How to get impulse response of an array of values aiming to apply fast fourier transform (fft) on impulse response array

Dear colleagues, I have a challenge in building or extracting the transfer function in the form of Laplace based on a signal plotted in time domain in a pdf paper/articles. So far i managed to evaluate manually the coordinates of the signal in time domain by the following code in Matlab:

t0=[0 20 30 48 63 77 100 110 150 180 200 233 300 330 370 400 450 500 600 700 800];% points of t are obtained from the signal in whalley's paper% 

y0=[0 0.5 1 1.5 2 2.5 2.875 3 3.375 3.5 3.55 3.57 3.6 3.57 3.55 3.52 3.48 3.44 3.35 3.25 3.15];% points of y are obtained from the signal in whalley's paper% 
figure 
plot(t0,y0,'o');

The question is how can I get the impulse response of the signal y0(t0) and how can I get the frequency response of the same signal for the same time duration. At the end I want to get the frequency response to plot bode diagram and to try to conclude the transfer function in the Laplace transform from Bode plot. Any suggestion from you ends?

Answer 1

[Star Strider]

You are doing ‘system identification’. MATLAB has an entire System Identification Toolbox devoted to it, and the Signal Processing Toolbox has the invfreqz function that will approximate it. This will be in discrete z-space. You can later use the bilinear transform to convert it to Laplace s-space. You will have to experiment to get the result you want.

[Basim Touqan]

Mr. Strider, Thank you for your suggestion which exactly fulfills what I need. The only problem that remains is that the data I have which is coordinates according to (t) vector and (y) vector [function: y=f(t)], while system identification toolbox requires measured data in terms of input (u) and output(y) data. I have tried to feed the system with (t) vector as inputs data but the outcome I obtained seemed not correct. Do you have an idea how can system identification toolbox accept do deal with my coordinates model or how can I convert coordinates model to input-output measured data model?

[Star Strider]

My pleasure.

You have to have some sort of known input signal to estimate the transfer function. If the signal you have is the output, the paper you got it from must mention the input somewhere.

The plot looks like the output of a second (possibly higher) order system in response to a step function with a magnitude of 3. It would be safe to assume a step function (beginning at t=0) as the input. So the input ‘u’ would simply be:

u = ones(size(t0))*3;

I would set the order for both the numerator and denominator at 5 to start, and then change the order until you get the best approximation. If you set separate orders for the numerator and denominator, be sure the numerator order is never larger than the denominator order.

[Basim Touqan]

Thank you sir, I think it works now, your assumption is correct, the paper is applying step input of 3. But I still have a problem, the time plot in toolbox is showing the signal up to the time second 18 while the original signal is plotted up to 800 seconds, any solution in this regards

[Basim Touqan]

Mr. Strider, Now I got regarding the proper timing, the orginal time intervals were not equal, I have obtained now t0 points with similar intervals of 25 seconds so the sampling time is 25 seconds too. I think now every thing is fine. Thank you so much for valuable suuport

[Star Strider]

My pleasure.

I apologise for the delay in responding. Life intrudes.