inflection point on sum of sinusoids

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Ben
Ben 2015-3-3
评论： Ben 2015-3-3
Hi, I have a problem that may have a neat solution but I'm not certain of the approach. I'd like to find the inflection points of a function which is the sum (superposition) of sinusoids (tidal data).Its formulated like this:
y = A1*sin(w1*t + p1) + A2*sin(w2*t + p2) + ... + An*sin(wn*t + pn)
where A1..An is the component Amplitude; w1..wn is the component frequency; p1..pn is the component phase. And each of these is known.
I think you can find the inflection by finding where the second derivative of y is zero. ie
y'' = 0 = -A1*w1^2*sin(w1*t +p1)+ A1*cos(w1*t +p1) + ..
Is there a fast way to determine the values of t (within a range) for which y'' =0? The timeseries is long (around 1 million points) so recreating the entire timeseries is time consuming.

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Andrew Newell
Andrew Newell 2015-3-3
If you define a vector A = [A1 A2 ... An], and analogous vectors w and p, then in MATLAB
y = sum(A.*sin(w*t+p));
and the second derivative is
ypp = -sum(A.*w.^2.*sin(w*t+p));
(yours is incorrect). Given A, w and p, you can define an anonymous function
f = @(t) -sum(A.*w.^2.*sin(w*t+p));
and then use fzero to find a root, e.g.,
inflection_point = fzero(f,[x0 x1])
for a solution between x0 and x1.
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Ben
Ben 2015-3-3
Thanks Andrew, for the code and for correcting my maths! I was hoping that it would give an array of inflection points but I can easily add a loop as the times between inflections are almost regular. Cheers

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