Biomedical Engineering Reference
In-Depth Information
2.4
Chapter 2.4
Basic signal processing
John Semmlow
2.4.1 Basic signals: the sinusoidal
waveform
terms, you have characterized the sine signal for all time.
The sine wave signal is rather boring: if you have seen one
cycle, you have seen them all. Moreover, because the
signal is completely defined by A and f p , if neither the
amplitude, A, nor the frequency, f p , changes over time, it
is hard to see how this signal could carry much in-
formation. These limitations notwithstanding, sine waves
(and cosine waves) are at the foundation of many signal
analysis techniques. In part, their importance stems from
their simplicity and the way they are treated by linear
systems. Sine wave-like signals can also be represented by
cosines, and the two are related.
A cos ð u tÞ¼A sin u t þ p
2
Signals are the foundation of information processing,
transmission, and storage. Signals alsoprovide the interface
with physiological systems and are the basis for commu-
nication between biological processes ( Figure 2.4-1 ).
Given the ubiquity of signals within and outside the body,
it should be no surprise that understanding at least the
basics of signals is fundamental to understanding, and
interacting with, biological processes.
A few signals are simple and can be defined analyti-
cally, that is as mathematical functions. For example,
a sinusoidal signal is defined by the equation:
xðtÞ¼A sin ð u p tÞ¼A sin ð 2p f p tÞ¼A sin 2p t
T
¼ A sin ð u t þ 90 degrees Þ
A sin ð u tÞ¼A cos u t p
2
¼ A cos ð u t 90 degrees Þ
[Eq. 2.4.4]
Note that the second representations [i.e., A sin(ut þ 90
degrees) and A cos(u t 90 degrees)] have conflicting
units: the first part of the sine argument, u t , is in radians,
whereas the second part is in degrees. Nonetheless, this
is common usage and is the form that is used throughout
this text.
A general sinusoid (as opposed to a pure sine wave or
pure cosine wave) is a sine or cosine with a general phase
term as shown in Eq. 2.4.5 :
[Eq. 2.4.1]
where A is the signal amplitude, or more accurately the
peak-to-peak amplitude, u p is the frequency in radians
per second, f p is the frequency in hertz, and T is the
period in seconds, and t is time in seconds. Recall that
frequency can be expressed in either radians or hertz (the
units formerly known as cycles per second) and are re-
lated by 2p:
u p ¼ 2p f p
[Eq. 2.4.2]
Both forms of frequency are used in the text, and the
reader should be familiar with both. The frequency in Hz
is also the inverse of the period, T:
f p ¼ 1
T
xðtÞ¼A sin ð u p t þ q Þ¼A sin ð 2p f p t þ q Þ
¼ A sin 2p t
T þ q or equivalently
[Eq. 2.4.5]
[Eq. 2.4.3]
xðtÞ¼A cos ð u p t q Þ¼A cos ð 2p f p t q Þ
¼ A cos 2p t
T q
The signal presented in Eq. 2.4.1 is completely defined
by A and f p (or w p , or T ); once you specify these two
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