Geometrical properties of optimal Volterra filters for signal detection

Volterra filters are a particular class of nonlinear filters defined by an extension of the concept of impulse response to the nonlinear case. Linear-quadratic filters are a special example of Volterra filters limited to the second order. In the first part of this paper it is shown that all the results recently published, valid in the linear-quadratic case, can be extended with the appropriate notations to Volterra filters of arbitrary order. In particular, the optimum Volterra filter giving the maximum of the deflection for the detection of a signal in noise is wholly calculated. In the second part several geometrical properties of optimal Volterra filters are investigated by introducing appropriate scalar products. In particular the concept of space orthogonal to the signal and the noise alone reference (NAR) property are introduced allowing a decomposition of the optimal filter that exhibits a relation between detection and estimation. Extensions to the infinite case and relations with the likelihood ratio are also investigated.

[1]  W. D. Stirling.  Least Squares Subject to Linear Constraints , 1981 .

[2]  S. Fakhouri.  Identification of the Volterra kernels of nonlinear systems , 1980 .

[3]  B. Picinbono,et al.  Detection and contrast , 1986 .

[4]  D. Kazakos.  Optimal constrained representation and filtering of signals , 1983 .

[5]  BERNARD C. PICINBONO.  A geometrical interpretation of signal detection and estimation (Corresp.) , 1980, IEEE Trans. Inf. Theory.

[6]  M. Schetzen,et al.  Nonlinear system modeling based on the Wiener theory , 1981, Proceedings of the IEEE.

[7]  R.N. Bracewell,et al.  Signal analysis , 1978, Proceedings of the IEEE.

[8]  P. Eykhoff,et al.  Some fundamental aspects of process-parameter estimation , 1963 .

[9]  Michel Bouvet,et al.  Constrained Wiener filtering , 1987, IEEE Trans. Inf. Theory.

[10]  Thomas Kailath.  Lectures on linear least-squares estimation , 1976 .

[11]  David J. Sakrison,et al.  Solution manual for problems in Communication theory : transmission of waveforms and digital information , 1968 .

[12]  P. Duvaut,et al.  Detection estimation et filtres de volterra , 1987 .

[13]  A. Journel,et al.  Kriging in terms of projections , 1977 .

[14]  Patrick Duvaut,et al.  Optimal linear-quadratic systems for detection and estimation , 1988, IEEE Trans. Inf. Theory.

[15]  P. Alper.  A consideration of the discrete Volterra series , 1965 .

[16]  Thomas Kailath,et al.  A projection method for signal detection in colored Gaussian noise , 1967, IEEE Trans. Inf. Theory.

[17]  Thomas Kailath,et al.  RKHS approach to detection and estimation problems-I: Deterministic signals in Gaussian noise , 1971, IEEE Trans. Inf. Theory.

[18]  B. Picinbono,et al.  NAR AGC adaptive detection of nonoverlapping signals in noise with fluctuating power , 1981 .