kohli@phony.crd.ge.com (Jim Kohli) (03/26/91)
After much ado trying to rectify differences observed
in PRONY when executed on a SUN vs. the test results
published in MARPLE, I have learned some rather important
things with respect to PRONY, and with respect to the test
data presented in the book.
Marple's test data is described as 64 samples of a
synthetically generated process of four complex sinusoidals in
colored noise. Figure 11.2 in Marple shows spectral estimates
calculated from PRONY results (on a VAX). Based on the
description of the data, one might suspect that the peaks at
.2, .1, -0.15, and probably -.23 represent the sinusoids, based
on the almost zero damping for each of these frequencies, and
that the other features of the plot are the result of the
colored noise.
With one major exception, the FREQUENCY results between the SUN
and the VAX agree to a limited extent. The major exception is that
on the VAX, PRONY finds a signal with the following characteristics,
not found on the SUN. The different frequency is bracketed with -->
and <--:
freq damp amp phase
-0.491626 -0.173706 0.135209 1.303734
-0.410073 -0.102563 0.591154 0.795532
-0.385892 -0.058281 0.514301 3.113064
-0.313451 -0.035726 0.233482 0.620367
-0.272540 -0.117261 0.260784 -2.611305
-0.233155 -0.009720 0.024938 0.273678
-0.150005 -0.000079 0.094664 -0.983270
--> -0.064256 -1.342355 1.006159 -1.333549 <--
0.100011 -0.000088 0.108695 0.615236
0.201003 -0.000244 1.219258 1.037941
0.209137 0.005920 0.914097 1.644919
0.300430 -0.038579 0.381632 0.196286
0.337652 -0.065558 0.506959 1.652484
0.382312 -0.069968 0.208304 -1.503253
0.426402 -0.115183 0.204989 0.953958
PRONY finds the following signals on a SUN, with the marked signal
signal absent from the VAX:
freq damp amp phase
-0.482901 -0.279896 0.621455 0.942346
-0.403843 -0.137851 1.503032 0.764551
-0.388234 -0.078389 1.087642 -2.980464
-0.312817 -0.033692 0.236659 0.591031
-0.266322 -0.095113 0.178823 3.079086
-0.230789 -0.012705 0.025987 -0.231236
-0.150006 -0.000004 0.093438 -0.941405
0.100017 -0.000093 0.107556 0.646416
0.201009 -0.000340 1.239947 1.045187
0.209128 0.006438 0.901616 1.658794
0.301765 -0.046087 0.476437 -0.002271
0.340582 -0.094534 0.781240 1.371097
0.385687 -0.107533 0.251470 -1.824027
--> 0.417867 -1.539567 2.514828 -1.838492 <--
0.427518 -0.185516 0.539178 0.856448
You may correctly infer from its damping factor that the spectral
estimate is quite flat at both of the odd points, and, in fact, these two
signals, on their respective machines, represent the greatest
damping assigned to a single signal component by almost a magnitude.
As I previously noted, the FREQUENCY components were generally in
limited agreement. The damping and amplitudes ranged quite widely.
So a few questions remain:
(1) Which machine (if either) is producing better results?
We could only KNOW that if we had the parameters which Marple
used to create the test data. I had assumed that since PRONY was
apparently developed and tested on a VAX, that the VAX results would
be good, but after modeling the data using the calculated PRONY parameters
I find that the sum of magnitude error is 16.26 on a VAX and 16.20 on
a SUN. The sum of X magnitudes was 75.77, so you can see that either this
model isn't that great, or PRONY isn't that great for this sort of analysis,
or the test data does not follow the assumptions of the model.
(2) What would happen if we tried to extract 14 (rather than 15)
signals from the test data?
The results from this experiment (included below) may be summarized
as follows:
(a) the frequencies match up pretty well, no anomolies; (b) the
damping factor matches only in order of magnitude ; (c) the amplitudes, in
some cases, don't even match order of magnitudes; (d) the same with the
phase information.
(3) Why is it that the other parameters don't match?
Beats the hell out of me. I could speculate about
variations in precision, but it doesn't seem right to me that
slight variations in precision can lead to such dramatic
variations in the damping, amplitude, and phase.
(4) What good is PRONY anyway?
As a spectral estimator, it's slow but gives reasonably good frequency
estimates. As an estimator of amplitudes, damping, and phases,
I find it impossible to trust based on this simple comparison.
In summary:
I am somewhat disillusioned about PRONY. I had expected it to minimally
be consistent between machines. The fact that it isn't worries me. I've
also done some other tests, with and without uniformly distributed low
power noise and find that PRONY generates USELESS damping, amp, and phase
information where the SNR is as high as 18! This is in contrast with a
noise-free dataset, which is correctly analyzed by PRONY on BOTH machines!
If anyone has more useful info or can tell me where I've gone wrong,
I'd appreciate it immensely. As would several others who seem to have
had similar problems.
Thanks billions and billions (of what?),
Jim Kohli
GE Medical Systems
Cc: KK,GG,TB,GAA,JB
PRONY analysis on a VAX, ip=14
freq damp amp phase
-0.483017 -0.298327 0.406169 -1.377244
-0.401362 -0.148488 1.586307 0.599750
-0.390590 -0.083145 1.128904 -2.758495
-0.311199 -0.044415 0.257460 0.226358
-0.266564 -0.168889 0.202475 -2.160322
-0.234425 -0.021196 0.048848 0.510255
-0.150022 0.000124 0.103155 -0.937507
0.100044 -0.000144 0.099705 0.572096
0.200781 0.000962 1.095562 1.064115
0.209305 0.003796 0.962849 1.549595
0.302695 -0.043607 0.383562 -0.088875
0.342033 -0.086267 0.654192 1.477895
0.387929 -0.083053 0.327827 -2.507180
0.433339 -0.176755 0.110752 -1.525994
PRONY analysis on a SUN, ip=14
freq damp amp phase
-0.488216 -0.190089 0.187805 -1.193899
-0.405863 -0.089196 0.652651 0.112258
-0.383956 -0.075260 0.553492 2.675876
-0.312014 -0.039341 0.244185 0.240129
-0.269939 -0.108541 0.135794 -2.000555
-0.232516 -0.006982 0.040439 0.255480
-0.150032 0.000152 0.103519 -0.948101
0.100042 -0.000142 0.099712 0.561036
0.200836 0.001228 1.102440 1.042306
0.209217 0.003993 0.971869 1.571278
0.300851 -0.047097 0.398952 0.138862
0.336671 -0.081288 0.573553 1.898996
0.381544 -0.067407 0.276133 -1.808118
0.428201 -0.114800 0.071015 -0.241177