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# Introduction

Fourier series is a useful orthonormal representation on $L 2 ⁢ 0 T L 2 0 T$ 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 $L 2 ⁢ 0 T L 2 0 T$ and have many nice properties.

# Basis Comparisons

Fourier series - $c n c n$ 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 $ei⁢ ω 0 ⁢t ω 0 t$

In Haar wavelet basis, the basis functions are scaled and translated versions of a "mother wavelet" $ψ⁢t ψ t$.

Basis functions $ψ j , k ⁢t ψ j , k t$ are indexed by a scale j and a shift k.

Let $∀0≤t Then $φ⁢t 2j2⁢ψ⁢2j⁢t−k φ⁢t2j2⁢ψ⁢2j⁢t−k j∈ℤ∧(k= 0 , 1 , 2 , … , 2 j - 1 ) φ t 2 j 2 ψ 2 j t k j ℤ k 0 , 1 , 2 , … , 2 j - 1 φ t 2 j 2 ψ 2 j t k$

1

Let $ψ j , k ⁢t=2j2⁢ψ⁢2j⁢t−k ψ j , k t 2 j 2 ψ 2 j t k$

Larger $jj$ → "skinnier" basis function, $j=012… j 0 1 2 …$, $2j 2 j$ shifts at each scale: $k= 0 , 1 , … , 2 j - 1 k 0 , 1 , … , 2 j - 1$

Check: each $ψ j , k ⁢t ψ j , k t$ has unit energy

$(∫ ψ j , k 2tdt=1)⇒( ∥ ψ j , k ( t ) ∥ 2 =1) t ψ j , k t 2 1 ∥ ψ j , k ( t ) ∥ 2 1$
2

Any two basis functions are orthogonal.

Also, $ψ j , k φ ψ j , k φ$ span $L 2 ⁢ 0 T L 2 0 T$

# Haar Wavelet Transform

Using what we know about Hilbert spaces: For any $f⁢t∈ L 2 ⁢ 0 T f t L 2 0 T$, we can write

Synthesis
$f⁢t=∑j∑k w j , k ⁢ ψ j , k ⁢t+ c 0 ⁢φ⁢t f t j j k k w j , k ψ j , k t c 0 φ t$
3
Analysis
$w j , k =∫0Tf⁢t⁢ ψ j , k ⁢tdt w j , k t 0 T f t ψ j , k t$
4
$c 0 =∫0Tf⁢t⁢φ⁢tdt c 0 t 0 T f t φ t$
5
Note:
the $w j , k w j , k$ are real
The Haar transform is super useful especially in image compression