Fitting a function to a sinusoidal curve

[bugatti79]

Hi Folks,

I have a curve that varies sinusoidally calculated from a numerical program as attached "Trace.png". I would like to fit this amplitude modulation expression to it.

$$f(t)=A[1+B \cos(\omega_1 t+ \phi)] \cos(\omega_2 t+ \theta)$$

I managed to adjust the parameters manually to get a very similar curve but couldn't exactly match it. Is there a mathematical technique I could use or is it possible at all?

Thanks

 

[Ackbach]

I would use Excel or LibreOffice Calc's Nonlinear Solver routine to solve this problem. Here's the setup:

Column A contains your time values.
Column B contains your data points.
Column C contains the various parameters you can vary, such as $A, B, \omega_1, \omega_2, \phi,$ and $\theta$.
Column D contains the formula you want to fit, all depending on values from Column C (you'll have to populate Column C with initial guesses). Don't forget to use, e.g.,

$C$1

to retrieve the value from C1.
Column E contains the square of the differences between Column B and Column D. Sum this column at the bottom.

That's the setup. Next, you invoke the Solver routine, and minimize the sum of squares at the bottom of Column E by varying the cells in Column C. If you're in LibreOffice Calc, you have to tell Calc to use a nonlinear solver, or else you'll get an error.

See if that works for you.

 

[bugatti79]

I would use Excel or LibreOffice Calc's Nonlinear Solver routine to solve this problem. Here's the setup:

Column A contains your time values.
Column B contains your data points.
Column C contains the various parameters you can vary, such as $A, B, \omega_1, \omega_2, \phi,$ and $\theta$.
Column D contains the formula you want to fit, all depending on values from Column C (you'll have to populate Column C with initial guesses). Don't forget to use, e.g.,

$C$1

to retrieve the value from C1.
Column E contains the square of the differences between Column B and Column D. Sum this column at the bottom.

That's the setup. Next, you invoke the Solver routine, and minimize the sum of squares at the bottom of Column E by varying the cells in Column C. If you're in LibreOffice Calc, you have to tell Calc to use a nonlinear solver, or else you'll get an error.

See if that works for you.

HI Ackbach

THanks, I will give it a go.

 

[bugatti79]

Hi Folks,

An additional idea cross my mind regarding the attachment "trace.png".

This curve comes from a numerical program which converts this trace to the frequency domain and plots all the harmonics $n_k \omega t$ and their amplitudes $a_n$, $b_n$.

What I would like to do is take these harmonics and their amplitudes and enter them into the Fourier series below to reproduce the same trace

$$x_t=d_0+d_1 \cos( n_1\omega t - \phi)+d_2 \cos( n_2\omega t - \phi)+d_3 \cos( n_3\omega t - \phi)+...$$

where $d_0=a_0/2$, $d_n=\sqrt (a_n^2+b_n^2)$ and $\phi_n=tan^{-1} (b_n/a_n)$

However, how would I determine $a_0$?

 

[CellCoree]

for your question. Fitting a function to a sinusoidal curve is a common problem in many scientific fields, including physics, engineering, and mathematics. There are several mathematical techniques that can be used to fit a function to a sinusoidal curve, depending on the specific parameters and data available.

One approach is to use a least squares fitting method, which involves minimizing the sum of the squared differences between the data points and the fitted curve. This method can be used to find the best-fit values for the parameters A, B, ω1, ω2, φ, and θ in your amplitude modulation expression.

Another approach is to use a Fourier series to approximate the curve. This involves representing the curve as a sum of sinusoidal functions with different frequencies and amplitudes. By adjusting the coefficients of the Fourier series, you can find the best-fit curve for your data.

It is also possible to use non-linear regression techniques, such as the Levenberg-Marquardt algorithm, to fit a function to your data. These methods involve finding the best-fit parameters by minimizing a cost function that takes into account both the data points and the function being fitted.

In conclusion, there are several mathematical techniques that can be used to fit a function to a sinusoidal curve. It may be helpful to consult with a mathematician or data analysis expert to determine the best approach for your specific data and parameters.