A windowed Fourier dictionary is constructed by translating
in time and frequency a time window g(t)g(t), of unit norm ∥g∥=1∥g∥=1,
centered at t=0t=0:
D
=
g
u
,
ξ
(
t
)
=
g
(
t

u
)
e
i
ξ
t
(
u
,
ξ
)
∈
R
2
.
D
=
g
u
,
ξ
(
t
)
=
g
(
t

u
)
e
i
ξ
t
(
u
,
ξ
)
∈
R
2
.
(5)The atom gu,ξgu,ξ is translated by u in time and by ξ in frequency.
The timeandfrequency spread of gu,ξgu,ξ is independent of u and ξ.
This means that each atom gu,ξgu,ξ corresponds to
a Heisenberg rectangle
that has a size σt×σωσt×σω
independent of its position (u,ξ)(u,ξ), as shown
by Figure 2.
The windowed Fourier transform
projects ff on each dictionary atom gu,ξgu,ξ:
S
f
(
u
,
ξ
)
=
〈
f
,
g
u
,
ξ
〉
=
∫

∞
+
∞
f
(
t
)
g
(
t

u
)
e

i
ξ
t
d
t
.
S
f
(
u
,
ξ
)
=
〈
f
,
g
u
,
ξ
〉
=
∫

∞
+
∞
f
(
t
)
g
(
t

u
)
e

i
ξ
t
d
t
.
(6)
It can be interpreted as a Fourier transform of ff at the frequency ξ,
localized by the window g(tu)g(tu) in the neighborhood of u.
This windowed Fourier transform is highly redundant and represents onedimensional signals
by a timefrequency image in (u,ξ)(u,ξ).
It is thus necessary to understand how to select many fewer
timefrequency coefficients that represent the signal efficiently.
When listening to music, we perceive sounds that have a
frequency that varies in time. Chapter 4 shows
that a spectral line of ff
creates highamplitude windowed Fourier coefficients Sf(u,ξ)Sf(u,ξ)
at frequencies ξ(u)ξ(u) that depend on time u.
These spectral components are detected and characterized by
ridge points, which are local maxima in this timefrequency plane.
Ridge points define a timefrequency approximation support
λ of ff with a geometry that depends on the timefrequency
evolution of the signal spectral components. Modifying the
sound duration or audio transpositions are implemented by modifying
the geometry of the ridge support in time frequency.
A windowed Fourier transform decomposes signals over waveforms that
have the same time and frequency resolution. It is thus effective
as long as the signal does not include structures having different
timefrequency resolutions, some being very localized
in time and others very localized in frequency. Wavelets address this
issue by changing the time and frequency resolution.