110
Biomedical Signal and Image Processing
processes can be approximated by some Gaussian process, such stationary Gaussian
processes are extremely important in signal processing applications, especially biomedical signal and image processing. For instance, in almost all processing techniques used for analysis of EEG, ECG, and EMG, the processes are assumed to be
stationary Gaussian. Hereafter, since strict and wide sense stationary Gaussian processes are the same; in referring to such processes, we only use the word stationary
without mentioning the details.
So far we have made several simplifying assumptions to create stochastic formulations that are more applicable to practical image and signal processing applications. However, even with the assumption of stationarity, it is practically impossible
to apply such formulations to some very important applications. At this point, we
need to make another assumption to narrow down our focus further to obtain a more
practically useful model of stochastic processes. The need and motivation for making more simplifying assumptions is as follows. Note that in real applications, the
PDF is not known and must be estimated from the observed data. In order to obtain
an estimate of the PDF or any statistical averages, we need to have access to several recordings of the process and then perform an ensemble averaging over these
recordings to estimate the desired averages. However, in almost all applications,
especially biomedical applications, very often only one signal reading is available.
For instance, clinics and hospitals capture only one ECG recording from a patient.
It is unreasonable to expect the clinics to collect a few hundred EEG recordings for
each patient so that we conduct our ensemble averaging to calculate PDF!
From the earlier discussion, it is clear that in many practical applications, one
recording is all we have, and, therefore, we need to calculate the averages such as
mean and variance from only one recording. Statistically speaking, this is not feasible unless further assumptions are made. The assumption that helps us with such
situations allows us to perform the averaging across time for only one recording and
to treat these averages as our ensemble averages. The stationary processes in which
the averages of any recording of the signal across time equal the ensemble averages
are called “ergodic processes.” Formally speaking, considering any outcome signal
x(w i , t) recorded for an ergodic process, we have
+∞
g t
( ) = E g x t =
( ( i , )) t
( ( ( )))
g x w t d
(6.21)
∫
−∞
Similarly, in discrete ergodic processes using only one recording of the stochastic
process, x(w i , n), we can use the following relation to calculate all ensemble averages
through averaging in time:
+∞
g n = E g x w n
( ( ( , ))) = ∑ ( ( i ,
(6.22)
( )
g x w n))
n=−∞
For ergodic processes, all averages such as mean and variance can be calculated using the aforementioned time averaging. Ergodicity is a practically useful
Biomedical Signal and Image Processing
processes can be approximated by some Gaussian process, such stationary Gaussian
processes are extremely important in signal processing applications, especially biomedical signal and image processing. For instance, in almost all processing techniques used for analysis of EEG, ECG, and EMG, the processes are assumed to be
stationary Gaussian. Hereafter, since strict and wide sense stationary Gaussian processes are the same; in referring to such processes, we only use the word stationary
without mentioning the details.
So far we have made several simplifying assumptions to create stochastic formulations that are more applicable to practical image and signal processing applications. However, even with the assumption of stationarity, it is practically impossible
to apply such formulations to some very important applications. At this point, we
need to make another assumption to narrow down our focus further to obtain a more
practically useful model of stochastic processes. The need and motivation for making more simplifying assumptions is as follows. Note that in real applications, the
PDF is not known and must be estimated from the observed data. In order to obtain
an estimate of the PDF or any statistical averages, we need to have access to several recordings of the process and then perform an ensemble averaging over these
recordings to estimate the desired averages. However, in almost all applications,
especially biomedical applications, very often only one signal reading is available.
For instance, clinics and hospitals capture only one ECG recording from a patient.
It is unreasonable to expect the clinics to collect a few hundred EEG recordings for
each patient so that we conduct our ensemble averaging to calculate PDF!
From the earlier discussion, it is clear that in many practical applications, one
recording is all we have, and, therefore, we need to calculate the averages such as
mean and variance from only one recording. Statistically speaking, this is not feasible unless further assumptions are made. The assumption that helps us with such
situations allows us to perform the averaging across time for only one recording and
to treat these averages as our ensemble averages. The stationary processes in which
the averages of any recording of the signal across time equal the ensemble averages
are called “ergodic processes.” Formally speaking, considering any outcome signal
x(w i , t) recorded for an ergodic process, we have
+∞
g t
( ) = E g x t =
( ( i , )) t
( ( ( )))
g x w t d
(6.21)
∫
−∞
Similarly, in discrete ergodic processes using only one recording of the stochastic
process, x(w i , n), we can use the following relation to calculate all ensemble averages
through averaging in time:
+∞
g n = E g x w n
( ( ( , ))) = ∑ ( ( i ,
(6.22)
( )
g x w n))
n=−∞
For ergodic processes, all averages such as mean and variance can be calculated using the aforementioned time averaging. Ergodicity is a practically useful