Introduction
A
N IMPORTANT
class of theoretical and practical
problems in communication and control is of a statistical nature.
Such problems are: (i) Prediction of random signals; (ii) separa-
tion of random signals from random noise; (iii) detection of
signals of known form (pulses, sinusoids) in the presence of
random noise.
In his pioneering work, Wiener [1]
3
showed that problems (i)
and (ii) lead to the so-called Wiener-Hopf integral equation; he
also gave a method (spectral factorization) for the solution of this
integral equation in the practically important special case of
stationary statistics and rational spectra.
Many extensions and generalizations followed Wiener’s basic
work. Zadeh and Ragazzini solved the finite-memory case [2].
Concurrently and independently of Bode and Shannon [3], they
also gave a simplified method [2] of solution.
Booton discussed
the nonstationary Wiener-Hopf equation [4]. These results are
now in standard texts [5-6]. A somewhat different approach along
these main lines has been given recently by Darlington [7]. For
extensions to sampled signals, see, e.g., Franklin [8], Lees [9].
Another approach based on the eigenfunctions of the Wiener-
Hopf equation (which applies also to nonstationary problems
whereas the preceding methods in general don’t), has been
pioneered by Davis [10] and applied by many others, e.g.,
Shinbrot [11], Blum [12], Pugachev [13], Solodovnikov [14].
In all these works, the objective is to obtain the specification of
a linear dynamic system (Wiener filter) which accomplishes the
prediction, separation, or detection of a random signal.
4
———
1
This research was supported in part by the U. S. Air Force Office of
Scientific Research under Contract AF 49 (638)-382.
2
7212 Bellona Ave.
3
Numbers in brackets designate References at end of paper.
4
Of course, in general these tasks may be done better by nonlinear
filters. At present, however, little or nothing is known about how to obtain
(both theoretically and practically) these nonlinear filters.
Contributed by the Instruments and Regulators Division and presented
at the Instruments and Regulators Conference, March 29– Apri1 2, 1959,
of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS.
NOTE: Statements and opinions advanced in papers are to be understood
as individual expressions of their authors and not those of the Society.
Manuscript received at ASME Headquarters, February 24, 1959. Paper
No. 59—IRD-11.
Present methods for solving the Wiener problem are subject to
a number of limitations which seriously curtail their practical
usefulness:
(1) The optimal filter is specified by its impulse response. It is
not a simple task to synthesize the filter from such data.
(2) Numerical determination of the optimal impulse response is
often quite involved and poorly suited to machine computation.