User's Manual - Cornell Lab of Ornithology - Cornell University

Normalized correlationsCanary's normalized correlation pre-normalizes the vectors x and y so thatthe maximum correlation magnitude value is 1. The formula for normalizedcorrelations is^R xy [n] = jjxjj jjyjj where jjxjj = p Pm x[m]2 : If we use the notation ^x = x=jjxjj we have^R xy [n] =R^x^y [n]:Correlations are best known in the signal processing literature as a theoreticaltool for analyzing the spectra of wide-sense stationary stochasticprocesses [3,5,9,10]. Their use as a pattern recognizer in deterministic signalsarises from the following relation for normalized correlations.^R xy [n] =1; 1 2 jj^x n ; ^yjj 2Therefore, a match between x n and y will result in a peak value of 1 in ^R xyat n. Avalue of ^R xy [n] = 0 means that x n and y are orthogonal.Filtered correlationsCanary's correlations are computed by taking the FFTs of the input vectors,multiplying one by the conjugate of the other, and computing the inverseFFT of the result. For ltered correlations, before multiplying the FFTs ofthe input signals, the frequency bins outside of the specied pass band arezeroed.For normalized ltered correlations, the norms jjxjj and jjyjj are computedfrom the FFTs after zeroing the stop bands.Complex envelopeThe complex envelope of a real signal is the magnitude of the corresponding\analytic" signal. The analytic signal corresponding to x[n] =a[n]cos(!n)isz[n] = a[n]e j!n= a[n]cos(!n)+ja[n]sin(!n)= x[n]+jH(x)[n]