# 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):

${\displaystyle \phi _{xz}(\tau )=\int _{-\infty }^{\infty }f(t)\phi _{xx}(\tau -t)dt,\;\tau >0}$.

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). ${\displaystyle \phi _{xx}(\tau )}$ is the autocorrelation of x and ${\displaystyle \phi _{xz}(\tau )}$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 ${\displaystyle f(t,\delta )}$ and time-varying correlation functions:

${\displaystyle \phi _{xz}(t,\tau )=\int _{-\infty }^{\infty }f(t,\delta )\phi _{xx}(\delta ,\tau )d\delta }$.

See Wiener filter and Lee (1960).