Re: First post for me
There's a rather lot of tweaky stuff to be discussed in this sentence, but lets start with the basics:
http://en.wikipedia.org/wiki/Convolution and following on with:
http://en.wikipedia.org/wiki/Convolution_theorem
The nature of convolution, and how it relates to the FT through the convolution theorem, get at the very heart of discrete time systems. The piecewise multiplication of weighted filter taps in the time domain gives the desired frequency domain behavior. This is one of the biggest AHA! moments one can have about the entire world of science and engineering, in my opinion. Realizing that pointwise multiplication of different sample values spaced at specific times (i.e. "taps") can than have this effect in the other domain (i.e. frequency response), is at the very core of essentially all we do in audio.
All transfer function analysis between two domains involves a "kernel" function. In FT analysis, the kernel function is a trigonometric one (i.e. sine or cosine). Sine and cosine are finite in frequency, but infinite in time. That is to say they have no start or stopping point when graphed out. Thus you trade certainty in one dimension for uncertainty in the other.
The tradeoff between the two domains is described by the famous Heisenberg uncertainty principle. To know the momentum of a particle precisely to, essentially, decompose it to a single wave, where the frequency of the sine wave is representative of the momentum. But, if you want to fix the particle precisely in time (i.e. space) you have to represent it as a the sum of many many different sine waves that combine together to provide an envelope function that represents the location of the particle. In the limit of this, the number of sine tones is infinite and the momentum data is traded for the precise envelope function. Thus physicists deal with similar behavior limitation familiar to those who make acoustic measurements.
The wavelet, by contrast, is a tradeoff kernel function. You trade resolution in one domain to learn more about the other. Heisenberg dictates you cannot know both at the same time, but you can get an idea of both simultaneously. This is the essence a wavelet-type kernel rather than trig kernel function in classic Fourier analysis.
Consider the FT complete trade of one domain for another, all time information for all frequency, while the wavelet tells you something about both domains up to the Heisenberg limit.
