- We first generated data with two sinewaves and background data from a second-order AR spectrum. In Figure 8.3, on top, we see the the raw spectrum (cyan), and the median filtered spectrum (magenta), the initial AR spectrum (red), and the detected sinusoids (asterisk).
- Next, we estimated the SINAR model parameters as described
(
`software/module_sinar.m`). On the bottom of Figure 8.3, we see again the the raw spectrum (cyan), the final AR spectrum (magenta), and the spectrum of the final sinusoidal component (green). - Next, we compressed the parameters (features) by
converting the real/imaginary amplitude pairs to just log-amplitude
using the two modules:
[z,jout]=module_c2r(z,jout,nsin+[1:2*nsin]); [z,jout]=module_log(z,jout,[ nsin+[1:nsin] 2*nsin+P+1]);

- We then re-synthesized the expanded parameters using the inverse
of the above chain:
zs=module_exp(z,0,[ nsin+[1:nsin] 2*nsin+P+1]); zs=module_c2r_synth(zs,nsin+[1:nsin]);

- Using the SINAR parameters from the above, we
re-synthesized the raw data using (
`software/module_sinar_synth.m`). - We then passed this raw data back into the chain
`software/module_sinar.m`,`software/module_c2r.m`,`software/module_log.m`and compared the features with the original set:0.335671813882368 0.335671815103722 -0.000000001221354 0.349466557731445 0.349466557694834 0.000000000036611 3.702689056085881 3.702689045510666 0.000000010575215 2.296026775397141 2.296026769375969 0.000000006021172 -1.248813788132684 -1.248813788102529 -0.000000000030155 0.748129007066103 0.748129007111796 -0.000000000045693 3.868379046063297 3.868379046072616 -0.000000000009318

Note that the features, including frequencies, amplitides, and AR coefficients, agree to many decimal places. - Next, we conducted an experiment to verify the
Cramér Rao lower bound (CRLB).
We generated data with a fixed parameters set, and measured the
estimation covariance of the ML parameter estimates.
We then compared this with the CRLB bound.
The CRLB was obtained by averaging the Fisher's Information matrices,
then inverting.
For frequency estimation, the result was:
Actual and estimated freq: ans = 0.332898504064205 0.332893584758298 0.362250453516612 0.362245648564146 ans = Freq CRLB and actual (stdev): ans = 1.0e-03 * 0.296121883718455 0.306671006314352 0.433985843610495 0.461426272795449

For amplitude estimation, the result was:ans = Ampl CRLB and actual (stdev): ans = 0.493213972877173 0.481101790137884 0.433333457180508 0.428876675859441 0.484875224240585 0.481079868355394 0.450742290250430 0.428879298178772

And, for AR parameters, the result was:AR CRLB, actual: ans = 0.001824998585736 -0.001277523563861 0.001916486591848 -0.001355315073806 -0.001277523563861 0.001824998585736 -0.001355315073806 0.001853675830711 CR bound for variance: 4.240074, actual stdev: 4.361447

All results agree to within about 10%. - Next, we conducted an acid test (See Section 2.3.8).
As a ``theoretical PDF", we generated data according to a circularly-stationary
process (Sections 9.1.4, 9.1.2) with a very peaked spectrum.
Note that in theory, a sinusoid in noise is not a circularly-stationary
process. But, for a very sharply-peaked spectrum, it is
an approximation. We did this because we could generate
data and compute the exact PDF of the circularly-stationary
process. In Figure 8.4, we show the spectral shape we used.
The spectal peak falls directly on an FFT bin, so depending on how
sharp the peak is, the adjacent bins can have little or no
spectral energy above the background.
If the data is generated in the frequencyy-domain
as independent exponentially-distributed bins
^{8.2}, the PDF computed using (9.7) is the exact PDF of the generated data. This is a perfect synthetic data / ``theoretical" PDF pair for the acid test of the SINAR model. In Figure 8.5, we show the results of the acid test. On the top is the SINAR model acid test results. On the bottom is the result for the`software/module_ar_mlx.m`module. Clearly the AR features alone produce a poor PDF estimate because the features are insufficient.**Figure 8.5:**Results of SINAR model acid test. Left: acid test using data with spectrum shown in Figure 8.4. Right: with just background AR spectrum. Top: SINAR model, bottom: AR model.

Baggenstoss 2017-05-19