113
Other Signal and Image Processing Methods
trivial example may seem too simple and obvious, there are so many other factors,
such as exercise and nutrition patterns whose potential effects on the function and
activities of the heart are constantly investigated using the cross-correlation function.
The cross-correlation function for the discrete processes is defined as follows:
+∞
r XY (m) =
x n y n
( ) ( − m p XY x n y n
))
(6.26)
) ( ( ), ( − m
∑
n=−∞
It is important to note that all autocorrelation and cross-correlation functions defined
earlier, even though calculated from random variables, are indeed deterministic signals. This means that we can apply the FT on these signals and investigate the frequency contents of the underlying stochastic processes. The FT of the correlation
signal is called power spectrum. For continuous stochastic processes, power spectrum is defined as follows:
R f = FT {rXX ( )
(6.27)
XX ( )
t }
and
R XY f =
{rXY ( )
( ) FT
t }
(6.28)
Similarly, for the discrete stochastic processes, the power spectral functions are
defined as follows:
R XX ( )
k = FT {rXX (m)}
(6.29)
and
R XY ( )
k = FT {rXY (m)}
(6.30)
For typical periodic signals, the overall shape of a power spectrum has two impulses
located at positive and negative frequency of oscillation. The peak on the positive
side is at the frequency of oscillation (one over the period of the signal), and the negative one is located simply at minus the frequency of oscillation.
An interesting power spectrum to study is that of the white noise. As discussed
earlier, the autocorrelation function of the white noise is an impulse. We also know
from the previous chapters that the FT of an impulse is a constant function that is
flat in all frequencies. This flat shape is the reason the white noise is called “white.”
This analogy is originated in optics where the perception of each color is caused by
an electromagnetic wave with a specific frequency, while white color, being a combination of all colors has all frequencies in it. Since the white noise’s flat frequency
spectrum has all frequencies in it, this noise is often referred to as the white noise.
Other Signal and Image Processing Methods
trivial example may seem too simple and obvious, there are so many other factors,
such as exercise and nutrition patterns whose potential effects on the function and
activities of the heart are constantly investigated using the cross-correlation function.
The cross-correlation function for the discrete processes is defined as follows:
+∞
r XY (m) =
x n y n
( ) ( − m p XY x n y n
))
(6.26)
) ( ( ), ( − m
∑
n=−∞
It is important to note that all autocorrelation and cross-correlation functions defined
earlier, even though calculated from random variables, are indeed deterministic signals. This means that we can apply the FT on these signals and investigate the frequency contents of the underlying stochastic processes. The FT of the correlation
signal is called power spectrum. For continuous stochastic processes, power spectrum is defined as follows:
R f = FT {rXX ( )
(6.27)
XX ( )
t }
and
R XY f =
{rXY ( )
( ) FT
t }
(6.28)
Similarly, for the discrete stochastic processes, the power spectral functions are
defined as follows:
R XX ( )
k = FT {rXX (m)}
(6.29)
and
R XY ( )
k = FT {rXY (m)}
(6.30)
For typical periodic signals, the overall shape of a power spectrum has two impulses
located at positive and negative frequency of oscillation. The peak on the positive
side is at the frequency of oscillation (one over the period of the signal), and the negative one is located simply at minus the frequency of oscillation.
An interesting power spectrum to study is that of the white noise. As discussed
earlier, the autocorrelation function of the white noise is an impulse. We also know
from the previous chapters that the FT of an impulse is a constant function that is
flat in all frequencies. This flat shape is the reason the white noise is called “white.”
This analogy is originated in optics where the perception of each color is caused by
an electromagnetic wave with a specific frequency, while white color, being a combination of all colors has all frequencies in it. Since the white noise’s flat frequency
spectrum has all frequencies in it, this noise is often referred to as the white noise.