Discrete Haar Wavelet Transform (DHWT)

• categories: Signal processing, Definition

The Discrete Haar Wavelet Transform is a simple wavelet transform used for signal analysis and data compression. It is defined by recursively computing averages and differences of a signal.

Definition

For a signal $x=x_1,x_2,\dots ,x_N$ with $N=2^k$ (a power of 2):

1. Pairwise averages:
   $a_i=\frac{x_{2i-1}+x_{2i}}{2},i=1,2,\dots ,\frac{N}{2}$
2. Pairwise differences:
   $d_i=\frac{x_{2i-1}-x_{2i}}{2},i=1,2,\dots ,\frac{N}{2}$

These steps are applied recursively to the averages ($a$) to form a hierarchical decomposition.

Key Properties

• Orthogonality: Haar wavelet basis functions are orthogonal.
• Compact Support: Haar wavelet has the smallest possible support.
• Fast Computation: Computable in $O(N)$ via the Fast Wavelet Transform (FWT).
• Piecewise Constant Approximation: Captures abrupt changes in signals efficiently.

Matrix Representation

The DHWT can be expressed as a matrix multiplication:
$y=Wx$
where $W$ is the Haar Wavelet Matrix, a product of scaling and wavelet filters.