Wavelet Transform

The Wavelet Transform is a method for analyzing a signal $x (t)$ in terms of both time and scale (or frequency), allowing the identification of localized features. The signal is expressed in a basis different from the Dirac delta basis, something similar to what happens to Fourier transform.

Definition

X (a, b) = \int_{- \infty}^{\infty} x (t) ψ_{a, b}^{*} (t) d t

$X (a, b)$ : the wavelet coefficients at scale $a$ and position $b$
$x (t)$ : the original signal
$ψ_{a, b}^{*} (t)$ : complex conjugate of the wavelet function at scale $a$ , translation $b$ .

Although we would hope something like

x (t) = \int X (a, b) ψ_{a, b} (t) d a d b

but this is wrong. What we have is, under suitable conditions on the wavelet $ψ$ , that one can reconstruct $x (t)$ as:

x (t) = \frac{1}{C_{ψ}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} X (a, b) ψ_{a, b} (t) \frac{d a d b}{a^{2}}

$C_{ψ}$ is the admissibility constant: $C_{ψ} = \int_{0}^{\infty} \frac{| \hat{ψ} (ω) |^{2}}{ω} d ω < \infty$ which ensures invertibility of the wavelet transform.
The measure $\frac{d a d b}{a^{2}}$ comes from the invariance under affine transformations.

The wavelet family is generated from a single function $ψ (t)$ , called the mother wavelet, by scaling and translating:

ψ_{a, b} (t) = \frac{1}{\sqrt{| a |}} ψ (\frac{t - b}{a})

$a$ : scale parameter (controls dilation/compression)
$b$ : translation parameter (controls time shift)

Meaning of $X (a, b)$

In the Fourier transform, the coefficient $\hat{f} (k)$ tells us how much of the frequency $k$ is present in the signal. In the wavelet transform, the coefficient $X (a, b)$ captures how much the signal $x (t)$ resembles the wavelet $ψ_{a, b} (t)$ , i.e., the feature at scale $a$ and time $b$ .
Think of $X (a, b)$ as a microscope focused at time $b$ , tuned to scale $a$ . The value of $X (a, b)$ :

is large (in magnitude) if there's a feature in $x (t)$ around time $b$ that matches the wavelet shape at scale $a$ ,
is small if no such structure is present there.

Common Wavelets $ψ (t)$

1. Morlet Wavelet (complex)

ψ (t) = π^{- 1 / 4} e^{i ω_{0} t} e^{- t^{2} / 2}

2. Mexican Hat (Ricker) Wavelet

ψ (t) = (1 - t^{2}) e^{- t^{2} / 2}

3. Haar Wavelet

ψ (t) = {\begin{cases} 1 & if 0 \leq t < \frac{1}{2} \\ - 1 & if \frac{1}{2} \leq t < 1 \\ 0 & otherwise \end{cases}