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: http://crydee.sai.msu.ru/~vab/Wavelet.rsc/Chirplet.dir/node7.html
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The family of chirplets is generated from the mother chirplet by applying simple parameterized mathematical operations to it. The parameters of these operations form an index into the chirplet family.
The operations corresponding to the coordinate axes of the chirplet transform parameter space are presented in Table 1. The operators will be explained as they are used. The general notion to keep in mind is that any combination of these operators results in a 2-D affine coordinate transformation in the TF plane, which may be represented using the homogeneous coordinates often used in computer graphicsfoley.
The continuous STFT may be formulated as an inner product of the signal with the family of functions given in (2):
S_t_c,f_c = g_t_c,f_c,(_t) | s(t)
where 
is a suitably-chosen (fixed) window size, and
s(t) is the original signal.
We use the Dirac inner product notation, defined by:
g | s = _-^ \; g^*(t) s(t) \; dt
where 
denotes the complex conjugate of g.
We use the vertical bar between the arguments
and absorb the conjugation into the first
element so that we can write 
by itself,
as an operator that acts on whatever follows,
in this case the signal, 
.
Suppose we take the Gaussian window, centered at t=0, with unit pulse duration, as given by:
We denote a time shift to the position 
, with an operator
that has a multiplicative
law of composition: 
(Table 1).
A frequency shift to the position 
consists of multiplying
the window by 
, which we will denote 
.
The single-operator notation (Table 1, second column)
consists of a pictorial icon depicting
the effect each operator has on the TF plane, even when the operator
is acting in the time-domain.
For example the symbol with the two up-arrows
indicates a uniform upward shift along the frequency axis,
of the time-frequency plane, for positive values of the parameter.
These pictorial icons are consistent with our observation that each of
these operators acts in the time domain to perform an area-preserving
affine
coordinate transformation in the time-frequency plane.
Using the new notation, we can re-write (4) as:
S_t_c,f_c = _\: t_c _f_c g \: \: s
where we have also eliminated the time coordinate, recognizing that
for any operator in the time-domain, there is an equivalent operator
in the frequency domain, or in the TF plane, or in whatever other
reasonable coordinate space one might wish to work in.
The multiplicative law of composition of the
operators is applied in the order that they appear from
right to left (e.g.
is applied first, and then 
is applied
to that result).
Note that these two operators do not commute.
Adopting the convention of applying the frequency-shift first,
and then the time-shift,
results in the term 
appearing
in the second exponent of (2).
Applying the operators in the reverse order would result in a different
phase-shift.
In order to form
a true group, we need a third parameter, 
,
to indicate the degree to which the two operators do not commute.
Such a group structure is known as the Heisenberg groupfolland.
If we are only interested in the magnitude of the TF plane (e.g. the
spectrogram) then we can simply consider the two-dimensional
(two-parameter) translational
group, and describe the operations in terms of this simpler group.
Both (4) and (7) are equivalent,
providing us with some measure of the signal energy around coordinates
, but (7) emphasizes the fact that the
STFT is a correlation between members of a two-parameter family
of time-shifted and frequency-shifted versions of the same primitive, g.
Using the simplified law of composition, we may compose a time shift
by with a frequency shift by , as follows:
_\: t_c _f_c = C_t_c,0,0,0,0 C_0,f_c,0,0,0 = C_t_c,f_c
where omissions from the parameter list of C indicate values of zero.
Equation 7) may be re-written, using the ``Composite notation'' (Table 1, third column):
S_t_c,f_c = C_t_c,f_c g(t) \: \: s(t)