We have seen that a field can represent a trajectory in either the time domain or the frequency domain. Since each has its advantages and disadvantages, often a combined representation is more suitable. In such a representation we have a time-varying spectrum.
The foundation for such a representation was laid fifty years
ago by Dennis Gabor, who also received the Nobel Prize for
his invention of holography. Gabor (1946) observed
that we perceive sound in terms of amplitude and pitch
simultaneously, that is, auditory perception is not entirely in the time domain
or the frequency domain.
He showed that any signal of finite duration and bandwidth
could be decomposed into a finite number of elementary
information units, which he called logons.
Each such unit controls the amplitude and phase of a
Gabor elementary function, which is an elementary signal
localized in time and frequency.
The relevance of this to motor control is that any motor
control signal has a calculable Gabor-information content,
which determines a finite number of coefficients necessary
and sufficient to generate that signal.
More precisely, at time t the measurement of a frequency
component f in a signal will require that the signal be
sampled for some finite duration 
.
Further, the uncertainty 
in the measured frequency
will be less the longer the signal is sampled.
Indeed, Gabor proves
(the so-called Gabor Uncertainty Principle).
(An intuitive presentation of the proof can be found in
MacLennan 1991.)
Therefore 
defines the maximum
possible definition of a (finite duration, finite bandwidth)
signal.
A signal of duration T and bandwidth F can be divided
into a finite number of elementary ``information cells'' of
duration 
and bandwidth 
, each localized
at a different time and frequency.
Each cell has an associated complex coefficient, which gives
the phase and amplitude of the signal in the corresponding
cell.
Let 
and 
; then there are
MN elementary information cells; in Gabor's terms, the
signal represents MN logons of information, namely, the
MN coefficients associated with the cells.
This is the most information that can be represented by the
signal, and these MN complex coefficients are sufficient to
regenerate the signal (which is its relevance for motor
control).
Let the cells be labeled (j, k) for 
and
.
Then cell (j,k) is centered at time 
and
frequency 
.
Each cell corresponds to a Gabor elementary function
localized to that time and frequency, one form of which is a
Gaussian-modulated sinusoid:
[
G_jk(t, ) =
[-(t;-;jt)^2^2]
[2k f (t;-;j t;-;)] ,
]
where
(the standard deviation of the Gaussian is 
).
A signal 
is then a superposition of these
elementary functions with amplitudes 
and phase
delays 
:
[ (t) =
_j=0^M-1 _k=0^N-1
_jk G_jk(t , _jk) .
]
The coefficients 
and 
are determined
uniquely by the signal 
.
The Gabor representation shows us how a signal can be
generated from the control coefficients 
and
:
during the jth time interval of length 
we use
the coefficients to control a bank of Gaussian-modulated
sinusoid generators (at frequencies 
);
controls the amplitude of generator k and
controls its phase.
Although the clocking out at discrete time intervals of the
coefficients is not impossible, it may seem a little
unnatural.
This can be avoided by replacing the discrete matrices
and 
by continuous fields.
In this approach the Gabor elementary function generators
operate on a continuum of frequencies in the signal's
bandwidth:
[
G_(t,) =
[-(t - )^2^2]
[2(t - - )] ,
]
The output signal is then generated by an integration:
[
(t) =
_0^T _0^F
_
G_(t , _)
.
]
In fact, the output can be generated by a temporal
convolution of the control fields and a bank of Gabor signal
generators, but the details will not be presented here.
It might be objected that the control fields 
and
would occupy more neural space than either a direct or
Fourier representation, but the control fields are relatively
low resolution and may be represented more compactly.
The inequality 
gives the
tradeoff in required resolution between the time and
frequency axes of the control fields.
Unlike the Fourier representation, the Gabor representation allows
frequency content and rate to be controlled independently.
Thus the amplitude and phase fields ( 
) can be ``clocked
out'' at a different rate from that at which they were stored, or even
at a varying rate, without affecting the moment to moment frequency
content of the signal.
Conversely, shifting the representing fields ( 
) along the
frequency axis shifts the frequency content of the signal, but does not
affect its duration or the time-evolution of its spectrum.
That is, the rate or time-evolution of the signal can be controlled
independently of the frequency band in which it is expressed.