Biology Reference
In-Depth Information
E
XERCISE
9-2
Show that the functions h(t)
¼
sin(2
p
t/5) and r(t)
¼
cos(2
p
t/5) have
period T
¼
5.
To generalize the two previous exercises, if L is any real number, the
functions h(t)
t/L) have period L.IfL is the
period of a function, its frequency (often denoted by
¼
sin(2
p
t/L) and r(t)
¼
cos(2
p
n
) is defined
by
n ¼
1/L. The functions h(t)
¼
sin(2
p
t/L) and r(t)
¼
cos(2
p
t/L) have
period L and frequency
n
.
Often, the phenomenon in question may not be truly periodic but,
instead, result from the compound effect of several factors, where each
factor may itself be nearly periodic with different factors having
different periods. For instance, to study the temperature of the
Chesapeake Bay coastal water, we would have to consider daily and
yearly cycles. We could expect that any measurable function of time G(t)
for the temperature will be a sum of (at least) two periodic functions
with periods equal to 24 hours and 365 days, respectively: G(t)
¼
D(t)
þ
Y(t). Furthermore, sun spot activity varies over an 11-year cycle; if its
effects on temperature are to be taken into account, the temperature may
be written as G(t)
¼
D(t)
þ
Y(t)
þ
S(t), with the function S(t) having a
period of 11 years.
When an observable function is a sum of periodic functions, or a sum of
multiples of periodic functions, we say that the observable function is a
linear combination of periodic components. In such cases, it is important
that the periods or, equivalently, the frequencies of the different
components, be identified from the observable function. This is
accomplished by Fourier transforming. When applied to the function G(t),
the Fourier transform generates a function f (
n
), called a power spectrum
function, where
denotes the frequency. For a given function G(t), the
power spectrum function gives a plot of the portion of a signal's power
(energy per unit time) falling within given frequency bins (see Grafakos
[2004] for the mathematical details).
n
The major practical problem is that the analytic form of the function
G(t) is rarely available. In most cases, we only have a set of discrete
measurements of G(t). Under such conditions, discrete approximations
of the Fourier transform function are used as an approximation to the
continuous Fourier transform. Various numerical methods are available
for solving the problem. The most common of these methods is an
algorithm developed by Tukey and Colley in 1965 called Fast Fourier
Transform (FFT). Computer software systems, such as MATLAB and
BERKELEY MADONNA, provide implementations of the algorithm.
Our goal here is to describe how information about f (
n
) can be used
