1805
Joseph
Fourier (French
mathematician and
physicist) asserted Fourier analysis
in 1805. It’s one of the most important tool
in mathematical analysis.

Fig.1 Joseph
Fourier (17681830)

For any 2π periodical
function f(x) can be expressed as the
following equations.
1909
Alfréd Haar(a
Hungarian mathematician) found another basis
function to analysis a signal as follows.
The function basis h(x) is called Haar function, and
it’s the first mention of “wavelets” at that
time.

Fig.2
Haar wavelet 
1930
Paul Levy found
Haar basis function is superior to the
Fourier basis functions for studying small
complicated details in the Brownian motion.
1946
Dennis Gabor was Hungarianborn electrical
engineer who won the Nobel Prize for Physics
in 1971 for his invention of holography. He
is considerd as the father
of holography. His other work included
research on highspeed oscilloscopes,
communication theory, physical optics, and
television.
In order to study
the spectral behavior of an analog signal
from its Fourier
transform, full
knowledge of the signal in the timedomain
must be acquired. The deficiency of Fourier
transform in timefrequency analysis was
observed by
Dennis
Gabor, who, in1946, introduced a
timelocalization “window function” to
extract local information of the Fourier
transform of the signal.
He suggested
representing a signal with time and
frequency as coordinates.The method is short
time Fourier transform,and the discussion of
STFT is Gabor transform.


Fig.3 Dennis Gabor(19001979) 
Fig.4 Alfréd Haar(18851933)

1975
Ronald Coifman and Guido Weiss interpreted
Lusin's theory in terms of atoms and
molecules which were to form the basic
building blocks of a function space. The
first synthesis of these theories leading up
to wavelet analysis.
1984
Morlet tried Windowed Fourier Analysis
while working for an oil
company. Oil companies searched for
underground oil by sending impulses into the
ground and analyzing their echoes. These
echoes could be analyzed to tell how thick a
layer of oil underground would be. Fourier
Analysis and Windowed Fourier Analysis were
used to analyze these echoes. However,
Fourier Analysis was a timeconsuming
process so Morlet began to look elsewhere
for a solution.
When he worked with Windowed Fourier
Analysis he discovered that keeping the
window fixed was the wrong approach. He did
exactly the opposite. He kept the frequency
of the function (number of oscillations)
constant and changed the window. He
discovered that stretching the window
stretched the function and scrunching or
squeezing the window compressed the
function. This concept of Morlet’s was
touched with the Haar wavelets from earlier.
Morlet had made an impact on the history
of wavelets; however, he wasn’t satisfied
with his efforts. In 1981, Morlet teamed up
with a man named Alex Grossman. They
developed continuous wavelet transform
in1984.


Fig.5 Jean Morlet 
Fig.6 Morlet wavelet 
1985
Yves Meyer discovered the first smooth
orthogonal wavelet basis functions with
better time and frequency localization.
1986
Stephane Mallat( a former student of
Yves Meyer) collaborated with Yves Meyer to
develop multiresolution analysis theory (MRA),
DWT, wavelet construction techniques.


Fig.7 Yves Meyer

Fig.8 Stephane Mallat

1987
The final stage of developing wavelet,
Ingrid Daubechies found
compact and orthogonal wavelets. Daubechies
wavelets are a family of orthogonal wavelets
defining a discrete wavelet transform and
characterized by a
maximal number of vanishing moments for
some given support.
Daubechies wavelets were not
only orthogonal (like Meyer’s) but which
could be implemented using simple digital
filtering ideas, in fact, using short
digital filters. The new wavelets were
almost simple to program.


Fig.9 Ingrid Daubechies 
Fig.10 Daubechies wavelet 
With the ideal of
Mallat and Daubechies, wavelet transform
could be computed on personal computers
quickly and easily.
1993
David Edward Newland in 1993 introduced
the harmonic wavelet transform, It combines
advantages of the
shorttime Fourier transform and the
continuous wavelet transform. It can be
expressed in terms of repeated
Fourier transforms.

Fig.11
David Edward Newland 
Post 1993
Wavelet transform
is a very popular tool in various field and
keeps developing. 