2.11 Spectral detection

From the results presented in the previous sections, the colored noise problem was found to be pervasive, but required a computationally difficult detector.The simplest detector structure occurs when the additive noise is white; this notion leads to the idea of whitening theobservations, thereby transforming the data into a simpler form (as far as detection theory is concerned). However, therequired whitening filter is often time-varying and can have a long-duration unit-sample response. Other, more computationallyexpedient, approaches to whitening are worth considering. An only slightly more complicated detection problem occurs when wehave a diagonal noise covariance matrix, as in the white noise case, but unequal values on the diagonal. In terms of theobservations, this situation means that they are contaminated by noise having statistically independent, but unequal variancecomponents: the noise would thus be non-stationary. Few problems fall directly into this category; however, the colorednoise problem can be recast into the white, unequal-variance problem by calculating the discrete Fourier Transform (DFT) ofthe observations and basing the detector on the resulting spectrum. The resulting spectral detectors greatly simplify detector structures for discrete-time problems if the qualifying assumptions described in sequel hold.

Let $W$ be the so-called $(L, L)$ "DFT matrix" $$W=\begin{pmatrix}1 & 1 & 1 & & 1\\ 1 & W & W^2 & & W^{(L-1)}\\ 1 & W^2 & W^4 & & W^{(2(L-1))}\\ & & & & \\ 1 & W^{(L-1)} & W^{(2(L-1))} & & W^{((L-1)(L-1))}\\ \end{pmatrix}$$ where $W$ is the elementary complex exponential $e^{-(i\frac{2\pi }{L})}$ . The discrete Fourier Transform of the sequence $r(l)$ , usually written as $R(k)=\sum_{l=0}^{L-1} r(l)e^{-(i\frac{2\pi lk}{L})}$ , can be written in matrix form as $R=Wr$ . To analyze the effect of evaluating the DFT of the observations, we describe the computations in matrix form foranalytic simplicity. The first critical assumption has been made: take special note that the length of the transform equals the duration of the observations. In many signal processing applications, thetransform length can differ from the data length, being either longer or shorter.The statistical properties developed in the following discussion are critically sensitive to the equality of these lengths. Thecovariance matrix $K_R$ of $R$ is given by $W{K}_r(W)$ . Symmetries of these matrices - the Vandermonde form of $W$ and the Hermitian, Toeplitz form of $K_r$ - leads to many simplifications in evaluating this product. The entries no the main diagonal aregiven by

In the simplest case, the covariance matrix of the discrete Fourier Transform of the observations can be well approximatedby a diagonal matrix. $$K_R=\begin{pmatrix}{}_0^2 & 0 & & 0\\ 0 & {}_1^2 & 0 & \\ & 0 & & 0\\ 0 & & 0 & {}_{L-1}^2\\ \end{pmatrix}$$ The non-zero components ${}_k^2$ of this matrix constitute the noise power spectrum at the various frequencies. The signal component of thetransformed observations $R$ is represented by $S_i$ , the DFT of the signal $s_i$ , while the noise component has this diagonal covariance matrix structure.

• The transform length equals that of the observations. In particular, the observations cannot be "padded" to forcethe transform length to equal a "nice" number (like a power of two).
• The noise's correlation structure should be much less than the duration of the observations. Equivalently, a narrowcorrelation function means the corresponding power spectrum varies slowly with frequency. If either condition fails tohold, calculating the Fourier Transform of the observations will not necessarily yield a simpler noise covariance matrix.

Sinusoidal signals are particularly well-suited to the spectral detection approach. If the signal's frequency equals one of the analysis frequencies in the FourierTransform ( ${}_0=\frac{2\pi k}{L}$ for some $k$ ), then the sequence $S_i(k)$ is non-zero only at this frequency index, only one term in the sufficient statistic's summation need be computed,and the noise power is no longer explicitly needed by the detector (it can be merged into the threshold). $$\Re ((R)K_R^{(-1)}{S}_i)-\frac{1}{2}(S_i)K_R^{(-1)}{S}_i=\frac{\Re (\overline{R(k)}{S}_i(k))}{{}_k^2}-\frac{1}{2}\frac{\left|S_i(k)\right|^2}{{}_k^2}$$ If the signal's frequency does not correspond to one of the analysis frequencies, spectral energy will be maximal at thenearest analysis frequency but will extend to nearby frequencies also. This effect is termed "leakage" and has been wellstudied. Exact formulation of the signal's DFT is usually complicated in this case; approximations which utilize only themaximal-energy frequency component will be sub-optimal ( i.e. , yield a smaller detection probability). The performance reduction may be small, however,justifying the reduced amount of computation.