SIGNAL PROCESSING - Biomechanics and Motor Control of Human Movement - page 19

(T cos (ωt )

sin (ωt ))

E 2
2 T

1
2 ω ( sin ( 2 ωT

R xx (τ ) = − 0 ) − + ωτ) −

Since T is one period of sin (ωt) , ∴ sin ( 2 ωT + ωτ) − sin (ωτ ) = 0 for all τ .

E 2
2

∴ R xx (τ ) = cos (ωτ ) (2.6)

E 2

Similarly if x (t ) = E cos (ωt ) also R xx (τ ) =

2 cos (ωτ )
Note that Equation (2.6) is an even function as predicted by property #2;
aplotofthis R xx (τ ) after normalization is presented in Figure 2.3.
This property is useful in detecting the presence of periodic signals buried
in white noise. White noise is defined as a signal made up of a series of
random points, where there is zero correlation between the signal at any
point with the signal at any point ahead of or behind it in time. Therefore, at
any τ 1. Thus, the autocorrelation of

white noise is an impulse, as shown in Figure 2.4.
If we have a signal, s(t ) , with added noise, n(t ) , we can express x (t ) =
s(t ) + n(t ) , and substituting in the numerator of Equation (2.2) we get:

= 0 R xx (τ ) = 0 and at τ = 0 R xx (τ ) =

T

R xx (τ ) =

( s(t ) + n(t ) )( s(t + τ) + n(t + τ) ) dt

0

T T

= s(t )s(t + τ)dt + n(t )s(t + τ)dt

0 0

T T

+ s(t )n(t + τ)dt + n(t )n(t + τ)dt

0 0

Since the signal and noise are uncorrelated, the 2 nd and 3 rd terms will = 0.

∴ R xx (τ ) = R ss (τ ) + R nn (τ ) (2.7)

Property #4. As seen in property #3 the frequency content of x (t ) is present
in R xx (τ ) . The power spectral density function is the Fourier transform of
R xx (τ ) ; more will be said about this in the next section on frequency analysis.
However, it is sometimes valuable to use the autocorrelation function to
identify any periodicity present in x (t ) or to identify the presence of an
interfering signal (e.g., hum) in our biological signal. Even if there were no
periodicity in x (t ) , the duration of R xx (τ ) would give an indication of the
frequency spectra of x (t ) ; lower frequencies result in R xx (τ ) remaining above
zero for longer phase shifts, while high frequencies tend to zero for small
phase shifts.