functions of variable bandwidth · functions of variable bandwidth roza aceska...

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Weightedmodulationspaces

Properties

Bandwidthshift

Summingbandlim-itedfunctions

what wecan dowith band-limitedfunctions

Reprodu-cingkernel

Summary

Numerical Harmonic Analysis Group

Functions of Variable Bandwidth

Roza [email protected]

Roza Aceska [email protected] Functions of Variable Bandwidth

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Bandwidthshift

Reprodu-cingkernel

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Definitions and notationsVariable bandwidth strip in the TF domainDilation-invariance of the corresponding spaceShifting the bandwidthReproducing kernel

Roza Aceska http://nuhag.eu

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Definitions, notations, ideas

Essential TF support concentrated within a strip withvariable bandwidth

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Definitions, notations, ideas

m−variable bandwidthSTm = (x, ξ) ∈ Rd × Rd : |ξ| ≤ m(x), m(x) > 1 .

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Definitions and notations

Weighted modulation spaces M2w(Rd)

‖f‖2 :=∫∫|Vgf(x, ξ)|2w2(x, ξ)dxdξ <∞, (1)

w(x, ξ) is a moderate weight.

STFT of a signal f , given a Schwartz window g:

Vgf(x, ξ) =∫f(t)g(t− x)e2πit·ξdt = 〈f,MξTxg〉

Moderate weight w

w(z1 + z2) ≤ cv(z1)w(z2) for all zi = (xi, ξi) ∈ R2d, (*)v-submultiplicative weight.

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‖f‖2 :=∫∫|Vgf(x, ξ)|2w2(x, ξ)dxdξ <∞, (1)

Moderate weight w

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‖f‖2 :=∫∫|Vgf(x, ξ)|2w2(x, ξ)dxdξ <∞, (1)

Moderate weight w

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Variable bandwidth m(x) ≥ 1, choose s ≥ 1

Weight

wm(x, ξ) =

1, |ξ| ≤ m(x) + 1(|ξ| −m(x))s, |ξ| ≥ m(x) + 1

Property

The weight (2) is s−moderate under suitable conditions.(eg. m′ is bounded or m has bounded variations)

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Variable bandwidth m(x) ≥ 1, choose s ≥ 1

Weight

wm(x, ξ) =

1, |ξ| ≤ m(x) + 1(|ξ| −m(x))s, |ξ| ≥ m(x) + 1

Property

The weight (2) is s−moderate under suitable conditions.(eg. m′ is bounded or m has bounded variations)

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Bandwidth shift: cosine-like band

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DefinitionThe space of functions of variable bandwidth is a weightedmodulation space

Bm(x)(Rd) := M2wm(Rd) = f | Vgf ∈ L2

wm. (3)

Example:

Band-limited functions are in Bm(x)(Rd).

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band-limited functions

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Dilation invariance

Proposition

Let b(λ) be such that |b(λ)| ≤ c |λ| , for λ ∈ R and aconstant c. If the moderate weight w fulfills the inequality

|w(λ−1x, λξ)| ≤ b(λ)w(x, ξ), (4)

then the weighted modulation space M2w is dilation

invariant.More precisely,

‖f(λ·)‖ ≤ b(λ)cg,λ‖f‖. (5)

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Sketch of proof:

For f1(t) = f(λt) and gλ(t) = g( tλ):

|Vgf1(x, ξ)| = 1λ |Vgλf1(λx, ξλ)|.

|Vgf1(x, ξ)|2w2(x, ξ) = λ−2|Vgλf1(x, ξ)|2w2(xλ−1, λξ).

Suppose |w( xλ , λξ)| ≤ b(λ)w(x, ξ), b2(λ)/λ2 -bounded. Then∫∫

|Vgf1(x, ξ)|2w2(x, ξ)dxdξ

≤ λ−2b2(λ)∫∫|Vgλf(x, ξ)|2w2(x, ξ)dxdξ︸︷︷︸

≈‖f‖

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Sketch of proof:

Suppose |w( xλ , λξ)| ≤ b(λ)w(x, ξ), b2(λ)/λ2 -bounded.

Then∫∫|Vgf1(x, ξ)|2w2(x, ξ)dxdξ

≈‖f‖

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Sketch of proof:

≈‖f‖

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Sketch of proof:

≈‖f‖

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Dilation invariance

Corollary

If m(x) is a bounded function and wm is a moderate weight,then Bm(x) is a dilation invariant space.

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Bandwidth shift: cosine-like band

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Bandwidth shift

|ξ| −m(x) > 1, but |ξ| −m(x)− h(x) ≤ 1;

wm(x, ξ) = (|ξ| −m(x))s, wm+h(x, ξ) = 1.

As 0 ≤ |ξ| −m(x) ≤ h(x) + 1,

2−swm+h(x, ξ) < 1 < wm(x, ξ) ≤ (h(x) + 1)s ≤ 2s =2swm+h(x, ξ).

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Bandwidth shift

Proposition

Let |h(x)| < 1 for all x ∈ R , and assume that m(x)generates a moderate weight wm. Then m(x) + h(x) alsogenerates a moderate weight wm+h and for all (x, ξ) ∈ R2d

2−swm+h(x, ξ) ≤ wm(x, ξ) ≤ 2swm+h(x, ξ).

s is the power of wm.

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LemmaMoving the variable bandwidth m(x) for a step |h(x)| < 1generates equivalent norms:

2−s‖Vgf · wm+h‖2 ≤ ‖Vgfwm‖2 ≤ 2−s‖Vgfwm+h‖2. (6)

Comment- finite number of shifting the bandwidth

Corollary

B(m+h)(x) ' Bm(x) ' Hs, (7)

Hs(Rd) = f ∈ S′ :∫|f(ξ)|2(1 + |ξ|)sdξ <∞ - Bessel

potential space.

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Corollary

B(m+h)(x) ' Bm(x) ' Hs, (7)

potential space.

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Corollary

B(m+h)(x) ' Bm(x) ' Hs, (7)

potential space.

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Example

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Let l > 0, Ωl = [−l, l]d;A function f is said to be band-limited, if supp(f) ⊆ Ωl .

BΩl(Rd) - the space of band-limited functions with spectralsupport in Ωl.

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Let l > 0, Ωl = [−l, l]d;A function f is said to be band-limited, if supp(f) ⊆ Ωl .

BΩl(Rd) - the space of band-limited functions with spectralsupport in Ωl.

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Let l > 0, Ωl = [−l, l]d and let BΩl(Rd) be the space ofband-limited functions with spectral support in Ωs.

Proposition

BΩl(Rd) ≤ Bm(x)(Rd).

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Let l > 0, Ωl = [−l, l]d and let BΩl(Rd) be the space ofband-limited functions with spectral support in Ωs.

Proposition

BΩl(Rd) ≤ Bm(x)(Rd).

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ProblemConsider a sequence of bandlimited functions

fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)

Is f =∑

n fn ∈ Bm(x)?

Not always! How about∑

n fnψn? (ψn form a bupu)

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fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)

Is f =∑

n fn ∈ Bm(x)?

Not always!

How about∑

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fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)

Is f =∑

n fn ∈ Bm(x)?

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fn ∈ BΩn , Ωn = [−m(n),m(n)] . (8)

Is f =∑

n fn ∈ Bm(x)?

Six signals

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used tools

Vg (c1f1 + c2f2) = c1Vg (f1) + c2Vg (f2)

for disjointly compactly supported fn,∥∥Vg (∑n∈IN fn)∥∥ =

∑n∈IN ‖Vg (fn)‖

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used tools

Vg (c1f1 + c2f2) = c1Vg (f1) + c2Vg (f2)

for disjointly compactly supported fn,∥∥Vg (∑n∈IN fn)∥∥ =

∑n∈IN ‖Vg (fn)‖

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Theorem

TheoremGiven a bounded uniform partition of unity (ψn)n∈IN and asequence(cn) ∈ l∞, assume that

fn ∈ BΩn , Ωn = [−m(n),m(n)].

Thenf =

∑n∈IN

cnfnψn ∈ Bm (9)

and||∑n∈IN

cnfnψn||Bm ≤ ‖c‖∞∑n∈IN

‖fnψn‖Bm

. (10)

Six signals

The sum

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Reproducing kernel

A reproducing kernel for Bm is a R2d functionΦ(x, y) := φy(x)such that

φ ∈ Bm and

f(y) = 〈f,Φ(·, y)〉Bm (11)

for every f ∈ Bm.

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Example

The reproducing kernel for a Sobolev space Hs is

Φ(x, y) = φ(x− y) = Ty(F−1(w−2m )). (12)

See Feichtinger & Werther, 2002: Robustness of minimalnorm interpolation in Sobolev algebras.

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Example

The reproducing kernel for a Sobolev space Hs is

Φ(x, y) = φ(x− y) = Ty(F−1(w−2m )). (12)

See Feichtinger & Werther, 2002: Robustness of minimalnorm interpolation in Sobolev algebras.

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Variable bandwidth reproducing kernel

Ms(Rd) = f ∈ S′ :∫∫|Vgf(x, ξ)|2wm(x, ξ)2dxdξ ≤ ∞,

weight is equivalent to vs(x, ξ) = (1 + |ξ|)s.

Scalar product: 〈f, g〉M := 〈Vgfwm, Vghwm〉L2 .

What would Φ be so that f(y) = 〈f, φy〉M?

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〈f, φy〉M =

〈Vgfwm, Vg(φy)wm〉L2 = 〈e−2πixξT Vgf , Vg(Tyφ)w2m〉L2

(use Vgf(x, ξ) = e−2πixξVgf(ξ,−x) = e−2πixξT Vgf(x, ξ))

= 〈f , V ∗g T −1(Vg(φy)e2πixξw2

)〉L2 .

By uniqueness,

V ∗g T −1(Vg(φy)e2πixξw2

)= e2πiy·,

Φ(·, y) = φy(·) = V ∗g

(e−2πixξw−2

s (x, ξ)T Vg(e2πiy·)(x, ξ)).

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〈f, φy〉M =

)〉L2 .

By uniqueness,

)= e2πiy·,

Φ(·, y) = φy(·) = V ∗g

(e−2πixξw−2

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〈f, φy〉M =

)〉L2 .

By uniqueness,

)= e2πiy·,

Φ(·, y) = φy(·) = V ∗g

(e−2πixξw−2

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〈f, φy〉M =

)〉L2 .

By uniqueness,

)= e2πiy·,

Φ(·, y) = φy(·) = V ∗g

(e−2πixξw−2

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〈f, φy〉M =

)〉L2 .

By uniqueness,

)= e2πiy·,

Φ(·, y) = φy(·) = V ∗g

(e−2πixξw−2

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Weighted modulation spaceBandwidth shiftDilation invarianceSumming band-limited functionsReproducing kernelFuture work

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Bibliography

Grochenig, K.; 2001: Foundations of Time-FrequencyAnalysisFeichtinger, H.G.; 1981: On a new Segal algebraFeichtinger, H.,Werther, T.; 2002:Robustness ofminimal norm interpolation in Sobolev algebras.http://homepage.univie.ac.at/roza.aceska/papers.html

functions of variable bandwidth · functions of variable bandwidth roza aceska...

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