Fourier Transform challenge

I am trying to take the Fourier transform of the following time
function;


f(t) = exp(-t)/(exp(exp(-t)), from t = -inf to t= +inf


This function approaches 0 at both time extremes, and seems
well-behaved; numerical evaluation of the Fourier Transform integral
seems to converge, and give very interesting results (it looks like an
asymmetric window function, and seems to have excellent sidelobe
rolloff characteristics). However, it seems
very difficult to arrive at a closed-form solution, due to the
(exp(exp(-t)) function. Can anyone help? 


Thanks! 


Bob Adams

MATLAB:

fourier(exp(-x-exp(-x)))

ans =

gamma(1+i*w)

This is great; thanks!


Bob Adams

(You might have pointed out that this can't be accomplished using
MATLAB itself, but rather MATLAB's Symbolic Math Toolbox.)

you are welcome....
-------------------------------
> This is great; thanks!
>
>
> Bob Adams
> 

Actually, this result is a bit confusing in that it's another integral
instead of a nice simple function of frequency. I'm trying to
understand why this function has such a marked frequency-domain lowpass
characteristic. To see why it's intriuging, plug the following into
Matlab.

Bob Adams





clear all;
close all;
numpoints = 2^16;
half = 2^15 + 1;
tstart = -30;
tend = 30;
tstep = (tend-tstart)/numpoints;
fs = 1/tstep;
t = tstart:tstep:tend;

fstep = fs/numpoints;
f = 0:fstep:fs/2;
w = 2.0*pi.*f;


f = (exp(-t))./exp(exp(-t));
F = fft(f);
fl = length(f);
Fyy = F.* conj(F);
plot(t,f);
figure;
plot(w,10*log10(Fyy(1:half) + 1e-64));

ie Maple.

Naebad

Try this:

clc
close all
clear
T=-4:0.01:6;
y=exp(-T)./exp(exp(-T));
Y=abs(fft(y,2*length(y)));
plot(T,y)
title('Bobs signal')
figure
plot(fftshift(Y)./length(y))
title('FFT of Bobs signal')
figure
z=gausswin(length(y))
plot(T,z);
title('Gaussian Window')
figure
plot(fftshift(abs(fft(z,2*length(y))))./length(y))
title('FFT of Gaussian window')


I don't know.....but maybe litterature about Gaussian windows is relevant?

Cheers...


------------
> Actually, this result is a bit confusing in that it's another integral
> instead of a nice simple function of frequency. I'm trying to
> understand why this function has such a marked frequency-domain lowpass
> characteristic. To see why it's intriuging, plug the following into
> Matlab.
>
> Bob Adams
>
>
>
>
>
> clear all;
> close all;
> numpoints = 2^16;
> half = 2^15 + 1;
> tstart = -30;
> tend = 30;
> tstep = (tend-tstart)/numpoints;
> fs = 1/tstep;
> t = tstart:tstep:tend;
>
> fstep = fs/numpoints;
> f = 0:fstep:fs/2;
> w = 2.0*pi.*f;
>
>
> f = (exp(-t))./exp(exp(-t));
> F = fft(f);
> fl = length(f);
> Fyy = F.* conj(F);
> plot(t,f);
> figure;
> plot(w,10*log10(Fyy(1:half) + 1e-64));
> 

Actually, in retrospect the gamma() solution is not that useful to me;
I need something that directly gives H(w) without need to evaluate an
integral.
 It's strange that the function exp(-x)/exp(exp(-x)) would have such a
dramatic frequency-domain characteristic; it rolls off very steeply at
higher frequencies(as evidenced by taking the FFT of this function,
using a very dense time-sampling grid). It seems like there is a hole
to be filled here, and it smells kind of fundamental to me.

This formula popped up in my search to find a "logarithmic sampling
theorom"; that is, are there any functions that can be completely
reconstructed from samples that occur on log(integer) time points. The
function exp(-x)/exp(exp(-x)) seems to play the same role for
log-sampled signals that sin(x)/x plays in the reconstruction of
linearly-sampled signals. This interpretation comes from one of the
equations that equates the Reimann Zeta function with the series (SUM
of N^-s), which can be interpreted as a logarithmic time series if you
evaluate it on a vertical line in the complex plane.

robert.w.adams@verizon.net wrote:
> Actually, in retrospect the gamma() solution is not that useful to me;
> I need something that directly gives H(w) without need to evaluate an
> integral.

What are most functions, other than evaluation of integrals
or series, or numerical approximations thereof?  (other than
simple polynomials of course, but surely you can't be suggesting
that the FT of exp(-t)/(exp(exp(-t)) has a simple finite-degree
polynomial representation.)  So what are your constraints
regarding the form or representation of H(w) which you desire?


-- 
rhn A.T nicholson d.O.t C-o-M