Adaptive Detection of Known Signals in Additive Noise by Means of Kernel Density Estimators

Rolf Gustafsson
Responsor AB, Odengatan 5, SE-164 92 Kista, Sweden

Ola Hössjer
Dept of Mathematical Statistics, Lund University Box 118, SE-221 00 Lund, Sweden

and Tommy Öberg


IEEE Trans. on Information Theory, vol 43, pp 1192-1204, July 1997. © 1997 IEEE.

Abstract:

We consider the problem of detecting known signals contaminated by additive noise with a completely unknown probability density function f.

To this end, we propose a new adaptive detection rule. It is defined by plugging a kernel density estimatior of f into the maximum a posteriori (MAP) detector. The estimate can either be computed off-line from a training sequence or on-line simultaneously with the detection.

For the off-line detector, we prove that the (asymptotic) error probability for weak signals converges to the minimal error probability of the MAP detector as the number of training data tends to inifinity, and we establish rates of convergence and the optimal choice of bandwith order for a certain class of noise densities.

In a Monte Carlo study, the off-line plug-in MAP detectors are compared with the L¹ and L²-detecors for various noise distributions. When the training sequence is long enough, the plug-in detectors have excellent performance for a wide range of distributions, whereas the L²-detecor breaks down for heavy-tailed distributions and the L¹-detector for distributions with little mass around the origin.