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 1sided.
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 
Fstat 
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 
tvalue 
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 
tvalue 
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 
tvalue 
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 insitu 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 multistep linear procedure that produces a suboptimal solution. To see if this fit can be improved, we will use AutoSignal's nonlinear fitting with a damped sinusoid model.
Close the Numeric Summary.
Click the NonLinear 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 
Fvalue 
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 
tvalue 
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 
tvalue 
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 
tvalue 
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 nonnormal trend in the residuals.
Close the Residuals Window.
Click OK to close the NonLinear 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 nonlinear optimization. In only 1 of 20 white noise data sets would this measure of nonnormality 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 nonSVD 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 leastsquares sense than that realized from the ARbased sinusoidal fits where this constraint is not present.
The Prony algorithm is also able to fit nonoscillating 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 nonlinear optimization is invoked from the Prony procedure, mixed models are automatically fitted. If a component in the Prony fit produces a frequency less than 1e8*Nyquist, it is treated as an exponential decay component in the nonlinear 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 nonlinear 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 
% 
1.01552e12 
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 
Fstat 
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 
tvalue 
95% 
Confidence Limits 
P >t 
Ampl 
3.98682529 
63269.7649 
6.3013e05 
1.248e+05 
1.2481e+05 
0.99995 
Freq 
1.016e12 
1.8449e+07 
5.504e20 
3.639e+07 
3.6395e+07 
1.00000 
Phase 
1.56344920 
2.16e+06 
7.2384e07 
4.261e+06 
4.2609e+06 
1.00000 
Damp 
10.0290423 
8.5171e+05 
1.1775e05 
1.68e+06 
1.6801e+06 
0.99999 
Exp Damped Sine Component 2
Parm 
Value 
Std Error 
tvalue 
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 
tvalue 
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, nonlinear optimization must be used.
Close the Numeric Summary.
Click the NonLinear 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 
Fvalue 
0.99990732 
0.99990293 
0.01862295 
2.5759e+05 
Data Power 
Model Power 
Error Power 
0.3573762239 
1.0024353366 
3.312078e05 
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 
tvalue 
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 
tvalue 
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 
tvalue 
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. 