Dictionary:Wiener-Hopf equations

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(wē’ n∂r hōpf)

1. The Wiener-Hopf equation of the first kind is an integral equation in the unknown f(t):

.

This equation is the necessary and sufficient condition for minimizing the mean-square error between a desired output z(t) and the actual output y(t) which results from passing an input x(t) through a causal filter with an impulse response f(t). is the autocorrelation of x and is the crosscorrelation of z and x. When digital processing is involved, this equation becomes the normal set of linear simultaneous equations (normal equations).

2. The Wiener-Hopf equation of the second kind which applies to a nonstationary input involves a time-varying filter and time-varying correlation functions:

.

See Wiener filter and Lee (1960).