2.4 White gaussian noise

By far the easiest detection problem to solve occurs when the noise vector consists of statistically independent, identicallydistributed, Gaussian random variables. In this book, a white sequence consists of statistically independent random variables. The white sequence's mean isusually taken to be zero

Each term in the computations for the optimum detector has a signal processing interpretation. When expanded, the term $s_i^T{s}_i$ equals $\sum_{l=0}^{L-1} s_i(l)^2$ , which is the signal energy $E_i$ . The remaining term - $r^T{s}_i$ - is the only one involving the observations and hence constitutes the sufficient statistic ${}_i(r)$ for the additive white Gaussian noise detection problem. $${}_i(r)=r^T{s}_i$$ An abstract, but physically relevant, interpretation of thisimportant quantity comes from the theory of linear vector spaces. There, the quantity $r^T{s}_i$ would be termed the dot product between $r$ and $s_i$ or the projection of $r$ onto $s_i$ . By employing the Schwarz inequality, the largest value of this quantity occurs when these vectors are proportional to eachother. Thus, a dot product computation measures how much alike two vectors are: they are completely alike when they areparallel (proportional) and completely dissimilar when orthogonal (the dot product is zero). More precisely, the dotproduct removes those components from the observations which are orthogonal to the signal. The dot product thereby generalizesthe familiar notion of filtering a signal contaminated by broadband noise. In filtering, the signal-to-noise ratio of abandlimited signal can be drastically improved by lowpass filtering; the output would consist only of the signal and"in-band" noise. The dot product 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).