Saturday, February 26, 2011

Quantized Redshifts. VII. Common Power Spectral Density "Bloopers"

A surprising number of researchers seem to regard the PSD (power spectral density) function as some kind of magical tool that 'finds frequencies'.
The PSD is a very powerful tool, but one with which you can 'shoot your own foot off' if you are not careful.

I've laid out the basic mathematics of the PSD, and this will enable us to explore a few basic claims that have been posed in the creationist and other literature.

Can the Power Spectral Density identify a "Center" of a distribution?

First we should make clear that
  1. In Big Bang cosmology (BBC), in the underlying Friedmann-Robertson-Walker metric,  every point is a center.  No matter where the observer is located, they will interpret the universe around them as being centered on them.  See Misconceptions about the Big Bang (Scientific American), "Is it possible to point to a direction in the sky and say "that way is the center of the universe, where the Big Bang started?"
  2. In addition, because of the finite speed of light (which many creationists try to ignore), when we look into the distant cosmos, we are looking into the distant past.  Since BBC initiates a starting point from which structures grow and evolve, the further out we observe, the universe will begin to look different as we see younger and younger cosmological structures.
The popular retort that the universe will look different from different locations works only in a case where the universe is static (not evolving) and the speed of light is very large or infinite (which violates loads of atomic spectra we observe in distant galaxies as well as using redshift as a measure of anything).  Myself and others have explored the 'decaying speed of light' claims of Barry Setterfield elsewhere (A Changing Speed of Light?).

I have no doubt that many Big Bang opponents will keep repeating this lie.

It has been suggested that the very existence of periodicities in extragalactic Power Spectra is evidence of a geocentric distribution.  Russ Humphreys explored this, suggesting that broader peaks in the PSD implied that the Earth was actually slightly offset from the center of the Universe in Humphreys' strangely titled "Our galaxy is the centre of the universe, ‘quantized’ redshifts show".

The major problem with such a claim is that it can be rigorously demonstrated that the PSD is independent of the location of the 'center' of a dataset, and therefore cannot identify the location of the 'center'.   I hinted at this by my question in my previous post (Quantized Redshifts. V. Exploring the PSD with Simple Datasets).  How do you prove this?

Consider the 1-dimensional case of a some function, f(x) with a 'center' defined as x=0.  Now displace that function on the x-axis by the distance, a, so the 'center' is now at x=a.  This defines a new function, f':
f'(x') = f(x-a)
because x'=x-a.
To see how this affects the PSD, first, take the Fourier transform of both of these functions:
and

So we see that the Fourier transform of the original function gets multiplied by a complex (real+imaginary) factor
when displaced in position.  But the Power Spectral Density is the Fourier transform multiplied by its own complex conjugate
so,

Therefore the PSD of the original function is EXACTLY the same as the PSD of the function with the displaced center.  It is impossible for the PSD of a distribution to be used to identify its center.  Potentially, one could use the Fourier transform itself to identify a value for the complex offset value found above, but the only way to do this would be to compare the Fourier transforms with a specific model distribution.

This proof is easily extended to three- and four-dimensional transforms.

The bottom line is the PSD cannot identify the center of a distribution.

Adding & Subtracting PSDs

Consider a signal, s(t), that is the sum of two signals, s1(t) and s2(t)
Because the Fourier Transform is linear, it is easy to show
But what about the PSD?  From the definition of the PSD:
We see that the PSD of two summed signals are not necessarily equal to the sums of the individual PSDs.  There is an additional component which is a type of mixed product of the Fourier transforms of the two input signals. There are useful cases where you CAN demonstrate the mixed term is zero, for example, if you are collecting signal from a source and there is an uncorrelated (random) background signal (see Effects of background counts in RMS normalization).

But in general, this mixed product is not automatically zero.

A similar relationship holds if we wish to subtract one signal from another.

Some researchers attempt to split the PSD into two additive components to justify subtraction of the components they perceive to be part of the distribution of the data.  This is an attempt to leave only the oscillating components.  The problem is the distribution they wish to subtract often has significant variation as well. It is the responsibility of the researcher to demonstrate that the input components have no correlations, or are correlated in a simple way, but I have not seen this done.

The errors described above are made in a number of papers claiming quantized redshifts and represent a basic misunderstanding of how the Fourier transform and PSD works.  These are basic errors that anyone who has who taken a course in mathematical methods for physicists (wikipedia) should not make.  One might expect a first-year graduate student or a neophyte to make these types of errors, but not someone with long experience with the PSD. 

But what should really be an embarrassment to some journals is that their peer-reviewers did not catch these errors either!

Next Weekend: An Uncommon Power Spectral Density "Blooper"

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