BSA Flow Software Installation and User's Guide - CSI

This definition includes the option that u(t) and/or v(t) might be complex. In
actual measurements, they are both real, so the complex conjugate v*(t)
equals v(t). Furthermore we assume that both u(t) and v(t) are wide-sense
stationary, meaning that the correlation function in (7-54) depends only on
the lag time τ=t2-t1 rather than on the exact values of t1 and t2.
With these simplifications the correlation function Ruv(τ) can be defined as:
R
uv
T
1
( τ) = lim ut ( ) vt ( + τ)
dt
T→∞
T ∫
0
(7-55)
-where the integration limits 0→T reflects that real measurements always
take place over a finite time-span.
Since both u(t) and v(t) are real, the correlation Ruv(τ) is also a real function.
Covariance For simplicity the literature often assumes that the signals u(t) and v(t) have
zero mean, and if they don’t calculations are performed on the fluctuating
part of the signal, meaning that the mean value is subtracted before
correlations are calculated.
Strictly speaking this is not correlation, but covariance, Cuv(τ):
Auto-
&
Cross-
-correlation
&
-covariance
C
uv
T
1
( τ) = lim [ ut ( ) − u][ vt ( + τ)
vdt ]
T→∞
T ∫
− (7-56)
0
-which again assumes that both u(t) and v(t) are real and stationary.
From (7-55) and (7-56) it is easy to show the following simple relation
between correlation and covariance:
R ( τ) = C ( τ)
+ uv
(7-57)
uv uv
Unfortunately much of the literature does not distinguish, and uses the term
correlation, even if covariance is actually being calculated.
From (7-57) it is obvious that if either u(t) or v(t) have zero mean,
covariance and correlation will indeed be identical, but otherwise they will
not, so you should be aware of the difference.
Ruv(τ) and Cuv(τ) are called cross-correlation and cross-covariance when
u(t) and v(t) represent 2 independent measurements, and autocorrelation and
autocovariance when u(t) and v(t) represent the same signal.
In the latter case the symbols Ruu(τ) and Cuu(τ) or simply R(τ) and C(τ) are
often used to designate auto-correlation and auto-covariance respectively.
Note that Ruu(-τ)=Ruu(τ) and Cuu(-τ)=Cuu(τ), meaning that autocorrelation
and autocovariance are even functions. This is not the case for
crosscorrelation and crosscovariance, since from (7-55) we get
Ruv(-τ)=Rvu(τ)≠Ruv(τ).
Note also that for τ=0; the autocovariance equals the variance of u(t):
2 2
{ () } σ uu {}
C( 0)
= E u t − u = = V u (7-58)
Similarly the autocorrelation for τ=0 corresponds to the average signalintensity:
7-122 BSA Flow Software: Reference guide