Lee:

The marginal distributions for the parameters never contain as much

information as the joint probability distributions except those

conditions where the underlying problem is truly separable. It is

almost always better from an information theoretic and probabilistic

point of view to use the joint conditional distributions for the

parameters given the observations and do the estimation jointly.

In this case however, we are actually making measurements on two

independent signals, and the observations are r1,r2 as you say. It

would be better in this case, since they are truly separated, to

measure each independently. The joint observation process would arise

if John measured the difference signal through a mixer or time interval

counter, etc.

Achilleas identified incorrect parameters given John’s statement of the

problem. Achilleas used amplitude and phase as the parameters and the

original statement of John’s problem has constant 1V peak to peak as the

amplitude and the unknowns are frequency and phase. Your formulation

of the problem is correct, but it is more general that John’s statement

of the problem since you include A1,A2 as (possibly) different when

John’s statement of the problem is that A1=A2=1.

r(t) = sin(wt+phi)+n(t)

and determining w and phi in the presence of noise n(t) is just about

the oldest problem in town. Let us consider John’s original problem

given the system he claims he has. Since John’s statement is that he

is doing the measurements on each separately using a coherent system,

he can repeatedly estimate w and phi using FFT’s and downsampling.

One way to reduce the impact of the noise given a fixed size FFT, is to

use the coherence as stated and to do long term autocorrelations, where

the autocorrelations are computed using FFT’s and then simply added,

complex bin by bin. This coherent addition of the correlations will

produce a very long term autocorrelation where accuracy of the estimates

from this process goes up like N where N is the number of FFT’s added.

THIS ASSUMES THE SYSTEM IS REALLY COHERENT FOR THAT N * FFTsize SAMPLES

and THE NOISE REALLY IS ZERO MEAN GAUSSIAN. Phase noise, drift,

non-Gaussian noise, etc. will destroy this coherence assumption and the

Gaussian properties we use in the analysis. He can reduce the ultimate

computational complexity by mixing, downsampling and doing the FFT

again and then mixing, downsampling and doing the FFT again, etc. until

the final bin traps his w to sufficient accuracy for his needs and then

phi is simply read off from the FFT coefficient. The mixing and

downsampling would be a usable approach. Careful bookkeeping must be

done on the phase group delay through the downsampling filters should

this approach be used or phi will be in error by the neglected phase

group delay.

This is one approach that I believe John can take and it is pretty

simple to put together even if it is not necessarily the most

computationally benign. He can grab tons of samples and do this in

Octave on his favorite Linux computer. In the case the signals are not

actually 1V pk-pk, this will also yield the amplitude since the power

of the sinusoid as measured by the FFT’s in use above will yield the

result for the amplitude. If this is to be done real time, then a

cascade of power of 2 reductions in sample rate and frequency offset can

be done until the parameters are trapped sufficiently for the exercise.

Bob

Lee P. wrote:

s(t) = s(t;A1,A2,t1,t2) = A1 sin(w (t-t1)) + A2 sin(w (t-t2))

technique would converge faster if only three parameters needed to be

http://lists.gnu.org/mailman/listinfo/discuss-gnuradio

–

Robert W. McGwier, Ph.D.

Center for Communications Research

805 Bunn Drive

Princeton, NJ 08540

(609)-924-4600

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