Biomedical Engineering Reference
In-Depth Information
to integration and the use of a continuous variable,
t,
in
place of the discrete integer,
k.
These conversion re-
lationships are generally applicable, and most digital-
domain equations can be transferred to continuous or
analog equations in this manner. In this text, usually the
reverse operation is used: the continuous-domain equa-
tion is developed first, then the corresponding digital-
domain equation is derived by substitution of summation
for integration and an integer variable for the continuous
time variable.
Although the average value is a basic property of
a signal, it does not provide any information about the
variability of the signal. The root-mean-squared (RMS)
value is a measurement that includes the signal's vari-
ability and its average. Obtaining the RMS value of
a signal is just a matter of following the measurement's
acronym in reverse: first squaring the signal, then taking
its average, and finally taking the square root of this
average:
Hence, there is a proportional relationship between the
peak-to-peak amplitude of a sinusoid (
A
in this example)
and its RMS value: specifically, the RMS value is 1
=
2
p
times the peak-to-peak amplitude, rounded in this text
to 0.707. This relationship is only true for sinusoids. For
other waveforms, the application of Eq.
2.4.15
or
Eq.
2.4.16
is required.
A statistical measure related to the RMS value is
the
variance,
s
2
.
The variance is a measure of signal
variability regardless of its average. The calculation of
variance for discrete and continuous
signals
is as
follows:
ð
T
s
2
¼
1
T
ðxðtÞxÞ
2
dt
[Eq. 2.4.17]
0
N
1
X
N
1
s
2
¼
ðx
k
xÞ
2
[Eq. 2.4.18]
k¼
1
xðtÞ
rms
¼
1
xðtÞ
2
dt
1
=
2
ð
T
where
x
is the mean or signal average. In statistics, the
variance is defined in terms of an estimator known as
the
expectation
operation applied to the probability
distribution function of the data. Because the distri-
bution of a signal is rarely known in advance, the
equations given here are used to calculate variance in
practical situations.
The
standard deviation
is another measure of a
signal's variability and is simply the square root of the
variance:
[Eq. 2.4.15]
T
0
The discrete form of the equation can be obtained by
following the simple rules described above.
x
rms
¼
1
1
=
2
N
X
N
x
k
[Eq. 2.4.16]
k¼
1
Example 2.4.4: Find the RMS value of the sinusoidal
signal:
1
=
2
s
¼
1
T
ð
T
ðxðtÞxÞ
2
dt
[Eq. 2.4.19]
xðtÞ¼A
sin
ð
u
p
tÞ¼A
sin
ð
2p
t=TÞ
0
Solution:
Because this signal is periodic, with each period
the same as the previous one, it is sufficient to apply the
RMS equation over a single period. (This is true for most
operations on sinusoids.) Neither the RMS value nor
anything else about the signal will change from one
period to the next. Applying Eq.
2.4.15
:
s
¼
1
ðx
k
xÞ
2
1
=
2
N
1
X
N
[Eq. 2.4.20]
k¼
1
In determining the standard deviation and variance
from discrete or digital data, it is common to nor-
malize by 1/
N
1 rather than 1/
N
. This is because
the former gives a better estimate of the actual stan-
dard deviation or variance when the data being used in
the calculation are samples of a larger data set that has
a normal distribution (rarely the case for signals). If
the data have zero mean, the standard deviation is the
same as the RMS value except for the normalization
factor in the digital calculation. Nonetheless, they are
from different traditions (statistics versus measure-
ment) and are used to describe conceptually different
aspects of a signal: signal magnitude for RMS and
signal variability for standard deviation.
Figure 2.4-5
shows the EEG data in
Figure 2.4-4
with positive and
xðtÞ
rms
¼
1
T
ð
T
xðtÞ
2
dt
1
=
2
¼
1
T
ð
T
A
sin
2p
t
T
p
2
dt
1
=
2
0
0
¼
1
T
cos
2p
t
T
sin
2p
t
T
þ
p
t
T
1
=
2
T
A
2
2
p
0
¼
A
2
2
p
ð
cos
ð
2p
Þ
sin
ð
2p
Þþ
p
þ
cos 0 sin 0
Þ
1
=
2
¼
A
2
p
2p
1
=
2
¼
A
2
2
1
=
2
A
2
¼
p
y 0
:
707
A
