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I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In particular consider the following extract:

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It talks about how to change a vector from one basis to another but it never rigorously defines what a wavelet is (by the way, I did understand that extract I included, just not the concept of "wavelet"). From my understanding, some special basis are called wavelets (for some special reason). But which basis are we allowed to call wavelets? I would assume that they have something to do with linear algebra and oscillation/sinusoidal functions but I don't really see what the relation between the two is.

To look for an alternative explanation I went to wikipedia and the initial paragraph starts as follows:

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have specific properties that make them useful for signal processing. Wavelets can be combined, using a "reverse, shift, multiply and integrate" technique called convolution, with portions of a known signal to extract information from the unknown signal.

With that description it makes me feel that wavelets are actually functions. However, I've had difficulty understanding this precisely, specially when trying to relate it to linear algebra. I guess I am having a hard time connecting the three, wavelets, linear algebra and their relations to sinusoidal functions (if there is any relation to them).

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  • $\begingroup$ @mvw I apologize if this is a simple question, but I am not familiar with the concept of localization (or maybe I need more context to infer/guess what it means) but what does that mean and what does it have to do with wavelets? Maybe you have a link I can read about localization? $\endgroup$
    – Charlie Parker
    Jun 4, 2015 at 1:57
  • $\begingroup$ You can extend the concept of a vector $x$ as span of some base vectors $x = \sum_i x_i b_i$ to functions. Thus a function being a span of (infinite many) base functions $f(x) = \sum_i c_i b_i(x)$. Example 1: Taylor and the base functions are the polynomials $x^i$: $f(x) = \sum_k a_k x^k$. $\endgroup$
    – mvw
    Jun 4, 2015 at 1:59
  • $\begingroup$ Example 2: Is the Fourier decomposition where the base functions are sine and cosine waves $f(x) = \sum_k a_k \sin(k x) + b_k \cos(k x)$. $\endgroup$
    – mvw
    Jun 4, 2015 at 2:02
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    $\begingroup$ If you plot some of those sines and cosines you see that they are defined over the whole axis. This is physical ok. But often $f$ is a signal which is only non-zero on a finite piece of the $x$-axis, e.g. from $x = -7$ to $x = 25$ and zero before $-7$ and zero after $25$. The Fourier decomposition needs a lot of coefficients to model such localized (only non-zero on a finite part or more mathematical: have finite support) functions. So wavelets are base functions that are localized themselves and thus are better suited for modelling localized $f$. That was the incentive I think. $\endgroup$
    – mvw
    Jun 4, 2015 at 2:06
  • $\begingroup$ Daubechies or Mallat would be better reference than Strang, in this case. $\endgroup$
    – J. M. ain't a mathematician
    Jun 22, 2016 at 5:29

3 Answers 3

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The idea being discussed doesn't need a rigorous definition of wavelet, or even to use wavelets at all; the notion of a wavelet is, I think, mainly a clever way of generating a convenient basis in a systematic fashion starting from a basic shape. (the particular basis you listed could probably stand to be scaled so as to be orthonormal basis rather than just an orthogonal basis)

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  • $\begingroup$ the discrete wavelet transforms are change of basis of the form $x(n) = \sum_{k\ge 0} y_k \ast \varphi_k(n)$ where $\ast$ is the (discrete) convolution, $x(n)$ is the input signal, $\psi_k(n) = \varphi_0(n 2^k)$ (the mother wavelet stretched) and $y_k$ are the wavelet transformed coefficients. As is, it is an octave filter-bank transform, and the DWT is when we require $y_k(n) = 0$ to be non-zero only if $2^k | n$ (i.e. the DWT is a decimated version of an octave filter-bank) . $\varphi_0(n)$ is called the mother wavelet $\endgroup$
    – reuns
    Jun 22, 2016 at 6:04
  • $\begingroup$ finally, we can write it as $x(n) = \sum_k \sum_m y_{k,m} \varphi_0(2^k(n-m))$ i.e. the input signal $x$ is represented as a weigthed sum of the stretched and shifted mother wavelet. $\endgroup$
    – reuns
    Jun 22, 2016 at 6:05
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Properties of wavelet: Wavelets have localized property w.r.t both frequency and space variable. It oscillates depending its vanishing moment.

Application to signal processing: Fourier transform can not give information of signal with irregular structure. Actually Fourier transform gives the global information while wavelet transform gives the local information due to its good localization character in space and frequency both(when we choose the compactly supported wavelet).

The local frequency of such oscillating signal are measured from the wavelet transform(continuous). But here the discrete wavelet transform is used. You can use https://en.wikipedia.org/wiki/Discrete_wavelet_transform. Thank you for the question.

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  • $\begingroup$ What does the term "localized" mean in this context? $\endgroup$
    – Charlie Parker
    Jun 18, 2016 at 15:56
  • $\begingroup$ In this context it usually means that the support of the wavelet is compact. $\endgroup$
    – Jürgen Sukumaran
    Jun 20, 2016 at 23:07
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The DFT has global self-similarity properties. That is what allows for the FFT factorizations. However each factor in the FFT spreads to affect the whole signal space.

To derive a DWT is to find or build a family of functions which have local self-similarity properties to allow for a similar factorization which spreads more locally over the signal space.

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