The autocorrelation function of a signal is defined as

(2.7.1)

where Tm is the integration time. The cross-correlation function of two signals is defined as

(2.7.2)

Clearly, if the signal at some time is strongly dependent on the same signal or another signal т seconds earlier, then the autocorrelation function or the cross-correlation function will be large for a delay t.

The autocorrelation function of a signal with period T or the cross-correla­tion function of two signals each with period T is a periodic function with period T.

It is important to develop a relation between the correlation functions and the Fourier coefficients for a signal. Consider first a periodic signal. It may be represented as

Then

or

Cі i(t) = (1/rj ехр[у(ш, + mk)t] dt

If Tm is an integer multiple of the period T, then this reduces to

Сц(т)= X Qc-texP(j’M

The Fourier transform of the autocorrelation function is

^{Сц(т)}Юк = (1/Г) Сц^ехрї-ушцтМт

That is, the Fourier transform of the autocorrelation function at frequency oik is the component of the power spectrum of the signal at frequency cok. The relation

= CkC_k (2.7.3)

is called Wiener’s theorem for periodic signals.

A similar relation may be developed for the cross-correlation function. If the Fourier series for x2(t) is

oo

x2U) = £ Dk exp(ja>kt)

fc = — oo

then

F{C 12(т)}Ик = DkC_k (2.7.4)

The quantity DkC^k is the component of the cross-power spectrum at cok.

Similar results may be derived for nonperiodic signals. In this case, the autocorrelation function and cross-correlation function are defined as

Л * m/2

Cn(r)= lim (1/TJ x^tjx^t + T)dt

Tm~* ZC J ~Tm/2

{•Tm/2

C12(t) = lim (l/TJ xi(t)x2(t + t)dt

The relations between the correlation functions and the power spectra are

f{Cn(r)) = x10’a))x1(-/ft)) (2.7.7)

^{Ci2W} = x^j^x^-jco) (2.7.8)

where XiO’ct)) is the Fourier transform of xt(t), and x2(jco) the Fourier transform of x2(t).

1.8. Convolution

The convolution theorem may be used for inversion of a product of Laplace transforms. The convolution theorem is

L~ 1{x1(s)x2(s)} = f хД-фc2(t — T)dr (2.8.1)

J о

where xt(t) is the inverse Laplace transform of x^s), and x2(t) the inverse Laplace transform of x2(s).

Example 2.8.1. Invert the following, using the convolution theorem:

1 / 1

s + 1) (s T 2

Xi(t) =

x2(t — r) = e~2t’~x)

f(t) = L lF(s) = Г e xe 2(‘ x> dx = e 1 — e 21 J о

The convolution integral has an important application in connection with system transfer functions. The definition of the transfer function G(s) is given by Eq. (2.3.1):

G(s) = SO(s)/Sl(s)

Then the Laplace transform of the output involves the Laplace transform of a product

SO(s) = G(s) SI(s)

Using the convolution theorem, we can specify the output time response: <50(t) = f h(t — t) <5/(t) dr (2.8.2)

SO(t) = Г /і(т) Sl(t — т) dr (2.8.3)

J о

where

/i(t) = L-1{G(s)} (2.8.4)

Thus, if h(t) is known, then the convolution theorem may be used to furnish the output for any input function. The function h(t) is called the impulse response of the system. It is a fundamental dynamic property of the system, and its determination is often the objective in a dynamics test.