Approximation of Operators by Gabor Multipliers · OVERVIEW DEFINITIONS Time Frequency Representations STFT Multiplier / Gabor Multiplier PROBLEM DEFINITION AND MOTI-VATION Motivation

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




Motivation

SOMEANSWERS


FURTHERWORK

Outlook of the talk

Definitions.

Motivation and Questions.

Some answers.

Extension to Multi Window Setting.

???

Engelputzeder, Nina http://nuhag.eu

OVERVIEW

DEFINITIONS




Motivation

SOMEANSWERS


FURTHERWORK

DEFINITIONS

DEFINITIONS


OVERVIEW

DEFINITIONS




Motivation

SOMEANSWERS


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


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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)


OVERVIEW

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Motivation

SOMEANSWERS


FURTHERWORK

PROBLEM DEFINITION AND MOTIVATION

MOTIVATION


OVERVIEW

DEFINITIONS




Motivation

SOMEANSWERS


FURTHERWORK

Motivation

Filter time frequency content of a signal

Extension of the concept of diagonalisation of an operator


OVERVIEW

DEFINITIONS




Motivation

SOMEANSWERS


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.


OVERVIEW

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Motivation

SOMEANSWERS


FURTHERWORK

SOME ANSWERS

LET’S START


OVERVIEW

DEFINITIONS




Motivation

SOMEANSWERS


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.


OVERVIEW

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Motivation

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FURTHERWORK

Representation as STFT Multiplier andDiscretization


OVERVIEW

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Motivation

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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).


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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)


OVERVIEW

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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.


OVERVIEW

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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.


OVERVIEW

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Motivation

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FURTHERWORK

Which Operators can be well approximated?

Essentially underspread.

Smooth Kohn Nirenberg Symbol σ(G ) ∈W k2 .

Spreading Function in L2w

k .


OVERVIEW

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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)”


OVERVIEW

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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.


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FURTHERWORK


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)


OVERVIEW

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FURTHERWORK


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.


OVERVIEW

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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)


OVERVIEW

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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)


OVERVIEW

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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.


OVERVIEW

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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.


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FURTHERWORK


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.


OVERVIEW

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FURTHERWORK


Figure: Approximation of an underspread operator by Gabormultipliers. The results for different windows are compared independence of the lattice parameter α. n=144.


OVERVIEW

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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.


OVERVIEW

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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


OVERVIEW

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Motivation

SOMEANSWERS


FURTHERWORK

Error for sampled symbol in operator norm

Error Matrix


OVERVIEW

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Motivation

SOMEANSWERS


FURTHERWORK

Which Signals are the bad ones?


OVERVIEW

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FURTHERWORK

INTRODUCING ADDITIONAL WINDOWS

INTRODUCING ADDITIONALWINDOWS


OVERVIEW

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Motivation

SOMEANSWERS


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.


OVERVIEW

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Motivation

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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.”


OVERVIEW

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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 .


OVERVIEW

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Motivation

SOMEANSWERS


FURTHERWORK

Further Work

Optimal number of windows for Multiple GaborMultipliers?

How to place them?


Approximation of Operators by Gabor Multipliers · OVERVIEW DEFINITIONS Time Frequency Representations STFT Multiplier / Gabor Multiplier PROBLEM DEFINITION AND MOTI-VATION Motivation

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