How can I draw the corresponding exponential decay curve to fit a damped sine wave?

I want to draw the exponential curve that fits the peaks of the damped signal. I have done this very crudely by plotting the x and y values of the peaks on the same figure as the damped signal, but is there a better way to do this, without having to search values manually on the graph. I am relatively new to matlab so any help would be appreciated. Thanks!

Answer 1

[John D'Errico]

You want to fit an exponential envelope function to the curve, essentially a least upper bound function. So the exponential mode would be something of the form...

Y_e = a + b*exp(-c*x)

If you know that the lower asymptote is exactly 1, then your model would be

Y_e = 1 + b*exp(-x)

We might estimate those coefficients by a simple log transformation, or we could use a nonlinear estimation. I'll show the log transformation. First, I'll create a damped sine wave curve as an example.

x = 0:100;
y = 1 + sin(x - pi/3).*exp(-0.2*x);

First, use only those points that fall above 1. So assuming vectors of points x and y...

ind = (y > 1);
x1 = x(ind);

% and transform y by a log transformation

y1 = log(y(ind) - 1);

A = [ones(numel(x1),1),x1(:)];

% estimate the model coeffs = lsqlin(A,y1(:),-A,-y1(:),[],[],[],[],[],optimset('algorithm','active-set','display','off'));

coeffs =
  -0.00091842
     -0.19999

Don't forget that when we logged the model, we transformed the coefficient b. We need to exponentiate

b = exp(coeffs(1))
b =
      0.99908

c = coeffs(2);

% now plot the data and the curve

plot(x,y,'ro',x,1 + b*exp(c*x),'b-)

I used lsqlin from the optimization toolbox to do the fit. The fundamental ideas in this solution were:

- Transform the problem to linearity.
- Solve the linear least squares problem, such that all residuals had the proper sign.

If you don't know the lower asymptote for the curve, then you would need to use a nonlinear optimization tool for the fit. The constraints would be such that the necessary optimizer would be fmincon.

[MC]

Thanks a lot! I tried that and I got exactly what I needed.

[John D'Errico]

I'm happy to have helped.

[Image Analyst]

John, what do you think about this approach using Hilbert: http://www.mathworks.com/matlabcentral/answers/168113#answer_163469

[John D'Errico]

I don't have hilbert. (Probably Signal Processing TB I presume?) It seems to work nicely enough, with the caveat of the Gibbs phenomena at the ends. It probably depends on one's goals for that envelope. Will those oscillations kill you? What will you use it for? Do you need a specific form for the solution?

For example, I did not mention that my SLM tools have inf and sup options in the fit, so they can fit least upper bound functions or greatest lower bound functions in the form of a spline. But if you really needed an exponential form, then a spline is useless.

Very often a solution is driven by your goals, and too often we, in the form of consultants, are not given those goals.

Answer 2

[Scott Webster]

I assume your damped sine wave data is experimental? (if you have an equation for it, then you should be able to get the parameters of your exponential from there)

Two ideas:

-curve fit the data with a custom damped sine wave function and then use the resulting parameters for the exponential to plot just that

-find the peak points (analyze derivatives?) and then just fit that with an exponential