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Re: [Phys-l] Integral Help



At 03:25 PM 4/17/2008, Josh Gates, you wrote:
It looks like I volunteered to help with an integral that's a little over
my head. A fellow ham is trying to write a program to autodecode Morse
code from an audio signal. This involves the Rayleigh fading channel and
some probability work, naming integrating over the phase to calculate
conditional probability. I think that I've verified his assertion of the
answer for N=2, but he wants higher orders - in particular N=5 - and I
don't think that I can help. I'd love to hear any responses, and will
forward them on to him.

The .doc file mentioned below can be downloaded at: [
http://www.wikiphys.org/1.doc ]http://www.wikiphys.org/1.doc

His statement of the problem:
===========================
I am working on the problem of CW
detection in the Rayleigh fading channel in the Bayesian framework. Since
CW
signals are incoherent, the probability density function must be
marginalized over the phase. This leads to the integral (1) in the attached
DOC file. The solution for N=2 is known (2), where I0 is the modified
Bessel
function of the first kind with order zero. I am interested in the solution
for higher N, in particular, for N=5, but I cannot find a way to take the
integral or to prove that a closed form solution does not exist. I will
appreciate your help with this.

73 Alex VE3NEA
======================

73 Josh KE4ERF

The international Morse code developed by Friedrich Gerke in 1848
from the railroad code of Morse and Vail had some features that
make it difficult to decode by machine means. This should be no
surprise, considering the time that speech recognition and
hand writing recognition have lately been in development.
Pattern recognition is after all, a human talent.

The principal difficulty revolves firstly round the variable
length of the code particles: one unit for a dot, one for a space,
three for a dash, seven for an inter-word space, and secondly,
the variable number of particles constituting one of the thirty
six principal letters and numbers.
The shortest letter, 'e' requires one dot, a time unit of one and
three time units of letter spacing. The longest numeral, zero,
requires five dash units, four spaces, and an inter symbol space,
a total of twenty two time units.

This range of transmission times for one symbol, 2:11 is unhelpful,
so that in commercial applications, Baudot and ASCII replaced IMC
with their constant length symbol formulations.

Current Morse Code readers are not inordinately expensive: $200
will buy a keyer reader which can send from a keyboard, and decode
to a text display [MFJ-464 is one].
So why is a ham now interested in developing a decoding method?
The advertizing copy for a commercial reader may provide a clue:
"...nothing can clean and copy a sloppy fist, especially weak signals
with lots of QRM ..."
"A sloppy fist" in this context implies a variability in time units
given for code particles, and "QRM" refers to interference which
obscures the meaning of code particles.

The mention of Bayesian probability points to the desirability of
taking particles as a group and assigning the likelihood of the
ensemble's representing a particular symbol. In fact, taking a hint
from the most accomplished code readers, who say they recognize
an entire word as the base entity, it is evidently desirable to
pattern match against a dictionary which may not in fact comprise
the 100,000 words found in an English dictionary, but may well
need more than the thousand words used for Basic English.

Supposing that a word is represented by five letters on average, and that
noisy conditions may make only two or three of those letters available,
it is not hard to see that the human capability of extracting meaning from
context is also valuable. Selecting candidate words from a dictionary,
given any two of five letters, itself needs brute force.

Accordingly, remembering the success of the chess-playing programs,
which finally relied on brute force testing of possibilities, I suspect
that brute force (amply available in current PCs) would again overcome
refined analytical methods.


Brian Whatcott Altus OK Eureka!