where 
can be represented
as a complex Fourier series where and a finite-energy signal x(t) can be represented as a Fourier integral
where 
,where X(f) is called the Fourier transform of x(t).
In general, the Fourier transform is complex, having both a magnitude
|X| and a phase /_X. The squared magnitude of X gives the
power for a periodic signal and the energy density for a finite-energy signal.
This lets us speak about the power or energy of a signal in different frequency
bands.
It is common to refer to X as the spectrum of x. Physically, this makes more sense for periodic signals than for aperiodic signals. For
aperiodic signals such a speech, the usual practice is to snip out a short
time segment by multiplying x(t) by a window function w(t),
and to call the Fourier transform of w(t) x(t) the
short-term spectrum.*
When w(t) x(t) is sampled and an FFT is used to compute the short-term spectrum, the tacit assumption is that this segment is being periodically repeated. One should always keep this in mind when using FFT's for spectral analysis, where a poor choice for a window function can produce results that are quite different from what is desired.
[Two standard references on these topics are
Bracewell
for Fourier analysis and
Oppenheim and Schafer
for discrete-time processing. For a gentler introduction,
Steiglitz is highly recommended.]
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