This expression is simply the real convolution of the linear system weighting function with the autocorrelation function of the system input. If the Fourier transform of both sides of Equation 8.6 is taken, the well-known result is obtained:
Note that ®xy(jra) and H(jra) are complex, while ®xx(ra) is a real, even function. Equation 8.7 can be used to find the frequency response function of the system:
In practice, the relation above yields an estimate of H(jra) because of the finite data length of x and y. A measure of how well H(jra) is estimated is found in the coherence function, y2(ra), for the system (Bendat and Piersol, 1966). The coherence function is real, and is defined as
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