3.4 Comparison of Frequency Extraction Methods
3.4.1 Extracting an Isolated Resonance
In order to identify the limitation of extracting resonances in the presence of noise, random noise of known strength was added to a time signal of a damped sinusoid, in the form given by equation 3.6, of a known frequency of 193.548T Hz (1.55µm free-space wavelength) and varying Q factors rang- ing from 100 up to 108. The random noise was generated with a normal
probability density function with varying strengths down to a SNR of 40dB. Signal Parameters
Results pertaining to four noise levels - namely 100dB, 80dB, 60dB and 40dB are discussed in the following sections. The noise values were generated from a Gaussian random distribution with zero mean and a standard deviation of 10−5, 10−4, 10−3 and 10−2 times the amplitude of the sinusoidal signal for the
respective noise levels. The random number distribution used was Mersenne Twister psuedo random number generator, the cycle length of which is about 106000 [23].
Under noise, all of these methods provide different values for different sig- nals. Hence, a single reproducible value cannot be provided for a given set of signal parameters. A statistical parameter representing the central value and its dispersion (in statistical distribution sense) over many signals with the same signal parameters produces a better picture of the strengths and weaknesses of these methods. Median was used as the measure of central tendency, while inter-quartile range was used as the measure of statistical dispersion about the median. These two robust statistical measures together indicate the accuracy of extracting each parameter from a given time signal with a known signal to noise ratio.
In this section, the isolated resonance was extracted from the time sig- nal formed as discussed above. All methods were given 500 double precision floating point samples of the time signal sampled at 1fs increments. 100 such signals were generated from the same sinusoidal signal, by adding ran- dom numbers drawn from the same statistical dispersion, to carry out the
statistical analysis. Results with Noise
The medians of estimated resonant frequencies of all methods at all noise lev- els were at 193.548T Hz indicating that methods have no bias in extracting resonant frequencies. Therefore, given enough number of time responses of the same resonator with SNR more than 40dB, one could statistically evaluate the actual resonant frequency. Further, the statistical dispersion of resonant frequencies was negligible (< 1e − 8%) for all methods at −100dB, −80dB noise levels. At −60dB only the harmonic inversion method had an account- able statistical dispersion (still low at 2 × 105%). This along with dispersion
of frequency for all methods at −40dB noise is presented in figure 3.8.
1e-05 0.0001 0.001 0.01 0.1 1
100 1000 10000 100000 1e+06 1e+07 1e+08
Frequency Uncertainity % Q Factor H inv -60dB D Prony -40dB R Prony -40dB M Pencil-40dB H inv -40dB
Figure 3.8: The numerical dispersion of frequency extracted by all methods at −40dB and harmonic inversion at −60dB.
is higher for low Q factors. This is due to the fast decay making most time samples (towards the end of the time signal) with lower signal to noise ratio than that indicated (Harmonic inversion has the largest statistical dispersion when the Q factors are high).
The median of the estimated Q factor is given in figure 3.9, whereas the dispersion of estimated Q factor as a percentage of the median Q is given in figure 3.10. 10 100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09
100 1000 10000 100000 1e+06 1e+07 1e+08
Median Q factor
Q Factor
Figure 3.9: The median estimated Q factor from a time signal of 500fs at noise levels −100dB (dot-dash), −80dB (dotted), −60dB (dashed), −40dB (solid) extracted from difference Prony (red), recurrence Prony (green), matrix pencil (blue) and harmonic in- version (black) methods.
As can be seen from figure 3.9, at −100dB, the ratio of the estimated median to the actual Q factor is close to unity according to the gradient of the graph. At −80dB, the graph levels at about 107, at −60dB levels at
about 106 and at −40dB levels at about 105 indicating that the median error
in Q factor estimation is directly proportional to the noise level present in the signal.
0.0001 0.001 0.01 0.1 1 10 100 1000
100 1000 10000 100000 1e+06 1e+07 1e+08
Dispersion of Estimated Q factor % of median Q
Q Factor
Figure 3.10: The interquartile range (measure of dispersion of Q) of estimated Q factor against the actual Q factor at noise levels at −100dB (dot-dash), −80dB (dotted), −60dB (dashed), −40dB (solid) extracted using difference Prony (red), recurrence Prony (green), matrix pencil (blue) and harmonic inversion (black) methods. According to figure3.10, the statistical dispersion of estimated Q increases by ten folds when the noise level is increased by ten folds in general. the percentage of statistical dispersion of Q reaches 100% when close to the lev- elling off point in figure 3.9. The two graphs indicate that the the maxi- mum limit of the Q factor estimated from these methods is directly propor- tional to the noise level. Figure 3.10, shows no clear distinction as to which method is best/worst, according to the results seen so far, harmonic inversion is marginally worse than others.
The effect of increased time duration on Q factor estimation was observed by comparing a time signal of 1000fs duration with the same signal parame- ters and a noise level of −60dB. The median and inter-quartile range of these
100 1000 10000 100000 1e+06 1e+07 1e+08
100 1000 10000 100000 1e+06 1e+07 1e+08
Median Q factor
Q Factor
Figure 3.11: The median of estimated Q factor from a time signal of 500fs (dashed) and 1000fs (solid) at a noise level of −60dB extracted from difference Prony (red), recurrence Prony (green), matrix pencil (blue) and harmonic inversion (black) methods.
According to figures 3.11and 3.12, as a direct result of increasing the time duration by 2, both sets of curves have been shifted by a factor of 10. In other words, the maximum Q factor obtainable with a given accuracy is increased 10 times when the time duration is doubled.
Compared to FFT based extraction method presented, these methods have similar accuracy in estimating resonant frequency. But when estimating the Q factor (or imaginary part of frequency), complex frequency extraction meth- ods, clearly outperform the FFT method. This is knowing that the complex extraction methods has 65 times less time duration and much lower SNR of 100dB-40dB (noiseless case in FFT includes quantisation noise of the IEEE double precision floating point value which calculates to ≈ 300dB).
0.01 0.1 1 10 100 1000
100 1000 10000 100000 1e+06 1e+07 1e+08
Dispersion of Estimated Q factor % of median Q
Q Factor
Figure 3.12: The inter-quartile range of estimated Q factor from a time signal of 500fs (dashed) and 1000fs (solid) at a noise level of −60dB extracted from difference Prony (red), recurrence Prony (green), matrix pencil (blue) and harmonic inversion (black) methods.