Crosscorrelationto

Crosscorrelationto is not a standard term in published literature. It most plausibly refers to cross-correlation with respect to a reference signal or template, a common operation in signal processing used to assess similarity as one signal is shifted in time.

In discrete time, the cross-correlation of two sequences x[n] and y[n] is defined as r_xy[k] = sum over n of x[n]·y[n + k], where the sum includes only indices for which both signals are defined. Other common conventions use y[n − k], which simply reverses the lag direction. In continuous time, the cross-correlation is r_xy(τ) = ∫ x(t)·y(t + τ) dt, representing the degree of similarity as a function of lag τ. For finite-length signals, the resulting function is defined over a finite lag range and often computed with padding.

Normalization can yield the normalized cross-correlation, r_xy_norm[k] = r_xy[k] / sqrt(r_xx[0]·r_yy[0]), producing values typically in the interval [−1, 1], facilitating comparisons across signals. Cross-correlation is closely related to autocorrelation (where x = y) and to the cross power spectral density, via the Wiener–Khinchin theorem: the Fourier transform of r_xy(τ) equals the cross power spectral density S_xy(ω) for stationary processes. Efficient computation often uses the Fast Fourier Transform (FFT) by exploiting the convolution–correlation relationship.

Applications include time-delay estimation, template matching, pattern recognition, synchronization in communications, and seismology. Practical considerations include noise sensitivity, edge effects for finite data, and the choice of normalization, which can affect the interpretation of peak values. See also cross-correlation, autocorrelation, convolution, and power spectral density.