constitutes the sufficient statistic ϒ i ⁢r ϒ i r for the additive white Gaussian noise detection problem. ϒ i ⁢r=rT⁢ s i ϒ i r r s i An abstract, but physically relevant, interpretation of this important quantity comes from the theory of linear vector spaces. In that context, the quantity rT⁢ s i r s i would be termed the projection of rr onto s i s i . From the Schwarz inequality, we know that the largest value of this projection occurs when these vectors are proportional to each other. Thus, a projection measures how much alike two vectors are: they are completely alike when they are parallel (proportional to each other) and completely dissimilar when orthogonal (the projection is zero). In effect, the projection operation removes those components from the observations which are orthogonal to the signal, thereby generalizing the familiar notion of filtering a signal contaminated by broadband noise. In filtering, the signal-to-noise ratio of a bandlimited signal can be drastically improved by lowpass filtering; the output would consist only of the signal and "in-band" noise. The projection serves a similar role, ideally removing those "out-of-band" components (the orthogonal ones) and retaining the "in-band" ones (those parallel to the signal).

The projection operation can be expanded as rT⁢ s i =∑ l =0L−1r⁢l⁢ s i ⁢l r s i l 0 L 1 r l s i l another signal processing interpretation emerges. The projection now describes a finite impulse response (FIR) filtering operation evaluated at a specific index. To demonstrate this interpretation, let h⁢l h l be the unit-sample response of a linear, shift-invariant filter where h⁢l=0 h l 0 for l<0 l 0 and l≥L l L . Letting r⁢l r l be the filter's input sequence, the convolution sum expresses the output. r⁢k*h⁢k=∑ l =k−(L−1)kr⁢l⁢h⁢k−l r k h k l k L 1 k r l h k l Letting k=L−1 k L 1 , the index at which the unit-sample response's last value overlaps the input's value at the origin, we have r⁢k*h⁢k| k =L−1=∑ l =0L−1r⁢l⁢h⁢L−1−l k L 1 r k h k l 0 L 1 r l h L 1 l Suppose we set the unit-sample response equal to the index-reversed, then delayed signal. h⁢l= s i ⁢L−1−l h l s i L 1 l In this case, the filtering operation becomes a projection operation. r⁢k* s i ⁢L−1−k| k =L−1=∑ l =0L−1r⁢l⁢ s i ⁢l k L 1 r k s i L 1 k l 0 L 1 r l s i l Figure 1 depicts these computations graphically.

Figure 1: The detector for signals contained in additive, white Gaussian noise consists of a matched filter, whose output is sampled at the duration of the signal and half of the signal energy is subtracted from it. The optimum detector incorporates a matched filter for each signal compares their outputs to determine the largest.

The sufficient statistic for the i th i th signal is thus expressed in signal processing notation as r⁢k* s i ⁢L−1−k| k =L−1− E i 2 k L 1 r k s i L 1 k E i 2 . The filtering term is called a matched filter because the observations are passed through a filter whose unit-sample response "matches" that of the signal being sought. We sample the matched filter's output at the precise moment when all of the observations fall within the filter's memory and then adjust this value by half the signal energy. The adjusted values for the two assumed signals are subtracted and compared to a threshold.

To compute the performance probabilities, the expressions should be simplified in the ways discussed in previous sections. As the energy terms are known a priori they can be incorporated into the threshold with the result ∑ l =0L−1r⁢l⁢( s 1 ⁢l− s 0 ⁢l) ≷ ℳ 1 ℳ 0 σ2⁢lnη+ E 1 − E 0 2 l 0 L 1 r l s 1 l s 0 l ≷ ℳ 1 ℳ 0 σ 2 η E 1 E 0 2 The left term constitutes the sufficient statistic for the binary detection problem. Because the additive noise is presumed Gaussian, the sufficient statistic is a Gaussian random variable no matter which model is assumed. Under ℳ i ℳ i , the specifics of this probability distribution are ∑ l =0L−1r⁢l⁢( s 1 ⁢l− s 0 ⁢l)∼N⁢mivari l 0 L 1 r l s 1 l s 0 l N mi vari where the mean and variance of the Gaussian distribution are given respectively by mi=∑ s i ⁢l⁢( s 1 ⁢l− s 0 ⁢l) mi s i l s 1 l s 0 l vari=σ2⁢∑ s 1 ⁢l− s 0 ⁢l2 vari σ 2 s 1 l s 0 l 2 Note that the variance does not depend on model. The false-alarm probability is given by P F =Q⁢σ2⁢lnη+ E 1 − E 0 2−m0var12 P F Q σ 2 η E 1 E 0 2 m0 var 1 2 The signal-related terms in the numerator of this expression can be manipulated so that the false-alarm probability of the optimal white Gaussian noise detector is succinctly expressed by P F =Q⁢lnη+12⁢σ2⁢∑ s 1 ⁢l− s 0 ⁢l21σ⁢∑ s 1 ⁢l− s 0 ⁢l212 P F Q η 1 2 σ 2 s 1 l s 0 l 2 1 σ s 1 l s 0 l 2 1 2

Note that the only signal-related quantity affecting this performance probability (and all of the others as well) is the ratio of the energy in the difference signal to the noise variance. The larger this ratio, the better (i.e., smaller) the performance probabilities become. Note that the details of the signal waveforms do not greatly affect the energy of the difference signal. For example, consider the case where the two signal energies are equal ( E 0 = E 1 =E E 0 E 1 E ); the energy of the difference signal is given by 2⁢E−2⁢∑ s 0 ⁢l⁢ s 1 ⁢l 2 E 2 s 0 l s 1 l . The largest value of this energy occurs when the signals are negatives of each other, with the difference-signal energy equaling 4⁢E 4 E . Thus, equal-energy but opposite-signed signals such as sine waves, square-waves, Bessel functions, etc. all yield exactly the same performance levels. The essential signal properties that do yield good performance values are elucidated by an alternate interpretation. The term ∑ s 1 ⁢l− s 0 ⁢l2 s 1 l s 0 l 2 equals ∥ s 1 − s 0 ∥2 s 1 s 0 2 , the L2 L 2 norm of the difference signal. Geometrically, the difference-signal energy is the same quantity as the square of the Euclidean distance between the two signals. In these terms, a larger distance between the two signals means better performance.