Fourier series is a
especiallly for inputs into LTI systems. However, it is ill
suited for some applications, i.e. image processing (recall
Wavelets, discovered in the last 15 years, are another
kind of basis for
and have many nice properties.
Fourier series -
give frequency information. Basis functions last the entire
Wavelets - basis functions give frequency info but are
local in time.
In Fourier basis, the basis functions are harmonic
In Haar wavelet basis, the
basis functions are scaled and
translated versions of a "mother wavelet"
are indexed by a scale j and a
Larger → "skinnier" basis
shifts at each scale:
has unit energy
Any two basis functions are orthogonal.
Haar Wavelet Transform
Using what we know about Hilbert spaces: For any
we can write
The Haar transform is super useful especially in
Haar Wavelet Demonstration