Devices involving the principles of Figs. 6 12 and 6 13 are usually restricted to sinusoidal excitation experiments A technique that is more generally applicable in transfer — function determinations involves the use of cross correla­tion. In addition to being used with sinusoidal excitation, cross correlation is commonly used with self-induced noise or types of excitation other than sinusoidal.

The method consists in determining the cross­correlation function, Сху(т), over a range of the time-lag r Either the digital definition, Eq 6.9, or the continuous integral (Table 6.1) definition may be used, depending on the experimental approach It is convenient to handle only the fluctuating parts of variables, i. e., x(t) — x and y(t) — y, such as in Eq. 6.10 After Сху(т) is known, either from an on-line or off-line device, it must be Fourier analyzed to obtain the cross spectrum, Pxy(f) (see Table 6 1), and the transfer function,

PXy(f)

G(f) = T^0 (6 35)

where Px(f) is the spectrum of the input variable, x(t)

The three operations of time delay, multiplication, and integration required to obtain Cxy(r) are indicated in Fig 6 14 Here x(t) may be either a fluctuating signal in the system or an excitation signal The operations shown have been done on-line4 1 for pseudorandom excitation by using “0” and “1” signals (read at the appropriate values of r from a tape containing the input sequence) in a simple switching multiplier

Frequently, however, the two signals x(t) and y(t) are recorded on a frequency-modulation (f-m) magnetic-tape system.40 95 1 00 Off-line playback is carried out using tape heads that are displaced to give the y(t) and x(t — r) input to the multiplier—integrator combination of F’lg 6 14 Analog-computer components are typically used to perform the operations required to give Cxy(r) In the special case where x(t) = y(t), the autocorrelation function may be obtained m this manner.