Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 30

Fig. 1.19 Pole-zero diagram
of the system
H(s) = s(s ? 1)/(s ? 2) 2
(s ? 3)

ω

j

s − plane

σ x x

−3 −2 −1 0

R sr ¼ Z

1

s ð k Þ r ð s þ k Þ dk ð 1 : 25 Þ

1

When the two signals being correlated are different it is common to refer to the
correlation integral as the cross-correlation function, If the two signals are the
same, the integral is typically referred to as the auto-correlation function.
This cross-correlation function is of the same form as the convolution integral
in ( 1.3 ), but the argument of the second function has a ''+'' sign in place of a ''-''
sign and also incorporates the time-delay s instead of the true time variable, t. Note
also that R sr (s) = R rs (s). The correlation gives an indication of how the two sig-
nals are ''related'' to each other when the time difference between them is s.In
other words, it is a measure of similarity between two signals separated by s.If
r(t) equals s(t), the correlation is referred to as the auto-correlation function of the
signal s(t), and is denoted by R s (s). For a periodic deterministic signal x(t) with
period T o , the auto-correlation function is defined as:

Z

T o

R x ð s Þ¼ 1
T o

x ð k Þ x ð s þ k Þ dk

0

There are also correlation formulas for random signals that will be defined later.
It can be shown that the auto-correlation function has the following properties
(see Tutorial 26):

P1: R x (s) is even (or symmetric about time-delay axis), R x (s) = R x (-s).
P2: R x (s) always has its absolute maximum at the origin, s = 0.
(see, for example, Fig. 1.20 , which shows the autocorrelation function of a non-
periodic signal).
The correlation integral between two signals tends to have a large maximum
value at s = 0 if the signals are exactly the same (auto-correlation), and has a finite
non-zero value over some range of s when the two signal are somewhat similar.It
tends to be approximately zero if the two signals are very dissimilar (as would be
the case, say, if one of the signals was a deterministic signal and the other was