RKHS approach to detection and estimation problems--I: Deterministic signals in Gaussian noise
- 1 September 1971
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 17 (5), 530-549
- https://doi.org/10.1109/tit.1971.1054673
Abstract
First it is shown how the Karhunen-Loève approach to the detection of a deterministic signal can be given a coordinate-free and geometric interpretation in a particular Hilbert space of functions that is uniquely determined by the covariance function of the additive Gaussian noise. This Hilbert space, which is called a reproducing-kernel Hilbert space (RKHS), has many special properties that appear to make it a natural space of functions to associate with a second-order random process. A mapping between the RKHS and the linear Hilbert space of random variables generated by the random process is studied in some detail. This mapping enables one to give a geometric treatment of the detection problem. The relations to the usual integral-equation approach to this problem are also discussed. Some of the special properties of the RKHS are developed and then used to study the singularity and stability of the detection problem and also to suggest simple means of approximating the detectability of the signal. The RKHS for several multidimensional and multivariable processes is presented; by going to the RKHS of functionals rather than functions it is also shown how generalized random processes, including white noise and stationary processes whose spectra grow at infinity, are treated.Keywords
This publication has 32 references indexed in Scilit:
- A general likelihood-ratio formula for random signals in Gaussian noiseIEEE Transactions on Information Theory, 1969
- Examples of optimum detection of Gaussian signals and interpretation of white noiseIEEE Transactions on Information Theory, 1968
- A recursive approach to signal detectionIEEE Transactions on Information Theory, 1968
- Radon-Nikodym Derivatives of Gaussian MeasuresThe Annals of Mathematical Statistics, 1966
- Randon-Nikodym Derivatives of Stationary Gaussian MeasuresThe Annals of Mathematical Statistics, 1964
- Lower Bounds for Minimum Covariance Matrices in Time Series Regression ProblemsThe Annals of Mathematical Statistics, 1964
- On Positive-Definite Integral Kernels and a Related Quadratic FormTransactions of the American Mathematical Society, 1958
- Integral representations of positive definite functions. IITransactions of the American Mathematical Society, 1954
- Integral representations of positive definite functionsTransactions of the American Mathematical Society, 1953
- Le Mouvement Brownien PlanAmerican Journal of Mathematics, 1940