3.3 Complex Symmetry Properties
Figure
3.3b. Symmetry
properties of an odd function, o(t).
Figure
3.3a. Symmetry
properties of an even function,
e(t).The symmetry properties inherent in complex Fourier transform pairs are very useful in practical applications. Symmetry refers to the even and odd parts of s(t) or S(f) in the time or frequency domains, respectively. A function, e(t) has even symmetry if it is a mirror image (symmetric) about the zero axis, i.e., e(-t) = e(t); a function has odd symmetry if the reflection about zero has an opposite sign (antisymmetric) where o(-t) = -o(t). Even and odd symmetries are illustrated in Figure 3.3a and b, respectively.
An arbitrary function, s(t) can always be separated into odd and even parts. These parts are, in general, complex, which leads to several complex, symmetry combinations for the Fourier transform pairs as described by Bracewell (1965). We need not consider all of them because everything we deal with in applied geophysics is real in the time (or space) domain. Such signals transform into functions that have real, even parts and imaginary, odd parts in the frequency domain. Figure 3.4a shows these relations nicely using the visualization technique presented by Bracewell (1995) which allows both real and imaginary parts of a function to be plotted on one graph in either domain.
A function whose real part is even and imaginary
part is odd, is called a Hermitian function
irrespective of whether it’s in
the time or frequency domain. Such a
function in the frequency domain would
have an amplitude spectrum that is even
and a phase spectrum that is odd. Since
a real function in the time (or space)
domain generates a Hermitian function
in the frequency domain, real geophysical
signals, s(t) that are even have Fourier
transforms that are real and even (Figure
3.4b). And, a signal, s(t) that’s
real and odd has a Fourier transform
that’s imaginary and odd (Figure
3.4c). Knowledge of these complex symmetries
is very useful in practical applications
of spectral analysis.
Figure
3.4a. Symmetry properties of
Fourier transform pairs when a real
signal, s(t) is arbitrary, neither
even nor odd. The Fourier
transforms are: Hermitian. Double-ended
arrows indicate Fourier transform
pairs.
Figure
3.4b. Symmetry properties of
Fourier transform pairs when a real
signal, s(t) is is an even function.
The Fourier transforms are: real,
even. Double-ended arrows indicate
Fourier transform pairs.
Figure
3.4c. Symmetry properties of
Fourier transform pairs when a real
signal, s(t) is is an odd function.
The Fourier transforms are: imaginary,
odd; respectively. Double-ended arrows
indicate Fourier transform pairs.
