TITLE:      A novel short-space/time Fourier transform for interpretation of
            electromagnetic data
AUTHORS:    S. L. Dvorak, M. Sheikh, D. G. Dudley, and M. W. Marcellin
CONFERENCE: Progress in Electromagnetic Research Symposium, Seattle, 
	    Washington, July 1995.
ABSTRACT:

It has been demonstrated that the short-time Fourier transform (STFT),
\begin{eqnarray*}
{F}_{STFT}(t,\omega;{\bf r})=\int_{-\infty}^\infty h(t-t')F(t',{\bf
r})e^{-j\omega t'}d
t',
\label{stft}
\end{eqnarray*}
is useful for the interpretation of electromagnetic data [1,2].
A plot of the STFT (i.e., a spectrogram) simultaneously provides information 
about the time and frequency behavior of the signal. In essence, the STFT 
allows for the visualization of the variables in a two-dimensional (2-D) 
hyperspace, i.e., $t$ and $\omega$. Since electromagnetic fields involve both 
temporal and spatial variations, in general, these variables form an 8-D 
hyperspace, i.e., the six additional variables include the spatial variables 
$(x,y,z)$ and the spatial-frequency variables $(k_x,k_y,k_z)$. Short-space 
Fourier transforms have been employed to visualize a single spatial variable 
and the corresponding spatial frequency [2]. However, there is currently no 
technique  which can simultaneously handle the visualization of anything larger 
than a 2-d hyperspace.
 
In order to provide additional insight for the interpretation of 
electromagnetic data, we discuss a novel short-space/time Fourier transform 
(SSTFT) in this presentation. For the purpose of demonstration, we apply the 
SSTFT to the problem of transient wave propagation in a waveguide. In this 
problem, the space and time variables are related by $\chi=\sqrt{t^2-(z/v)^2}$,
where $z$ denotes the longitudinal coordinate in the waveguide. We will 
demonstrate that the new SSTFT, 
\begin{eqnarray*}
{F}_{SSTFT}(\chi,\Omega;\xi)=\int_{-\infty}^\infty h(\chi-\chi')F(
\chi'\cosh\xi,v\chi'\sinh\xi)e^{-j\Omega \chi'}d
\chi',
\label{sstft}
\end{eqnarray*}
provides simultaneous information about the temporal-frequency and 
spatial-frequency content of a signal for early and late times. The advantages 
of the SSTFT, as compared with the STFT, will also be discussed.
 
REFERENCES:

[1] K. F. Casey, D. G. Dudley, M. R. Portnoff, ``Radiation and
Dispersion Effects from Frequency-Modulated (FM) Sources,''
{\it Electromagnetics}, pp. 349--376, 1990.
 
[2] L. Carin and L. B. Felsen, ``Wave-Oriented Data Processing
for Frequency- and Time-Domain Scattering by Nonuniform Truncated arrays,''
{\em IEEE Antennas and Propagat. Mag.}, June 1994.
\end{document}