Geoscience Reference
In-Depth Information
The computation of the inverse FFT is very similar to the forward FFT because of the
identical nature of the two. Estimation of power spectrum of a signal can be done by means
of FFT.
5. The concept of spectra
Though we have mentioned spectra or spectrum in our previous discussions, we shall
formally explain it here. The word spectrum (plural: spectra) is used to describe the
variation of certain quantities such as energy or amplitude as a function of some parameter,
normally frequency or wavelength. Optical spectrum of white light (colour spectrum)
dispersed by a glass prism or some other refractive bodies (such as water) is a good
example.
When a signal is expressed as a function of frequency, it is said to have been transformed
into a frequency spectrum. Thus, mathematically, a time-domain signal, f(t) can be
expressed by F(ω), where ω represents angular frequency (ω = 2πf; f being the linear
frequency). The function F(ω) is in general complex and may be represented by
1.
The sum of real and imaginary parts:
F
(
ω
) =
a
(
ω
) +
ib
(
ω
)
2.
The product of real and complex parts:
(
)
=
|
()
|
()
Where
|()|=
()+
()
and
(
)
=
(
)
()
+2;=0,±1,±2,±3,…
The modulus
|()|
is normally called the amplitude spectrum and the argument
()
is
called the phase spectrum.
5.1 Spectral analysis of periodic functions
In modern analysis, the time function may be expressed through certain mathematical
transformation into a function of frequency. The discrete Fourier transform views both the
time domain and the frequency domain as periodic. This can be confusing and inconvenient
since most of the real time signals are not periodic.
The most serious consequence of time domain periodicity is time domain aliasing.
Naturally, if we take time domain signal and pass it through DFT, we find the frequency
spectrum. If we could immediately pass this frequency spectrum through the inverse DFT to
reconstruct the original time domain signal, we are expected to recover this signal, save for
spill over from one period into several periods - a problem of circular convolution.
Periodicity in the frequency domain behaves in much the same way (as frequency aliasing),
but is more complicated.
5.2 Spectral analysis of aperiodic functions
Equations (14) and (15) or any of their similar versions give the Fourier transform pair for a
periodic function. For non-repetitive (or aperiodic) signal, the period T→
∞
and the Fourier
transform pair are expressed as
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