Gabor Frames. Karlheinz Gröchenig. Faculty of Mathematics, University of Vienna.

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1 Gabor Frames Karlheinz Gröchenig Faculty of Mathematics, University of Vienna HIM Bonn, January 2016 Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

2 Outline 1 Gabor Systems in Signal Processing 2 Coarse Structure of Gabor Frames 3 Fine Structure of Gabor Frames 4 Mysteries 5 References Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

3 Gabor Systems in Signal Processing Time-Frequency Shifts Translation operator: T x f (t) = f (t x) Modulation operator M ξ f (t) = e 2πiξ t f (t) Time-frequency shift (phase-space shift): z = (x, ξ) R 2d, t R d π(z)f (t) = e } 2πiξ t {{} f (t x) }{{} M ξ T x f (t) π(z) is unitary on L 2 (R d ) and an isometry on L p (R d ) Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

4 Gabor Systems in Signal Processing Gabor Systems Fix lattice Λ = AZ 2d for 2d 2d-matrix A with det A 0, Fix window g L 2 (R d ), g 0 G(g, Λ) = {π(λ)g : λ Λ} Gabor system Rectangular lattice Λ = αz d βz d Separable lattice Λ = PZ d QZ d, P, Q GL (d, R) Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

5 Gabor Systems in Signal Processing Gabor Frames G(g, Λ) is a Gabor frame, if for some A, B > 0 A f 2 2 λ Λ f, π(λ)g 2 B f 2 2 f L 2 (R d ) Equivalently, the frame operator Sf = λ Λ f, π(λ)g π(λ)g is invertible on L 2 (R d ), since A f 2 2 = Sf, f = λ f, π(λ)g π(λ)g, f B f 2 2 Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

6 Gabor Systems in Signal Processing Gabor Riesz Sequences G(g, Λ) is a (Gabor) Riesz sequence, if for some A, B > 0 A c 2 2 λ Λ c λ π(λ)g 2 2 B c 2 2 c l 2 (Λ) Equivalently, the Gramian (Gc) λ = µ Λ π(µ)g, π(λ)g c µ is invertible on l 2 (Λ), since A c 2 2 λ Λ c λ π(λ)g 2 2 = Gc, c B c 2 2 Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

7 Gabor Systems in Signal Processing Gabor Expansions Lemma If G(g, Λ) = {π(λ)g : λ Λ} is a frame, then there exists a γ L 2 (R d ), e.g., γ = S 1 g, such that f = f, π(λ)g π(λ)γ =, π(λ)γ π(λ)g λ Λ λ Λ f with unconditional convergence of the series in L 2 (R d ). Proof: Sπ(λ) = π(λ)s λ Λ f = S 1 Sf = λ Λ f, π(λ)g π(λ)s 1 g Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

8 Gabor Systems in Signal Processing Riesz Sequences Assume that G(g, Λ) is a Riesz sequence. Transmit signal f = µ Λ c µπ(λ)g. At receiver compute correlations y λ = f, π(λ)g = µ Λ c µ π(µ)g, π(λ)g = (Gc) λ so y = Gc. Consequently c = G 1 Gc = G 1 y Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

9 Gabor Systems in Signal Processing Mathematical Problems Find conditions on g and Λ, such that G(g, Λ) is a frame. Find characterizations of Gabor frames Find (classes of) examples Given g, characterize all lattices Λ, such that G(g, Λ) is a frame. Relevance and Relations to other fields? Gabor analysis Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

10 Coarse Structure of Gabor Frames Duality of Gabor Systems ( ) 0 I Definition: Let J =. If Λ = AZ I 0 2d is a lattice, the lattice Λ = J (A T ) 1 Z 2d is called the adjoint lattice. Theorem (Janssen, Ron-Shen, Feichtinger-Kozek) Let g L 2 (R d ), g 0 and Λ R 2d be a lattice. TFAE: (i) G(g, Λ) is a frame. (ii) G(g, Λ ) is a Riesz sequence. (iii) There exists a dual window γ L 2 (R d ), such that G(γ, Λ ) is Bessel and γ satisfies the biorthogonality condition (vol(λ)) 1 γ, π(µ)g = δ µ,0 µ Λ. Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

11 Coarse Structure of Gabor Frames Duality II A f 2 2 f, π(λ)g 2 B f 2 2 f L 2 (R d ) λ Λ if and only if A c 2 2 c µ π(µ)g 2 2 B c 2 2 c l 2 (Λ ) µ Λ Keywords for proof: gymnastics of time-frequency shifts, orthogonality relations for short-time Fourier transform, Poisson summation formula applied to spectrogram. Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

12 Coarse Structure of Gabor Frames Poisson Summation Formula f (k) = ˆf (k) for f S(R d )/M 1 (R d ). k Z d k Z d Note: If h(z) = f (Az), then ĥ(ζ) = det A 1ˆf ((A T ) 1 ζ). Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

13 Coarse Structure of Gabor Frames Characterization of Gabor Frames for Rectangular Lattices Lemma Let g L 2 (R d ) and α, β > 0. TFAE: (i) G(g, α, β) is a frame. (ii) There exist A, B > 0, such that for all c l 2 (Z d ) and almost all x R d A c 2 2 j Z d k Z d c k g(x + αj k β 2 B c 2 2. Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

14 Coarse Structure of Gabor Frames Frame Set Definition Given g L 2 (R d ) fixed. Then F full (g) = {Λ lattice : G(g, Λ) is frame} is called the full frame set of g, and F(g) = {(α, β) R 2 + : G(g, αz d βz d ) is frame} is called the reduced frame set of g Likewise R full (g) = {Λ lattice : G(g, Λ) is Riesz sequence } and F(g) = {(α, β) R 2 + : G(g, αz d βz d ) is Riesz sequence } Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

15 Coarse Structure of Gabor Frames Modulation Spaces A function g belongs to the modulation space M 1 (R d ) (Feichtinger s algebra), if R 2d g, π(z)g dz < or equivalently, if for fixed ψ S(R d ), ψ 0 Lemma R 2d g, π(z)ψ dz < For f M 1 (R d ) the Poisson summation formula is valid. k Z d f (k) = k Z d ˆf (k) f M 1. Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

16 Coarse Structure of Gabor Frames Theorem Coarse Structure Main Theorem Assume that g M 1 (R d ). Then F full (g) is an open subset of {Λ lattice : vol (Λ) < 1} and contains a neighborhood of 0. Likewise, F(g) is an open subset of {(α, β) R 2 + : αβ < 1} and contains a neighborhood of (0, 0) β 6 5 NO YES α Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

17 Fine Structure of Gabor Frames Examples/Questions Let g(t) = te πt2 (first Hermite function) Is G(g, 2 3Z Z) a frame? Is G(g, Z Z) a frame? [Are the points (2/3, 1) and ( , 1) in F(g)?] Let g(t) = χ [ 1/2,1/2] χ [ 1/2,1/2] = (1 x ) +. Is G(g, 2 3Z Z) a frame? Is G(g, 1 7Z Z) a frame? Is G(g, 1 7Z Z) a frame? [Are (2/3, 1), (1/7, 2), (1/7, ) F(g)?]?? Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

18 Fine Structure of Gabor Frames Precise Results about Gabor Frames in 1 D 1 Lyubarski-Seip for Gaussian g(t) = e at2 G(g, Λ) is frame vol(λ) < 1 2 Janssen-Strohmer for hyperbolic cosine g(t) = (cosh at) 1 G(g, αz βz) is frame αβ < 1 3 Janssen for exponential g(t) = e a t G(g, αz βz) is frame αβ < 1 4 Janssen for one-sided exponential function g(t) = e at χ R +(t) G(g, αz βz) is frame αβ 1 5 g(t) = (1 + at 2 ) 1 G(g, αz βz) is frame αβ < 1 6 g(t) = (1 iat) 1 for a > 0 G(g, αz βz) is frame αβ 1 Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

19 Fine Structure of Gabor Frames Weierstrass Sigma Function Interpolation problem G(µ) = δ µ,0 is solved by σ Λ (z) = µ Λ,µ 0 ( 1 z ) e z µ + z2 2µ 2 µ Weierstrass sigma-function Classical result: Theorem For suitable a C, σ Λ (z)e az2 = O(e πs(λ) z 2 /2 ). Λ = AZ 2d Corollary (a) Interpolation problem G(µ) = δ µ,0, µ Λ possesses a solution G F, if and only if s(λ) = det A < 1. (b) G(ϕ, Λ) is a frame if and only if s(λ) < 1 (Lyubarski, Seip 90). Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

20 Fine Structure of Gabor Frames Totally Positive Functions Definition: A non-zero function g is totally positive, if for x 1 < x 2 < < x N and y 1 < y 2 < < y N,N N det [ g(x i y j ) ] 1 i,j n 0 Examples: g(t) = e at2, (e at + e at ) 1, e a t, e at χ R +(t) AHA Schoenberg: g : R R is totally positive, if and only if ˆf (ξ) = Ce γξ 2 +2πiδξ (1 + 2πiδ ν ξ) 1 e 2πiδνξ, ν=1 with real parameters C, γ, δ, δ ν satisfying C > 0, γ 0, 0 < γ + δν 2 <. ν=1 Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

21 Fine Structure of Gabor Frames Gabor Frames and Totally Positive Functions Definition: g is totally positive of finite type M, if ˆf M (ξ) = C (1 + 2πiδ ν ξ) 1 e 2πiδνξ, ν=1 Theorem (KG, J. Stöckler) Assume that g is a totally positive function of finite type M 2. Then G(g, Λ) is a frame, if and only if αβ < 1. Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

22 Figure 1: AsketchoftheframesetforB n, n =2,for0 < ab < 1. Red,pinkandpurpleindicate Karlheinz Gröchenig (a, b)-values, (Vienna) where G(B 2, a, b) Gabor is notframes a frame. and Alltheir other Mysteries colors indicate the frame property. January The / 30 Indeed, F(B 1) is not open and (a, b) =(1, 1) F(B 1). Mysteries The main messages of this note are that the B-spline conjecture is false and that the frame set for B-splines of all orders must have a very complicated structure, sharing several similarities with F(B 1). Splines Kloos and Stöckler [11] and Christensen, Kim, and Kim [4] reported positive results to support of the frame set conjecture for B-splines, adding new parametervalues(a, b) R 2 + to the known parts of F(B n); thesenewvaluesareillustratedinfigure1forthecase of B 2. Along the same lines, we verify the frame set conjecture for B-spline for a new g = χ [0,1] χ [0,1] (n + 1-times) 3 b n n 2 n 3n 2 a

23 Mysteries Hermite Functions Theorem (KG, Lyubarski) d n h n = c n e πx 2 dx n (e 2πx 2 ) If vol(λ) < 1 n+1, then G(h n, Λ) is a frame. However Proposition (Lyubarski, Nes) If g L 2 (R) is odd and αβ = 1 1 N G(g, αz βz) is NOT a frame. for N = 2, 3,..., then Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

24 Mysteries Hermite Functions β NO YES α Figure: Possible frame set of odd function Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

25 Mysteries Frame Bounds Estimates for frame bounds for Λ = αz 2 A(α) f 2 2 f, π(λ)ϕ 2 B(α) f 2 2 λ αz 2 f L 2 (R d ) Theorem (Borichev, KG, Lyubarski.) For 1/2 α < 1 c B(α) C c(1 α 2 ) A(α) C(1 α 2 ) Can be extended to other windows. Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

26 Mysteries Frame Bounds II Estimates for frame bounds for Λ = αz 2 Let φ(t) = e πt2 and A(Λ) = S 1 1 and B(Λ) = S be the optimal frame bounds in A f 2 2 λ Λ f, π(λ)g 2 B f 2 2 f L 2 (R d ) Conjecture (Strohmer 2001): (i) Among all rectangular lattices with αβ = σ < 1, the condition number B(Λ)/A(Λ) is minimized by the square lattice σz 2. (Proved by Faulhuber/Steinerberger for σ = N 1, N = 2, 3,... ) (ii) Among all lattices with vol(λ) = σ < 1, the condition number B(Λ)/A(Λ) is minimized by the hexagonal lattice. (open) Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

27 Mysteries Further Directions Zak transform methods Gabor frames and function spaces (characterizations of modulation spaces) Gabor frames and pseudodifferential operators (almost diagonalization of pseudodifferential operators with Gabor frames) Gabor frames on finite Abelian groups Deformation results Gabor frames and Schrödinger equation... Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

28 References References K. Gröchenig. The mystery of Gabor frames. J. Fourier Anal. Appl., 20(4): , H. G. Feichtinger and W. Kozek. Quantization of TF lattice-invariant operators on elementary LCA groups. In Gabor analysis and algorithms, pages Birkhäuser Boston, Boston, MA, H. G. Feichtinger and F. Luef. Wiener amalgam spaces for the fundamental identity of Gabor analysis. Collect. Math., (Vol. Extra): , K. Gröchenig. Foundations of time-frequency analysis. Birkhäuser Boston Inc., Boston, MA, K. Gröchenig and Y. Lyubarskii. Gabor (super)frames with Hermite functions. Math. Ann., 345(2): , K. Gröchenig and J. Stöckler. Gabor frames and totally positive functions. Duke Math. J, 162(6): , C. Heil. History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl., 13(2): , Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

29 References A. J. E. M. Janssen. Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl., 1(4): , A. J. E. M. Janssen. Some Weyl-Heisenberg frame bound calculations. Indag. Math., 7: , A. J. E. M. Janssen. Zak transforms with few zeros and the tie. In Advances in Gabor Analysis. Birkhäuser Boston, Boston, MA, J. Lemvig On some Hermite series identities and their applications to Gabor analysis, and Counterexamples to the B-spline conjecture for Gabor frames. arxiv: and arxiv: A. Ron and Z. Shen. Weyl Heisenberg frames and Riesz bases in L 2 (R d ). Duke Math. J., 89(2): , Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30

30 References Links My homepage: Hans Feichtinger s homepage: Numerical Harmonic Analysis Group: < (contains most/all papers related to Gabor Analysis) The Large Time-Frequency Analysis Toolbox Karlheinz Gröchenig (Vienna) Gabor Frames and their Mysteries January / 30