DAMPED SINE AND EXPONENTIAL MODELING :

This tutorial covers the procedures within AutoSignal that model damped sines and exponential decays.

Generating A Test Signal For Damped Sine Modeling

Select the Generate Signal option in the Edit menu or Main toolbar.

For this tutorial, we will create a data stream consisting of three exponentially damped sinusoids and white noise.

Click Read and select the file tutor9a.sig from the Signals subdirectory.

The following signal expression is imported: AMP1=2
AMP2=3
AMP3=4
FREQ1=100
FREQ2=200
FREQ3=400
PHASE1=PI
PHASE2=PI/2
PHASE3=0
RATE1=50
RATE2=20
RATE3=10
F1=AMP1*EXP(-RATE1*X)*SIN(2*PI*X*FREQ1+PHASE1)
F2=AMP2*EXP(-RATE2*X)*SIN(2*PI*X*FREQ2+PHASE2)
F3=AMP3*EXP(-RATE3*X)*SIN(2*PI*X*FREQ3+PHASE3)
Y=F1+F2+F3

The X (time) values vary from 0 to 0.0995 with a 0.0005 sample increment. The Nyquist frequency is 1000. The first damped sinusoid is at frequency 100 and has the highest damping rate. The second damped sinusoid is at frequency 200 and has an intermediate damping rate. The final damped sinusoid has a frequency of 400 and the lowest damping rate. 10% Gaussian noise is added.

Click OK to process the current signal.

An AutoSignal graph is presented containing the 200 point generated data.

Click OK to accept the generated data. Click Yes when asked to update the main data table with the revised data.

Prony Modeling

The main algorithm in AutoSignal for estimating the parameters of damped sinusoids is the Prony procedure. AutoSignal offers a number of enhancements that improve the stability and noise resistance of this particular algorithm.

Select the Prony Spectrum option in the Spectral menu or toolbar. Be sure the algorithm is set to Dmp Svd. Set the model order to 60. Be sure the Allow real exp is not checked and that Full Range and Adaptive n are checked. Be sure the spectral plot is dB 1-sided.

Click the Graphically Select Signal and Noise Subspaces button.

The six eigenmodes of signal space are readily evident in the singular value plot.

Click on the 6th singular value in order to use the first six eigenmodes as signal and the remainder as noise.

Click OK to close the singular value plot and automatically update the signal space in the Prony procedure.

Click the Display Maxima button so that frequency labels appear at the three spectral peaks.

The frequencies have been determined to a good accuracy. In the Prony procedure, the spectral plot is secondary and created from the parametric fit. The primary focus of interest with this algorithm is the numeric summary.

Click the Numeric Summary button. Be sure the Add Complex Exponential Fit Summary and Add Complex Exponential Fit Details are checked in the Format menu. Select the 95% confidence limit. Inspect the Prony fit.

Exp Damped Sine Fit (Suboptimal)

 Frequency Amplitude Phase Damping Power % 98.910296672 1.8656065702 3.2600887110 51.039866501 0.0168316858 3.4890975390 200.02092335 3.1319048653 1.5288969890 20.913130098 0.1152637637 23.893418539 399.89097229 4.1597705055 0.0254296427 10.944288783 0.3503125554 72.617483922

 r² DOF r² Std Err F-stat 0.9907818304 0.9901902901 0.2230689922 1836.9549423

 Data Power Model Power Error Power Ratio 0.5074129771 0.5031525830 0.0046774189 107.57056322

Exp Damped Sine Component 1

 Parm Value Std Error t-value 95% Confidence Limits P >|t| Ampl 1.86560657 0.10274651 18.1573717 1.66292236 2.06829078 0.00000 Freq 98.9102967 0.62531849 158.175872 97.6767542 100.143839 0.00000 Phase 3.26008871 0.05500609 59.2677773 3.15158025 3.36859717 0.00000 Damp 51.0398665 3.95602352 12.9018107 43.2359665 58.8437665 0.00000

Exp Damped Sine Component 2

 Parm Value Std Error t-value 95% Confidence Limits P >|t| Ampl 3.13190487 0.06817085 45.9419951 2.99742677 3.26638296 0.00000 Freq 200.020923 0.11711999 1707.82906 199.789885 200.251962 0.00000 Phase 1.52889699 0.02263230 67.5537578 1.48425109 1.57354289 0.00000 Damp 20.9131301 0.72189054 28.9699462 19.4890836 22.3371766 0.00000

Exp Damped Sine Component 3

 Parm Value Std Error t-value 95% Confidence Limits P >|t| Ampl 4.15977051 0.05722499 72.6914999 4.04688490 4.27265611 0.00000 Freq 399.890972 0.05177939 7722.97625 399.788829 399.993116 0.00000 Phase 0.02542964 0.01354436 1.87750721 -0.0012888 0.05214811 0.06200 Damp 10.9442888 0.32723100 33.4451470 10.2987724 11.5898052 0.00000

The parameters for all three of the damped sinusoids have been recovered with a good measure of accuracy. The 0.99 r² value is an excellent goodness of fit, and is achieved because of the in-situ noise removal of the SVD procedure. The values you see will differ somewhat due to the influence of the white noise component.

Although both the frequencies and damping factors derive from the rooting of an AR forward prediction polynomial within the Prony algorithm, only the frequencies are determined with a consistently narrow relative confidence limit. In the specific data fitted here, the 95% confidence limits did not fully bracket the parameters. A Prony fit is a multi-step linear procedure that produces a suboptimal solution. To see if this fit can be improved, we will use AutoSignal's non-linear fitting with a damped sinusoid model.

Close the Numeric Summary.

Click the Non-Linear Optimization button. Be sure the Component Model is Sine, Exp Damped. Accept the defaults and click OK to initiate the iterative fitting. Click Review Fit when the fitting is complete.

Click on the Set Confidence/Prediction Intervals button. Be sure Prediction Intervals is checked and that a 95% Confidence is selected. Click OK to close the Intervals dialog.

The three different damped sinusoids are readily visualized. Note the tight confidence intervals about the fitted curve.

Click the Numeric Summary button. Be sure Fitted Parameters and Parameter Statistics are checked in the Options menu. Inspect the optimized fit.

Fitted Parameters

 r² Coef Det DF Adj r² Fit Std Err F-value 0.99101628 0.99043978 0.22021405 1885.33981

 Data Power Model Power Error Power 0.5074129771 0.5028657949 30.0045584572

 Comp Type Frequency Amplitude Phase Damping Power % 1 Sine Damped 99.1849219 1.92844537 3.25547199 49.1785094 0.35067376 3.95601003 1 Sine Damped 200.062465 3.09623949 1.52862318 20.7823575 0.11364115 23.5067821 1 Sine Damped 399.908124 4.10700350 0.01842826 10.5592699 0.35067376 72.5372079

Parameter Statistics

Comp 1 Sin Decay Exp

 Parm Value Std Error t-value 95% Confidence Limits Ampl 1.92844537 0.09949567 19.3822038 1.73217397 2.12471677 Freq 99.1849219 0.56486847 175.589412 98.0706269 100.299217 Phase 3.25547199 0.05149670 63.2170993 3.15388637 3.35705761 Rate 49.1785094 3.57309168 13.7635733 42.1300048 56.2270141

Comp 2 Sin Decay Exp

 Parm Value Std Error t-value 95% Confidence Limits Ampl 3.09623949 0.06709202 46.1491473 2.96388957 3.22858941 Freq 200.062465 0.11613690 1722.64340 199.833366 200.291564 Phase 1.52862318 0.02252509 67.8631243 1.48418877 1.57305759 Rate 20.7823575 0.71583627 29.0322779 19.3702540 22.1944610

Comp 3 Sin Decay Exp

 Parm Value Std Error t-value 95% Confidence Limits Ampl 4.10700350 0.05604802 73.2765139 3.99643966 4.21756733 Freq 399.908124 0.05074748 7880.35480 399.808017 400.008232 Phase 0.01842826 0.01344019 1.37113102 -0.0080847 0.04494122 Rate 10.5592699 0.32064725 32.9311101 9.92674107 11.1917988

Although the optimized fit shows only a modest improvement in the r² goodness of fit index, the frequencies, amplitudes, and damping factors are closer to the values used to generate the data. With the exception of the phase values, all of the underlying parameters are now bracketed by the 95% confidence interval.

Close the Numeric Summary window.

Assessing Residuals

When fitting any form of parametric model to data, it is important to determine if the model is properly specified. If a model is underspecified, components that are present in the data may not be accounted. If a model is overspecified, redundant parameters will harm the individual coefficient statistics.

Click on the View Residuals button. Be sure the SNP option is selected. It is the second from the end of the toolbar.

The stabilized normal probability plot should be similar to the following graph where the residuals are confirmed as being normally distributed. The blue line is a 90% critical limit. In 1 of 10 random Gaussian noise sets, a single SNP point will reach this 90% critical limit. The green line is a 95% limit, the yellow line a 99% limit, and the red line is a 99.9% limit.

When a component of sufficient power has not been accounted for within a parametric model, it will appear as a non-normal trend in the residuals.

Close the Residuals Window.

Click OK to close the Non-Linear Optimization Review.

We will now explore the SNP for the residuals from the suboptimum Prony fit.

Click on the View Residuals button.

The residuals from the Prony fit are not as normally distributed as those arising from the subsequent non-linear optimization. In only 1 of 20 white noise data sets would this measure of non-normality occur.

The SNP can thus reflect the degree to which a fit is suboptimal as well as indicating an inappropriate or incomplete model.

Close the Residuals window.

Component Count From Root Inspection

Although the singular values are generally sufficient for determining the number of damped sinusoids present in a data set, it is possible to inspect the complex roots of the AR polynomial as an additional tool for assessing the component count.

In order to see if additional components are present using this approach, you must either use a non-SVD algorithm or increase the signal subspace in the SVD so that additional components will be treated as signal and included in the fit.

Set the Signal Subspace to 10.

Click the Plot Roots button. Be sure Unit Circle is selected.

Select the Magnitude plot.

The forward prediction roots are plotted with the + symbol and the backward prediction roots with the o symbol. In this case, all of the forward prediction roots are within the unit circle. Six of the backward prediction roots are outside the unit circle and mirror in magnitude six of the forward prediction roots. This confirms a signal space of 6 and the 3 damped sine components.

Note that damped sinusoids are not found on or near the unit circle as is true of undamped sinusoids. The only way to assess the count of damped sinusoids using this analysis is the location difference between forward and backward prediction roots. Damped sinusoids approach the unit circle only as the damping factors (the exponential decay) approach zero.

For undamped sinusoids, the signal space can be inferred from the complex roots found with a magnitude very close to 1.0. This difference between the backward and forward prediction signal roots with damped sinusoids is not present for undamped sinusoids.

Click OK to close the Complex Roots window.

Click OK to close the Prony procedure.

Mixed Model Prony Fits

The damped algorithms in the Prony procedure are not limited to fitting damped sinusoids. If an undamped sinusoid is present, the algorithm will report a damping factor very close to zero. Because an undamped sinusoid is a specialization of a damped sinusoid (a damped sinusoid with zero rate of decay), it is possible to fit any mixture of damped and undamped sinusoids using the Prony algorithm.

The undamped algorithms use modifications to the Prony algorithm that enforce sinusoidal modeling. In this approach, the roots of the AR polynomial are forced to the unit circle. The undamped algorithms are offered primarily as a reference for the damped model fits. If a damped model fit results in very low damping factors, an undamped fit can be made to see if it is statistically equivalent. Since the undamped algorithms constrain all roots exactly to the unit circle, the results from the Prony procedure for undamped sinusoidal modeling will generally be less accurate in a least-squares sense than that realized from the AR-based sinusoidal fits where this constraint is not present.

The Prony algorithm is also able to fit non-oscillating exponential decays. In this instance, the frequency rather than the damping factor is estimated at or near zero. When Allow real exp is not checked, only sinusoids and damped sinusoids are processed. When Allow real exp is checked, the real roots at frequency zero that give rise to exponential decay components are also included in the Prony fit.

Keep in mind that real exponentials and undamped sinusoids are subsets of the complex exponentials (damped sinusoids) modeled by the Prony procedure. What is always being modeled are complex exponentials. A complex exponential reduces to a real exponential only when a real root results in a zero frequency. A complex exponential reduces to an undamped sinusoid only when a complex root falls on the unit circle. These reductions occur as part of the fitting procedure and are data dependent. They cannot be specified.

When AutoSignal's non-linear optimization is invoked from the Prony procedure, mixed models are automatically fitted. If a component in the Prony fit produces a frequency less than 1e-8*Nyquist, it is treated as an exponential decay component in the non-linear fitting. If a component's damping factor is such that the sinusoid decays less than 1% in amplitude across the sampling range, it is treated as an undamped sinusoid in the non-linear fitting. When a mixed model results from a Prony fit, the NL Optimization option must be used for valid confidence limits on the fitted parameters.

Generating A Test Signal For Mixed Complex Exponential Modeling

Select the Generate Signal option in the Edit menu or Main toolbar.

We will import a signal that is identical to the previous one, except that one of the three damped sinusoids is converted into an undamped sinusoid by assigning it a zero damping coefficient and another is converted into a real exponential. We will also also decrease the amount of noise added to insure that the undamped sinusoid in the signal is fitted as such.

Click Read and select the file tutor9b.sig from the Signals subdirectory.

The following signal expression is imported:

AMP1=2
AMP2=3
AMP3=4
FREQ1=100
FREQ2=200
PHASE1=PI
PHASE2=PI/2
RATE2=20
RATE3=10
F1=AMP1*SIN(2*PI*X*FREQ1+PHASE1)
F2=AMP2*EXP(-RATE2*X)*SIN(2*PI*X*FREQ2+PHASE2)
F3=AMP3*EXP(-RATE3*X)
Y=F1+F2+F3

The first component at frequency 100 is now an undamped sinusoid. The second component at frequency 200 remains a damped sinusoid. The third component contains the same amplitude and rate, but is now a real exponential. 1% Gaussian noise is added.

Click OK to process the current signal.

Click OK to accept the generated data. Click Yes when asked to update the main data table with the revised data.

Select the Prony Spectrum option. Check the Allow real exp box.

Click the Graphically Select Signal and Noise Subspaces button.

Note that the signal space has decreased from 6 to 5 since the real exponential is captured by only a single eigenmode.

Click on the 5th singular value to specify the signal space.

Click OK to close the singular value plot and automatically update the signal space in the Prony procedure.

The Prony energy spectrum now contains only two peaks. The exponential appears as the decay from frequency zero.

Click the Numeric Summary button and inspect the Prony fit.

Exp Damped Sine Fit (Suboptimal)

 Frequency Amplitude Phase Damping Power % -1.01552e-12 3.9868252864 1.5634492012 10.029042265 0.6847390464 69.152860926 100.00716525 1.9888303904 3.1370507338 0.0468264458 0.1949671737 19.690038003 199.99688195 3.0019583758 1.5506902365 0.0468264458 19.999671756 0.1104755848

 r2 DOF r2 Std Err F-stat 0.9997337391 0.9997166529 0.0318165121 64171.498330

 Data Power Model Power Error Power 0.3573762239 0.9951883962 10458.570167

Exp Damped Sine Component 1

 Parm Value Std Error t-value 95% Confidence Limits P >|t| Ampl 3.98682529 63269.7649 6.3013e-05 -1.248e+05 1.2481e+05 0.99995 Freq -1.016e-12 1.8449e+07 -5.504e-20 -3.639e+07 3.6395e+07 1.00000 Phase 1.56344920 2.16e+06 7.2384e-07 -4.261e+06 4.2609e+06 1.00000 Damp 10.0290423 8.5171e+05 1.1775e-05 -1.68e+06 1.6801e+06 0.99999

Exp Damped Sine Component 2

 Parm Value Std Error t-value 95% Confidence Limits P >|t| Ampl 1.98883039 0.00642072 309.751775 1.97616447 2.00149631 0.00000 Freq 100.007165 0.00887861 11263.8371 99.9896508 100.024680 0.00000 Phase 3.13705073 0.00320096 980.034609 3.13073632 3.14336515 0.00000 Damp 0.04682645 0.05587906 0.83799633 -0.0634041 0.15705698 0.40310

Exp Damped Sine Component 3

 Parm Value Std Error t-value 95% Confidence Limits P >|t| Ampl 3.00195838 0.00940777 319.093614 2.98340002 3.02051673 0.00000 Freq 199.996882 0.01641487 12183.8825 199.964501 200.029263 0.00000 Phase 1.55069024 0.00322937 480.183816 1.54431978 1.55706069 0.06200 Damp 19.9996718 0.10168875 196.675371 19.7990742 20.2002694 0.00000

The real exponential is listed as the first component since the ordering is by ascending frequency. The undamped sinusoid is listed as the second component. The unchanged damped sinusoid is the third component.

All of the parameters have been recovered with a good deal of accuracy and the goodness of fit values are superb. On the other hand, the confidence limits are invalid for all parameters in the first (exponential decay) component. The confidence limits are also very wide for the damped parameter of the undamped sinusoid. The Prony fit and its statistics reflect only the complex exponential model. To see the statistics for a mixed model, non-linear optimization must be used.

Close the Numeric Summary.

Click the Non-Linear Optimization button. To fit the mixed model, Sine, Exp Damped must be selected. Accept the defaults and click OK to initiate the iterative fitting. Click Review Fit when the fitting is complete.

The three different components are readily visualized. With the reduced noise level, the confidence intervals about the fitted curve are barely visible.

Click the Numeric Summary button and inspect the optimized fit.

Fitted Parameters

 r² Coef Det DF Adj r² Fit Std Err F-value 0.99990732 0.99990293 0.01862295 2.5759e+05

 Data Power Model Power Error Power 0.3573762239 1.0024353366 3.312078e-05

 Comp Type a1 a2 a3 a4 1 Real Exp 10.0001636 3.99748178 2 Sine 99.9992904 1.99689882 3.14274917 3 Sine Damped 199.996544 2.99975395 1.57001113 19.9551158

Parameter Statistics

Comp 1 Real Exp

 Parm Value Std Error t-value 95% Confidence Limits Ampl 3.99748178 0.00329399 1213.56685 3.99098450 4.00397906 Rate 10.0001636 0.01912000 523.021187 9.96245015 10.0378771

Comp 2 Sine

 Parm Value Std Error t-value 95% Confidence Limits Ampl 1.99689882 0.00187028 1067.70163 1.99320976 2.00058787 Freq 99.9992904 0.00516437 19363.3068 99.9891039 100.009477 Phase 3.14274917 0.00186399 1686.02879 3.13907251 3.14642583

Comp 3 Sine Damped

 Parm Value Std Error t-value 95% Confidence Limits Ampl 2.99975395 0.00549885 545.523474 2.98890766 3.01060023 Freq 199.996544 0.00959181 20850.7701 199.977624 200.015463 Phase 1.57001113 0.00188930 831.002888 1.56628456 1.57373770 Rate 19.9551158 0.05942892 335.781215 19.8378945 20.0723371

The goodness of fit has been further improved. Since the correct mixed model was automatically fitted, the confidence limits are now fully valid for all parameters in all three components.

Close the Numeric Summary window. Close the Prony procedure.

Multicomponent Exponential Decay Data

If you are fitting multiple simultaneous first order decays, as is often done when different radioactive labels are used in a single experiment, keep in mind that each of the components is likely to be highly correlated. The Prony method may not be able to resolve the individual components, especially when significant observation noise has masked the subtleties in the curve.

When the Prony method finds only a single real exponential when there are two known to be present, this is a good indication that there is not sufficient statistical information present within the data to resolve the individual components.

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