Fourier analysis (of a sound)

[Oliver]

Hi Moto,

If you really want to understand Fourier Transforms, you can't avoid the maths/physics of waves, signals, and systems.  This page has a fair introduction.

If you would like to see an intuitively satisfying example, with a periodic signal, have a look at this demonstration of Gibbs Phenomenon.  One can see how the summation of a series of sinusoidal waves, each with the appropriate magnitudes and phase shifts, can approximate a square wave.  You can also see how a band width limit (finite number of sine waves) affects the approximation, producing ringing in the reconstructed signal.

Examining a signal in a series of sliding windows doesn't really get around the nature of waves and signals, but is a functional way of examing signal characteristics.  The specific window functions also affect the nature of the spectral information being examined.  With something like digital audio, it is already characterised to a fair extent, and a sliding window is a very useful approach.

You may also like to look at the phenomenon of wave packets.  Wave packets can describe a non repetitive pulse in time/frequency (actually time and frequency are just one pair of inverse variables, it's true for all pairs of inverse variables, such as wavelength lambda, and spatial frequency).  Have a look at this link for a description (sorry I wasn't able to find a good animation for this one).  Sinusoidal signals covering a very wide bandwidth, can sum to form a localised pulse in time/space.

If you really want to explore this material, MATLAB is a really good tool, but is quite mathematical.  MATLAB has some GUI demonstration tools, which are quite good, but I can't recall if there is anything appropriate to this.

A really good book for this stuff is Signals & Systems, by Haykin and Van Veen..  I think you're at university now, right?  I think most electronic engineering libraries will have a copy of this, if you want to have a flip through it.  Some really good diagrams of signals in there.

You asked about Fourier as related to sound: interestingly enough, all waves have a set of properties in common, so the principles are trasnferable.  The same principles are relevent to the quantum mechanical wave model of matter, and has baring on concepts like the uncertainty principle.

Btw, you seem be thinking on a number of maths/physics/programming/engineering type questions; are you sure you're not being lured into an engineering career? :crazy:

-Oli

[Oliver]

What's wrong with an engineering career?

Well, it's certainly not without its good points, but there are plenty of headaches along the way, for the unsuspecting.

Some of my engineering classes had an over 40% failure rate, and very low levels of satisfaction.  A very common comment by people in my electronic engineering classes was "it's nothing like what you expect it to be" or "damn it, there's just too much maths".  For me, it wasn't too far form what I expected, but there were still plenty of headaches.  Anyway, I love physics, and tinkering, so it's still rewarding for me.

Though I find it interesting, I struggle to see how it is of use to a Speech & Language Therapist.

Yeah, I'm not too sure if you would need a deep understanding of Fourier analysis for that.  Maybe a couple of things to understand would be:

 *   Frequency content of a signal can be used to characterise a system which the signal comes from or a system which it passes through.  So, I guess there could be some application to dealing with speech impediments (is that what your course is about?).  I'm not sure it would be any better than using your own ear, though.

 *   All waves have a fundamentally sinusoidal nature (this is paraphrasing somewhat  ;-)  )

 *   Complex variables (complex number type variables, typically with i,j,\theta,f) being used in Fourier analysis, provide a mathematically convenient/powerful form to deal with signals and systems; if you don't need to work with them in detail, then don't worry too much about understanding things like the complex plain, or negative frequency components, which seem to deter a lot of beginners.

I'm guessing an understanding of accoustic propagation may be useful though, and I think it could be understood quite well without delving too much into things like Fourier analysis.

There's also an interesting historical side note to all this: Joseph Fourier was a member of Napolean's army, and if I remember correctly, developed notions used in Fourier analysis well before they could be practically applied for what they are used today.

-Oli