** Next:** Time-domain conjugate
** Up:** CORRELATION AND SPECTRA
** Previous:** Common signals

The **spectrum** of a signal is the magnitude squared
of the Fourier transform of the function.
Consider the real signal that is a delayed impulse.
Its *Z*-transform is simply *Z*; so the real part is
, and the imaginary part is .The real part is thus an **even function**
of frequency
and the imaginary part an **odd function**
of frequency.
This is also true of *Z*^{2} and any sum of powers
(weighted by real numbers),
and thus it is true of any time function.
For any real signal, therefore,
the Fourier transform has an even real part RE
and an imaginary odd part IO.
Taking the squared magnitude gives
(RE+*i*IO)(RE-*i*IO)= (RE)^{2} + (IO)^{2}.
The square of an even function is obviously
even, and the square of an odd function is
also even.
Thus, because the spectrum of a real-time
function is even, its values at plus
frequencies are the same as its values at minus frequencies.
In other words, no special meaning should be
attached to negative frequencies.
This is not so of
complex-valued signals.
Although most signals which arise
in applications are real signals,
a discussion of correlation and spectra is not mathematically
complete without considering
**complex-valued signal**s.

Furthermore, complex-valued signals arise in many different contexts.
In seismology, they
arise in imaging studies when the space axis is Fourier transformed,
i.e., when a two-dimensional function *p*(*t*,*x*)
is Fourier transformed over space to *P*(*t*,*k*_{x}).
More generally,
complex-valued signals arise where rotation occurs.
For example, consider two vector-component wind-speed
indicators:
one pointing north, recording *n*_{t}, and the
other pointing west, recording *w*_{t}.
Now, if we make a complex-valued time series *v*_{t}=*n*_{t}+*iw*_{t},
the magnitude and phase angle of the complex numbers
have an obvious physical interpretation:
corresponds to rotation
in one direction (counterclockwise),
and to rotation in the other direction.
To see why, suppose and .Then .The Fourier transform is

| |
(43) |

The integrand oscillates and averages out to zero, except
for the frequency .So the frequency function is a pulse at :
| |
(44) |

Conversely, if *w*_{t} were ,then the frequency function would be a pulse at ,meaning that the wind velocity vector is rotating the other way.

** Next:** Time-domain conjugate
** Up:** CORRELATION AND SPECTRA
** Previous:** Common signals
Stanford Exploration Project

10/21/1998