Date: Fri, 13 Oct 2000 10:18:22 -0400 (EDT)
From: Ed Schmahl <schmahl>
Subject: Why not irregular time bins?

To interested HESSI developers:

Need for calib_eventlists with irregular binning
------------------------------------------------
At the present time, the default HESSI software uses
regularly-spaced time bins for count-rate profiles.  There are many
cases, however, where irregular time bins are useful.  One important
case is that of "regularization" in which the sampling bins are
selected to be regular in phase, but irregular in time.  By "phase", I mean
the "phase_map_ctr" array within calibrated event lists. Right now,
if one wants the phase_map_center array to be regularly
spaced, it is necessary to take the default calib_eventlist structure,
ipated advantages:

(1) The HESSI signals become narrow band, that is, instead of
spectral power from 0 to f_max, the power is concentrated only
near f_max, and filtering of signals becomes reliable and 
interpretation of power spectra becomes simple.

(2) With narrow-band signals one can exploit the IDL 5.x HILBERT 
function, which leads to an almost immediate, bomb-proof, one-line
conversion of counts to visibilities:

     visibility = count + complex(0,1)*hilbert(count)

This can eliminate hundreds of lines of complex code--
hsi_calib_ev2vis.pro--for example. (Note that Hilbert.pro only 
works properly when the power spectrum is narrowband, which
means that regularization must be done before using it.)

(3) The narrow-band nature of the visibility produced after
regularization allows one to remove the
highest frequency by a  simple shift to phase center:

    vis_phase_ctr = visibility * exp(i * u*xc + v*yc)

The power is now concentrated at lower frequencies, which can't help
but improve mapping stability and speed.

(4) Havinach UV circle the
FFT can be used in one dimension (execution time = order N log N) while 
the regular FT (time = order N^2) must be used for the other dimension.
The time savings can be an order of magnitude for 64 x 64 arrays
and more than that for larger arrays.

Summary
-------
Given the above motivations, I suggest that in creating a calibrated 
event list (using the hsi_calib_eventlist object)
we should have the user option to input an arbitrary time bin
array of one's own, rather than the default regularly-spaced
time bins.  Then, if the user had pre-calculated the bins required 
to make equal increments in phase_map_ctr, the output structure
would be regularized, and have all the benefits described above.  

Later on, if it appears desirable, object-oriented code could be
implemented that would create the regularized binning automatically
from the map center location.


--EJS  Oct 13, 2000

PS:

Credits and kudos to Brian, who referred me to Norden Huang's work on empirical
mode decompositions.  Looking at Huang 1998 (Proc R Soc Lond A 454,
903) I encountered the Hilbert transform for the first time, and that
led me to thinking along the above lines.


ing at Huang 1998 (Proc R Soc Lond A 454,
903) I encountered the Hilbert transform for the first time, and that
led me to thinking along the above lines.