15.10 Haar wavelet basis

Introduction

Fourier series is a useful orthonormal representation on $L^2(\left[0 , T\right])$ especiallly for inputs into LTI systems. However, it is ill suited for some applications, i.e. image processing (recall Gibb's phenomena ).

Wavelets , discovered in the last 15 years, are another kind of basis for $L^2(\left[0 , T\right])$ and have many nice properties.

Basis comparisons

Fourier series - $c_n$ give frequency information. Basis functions last the entire interval.

Wavelets - basis functions give frequency info but are local in time.

In Fourier basis, the basis functions are harmonic multiples of $e^{i{\omega }_{0}t}$

In Haar wavelet basis , the basis functions are scaled and translated versions of a "mother wavelet" $\psi (t)$ .

Basis functions $\{{\psi }_{j,k}(t)\}$ are indexed by a scale j and a shift k.

Let $\forall , 0\le t< T\colon \phi (t)=1$ Then $\{\phi (t)2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)\colon j\in \mathbb{Z}\land (k=0,1,2,\dots ,2^j-1)\}$

Let ${\psi }_{j,k}(t)=2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)$

Larger $j$ → "skinnier" basis function, $j=\{0, 1, 2, \dots \}$ , $2^j$ shifts at each scale: $k=0,1,\dots ,2^j-1$

Check: each ${\psi }_{j,k}(t)$ has unit energy

Any two basis functions are orthogonal.

Also, $\{{\psi }_{j,k}, \phi \}$ span $L^2(\left[0 , T\right])$

Haar wavelet transform

Using what we know about Hilbert spaces : For any $f(t)\in L^2(\left[0 , T\right])$ , we can write

Haar wavelet demonstration