Frequency encoding generates a signal 
from its
(discrete or continuous) Fourier transform 
, which is
represented spatially.
Suppose we have a signal 
of duration T (or
periodic with period T); write it as a discrete Fourier
series:
(The number of coefficients n is determined by the Nyquist
frequency: twice the highest frequency in 
.)
The signal then is determined by the amplitude fields
and the phase fields
(together they constitute the discrete
Fourier transform 
).
The signal is generated by using them to control the
amplitude and phase of a ``bank'' of sinusoidal signal
generators, in accord with
Eq. 5.
(Of course, it's not essential that the signal generators be
sinusoidal, since the Fourier expansion can be done in terms
of any orthonormal basis.)
The approach is easily extended to the continuous Fourier
transform; write
[ _u(t) =
12
_-_max^_max _u e^-it
.]
Now define a one-dimensional field of signal generators,
,
implemented, perhaps, by pairs of neurons in quadrature phase;
then the signal is constructed by
[ _u(t) =
_-_max^_max _u _(t)
= _u(t), ]
which we may abbreviate 
.
The Fourier representation is especially appropriate when
frequency-domain transformations need to be applied to the
signal, or when the signal is periodic (since only one cycle
needs to be encoded).
If the Fourier representation is translated by 
along the
frequency axis, then the overall duration of one cycle changes
(so an increase of frequency leads to a decrease of duration and vice
versa).
Conversely, the duration of the signal cannot be changed without
changing its frequency content (since the fundamental frequency is the
reciprocal of the duration).
