>> For example, the Fourier transform allows you to construct any 
>> function from sine and cosine functions.
>>
> 
> This isn't quite true over the reals, even assuming you're only looking 
> for functions with a given period.  For example the function which is 
> zero everywhere except being 1 at a single point will generate the same 
> Fourier representation as the constant zero function since it will have 
> the same integrals.

O RLY?

My DSP textbook says the Fourier transform of the delta function yields 
an amplitude of 1 for all frequencies. (Whereas the Fourier transform of 
a zero signal would be a zero signal.)

> I think (no proof) that you can reconstruct any function up to the 
> addition of a function which is nonzero over an area of zero `volume' 
> though (assuming you don't count things like a delta functions).  Not 
> that it matters for what you're doing of course, but you seem like the 
> sort of chap who might find it interesting.
> 
>>
>> So, like, how do you tell if two functions are orthogonal? And how do 
>> you tell when a set of them is complete?
> 
> What are you using these function for?  There may be better or worse 
> ways to do things depending on what you want.

JPEG compression works by decomposing an image into a set of (2D) cosine 
waves, and then quantinising the data. In my opinion, cosines are not 
actually a particularly good choice for this. (Gibb's phenominon is very 
ugly to look at.) So I'd like to try it with other functions - but first 
I need to work out how...

(A similar thing could be said about most [lossy] audio codecs known to 
man. Of course, here Gibb's isn't a problem at all. Here the problem 
becomes time invariance...)

Post a reply to this message