Saturday, February 5, 2011

Quantized Redshifts. III. How is Fourier Series Related to the PSD?

Continuing with the introduction to Fourier analysis...

The amplitude of each term satisfies a relationship with the original function called the Parseval equality (Wikipedia).
Mathematical Note: some readers may not have the mathematical background to make sense of some f the symbols used in this equation or others in this blog (and particularly this series of posts).  The vertically stretched 'S' on the left hand side of the equation is the calculus symbol for integration (wikipedia), basically a method of summing little chunks of 'area' under a smoothly varying function.  The funky-looking 'E' on the right side of the equation is the Greek letter Sigma, and represents summation of a series of values (wikipedia).  On a personal note, it was curiosity about these symbols when I was in the eight-grade that motivated me to learn calculus on my own.  It's value extends far beyond science and mathematics.

Physically, we can think of this as an expression of conservation of energy, that the total energy in the signal, the square of the amplitude of the function, f(x), must be equal to the total energy in each frequency composing the signal.  If we plot the square of amplitude of each frequency (an^2+bn^2), we obtain a plot which is the equivalent of the power spectral density (PSD).
Note that the PSD for the square wave displays power not just in the zero frequency bin, but also has smaller peaks of power at higher frequencies in between frequencies of zero power!  Beyond the fact that the square wave repeats with a wavelength of 20, why would it have these additional peaks?!  It is our first clue that one must be very careful when interpreting the PSD.

We'll examine a few more cases, such as the sawtooth wave, were the Fourier coefficients are computed to be
A plot like those above yields:
Click to Enlarge
Due to the sharp discontinuity at L=20, we observe a small overshoot that is part of the Gibbs phenomenon.

Let's also look at the half-sine (rectifier) wave with Fourier coefficients

Click to Enlarge
Notice that in all these graphs, there is power in frequencies other than the zero (no frequency) bin.  With appropriate scaling, one would see there are non-zero amplitudes for values of n up to infinity.

Are the frequencies real?

One should be careful to define what one means by 'real'.  I'd be inclined to define 'real' in this case as if the frequencies found are representative of the underlying physical mechanism for forming the original signal as opposed to an artifact of the sampling.  And since all non-zero signals exhibit some sampling, there will always be sampling artifacts in the resulting PSD.

The Fourier series demonstrates that there is an alternative way to represent almost any finite function.  In many ways, it is as indistinguishable as saying that you could represent the number 10 by 3+2+5 or 4+6.  Think about the way you do pen-and-paper arithmetic, especially subtraction and division, and you'll realize these alternative representations are done all the time.  The Fourier series becomes useful by providing an additional set of mathematical 'tricks' which may facilitate solving some other problem.

For example, in the case of the half-sine (rectifier) example above, such as signal is usually generating by running a sine wave signal (such as A/C) through a non-linear device like a rectifier, which chops off the negative (reverse) part of the current.  In this case, the signal does not have to be formed by a summation of the frequencies in the Fourier series (though that would be an alternative way to form it).  In this case, I would be inclined to say the frequency components of the signal are not 'real', but simply an alternative representation.

But suppose you're examining a process where you expect to have some regular structures or variation?  Then the peaks might tell you more about the structure under examination, but one must exercise extreme care.

The major point of this section is to introduce the fact that non-zero power in the frequency bin of a given signal profile is NOT evidence of a real frequency indicating some intrinsic structure.  I will demonstrate more examples of this in the coming sections.

More online Fourier resources
Models were generated using Python v2.6, v2.7, numpy 1.5, scipy 0.8.  Plots generated using matplotlib 1.0.0

Next Weekend: The Fourier Transform, FFT and PSD, Oh My!


Rick DeLano said...

"I'd be inclined to define 'real' in this case as if the frequencies found are representative of the underlying physical mechanism for forming the original signal as opposed to an artifact of the sampling."

That's great.

So would Hartnett and Hirano.

So would the referees who have accepted their papers for publication at Astrophysics and Space Science and Physical Review D.

All of this review is wonderful stuff, but let us recall that you have now had going on six months to establish one single error in Hartnett and Hirano 2008, or Hirano 2010 for that matter, which relies upon the same analysis.

That means the Hartnett/Hirano analysis has now undergone two peer review processes, one for Astrophysics and one for Phys Rev.

In the meantime, five and a half months later, you are still warming up in the bullpen here.

Either your buildup is a bit too deliberate, or else you just don't have the goods.

Were I a wagering man.................

W.T."Tom" Bridgman said...

You seem to think that you are the only audience for this project.

That would be an error. I am exploring professional publication options.

Rick DeLano said...

Actually I think that would be great. The articles are very interesting and useful.

It seems too bad that you set about casting aspersions on Hartnett and Hirano's work, since you clearly have no basis whatsoever upon which to substantiate them.

Other than those unfortunate initial aspersions, the series has been quite good.

I believe this most likely concludes our business.

All the best.

W.T."Tom" Bridgman said...

I'm not surprised.

You claim your theory 'explains' everything but seems to predict NOTHING. This is the usual pattern when pseudoscience gets confronted with the REAL standards of science. They try to save their egos by claiming I have treated them or others poorly and running away.

But how to you treat someone with respect with they repeatedly, WILLFULLY, insist that 2+2=5 and offer evidence no better than their 'say so'.

So does this mean we'll never see your claimed geocentrist 'proof' that geocentrism can explain all the Lagrange points?