Re: Interpreting discrete Fourier transform.
- From: Greg Hansen <glhansen@xxxxxxx>
- Date: Tue, 12 Dec 2006 11:23:27 -0600
Pygmalion wrote:
Fellow physicians,
Discrete Fourier transform can be very useful to evaluate frequency and
amplitude of oscillation in a measured signal. Interpretation is
extremely simple if obtained spectral function has non-zero value for
only one frequency, e.g.
f = 4.2 Hz A = 0 mm
f = 4.3 Hz A = 40 mm
f = 4.4 Hz A = 0 mm
Here A is an absolute value of the Fourier transform, that is A =
sqrt((Im A)^2 + (Re A)^2). In that case oscillation has frequency of
4.3 Hz and amplitude of 40 mm.
What about more complex situations? When oscillation is not perfect,
values are obtained in narrow range around dominant oscillation, e.g.
f = 4.2 Hz A = 5 mm
f = 4.3 Hz A = 32 mm
f = 4.4 Hz A = 3 mm
How can one obtain total amplitude for this oscillation? Simply adding
amplitudes does not seem to be the answer. Also what is a theoretical
explanation for this interpretation?
Thanks for the answer,
Marko.
Here's how I would try to analyze it. You can't just add the components because the peak of one won't necessarily occur at the peak of the other. The discrete Fourier transform can be written as a sum of components with frequency and phase,
F(x) = sum_n a_n*cos(n*pi*x/L + phi_n)
How do you find the maximum value? The same way you find any other maximum-- by finding the location of the maximum and then plugging it to give the value. Find extrema with the first derivative.
dF/dx = sum_n a_n * (-1)n*pi/L sin(n*pi*x/L + phi_n) = 0
Solve for x. Use the second derivative to pick out maxima and minima. Plug x_maxes into F(x) to get the amplitude. There may be several maxima. Pick the highest one.
Even for just three frequencies, you'll have to solve it numerically.
Or it might be easier just to find the max of the untransformed wave by stepping through the data points one by one, or to do an inverse transform with the few components you're interested in, and find the max of that.
.
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- Interpreting discrete Fourier transform.
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