In some situations, it is desirable to measure the parameters of a linear system "online" without being able to use either an impulse input or a purely sinusoidal input with variable frequency. Under these conditions, one records simultaneously the system input, x(t), and its output, y(t), over a long period of time. Over that period of time, the system must remain stationary (i.e., the system parameters must remain fixed in time). x(t) can be a random signal, including broadband Gaussian noise. The system cross-correlation function, 9xy(x) is defined here as
(In practice, a cross-correlogram, $xy(x), is computed because of the finite length of data.) The convolution integral can be used to bring the system weighting function into the expression:
9xy (T) " T^ 2T J ) J ^h(V) x( + X - V) dV dt 83
Inverting the order of integration (a legitimate bit of mathematical legerdemain) yields:
9 xy (t) = J ^h(v) dv ^ 2T J Tx(l) x(l + x - v) dt 8.4
The second integral is the autocorrelation function of the input signal, defined as
Thus the cross-correlation function can finally be expressed as
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