Biomedical Engineering Reference
In-Depth Information
Sometimes a signal will be correlated with another
signal in its entirety, a process known as correlation, or
the closely related covariance. If the correlation between
the signal of interest and the reference is zero, it does not
necessarily mean the two signals have nothing in
common, but it does mean the signals are mathematically
orthogonal.
If the probing signal is short, a running correlation
known as cross-correlation may be appropriate.
Cross-correlation not only shows the match between the
probing signal and the signal of interest, but also where
that match is greatest. A signal can also be correlated
with shifted versions of itself, a process known as auto.
The autocorrelation function describes the period for
which a signal remains partially correlated with itself and
this relates to the structure of the signal. For example,
a signal consisting of random noise decorrelates
immediately, whereas a slowly varying signal will remain
correlated
Figure 2.4-16 The maximum cross-correlation between sine
waves and the electroencephalogram signal plotted as a function
of the sine wave's frequency. A peak is seen between 7 and 9 Hz,
which indicates the presence of an oscillatory pattern known as
the 'alpha wave.'
over
for
long
period. Correlation,
cross-correlation,
autocorrelation,
and the
related
covariances are all easy to imple in MATLAB.
Problems
r
¼
axcor(eeg,x); % Perform cross-
correlation
rmax (i)
¼
max(r); % Store max value
end
..labels and plot..
1.
Two 10-Hz sine waves have a relative phase shift of
30 degrees. What is the time difference between
them? If the frequency of these sine waves doubles,
but the time difference stays the same, what is the
phase difference between them?
2.
Convert
x
(
t
)
¼
6 sin(5
t
)
5 cos(5
t
) into a single
sinusoid [i.e., M sin(5
t þ q
)].
3.
Convert
x
(
t
)
¼
30 sin(2
t þ
50) into sine and cosine
components.
4.
Convert
x
(
t
)
¼
5 cos(10t
þ
30)
þ
2 sin(10t
20)
þ
6 cos(10
t þ
80) into a single sinusoid as in
Problem 2.
5.
Find the delay between
x
1
(
t
)
¼
cos(10t
þ
20) and
x
2
(
t
)
¼
sin(10
t
- 20).
6.
Equations
2.4.8
, 2.4.9, and 2.4.10 were developed
to convert a sinusoid such as cos(u
t
q) into a sine
and cosine wave and vice versa. Derive the
equations to convert between a sinusoid based on
the sine, sin(u
t þ
q) and a sine and cosine wave.
7.
Find the RMS value of the square wave with
amplitude of 1.0 V and a period 0.2 seconds.
8.
If a signal is measured as 2.5 V peak-to-peak and
the noise is measured as 28 mV RMS, what is the
SNR in decibels?
9.
Use Eq.
2.4.28
to find the correlation
(unnormalized) between sin(2p
t
) and cos(
2
p
t
).
10.
Use Eq.
2.4.28
to find the correlation between a
cosine and a square wave as shown below. This is
Results:
The result of the cross-correlations is seen in
Figure 2.4.16
and an interesting structure is seen to
emerge. Some frequencies show much higher correlation
with sine and EEG, indicating more sine wave content at
these frequencies. A particularly strong peak is seen in
the region of 7 to 9 Hz, indicating the presence of an
oscillatory pattern known as the
alpha wave.
2.4.5 Summary
The sinusoidal waveform is arguably the single most
important waveform in signal processing. Because of its
importance, it is essential to know the mathematics as-
sociated with sines, cosines, and general sinusoids, in-
cluding complex representations.
Several basic measurements apply to any signal
including mean value, RMS value, and variance or stan-
dard deviation. Although these measurements provide
some essential basic information, they do not provide
much information on signal content or meaning. A
common approach to obtaining more information is to
probe a signal by correlating it with one or more refer-
ence waveforms. One of the most popular probing signals
is the sinusoid.
