quick question about the fourier transformation

[CPL.Luke]

Recently I've been reading up on QM and came upon the lovely little thing known as the fourier transformation. I was able to tell that it involved calc, and my physics teacher explained what it did, but does anyone know exactly what level of mathmatics is involved in it? or the specific subjects needed to understand it?

[timo]

For QM it is probably sufficient to know that the FT is a base transform in a vector space of functions. Therefore, you should know Linear Algebra if you want to understand what you are doing.

[DQW]

And to perform an actual FT, you must know how to integrate.

[CPL.Luke]

got the integration bit under my belt already, working on some linear algebra now, thanks for the help

[Klaynos]

From my personal experiance practice LOTS of integration esspecially of trig functions, and integrating by parts etc...

[CPL.Luke]

ok I just spent last night reading up on a half ton of linear algebra (matrices and such) is anyone willing to give a brief tutorial on how to do the fourier integral?

[MetaFrizzics]

Sure I'll help you. The Fourier Transforms and integrals are actually really easy.

 

Start with the Fourier Theorem:

 

(It's brilliant, and really useful in electronics, especially audio etc.)

 

Simple version:

Any repeating waveform (of any shape) can be broken down into sine waves, which when added back up result in the original funky waveform again. What is so awesome and elegant is that the frequencies of all the (smaller) sinewave components are all multiples of the original frequency of the funky wave.

 

So, a sawtooth wave from an electric organ is made up of the original note, plus a note an octave higher (2xf) and a bit of the note an octave + musical 5th higher (3xf) and some amount of the note 2 octaves higher (4xf = 2x2xf) etc. All of these little sounds added to the original note create the complex wave shape you hear (and the speaker gets electronically as a voltage).

 

This means the lowest 'harmonic' in a fancy shaped wave is just the sinewave of that same frequency as the fancy wave.

 

To get all the possible shapes (these have to be actual sensible functions) of wave, you simply adjust the phase and amplitude (volume) of each harmonic to create the shape (and the distinct tone or timbre of an instrument). Thus a flute might be an almost pure sinewave, while an oboe could be a sawtooth. The difference is in the loudness and relative phase (synchronization) of the harmonics (sinewaves at various multiples of the frequency.)

 

What use is this knowledge, or ability to convert a complex wave back and forth between two or three ways of viewing it, writing it down etc? Let's see: If we make an 'additive' synthesizer, we have sinewave oscillators controlled by a keyboard (and ganged to each other to act as harmonics). This is literally a 'Fourier Analysis' synthesizer!.

 

Similarly, An AM radio signal can also be broken down into or viewed as a carrier plus side-bands of sub/super-harmonics. The audio is encoded in the overall amplitude envelope, but it is also contained in the sidebands. Its another way of looking at the same thing:

 

Radio >> Fourier

 

The varying radio carrier can be viewed as a set of sinewaves of various frequencies and amplitudes. In this case the Fourier Analysis method would view the 'period' of this wave *not* as the radio frequency, but as the lowest component of the audio signal!

[DQW]

To practice, find the fourier transforms of some of the commonly found functions : the constant, the dirac delta, the exponential (e^{-kx}), the gaussian (e^{-ax^2}), the sine ( sin(kx) ), the single-particle coulom potential (1/x), etc.

[MetaFrizzics]

Here's a link to a handy intro to the nuts and bolts of Fourier for signal processing:

Fourier Notes

Six nice pages on technique with C+ code

Fourier Theorem Handout

 

Java Applet that lets you see and hear Fourier Analysis in Action!

YOU set harmonics, see the resultant wave!

(Caution: if your soundcard gets stuck on, you may have to reboot or turn off speaker temporarily.)

 

Nice intro to Fourier Square Wave & pics

Hardcore Fourier Analysis Website ****

University Level Lecture

Fourier & ADD EEGs (yikes!)

How to use an FFT Spectrum Analyser

Nice Lecture introducing the Delta Function

34 page PDF file on Fourier Analysis - nice and meaty! ****

[MetaFrizzics]

Recommended Books:

 

Fourier Series Georgi Tolstov 1962 - Dover reprint $10

(thorough account with proofs and applications: older text)

...readily accessible to students in fields of physics and engineering...(back cover)

 

1 Trig Fourier series

2 Orthogonal systems

3 Convergence

4 Trig series with Decreasing Coefficients

5 Operators on Fourier Series

6 Summation of Trig Fourier

7 Double Fourier Series / Integrals

8 Bessel Functions & Fourier-Bessel series

9 Eigenfunction Method & Physics

 

Intro to Theory of Fourier's Series & Integrals -Carslaw 1950 -Dover $10

No one can properly understand FS and Integrals without a knowledge of what is involved in the convergence of infinite series and integrals...

1 Rational / Irrational Numbers

2 Infinite Sequences & Series

3 Functions of Single Variable Limits Continuity

4 The Definite Integral

5 Infinite Series (single variable)

6 Arbitrary parameters

7 Fourier's Series (with Deirichlets Cond.& Poisson)

8 Nature of Convergence & Fourier Constants

9 Approximations Gibb Phenomenon

10 Fourier's Integrals (w Sommerfeld's discussion)

Appendices Harmonic Analysis & Lesbesque's Theory of Definite Integral

Fourier Integral & Certain Applications - Norbert Wiener 1958 Dover $10

There are 3 more or less separate groups of ideas..pertaining to Fourier and Plancherel theorem: the notions of an absolutely convergent Fourier Series and of a Tauberian Theorem; and the concept of the spectrum. ...

1 Plancherels Theorem (Hermite Functions)

2 Tauberian Theorem

3 Special Tauberian Theorems

4 Generalized Harmonic Analysis

 

P.S., let me know if all this effort was worth it for anyone....thanks!

[MetaFrizzics]

Two more great little books on Fourier:

 

(1) A Student's Guide to Fourier Transforms J. F. James

 

"Showing a Fourier Xform to a physics student generally produces the same reaction as showing a crucifix to Count Dracula...It tends to be taught by theorist who use the methods themselves to solve otherwise intractable differential equations. The result is often a heavy load of mathematical analysis.

This need not be so. Engineers and physicists use Fourier quite differently: ...the transforms are done digitally and there is a minimum of math involved.

In spite of the forest of integration signs there is in fact very little integration done, and most of that is at a high-school level. ..." (Preface)

 

(2) An Introduction to Lebesgue Integration and Fourier Series Wilcox & Myers

It is our intention to motivate what we are doing. For example, the inadequacies of the Reimann integral are pointed out, ...The primary difference is that in Lebesgue's approach, "Lebesgue sums" are formed relative to an arbitrary partition of an interval containing the range of a bounded function, in contrast to Reimann's partitioning of the domain.... This leads naturally to the L^2 theory of Fourier Series.

 

These two are more modern, and contain some useful new methods as well.

[CPL.Luke]

sorry, I've been a way from this thread awhie and havn't had the chance to say thankyou,

 

I am off to the library to check out some of those books