🚀 go-pugleaf

In article <1769@mrsvr.UUCP> kohli@gemed.ge.com (Jim Kohli) writes:
>
>I've been attempting to use the routines in Marple's "Digital
>Spectral Analysis with applications" (a VERY fine book), but
>I'm having some problem coming up with a "rule of thumb" for
>automagically assigning a value for IP, the "dimension of data
>matrix" parameter, in the EIGENFREQ routine.  This parameter
>represents the rank of the sample autocorrelation matrix.
>
>Can someone tell me what the rationale is for not making this
>parameter the same dimension as the data being operated on?
>(other than reducing the amount of computations-- and if
>that is why, some criteria for determination of a good rule
>of thumb???).
>
>In Marple's example on 64 points of test data, the signal
>space dimension is set to be 11, but IP is 15 (as opposed
>to 64).
>
>thanks (as usual!),
>
>Jim Kohli
>
>GE Medical Systems
>Waukesha WI, USA

You are confused with IP (dimension of covariance matrix), number of
signals, and number of observations. For example, in Marple's example,
if you have only 64 points of data, it is known that the maximum
(allowable) dimension of covariance matirx is about 0.25*64=16. He
simply chose 15 instead of 16. If you choose bigger number for IP, from
the principle of parsimony, your estimates will be very unreliable.
Then the signal space dimension indicates the number of signals.
Therefore, IP should be greater than the number of signals. For
example, if we have the following oberved data described by:

y(t) =   5
	--
	\    a      -j ( 2 * pi * f  t)   + n(t)
	/     i   e                i
	--
	i=1

for t=1,..,100
and covariance matrix with IP=10 is 

                  *
      R   =  A S A  + pI
       xx

where p is the variance of noise n(t).
Then define

     ^       ^        ^
     C    =  R    -   p I
      xx      xx

Then the rank of the above  matrix is 5, which is number of signals.
This values is also called "signal space dimension." Finally, if you
determine the number of signals, you can estimate frequencies using
orthogonal principle of space and null spaces. If you want to study
about this, take any book of Linear Algebra. (in fact, Marple's book
is also a good book to review. See Chapters 2 and 3)
Note that MUSIC or EV in Marple's book are scanning procedures of null
space (noise space) with a specific metrics.


+----+----+----+----+----+----+----+----+----+----+----+----+----+
|  Stephen Oh         oh@csc.ti.com     |  Texas Instruments     |
|  Speech and Image Understandung Lab.  | Computer Science Center|
+----+----+----+----+----+----+----+----+----+----+----+----+----+