Biomedical Engineering Reference
In-Depth Information
Many biological signals are approximately periodic in nature. Signals associated with
the beating heart—blood pressure, blood velocity, electrocardiogram (ECG)—fall into this
category. However, due to intrinsic natural variability, noise, and/or the influence of other
functions, such as respiration, beat-to-beat differences are to be expected. Figure 11.2 is an
example of a blood pressure signal that has all of the described variability.
Blood pressure signals have many features that clinicians and researchers use to deter-
mine a patient's health. Some variables that are often measured include the peak pressure
while the heart is ejecting blood (systolic phase), the minimum pressure achieved while
the aortic valve is closed (diastolic phase), the peak derivative (dP/dt) during the early part
of the systolic phase (considered an indication of the strength of the heart), and the time
constant of the exponential decay during diastole (a function of the resistance and compli-
ance of the blood vessels). One way to determine variables of interest is to calculate the
variables or parameters for each beat in a series of beats and then report the means. This
is often not possible because noise from individual measurements makes it very difficult
to accurately determine the relevant biological parameters. An alternate approach is to first
average the signal measurements from separate trials,
N
1
N
x
ð
t
Þ¼
1
x
i
ð
t
Þ
,
ð
11
:
50
Þ
i
¼
such that a representative beat is obtained. If the signal is discrete, this average is repre-
sented by
N
1
N
x
ð
k
Þ¼
1
x
i
ð
k
Þ
ð
11
:
51
Þ
i
¼
Here,
x
i
(
t
), or
x
i
(
k
) for the discrete case, represents the
i
-th measured heartbeat signal out of
a total of
for the discrete case, represents the mean or
average waveform obtained following the averaging procedure. Substituting Eq. (11.50)
into (11.49) leads to
N
measurements. The signal
x
ð
t
Þ
,
x
ð
k
Þ
N
1
N
x
ð
t
Þ¼
x
ð
t
Þþ
1
n
i
ð
t
Þ¼
x
ð
t
Þþ
e
ð
t
Þ:
ð
11
:
52
Þ
i
¼
If the noise term,
), is random and independent from trial-to-trial, it can be shown that the
measurement error term in Eq. (11.52), e(
n
i
(
t
t
), which contains the influence of the noise,
approaches 0 as
) tends to be small. This
is a very powerful result! It tells us that we can effectively remove the noise by simply averag-
ing measurements from many trials. Essentially, if we average a sufficiently large number of
signal trials, the averaged signal closely approximates the true noiseless signal waveform.
Many biological acquisition systems are designed to calculate signal averages
(Eqs. (11.50) and (11.51)) as data are collected. The summation process is triggered by
a signal or a signal-related feature. The ECG signal, which has many sharp features,
is often used for heartbeat-related data. Figure 11.24 shows a signal-averaged blood
pressure waveform for the data shown in Figure 11.2. Figure 11.25 shows the signal-
averaging procedure for an auditory brainstem response (ABR) EEG measurement.
N
!1
.Thus,
x
ð
t
Þ
x
ð
t
Þ
for very large
N
,wheree(
t
