OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Numerical Harmonic Analysis Group
Approximation of Operators by GaborMultipliers
Nina, [email protected]
May 6, 2009
Nina, Engelputzeder [email protected] Approximation of Operators by Gabor Multipliers
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Outlook of the talk
Definitions.
Motivation and Questions.
Some answers.
Extension to Multi Window Setting.
???
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
DEFINITIONS
DEFINITIONS
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Time Frequency Representations
For g ∈ L2(Rd), g 6= 0 the Short Time Fourier Transform off ∈ L2(Rd) is defined as
Vg f =⟨f , π(λ)g
⟩=
∫ ∞−∞
f (t)g(t − x)e−2πiωdt (1)
For a Hilbert Schmidt operator H ∈HS the SpreadingFunction ηH ∈ L2(R2) is defined as
H =
∫ ∞−∞
∫ ∞−∞
ηH(b, η)π(b, η)dbdη (2)
Unitary Isomorphism: ||H||HS = ||ηH ||L2
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
STFT Multiplier / Gabor Multiplier
A STFT Multiplier S for g , f , h ∈ L2(R2) and m ∈ L∞(R2) isdefined as
Sf =
∫ ∞−∞
∫ ∞−∞
m(b, ν)Vg1f (b, ν)π(b, ν)g2dbdν (3)
A Gabor Multiplier G on the lattice Λ = (αZ× βZ) withsymbol m ∈ l∞(Z2)is defined as
Gf =∞∑−∞
∞∑−∞
m(m, n)Vg1f (mα, nβ)π(mα, nβ)g2 (4)
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
PROBLEM DEFINITION AND MOTIVATION
MOTIVATION
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Motivation
Filter time frequency content of a signal
Extension of the concept of diagonalisation of an operator
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Questions
Norm: Hilbert Schmidt Norm / Operator Norm.
Approximants: Class of operators to approximate.
Window: Optimal Window, Number of SynthesisWindows.
Symbol: Best Approximation / Sampling.
Lattice: Lattice Parameters, Rectangular Lattice /Quincunx Lattice.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
SOME ANSWERS
LET’S START
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Spreading Function of a Gabor Multiplier
The Spreading Function of a Gabor Multiplier is given by
η(G ) = Fs(m) · Vg1g2 (5)
with Fs(m) being the symplectic Fourier Transform of thesymbol m defined as
Fs(m) =
∫m(b, η)e2πi(ηt−ζb)dbdη. (6)
Fs(m) is periodic on the adjoint lattice Λ0.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Representation as STFT Multiplier andDiscretization
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Representation as STFT Multiplier andDiscretization
An operator A can be represented as a STFT Multiplier ifand only if supp(ηA) ⊆ supp(Vg1g2).
Split the error for the approximation of an operator by aGabor Multiplier ||A− Gm||HS = ||ηA − ηG ||L2 into
||(1− χΩ)ηA||L2 + ||χΩηA − ηG ||L2 (7)
with χ being the characteristic function andΩ = supp(Vg1g2).
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
The Link to TI-Spaces
The Kohn Nirenberg Symbol σ(K ) of an operator K withdistributional kernel κ is given by
σ(K )(x , ζ) =
∫Rd
κ(K )(x , x − t)e−2πiζtdt (8)
Unitary Gelfand triple isomorphism on (S0, L2,S ′
0)Translation covariant: σ(π2(λ)(K )) = Tλσ(K )
For a Gabor Multiplier:
σ(G ) = m ∗Λ σ(g1 ⊗ g2) (9)
Translation invariant space generated by σ(g1 ⊗ g2)
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Translation Invariant Spaces
Definition
A closed space S is called translation invariant if for fixedh > 0:
∀α ∈ hZd , ∀f ∈ S , f ∈ S ⇒ f (.+ α) ∈ S (10)
The translation invariant space generated by Φ ⊂ S is thesmallest shift invariant space containing Φ.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Approximation Order
Definition
We say that G := G(g1, g2,Λi ), Λi = αiZ2, providesapproximation order k if for every operator K with Kohn-Nirenberg symbol σ(K ) ∈W k
2 (R2).
dist(K ,G ) : = infG∈G||K − G ||HS (11)
= infσ(G)∈S
||σ(K )− σ(G )||L2 ≤ Cαki ||σ(K )||W k
2.(12)
W k2 denotes the Sobolev space of smoothness k.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Which Operators can be well approximated?
Essentially underspread.
Smooth Kohn Nirenberg Symbol σ(G ) ∈W k2 .
Spreading Function in L2w
k .
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Approximation Order
Theorem
The space of Gabor multipliers G(g i1, g
i2,Λ), Λ of the form
Λ = αiZ2, provides approximation order k if and only if thereexists a neighborhood Ω of zero such that
O :=
∑λ∈ 1
αiZ2\0 |Vg i
1g i
2((·+ λ))|2∑λ∈ 1
αiZ2 |Vg i
1g i
2((·+ λ))|21
(|.|2 + α2i )k∈ L∞(Ω), αi > 0
(13)and the boundedness is uniform ∀0 < αi ≤ α0 <∞.
Holtz/Ron: ”Approximation orders of shift-invariantsubspaces of W s
2 (Rd)”
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Finding the optimal window
Lemma
A Gabor Multiplier with windows g1, g2 in the modulationspace Mv
2p with vp(z) = (1 + |z |)p has minimum approximation
order p.
Lemma
The rate of convergence of a Gabor Multiplier with windowsg1, g2 ∈ Mv
2p with v being an exponential weight is exponential.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Finding the optimal window
Proof: If g1, g2 ∈ M2vp
, then Vg1g2 ∈ L2vp
. Therefore we canestimate
∑λ∈ 1
hZ2\0 |Vg1g2(λ+ z)|2∑
λ∈ 1hZ2 |Vg1g2(λ+ z)|2
=
∑λ∈ 1
hZ2\0 |Vg1g2(λ+ z)|2
|Vg1g2(z)|2 +∑
λ∈ 1hZ2\0 |Vg1g2(λ+ z)|2
≤
∫|y |≥ 1
h+z |Vg1g2(y)|2
|Vg1g2(y)|2(14)
≤
∫|y |≥ 1
h+|z|
|Vg1g2(y)|2(1+|y |)2p
(1+|y |)2p dy
|Vg1g2(z)|2≤ (
h
1 + |z |h)2p||Vg1g2||L2
v
Vg1g2(1/h)(15)
≤ C (h2 + |z |2)p. (16)
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Finding the optimal window
Lemma
Assume an operator A with spreading function ηA withsupp(ηA) ⊆ Ω with Ω being the fundamental region of theadjoint lattice. Then the optimal synthesis window g2 for thebest approximation of A by a Gabor Multiplier G ∈ G(g ,Λ) isgiven by the largest eigenvector of the localization operatorLΩ =
∑λ∈Ω
⟨f , π(λ)g1
⟩π(λ)g1.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Idea of the proof
If we assume supp(ηA) ⊆ Ω, then the error of the bestapproximation of an operator A by a Gabor multiplier is givenby
||ηA −ηA|Vg1g2|2∑
λ∈ 1hZ2 |Vg1g2|2(·+ λ)
||L2Ω (17)
Therefore we have to maximize
max || |Vg1g2|2∑λ∈ 1
hZ2 |Vg1g2|2(·+ λ)
||L2(Ω). (18)
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Idea of the proof
This is equivalent to∫Ω
|Vg1g2|2∑λ∈ 1
αZ2 |Vg1g2|2(·+ λ)
dλ =
∫Ω |Vg1g2(λ)|2dλ∫R2 |Vg1g2(λ)|2dλ
(19)
Expression also known as a measure of concentration of afunction (compare e.g. Ramathan Topilawa)
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Idea of the proof
Further:
∫Ω |Vg1g2(λ)|2dλ∫R2 |Vg1g2(λ)|2dλ
=
∫Ω
⟨g2, π(λ)g1
⟩⟨g2, π(λ)g1
⟩∫R2
⟨g2, π(λ)g1
⟩⟨g2, π(λ)g1
⟩ (20)
=
⟨ ∫Ω
⟨g2, π(λ)g1
⟩π(λ)g1, g2
⟩||g2||2
(21)
=
⟨LΩg2, g2
⟩||g2||2
, (22)
where LΩ denotes the localization operator with maskingregion Ω.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Eigenfunctions of localization operators
The eigenfunctions of localization operators with radialsymmetric domain are the Hermite functions.(Daubechies)The eigenfunctions of localization operators with squaredomain look as follows:
Figure: First 6 eigenvectors of a localization operator.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Finding the optimal window
Figure: Optimal windows for different symmetries of the lattice. Thefirst window is the Gaussian, the second window shows the firsteigenvector of a concentration operator on a square domain. Thethird window is the first eigenvector of a concentration operator on arectangular domain.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Finding the optimal window
Figure: Approximation of an underspread operator by Gabormultipliers. The results for different windows are compared independence of the lattice parameter α. n=144.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Best Approximation vs. Sampled Symbol
If F s(m) ⊆ Ω then
mb(x)−ms(x) = F s(m)(x)(1− |Vg1g2(x)|2∑λ∈Λ0 |Vg1g2(x+λ)|2 )
Figure: Approximation of an STFT Multiplier with Gabor Multipliers.Analysis and Synthesis Windows are Gaussians.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Error for sampled symbol in operator norm
Idea: Calculate the errors for pure frequenciessupm ||Gm − Gm||Op =
supm
sup||f ||=1
||∑λ∈R2d
m(λ)(gλ ⊗ gλ)f − αβ∑λ∈Λ
m(λ)(gλ ⊗ gλ)f ||
supm
sup||f ||=1
||∑λ∈R2d
∑a
m(a)e2πi aλN Pλf−αβ
∑λ∈Λ
∑a
m(a)e2πi aλN Pλf ||
supm
sup||c||=1
||∑k
∑a
ckm(a)
∑λ∈R2d
e2πi aλN Pλ − αβ
∑λ∈Λ
e2πi aλN Pλ
χk ||
||Gm−Gm||Op ≤∑a
|m(a)|·||∑λ∈R2d
e2πi aλN Pλ−αβ
∑λ∈Λ
e2πi aλN Pλ||Op
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Error for sampled symbol in operator norm
Error Matrix
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Which Signals are the bad ones?
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
INTRODUCING ADDITIONAL WINDOWS
INTRODUCING ADDITIONALWINDOWS
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Limits of Gabor Multipliers and STFT Multipliers
Limited number of parameters for Gabor Multipliers.Badly conditioned problem for STFT Multipliers.
Figure: Approximation problem of an operator by a STFT multiplierwith Gaussian window. The Condition number of the problemincreases with increasing spreading support of the operator.
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Definition of a Generalized Gabor Multiplier
Definition
For an analysis window g ∈ S0 and a family of reconstructionwindows hj ∈ S0 and symbol functions mj a Multiple GaborMultiplier is defined as
M =∑
j
Gj =∑
j
∑λ∈Λ
mj(λ)Vg f (λ)π(λ)hj (23)
Dorfler/Torresani ”On the time frequency representationof operators and generalized Gabor multiplierapproximations.”
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Generalized Gabor Multipliers
If we have an operator H ∈HS, then H can berepresented as
H =∑λ∈Λ
∑λ′∈Λ
⟨H, (gλ ⊗ hλ′)
⟩(gλ ⊗ hλ′). (24)
(gλ ⊗ hλ′)(λ,λ′)∈Λ×Λ is a frame for the space of HilbertSchmidt operators HS(L2)
Tλσ(g ⊗ π(λ′)h)
(λ,λ′)∈Λ×Λis a frame for L2(R× R)
0 < A ≤∑
λ∈Λ |Vg (π(λ)h)|2 ≤ B <∞ almost everywhere
on Rd × Rd .
Engelputzeder, Nina http://nuhag.eu
OVERVIEW
DEFINITIONS
Time FrequencyRepresentations
STFT Multiplier/ GaborMultiplier
PROBLEMDEFINITIONAND MOTI-VATION
Motivation
SOMEANSWERS
INTRODUCINGADDI-TIONALWINDOWS
FURTHERWORK
Further Work
Optimal number of windows for Multiple GaborMultipliers?
How to place them?
Engelputzeder, Nina http://nuhag.eu