“Chirps” everywhere Patrick Flandrin* CNRS — ´ Ecole Normale Sup´ erieure de Lyon *thanks to Pierre-Olivier Amblard (LIS Grenoble), Fran¸ cois Auger (Univ. Nantes), Pierre Borgnat (ENS Ly on), Eric Chassande-Motti n (Obs. Nice), Franz Hlawatsch (TU Wien), Paulo Gon¸ calv` es (INRIAlp es), Olivier Michel (Univ. Nice) and Jeffrey C. O’Neill (iConverse)
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Time-frequency interpretation — Unitarity of a time-frequency
distribution ρx(t, f ) guarantees the equivalence:
|x, y|2 = ρx, ρy.
Chirps — Unitarity + localization ⇒ detection/estimation via
path integration in the plane.
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Time-frequency detection?
Language — Time-frequency offers a natural language for deal-ing with detection/estimation problems beyond nominal situa-
tions.
Robustness — Uncertainties in a chirp model can be incorporatedby replacing the integration curve by a domain (example of post-newtonian approximations in the case of gravitational waves).
time
freque
ncy
gravitational wave
?
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Doppler tolerance
Signal design — Specification of performance by a geometrical
interpretation of the time-frequency structure of a chirp.
time
frequency
linear chirp
time
frequency
hyperbolic chirp
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time
frequency
bat echolocation calls (+ echo)
time
frequency
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modeling
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Chirps and “atomic” decompositions
Fourier — The usual Fourier Transform (FT) can be formally
written as (F x)(f ) := x, ef , with ef (t) := exp{i2πf t}, so that:
x(t) = +∞
−∞x, e
f e
f (t) df.
Extensions — Replace complex exponentials by chirps, consid-
ered as warped versions of monochromatic waves, or by “chirplets”
(chirps of short duration) ⇒ modified short-time FTs or wavelet
transforms modifiees .
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Modified TFs — Example
Mellin Transform — A Mellin Transform (MT) of a signal x(t) ∈L2(IR+, t−2α+1dt) can be defined as the projection:
(Mx)(s) := +∞
0x(t) t−i2πs−α dt = x, c.
• Analysis on hyperbolic chirps c(t) := t−α
exp{i2πs log t}.
• ϕc(t)/2π = s/t ⇒, the Mellin parameter s can be interpreted
as a hyperbolic chirp rate .
• The MT can also be seen as a form of warped FT, since
x(t) := e(1−α)t x(et) ⇒ (Mx)(s) = (F x)(s).
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“Chirplet” decomposition — An example
signal + noise 8 atoms
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Chirps and self-similarity
Dilation — Given H, λ > 0, let DH,λ be the operator acting on
processes {X(t), t > 0} as (DH,λX)(t) := λ−H X(λt).
Self-similarity — A process {X(t), t > 0} is said to be self-similar
of parameter H (or “H -ss”) if, for any λ > 0,
{(DH,λX)(t), t > 0}d
= {X(t), t > 0}.
Self-similarity and stationarity — Self-similar processes and sta-tionary processes can be put in a one-to-one correspondence
(Lamperti, ’62).
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Lamperti
Definition — Given H > 0, the Lamperti transformation LH
acts
on {Y (t), t ∈ IR} as:
(LH Y )(t) := tH Y (log t), t > 0,
and its inverse L
−1
H acts on {X(t), t > 0} as :(L−1
H X)(t) := e−Ht X(et), t ∈ IR.
Theorem — If {Y (t), t ∈ IR} is stationary, its Lamperti transform{(LH Y )(t), t > 0} is H -ss. Conversely, if {X(t), t > 0} is H -ss, its
(inverse) Lamperti transform {(L−1H X)(t), t ∈ IR} is stationary.
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tone
↑
Lamperti
↓
chirp
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“Spectral” representations
Fourier — (Harmonisable) stationary processes admit a spectral
representation based on Fourier modes (monochromatic waves):