Biomedical Engineering Reference
In-Depth Information
assuming the time series runs from
t
= 1 to
t
=
T
. (Rather than separating out
the sine and cosine terms, it is easier to use the complex-number representation,
via
e
i
R
= cos R +
i
sin R.) The inverse Fourier transform recovers the original
function:
1
y
=
,
y
[21]
2
QO
t
1
T
1
i
=
y
e
y
.
[22]
T
t
O
T
O
=
0
The Fourier transform is a linear operator, in the sense that (
x
+
y
) =
x
+
y
.
Moreover, it represents series we are interested in as a sum of trigonometric
functions, which are themselves solutions to linear differential equations. These
facts lead to extremely powerful frequency-domain techniques for studying lin-
ear systems. Of course, the Fourier transform is always
valid
, whether the sys-
tem concerned is linear or not, and it may well be useful, though that is not
guaranteed.
2
, is called
the
spectral density
or
power spectrum
. For stationary processes, the power
spectrum
f
(O) is the Fourier transform of the autocovariance function
C
(U) (a
result called the Wiener-Khinchin theorem). An important consequence is that a
Gaussian process is completely specified by its power spectrum. In particular,
consider a sequence of independent Gaussian variables, each with variance T
2
.
Because they are perfectly uncorrelated,
C
(0) = T
2
, and
C
(U) = 0 for any Uv 0.
The Fourier transform of such a
C
(U) is just
f
(O) = T
2
, independent of O—every
frequency has just as much power. Because white light has equal power in every
color of the spectrum, such a process is called
white noise
. Correlated proc-
esses, with uneven power spectra, are sometimes called
colored noise
, and there
is an elaborate terminology of red, pink, brown, etc., noises (53, ch. 3).
The easiest way to estimate the power spectrum is simply to take the Fou-
rier transform of the time series, using, e.g., the fast Fourier transform algorithm
(54). Equivalently, one might calculate the autocovariance and Fourier trans-
form in that manner. Either way, one has an estimate of the spectrum, which is
called the
periodogram
. It is unbiased, in that the expected value of the perio-
dogram at a given frequency is the true power at that frequency. Unfortunately,
it is not consistent—the variance around the true value does not shrink as the
series grows. The easiest way to overcome this is to apply any of several well-
known smoothing functions to the periodogram, a procedure called
windowing
(55). (Standard software packages will accomplish this automatically.)
The Fourier transform takes the original series and decomposes it into a
sum of sines and cosines. This is possible because
any
reasonable function can
be represented in this way. The trigonometric functions are thus a
basis
for the
space of functions. There are many other possible bases, and one can equally
The squared absolute value of the Fourier transform,
f
() |
O
=
y
O
|
