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on the eigenfunctions and eigenvalues of the fourier transform [Jun. 22nd, 2011|02:37 pm]
Tobin Fricke's Lab Notebook
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Recently my interest was piqued in the eigenfunctions of the Fourier transform. The Fourier transform is a linear operator on a space of functions, so it has eigenvectors and eigenvalues: functions who are their own Fourier transform. I think many people know that "the Fourier transform of a Gaussian is a Gaussian", but the other eigenfunctions are not so well known.

It is relatively easy to show that taking the Fourier transform four times in succession is an identity operation. If you remember that taking the complex conjugate in the frequency domain is equivalent to time reversal in the time domain, this is easy to see. Taking the Fourier transform twice gives you time reversal, so taking it four times gives you the identity.

This means that the eigenvalues of the Fourier transform have to obey x^4 = 1, i.e. the eigenvalues of the Fourier transform are the 4th roots of unity: {1, i, -1, -i} (ref) This, of course, agrees with our knowledge that the Fourier transform is unitary. This is Parseval's theorem: the RMS of a function and its transform are equal.

With each eigenvalue we can associate a set of eigenfunctions who have that eigenvalue. Call these sets H0, H1, H2, H3. The Fourier transform of a function in Hn is just that function times an eigenvalue of i^n. We can define the Fourier transform in terms of these sets:

F{f} = f |H0⟩⟨H0| + i f |H1⟩⟨H1| i f |H2⟩⟨H2| - i f|H3⟩⟨H3|

where |Hn⟩⟨Hn| is the projection operator onto the subspace Hn.

It is curious to me that there are only four eigenvalues, and thus these spaces must be very big. There must be very many ways to parametrize each of the families Hn--very many different bases.

It turns out that one such basis is very familiar to physicists: Hermite-Gauss functions (i.e. a Gaussian multiplied by a Hermite polynomial) are eigenfunctions of the Fourier transform. These show up very often in physics; two particular examples come to mind:

1. the energy eigenstate wavefunctions of the quantum simple harmonic oscillator
2. the Hermite-Gauss modes of laser resonators

In his book "Fourier Analysis", Javier Duoandikoetxea (what a name!) (page 22, available on the amazon preview, search for "Eigenfunctions") tells us that the Hermite-Gauss functions provide a complete basis for L^2, i.e. the space of square-integrable functions:

h_n(x) = ((-1)^n / n!) exp( π x^2) (d/dx)^n exp(-π x^2)

F{h_n} = (-i)^n h_n

Let e_n be a normalized version of h_n:

e_n = h_n / || h_n || = Sqrt[(4π)^(-n) Sqrt[2] n!] h_n

Then {e_n} is an orthonormal basis for L^2, and the Fourier transform may be written

F{f} = sum of (-i)^n ⟨f|en⟩ over n in ℤ

[Duoandikoetxea says that this is the approach taken by Norbert Wiener in "The Fourier Integral and Certain of its Applications".]

Why do these Hermite-Gauss functions show up in the physical situations mentioned earlier? If we transform to unitless variables, the Hamiltonian of the harmonic oscillator is simply:

H = x^2 + p^2

This equation is symmetric under interchange of x and p. Finding the energy eigenstates means solving for the eigenstates ψ such that E ψ = H ψ where E is a scalar. Symmetry under interchanging x and p means the coordinate-space and momentum-space representations of the wavefunction must be the same. How do we transform the wavefunction from position space to momentum space? We take the fourier transform. Thus the energy eigenstates of the harmonic oscillator must also be eigenfunctions of the Fourier transform.

But why these Hermite-Gauss functions in particular? We can take any linear combination of functions all in the same subspace Hn and get an eigenfunction of the Fourier transform, but this won't in general be an energy eigenstate of the SHO. After all, the Fourier transform has only four eigenvalues, but the SHO Hamiltonian has an entire ladder of eigenvalues.

What other properties are needed to uniquely define the Hermite-Gauss functions? (Related question on math.SE: How do I compute the eigenfunctions of the Fourier Transform?)
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latexdiff [Jun. 14th, 2011|10:55 pm]
Tobin Fricke's Lab Notebook

Here's a great little tool: latexdiff. It takes two LaTeX files and produces a third LaTeX file showing deletions in struck-through red and additions in underlined blue:

Usage couldn't be easier:
latexdiff oldversion.tex newversion.tex > diff.tex
pdflatex diff.tex
If you run Ubuntu, you can get it with a simple "sudo apt-get install latexdiff".
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phase modulation phasor animation [Jun. 8th, 2011|07:14 pm]
Tobin Fricke's Lab Notebook
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Hey, here's an animation showing how phase modulation works in the phasor picture. Specifically, this shows phase modulation with a sinusoidal modulation waveform and a modulation depth of π/4 radians. The blue line segments represent the phasors at the carrier and the harmonics of the modulation frequency.

The stationary horizontal blue line segment represents the carrier. Ordinarily, it would be spinning around the phasor diagram very quickly--at whatever the carrier frequency is. The phasor picture is more useful, though, if we choose a "co-moving reference frame" in which the carrier is stationary. The upper sidebands of the carrier then appear to move clockwise while the lower sidebands rotate anti-clockwise.

The next two blue line segments (after the carrier) are the first-order sidebands. Notice how they are phased such that their sum is always perpendicular to the carrier, and, thus, to first order, they don't change the amplitude of the resultant phasor. For big modulation depths, such as here, they do change the amplitude a little, so the 2nd and higher order sidebands are needed to correct this.

some things that would be cool:
* re-implement in javascript (processing.js?)
* add an interactive modulation-depth slider
* allow user to try out other modulation waveforms

matlab source: https://gist.github.com/1015769
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swimming in air [May. 3rd, 2011|10:10 am]
Tobin Fricke's Lab Notebook
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In the current PRL: "Paddling Mode of Forward Flight in Insects."

See also the old favorite "Life at low Reynolds number".
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Livejournal strips Data URIs [Apr. 26th, 2011|04:05 pm]
Tobin Fricke's Lab Notebook
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I used to use the data URI scheme to embed small PNG renderings of LaTeX formulas in my LJ entries (such as here). I used the little webeqn script written by four to do this automatically.

Alas, it seems LJ now strips these URLs away. So it doesn't work anymore.

Another item to file under "death of livejournal," I guess.
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acronym capitalization [Apr. 26th, 2011|03:44 pm]
Tobin Fricke's Lab Notebook
I don't understand the journal conventions for capitalizing acronyms or putting them in lower case. Personally, I find lower-case acronyms quite awkward (especially "if" - Intermediate Frequency) but for some reason the editors sometimes insist.

Here are some examples from a Phys Rev D paper.

Upper-case acronyms:

LO (local oscillator)
GW (gravitational Wave)
PM (phase modulation)
BS (beam splitter)
SR (signal recycling)
QND (quantum non-demolition)
RSE (resonant sideband extraction)</div>

Lower-case acronyms:

rf (radio frequency)
if (intermediate frequency)
dc (direct current)

They sometimes insist on lower-case even when one of these acronyms is the first word in a title, i.e. "dc readout experiment at the Caltech 40m prototype interferometer" (in CQG).
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technical reports in citeulike [Apr. 20th, 2011|04:55 pm]
Tobin Fricke's Lab Notebook
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I use citeulike (previously mentioned here) to keep track of papers and to automatically produce a BibTex database for use when writing papers with LaTeX. It's great.

LIGO produces very many internal documents which I frequently reference; each of these documents is assigned a unique serial number, like "LIGO-T970084-00" (an excellent, highly Google-friendly practice). I've been entering these into citeulike as technical reports, which is appropriate, but one annoyance is that I did not know how to enter this document serial number into citeulike.

Solution: Citeulike uses the "Issue" field of its database to produce the "Number" field in a BibTex file. Simply enter the DCC (serial) number in this field! The BibTex @techrep class will use the number field if it is provided.
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displaced Hermite-Gauss mode [Mar. 25th, 2011|01:10 am]
Tobin Fricke's Lab Notebook
[Current Mood |mesmerized]

c.f. http://math.stackexchange.com/q/28719/2191
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6.453 Quantum Optical Communication [Mar. 21st, 2011|02:10 pm]
Tobin Fricke's Lab Notebook
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The lecture notes from this course at MIT look super useful:

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phase modulation II [Mar. 16th, 2011|04:04 pm]
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Suppose we have a signal consisting of a carrier (at frequency ω and with unit amplitude) and two sidebands, of amplitudes a (lower) and b (upper), separated from the carrier by a frequency Ω:

E(t) = (1 + a exp(-i Ω t) + b exp(i Ω t)) exp(i ω t)

To find the power in this signal, we take the modulus squared:

P = E*E
where * is the complex conjugate.

P = (1 + |a|^2 + |b|^2)
+ (a* + b) exp(-i Ω t) + (a + b*) exp(i Ω t)
+ ab* exp(-2 i Ω t) + a*b exp(2 i Ω t)

The condition for the 1Ω variation in the power to vanish is a=-b*, i.e. the real parts of the amplitudes of the sidebands must be opposite, and the imaginary parts must be equal. So we can extract the amplitude and phase modulation indicies:

m_AM = (a + b*)
m_PM = (a - b*)

What is the condition for the 2Ω signal to vanish? With just two sidebands, it will always be present (though at second order in the sideband amplitude). In true phase modulation, the 2Ω signal is cancelled by the interaction of (the infinite number of) higher-order sidebands. As best I can tell, there is no simple arrangement of this cancellation other than via a magical property of the Bessel functions.
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