An Exact Solution for Sparse Sampling for Optimal Detection of Known Signals in Gaussian Noise

Detection of known signals of interest that are embedded in colored noise involves whitening the received samples and matched-filtering. In many applications, due to computational constraints, it is critical to select only a subset of the received samples for detection. This letter addresses the problem of selecting only a given number of temporal or spatial samples while maximizing detection performance for deterministic signals in first-order autoregressive Gaussian noise. The direct solution of this entails a combinatorial search, where the deflection coefficient is evaluated for each possible combination of sparse samples. This approach is infeasible when the number of samples is large since the number of possible combinations increases factorially with the number of samples. We present an efficient method to whiten Gaussian noise samples and express deflection coefficient in a form that is amenable to dynamic programming. Exploiting dynamic programming, we specify a feasible and efficient procedure to find optimal sparse samples where the number of computational steps increases linearly with the number of samples. Also, conditions under which uniform sampling is optimal is given.