Introduction Fourier series is a useful orthonormal representation on 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 and have many nice properties.

Basis Comparisons Fourier series - c n give frequency information. Basis functions last the entire interval.

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

Wavelet basis functions In Fourier basis, the basis functions are harmonic multiples of ω 0 t

basis 1 T ω 0 n t In Haar wavelet basis, the basis functions are scaled and translated versions of a "mother wavelet" ψ t .

Basis functions ψ j , k t are indexed by a scale j and a shift k. Let 0 t T φ t 1 Then 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

ψ t 1 0 t T 2 -1 0 T 2 T

Let ψ j , k t 2 j 2 ψ 2 j t k

Larger j → "skinnier" basis function, j 0 1 2 … , 2 j shifts at each scale: k 0 , 1 , … , 2 j - 1 Check: each ψ j , k t has unit energy

t ψ j , k t 2 1 ∥ ψ j , k ( t ) ∥ 2 1 Any two basis functions are orthogonal.

Same scale Different scale Integral of product = 0 Also, ψ j , k φ span L 2 0 T

Haar Wavelet Transform Using what we know about Hilbert spaces: For any f t L 2 0 T , we can write Synthesis f t j j k k w j , k ψ j , k t c 0 φ t Analysis w j , k t 0 T f t ψ j , k t c 0 t 0 T f t φ t the w j , k are real The Haar transform is super useful especially in image compression This demonstration lets you create a signal by combining Haar basis functions, illustrating the synthesis equation of the Haar Wavelet Transform. See here for instructions on how to use the demo.