Oyy (ra) is the autocorrelogram of the system output, Oxx (ra) is the autocorrelogram of the input, and, of course, Oxy (jra) is the system cross-correlogram, all calculated with finite length data. Note that 0 3 y2(ra) 3 1. A coherence function approaching unity means that the system has constant parameters, is linear, and the inputs are stationary and clearly defined random signals of great length. When y2(ra) is less than unity it can mean (1) the system is nonlinear and/or time-variable; (2) extraneous noise is present in the measurement of x(t) and y(t); (3) the output y(t) not only depends on x(t), but also on other (hidden) inputs. In general, y2(ro) will drop off at high frequencies because of windowing and aliasing effects on the sampled x(t) and y(t).

Often linear systems are "identified" or characterized using an input of broadband Gaussian noise. Such noise can be assumed to be white if its frequency spectrum is flat and extends well beyond the frequencies where 20 log|H(jra)| is down by 40 dB. If the white assumption can be justified, then one can let ®xx(ra) = K mean-squared units/rad/s. The inverse Fourier transform of ®xx(ra) is the autocorrelation function, 9xx(x). In the case of white noise, 9xx(x) = K8(x), that is, a delta function of area K. This means that ideally, H(jra) = ®xy(jra)/K.

Suppose a system is described by having two inputs, x(t) and u(t), each of which is acted on by linear systems (LS) having frequency response functions, Hx(jra) and Hu(jra), respectively. The LS outputs, yx(t) and yu(t) are added to make y(t), the overall system output (See Figure 8.1). By superposition, the autopower spectrum of y(t) can be written

®yy(ra) = ®xx(ro)|Hx(jro)|2 + ®u»|Hu(jra)|2 8.10

The output cross-power spectra are given by the simultaneous vector equations:

®xy(jro) = Hx(jra) ®xx(ro) + Hu(jra) ®xu(j®) 8.11A

®Uy(jra) = Hx(jra) ®ux(ro) + Hu(jra) ®uu(jra) 8.11B

FIGURE 8.0-1 Block diagram of a linear system with a common (summed) output.

One can solve for the transfer functions, Hx(jra) and Hu(jra) using Cramer's rule. Thus,

xy ((")fruuN - uy ((")frxu (j") [ xx uu (")-fr ux ((")fr xu ((")]

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