Partially Known Signal Waveform 1.4 2003/06/13 2003/09/15 14:03:51.788 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Kyle Clarkson kclarks@rice.edu Elizabeth Gregory lizzardg@rice.edu Kevin Duh kevinduh@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Jeffrey Silverman jsilv@rice.edu The nominal signal waveform is known, but the actual signal present in the observed data can be corrupted slightly. Using the minimax approach, we seek the worst possible signal that could be present consistent with constraints on the corruption. Once we know what signal that is, our detector should consist of a filter matched to that worst-case signal. Let the observed signal be of the form s l s o l c l : s o l is the nominal signal and c l the corruption in the observed signal. The nominal-signal energy is E o and the signal corruption is assumed to have an energy that is less than E c . Spectral techniques are assumed to be applicable so that the covariance matrix of the Fourier transformed noise is diagonal. The signal-to-noise ratio is given by k S k 2 σ k 2 . What corruption yields the smallest signal-to-noise ratio? The worst-case signal will have the largest amount of corruption possible. This constrained minimization problem - minimize the signal-to-noise ratio while forcing the energy of the corruption to be less than E c - can then be solved using Lagrange multipliers. S k k S k 2 σ k 2 λ k C k 2 E c By evaluating the appropriate derivatives, we find the spectrum of the worst-case signal to be a frequency-weighted version of the nominal signal's spectrum. S w k λ σ k 2 1 λ σ k 2 S o k where k 1 1 λ σ k 2 2 S o k 2 E c The only unspecifies parameter is the Lagrange multiplier λ, with the latter equation providing an implicit solution in most cases. If the noise is white, the σ k 2 equal a constant, implying that the worst-case signal is a scaled version of the nominal, equaling S w k 1 E c E o S o k . The robust decision rule derived from the likelihood ratio is given by k L 1 0 R k S o k ≷ ℳ 0 ℳ 1 γ By incorporating the scaling constant 1 E c E o into the threshold γ, we find that the matched filter used in the white noise, known-signal case is robust with respect to signal uncertainties. The threshold value is identical to that derived using the nominal signal as model ℳ 0 does not depend on the uncertainties in the signal model. Thus, the detector used in the known-signal, white noise case can also be used when the signal is partially corrupted. Note that in solving the general signal corruption case, the imprecise signal amplitude situation was also solved. If the noise is not white, the proportion between the nominal and worst-case signal spectral components is not a constant. The decision rule expressed in term of the frequency-domain sufficient statistic becomes k L 1 0 λ σ k 2 1 λ σ k 2 R k S o k ≷ ℳ 0 ℳ 1 γ Thus, the detector derived for colored noise problems is not robust. The threshold depends on the noise spectrum and the energy of the corruption. Furthermore, calculating the value of the Lagrange multiplier in the colored noise problem is quite difficult, with multiple solutions to its constraint equation quite possible. Only one of these solutions will correspond to the worst-case signal.