# Dictionary:Wiener-Hopf equations

(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).