# How to generalise the Fourier transform?

The Fourier transform takes a signal and splits it into a series of sine and cosine waves.

I am told that it's supposed to be possible to split a signal into some other set of functions. My question is: How do you do this?

I'm presuming that the set of functions you use would have to have certain properties for this to work. (E.g., you have to have "enough" different functions to capture all the information of the original signal.) How do you figure out whether your set of functions is suitable? And then how do you do the actual splitting?

• Begin with Fourier series rather that Fouroer transforms. Look for "orthonormal set" for the properties needed and "complete orthonormal set" to see if you have "enough" different functions to capture all the information. The "splitting" is done exactly the way it is done for Fourier series except that you integrate $x(t)\psi_n(t)$ rather than $x(t)\cos(n\omega t)$. – Dilip Sarwate Apr 27 '12 at 13:15
• One example of another (discrete in both domains) transform with non-sinusoidal basis functions is the Hadamard transform. – Jason R Apr 27 '12 at 14:26
• When you say generalize, you have to be more specific (pun not intended). The fractional Fourier Transform for example, is said to generalize the Fourier transform with respect to an otherwise hidden parameter in the conventional Fourier transform. As Dilip has pointed out, if you're referring to the basis you need to find a suitable kernel. Mathematically, this means a "complete orthonormal set." Functionally, this means a kernel that will sparsely represent your signal and provide meaningful information. – Bryan Apr 27 '12 at 15:25
• Wavelet transforms? – CyberMen May 7 '12 at 18:36

The Fourier Transform is just one of so many different transforms that alter the representation of (usually) a time-series from the time domain, to another domain (usually a frequency domain but other representations exist for other transforms such as time / frequency, time / scale and others).

You can find much more information about transforms in general from this Wikipedia list of articles that lists some popular and often used transforms. (You might want to focus on the Discrete and Integral Transforms at first)

Alternatively, you can check out this recent discussion on how the Wavelet transform, achieves a decomposition similar to that of the Fourier transform.

Finally, when you have the luxury to have acquired simultaneously many different time series from the same phenomenon you can even employ techniques such as Principal Component Analysis (PCA) and Independent Component Analysis (ICA) which go to the point of transforming a signal to a sum of elementary waveforms that are actually extracted from the signal itself (rather than being pre-set as it is done in the Fourier (and related transforms) or Wavelets).

• Transforms are a specified version of wavelets. Basically, a wavelet is a more generalized form of a Fourier transform. Essentially, anything can be decomposed into a sum of time-shifted, scaled versions of an oscillating function. – CyberMen May 1 '12 at 19:22

In addition to the answers given here, I should add that there are situations in which uniqueness of decomposition or even completeness are not the most sought-after properties. Instead, a "compact" description with as little coefficients as possible is sought, and to this effect, it is useful to have a decomposition basis not bound to one single "family" of elements (say sine waves). In such situations, you can really put whatever you want in the basis you'll use for your decomposition, and the decomposition itself is performed using the Matching Pursuit algorithm. This proves to be well suitable for audio signals, which can exhibit both very steady, sustained segments (the long, decaying, almost pure sound of a vibraphone note), but also transient part (the very wide-band burst of energy at the beginning of the note).

• +1 for the interesting link to Matching Pursuit Algorithm. – MathematicalOrchid May 1 '12 at 8:40

The Fourier transform is one of the many ways to express a function as a weighted sum of some other functions, often called basis functions. This can be done for two reasons

1. The basis function can have physical meaning and/or give some insight into the nature of the original function. The basis function can be viewed as the "constituent components" of the function.
2. It can make the math easier. Instead of doing some operation on the function, you can split it into basis functions, operate on the basis functions and then out it back together again.

The Fourier Transform is popular as it does both: the basis functions are sine wave with the parameter "frequency" have well defined physical meaning and they are also invariant to linear time invariant systems. I.e. sine wave in gives sine wave out. Both properties are quite useful. The Fourier Transform is my no means the only way of doing this. Any set of linear independent functions can be used. Popular are orthonormal bases as it makes the actual transforms real easy.

• That much I understand. What I don't understand is exactly what constitutes a "set of linear independent functions". – MathematicalOrchid May 1 '12 at 10:20
• Some background in linear algebra would help you understand what he means. Here is an article at Wikipedia that might help a bit. – Jason R May 1 '12 at 12:45
• Think about your basis functions as "building blocks" that you combine to create the more complicated functions. Linear independence means that you cannot build any basis function as a combination of other basis functions. They have to be uniquely different. – Hilmar May 1 '12 at 23:44