Gabor Frame Sets of Invariance - A Hamiltonian Approach to Gabor Frame Deformations
aa r X i v : . [ m a t h . F A ] J a n GABOR FRAME SETS OF INVARIANCE - A HAMILTONIAN APPROACHTO GABOR FRAME DEFORMATIONS
MARKUS FAULHUBER
Abstract.
In this work we study families of pairs of window functions and lattices whichlead to Gabor frames which all possess the same frame bounds. To be more precise, for everygeneralized Gaussian g , we will construct an uncountable family of lattices { Λ τ } such that eachpairing of g with some Λ τ yields a Gabor frame, and all pairings yield the same frame bounds.On the other hand, for each lattice we will find a countable family of generalized Gaussians { g i } such that each pairing leaves the frame bounds invariant. Therefore, we are tempted to speakabout Gabor Frame Sets of Invariance . Introduction and Notation A Gabor frame (or
Weyl-Heisenberg frame ) for L ( R d ) is generated by a (fixed, non-zero) window function g ∈ L ( R d ) and an index set Λ ⊂ R d . It is denoted by G ( g, Λ) and consists of time-frequency shifted versions of g .We denote by λ = ( x, ω ) ∈ R d × R d a point in the time-frequency plane and use the followingnotation for a time-frequency shift by λ :(1.1) π ( λ ) g ( t ) = M ω T x g ( t ) = e πiω · t g ( t − x ) , x, ω, t ∈ R d . The operators involved in Equation (1.1) are the translation operator (1.2) T x g ( t ) = g ( t − x )and the modulation operator (1.3) M ω g ( t ) = e πiω · t . The latter one shifts a function in the Fourier or frequency space, hence the name time-frequencyshift for the composition of the mentioned operators.
Remark.
The point “ · ” denotes the Euclidean inner product of two column vectors, i.e. ω · t = h ω, t i = ω T t (e.g. in Equation (1.1)). Also, we will use the notation x = x · x , Sx · y = x T S T y and Sx = x T S T x for x, y ∈ R d and S a d × d matrix. Remark.
We note that the translation and modulation operator do not commute in general, infact(1.4) M ω T x = e πiω · x T x M ω . This formula is closely related to the commutation relations in quantum mechanics.Let Λ be an index set. The time-frequency shifted versions of the window g with respect to Λare called atoms and the set(1.5) G ( g, Λ) = { π ( λ ) g | λ ∈ Λ } is called a Gabor system . G ( g, Λ) is called a frame if it fulfills the frame property (1.6) A k f k ≤ X λ ∈ Λ |h f, π ( λ ) g i| ≤ B k f k , ∀ f ∈ L ( R d ) Mathematics Subject Classification.
Key words and phrases.
Frame Bounds, Gabor Frame, Hamiltonian Deformation.Austrian Science Fund (FWF): [P26273-N25]. for some positive constants 0 < A ≤ B < ∞ called frame constants or frame bounds . The indexset Λ ⊂ R d is called a lattice in the time-frequency plane if and only if there exists an invertible(non-unique) 2 d × d matrix S , in the sense that Λ = S Z d . The volume of the lattice is definedas(1.7) vol (Λ) = | det( S ) | and the density of the lattice is given by(1.8) δ (Λ) = 1 vol (Λ) . For more details on frames, Gabor frames and time-frequency analysis we refer to [3], [12], [14],[16], [19].
Remark.
The reader familiar with the topic of time-frequency analysis will note that we restrictedourselves to the Hilbert space case. The spaces usually involved when it comes to time-frequencyanalysis are the modulation spaces (1.9) M p ( R d ) = { f ∈ S ′ ( R d ) | k f k pM p < ∞} , where(1.10) k f k pM p = Z R d Z R d |h f, M ω T x g i| p dx dω, with (non-zero) fixed g ∈ S ( R d ) (see e.g [16], [17]). We note that M p ( R d ) is independent of thechoice of g ∈ S ( R d ) and furthermore, M ( R d ) = L ( R d ). In time-frequency analysis the spaces M ( R d ) and its dual space M ∞ ( R d ) replace in a natural way the Schwartz space S ( R d ) and itsdual space, the space of tempered distributions S ′ ( R d ) usually used in the field of analysis. In time-frequency analysis the test or window functions are often assumed to be in Feichtinger’s Algebra S ( R d ) = M ( R d ), whose dual space S ′ ( R d ) = M ∞ ( R d ) is the natural space of distributions intime-frequency analysis. For further reading on the topic of modulation spaces we refer to [11],[16], [17].From Equation (1.6) we see that the frame bounds A and B depend on the window g andthe lattice Λ. Fixing the density of the lattice, a question arising is how far one can deform thewindow or the lattice or both without destroying the frame property. Results in this directionare usually called perturbation or deformation results and some are given in [8], [13], [17], [18].Usually, results concerning deformations of Gabor frames do not describe the behavior of theframe bounds explicitly, but rather state whether the frame property is kept at all or not. We willpresent some perturbation results where not only the frame property is kept, but also the framebounds.Although many of the definitions and well-known, general theorems are stated for L ( R d ), wewill state our results only for d = 1.This work is structured as follows. In Section 2 we recall the basic properties of the symplecticgroup, in Section 3 we recall the corresponding properties of the metaplectic group as well as theinterplay between the two mentioned groups. Finally, in Section 4 we will state and proof themain theorem of this work and present some examples.2. The Symplectic Group
As we want to describe a lattice by a matrix one possible way of describing the deformationprocess is by multiplying the generating matrix with another matrix from the left. In particular,all our deformations will by carried out by symplectic matrices. Hence, in this section we wantto recall some basic facts about the symplectic group and its elements, the symplectic matrices.They are widely used in Hamiltonian mechanics and also serve as tools in time-frequency analysis[6], [15], [16]. All results of this section can be found in full generality in de Gosson’s book [6].We will also list further references at many points. Although we will only treat the 1-dimensionalcase, most results can be formulated verbatim for d >
ABOR FRAME SETS OF INVARIANCE 3
Symplectic Matrices.Definition 2.1.
A matrix S ∈ GL (2 d, R ) is called symplectic if and only if(2.1) SJS T = S T JS = J, where J = (cid:18) I − I (cid:19) , 0 is the d × d zero matrix and I is the d × d identity matrix. J is calledthe standard symplectic matrix . Remark.
We note that condition (2.1) is redundant. Actually, we have(2.2)
SJS T = J ⇐⇒ S T JS = J. From (2.1) we conclude that all symplectic matrices S ∈ Sp (2 d, R ) must have determinant equalto ±
1. In fact, if S ∈ Sp (2 d, R ) then det ( S ) = 1, see [6], [7], [16]. Also, Sp (2 d, R ) is a subgroupof SL (2 d, R ) and in the case d = 1 we have Sp (2 , R ) = SL (2 , R ). In all other cases where d > Sp (2 d, R ) is a proper subgroup of SL (2 d, R ). Lemma 2.2.
The set of all symplectic matrices forms a group denoted by Sp (2 d, R ) .Proof. Let S , S ∈ Sp (2 d, R ). It follows from Equation (2.1) that the product S S ∈ Sp (2 d, R ).Taking the inverse of the double equality in (2.1) and using the fact that J − = − J we see that S − ∈ Sp (2 d, R ) if S ∈ Sp (2 d, R ). (cid:3) It is convenient and commonly used write symplectic matrices as block matrices in the followingform(2.3) S = (cid:18) A BC D (cid:19) , where A, B, C, D are d × d matrices. With this notation we have the following formula for theinverse of a symplectic matrix(2.4) S − = (cid:18) D T − B T − C T A T (cid:19) . In the case d = 1 this reduces to the well-known inversion formula for a matrix S belonging to SL (2 , R ), as A, B, C, D ∈ R are scalars.2.2. Free Symplectic Matrices.
We will now introduce the building blocks of the symplecticgroup, the free symplectic matrices and state that any symplectic matrix is the product of thesebuilding blocks [6], [9].
Definition 2.3.
We call a symplectic matrix S = (cid:18) A BC D (cid:19) ∈ Sp (2 d, R ) a free symplectic matrix if B is invertible. Definition 2.4.
Let P be a symmetric d × d matrix and let L be an invertible d × d matrix. Wedefine the following 2 d × d matrices J = (cid:18) I − I (cid:19) (2.5) V P = (cid:18) I − P I (cid:19) (2.6) M L = (cid:18) L − L T (cid:19) (2.7)which all belong to Sp (2 d, R ). We call them generator matrices for the free symplectic matrices.The name generator matrix is justified by the following propositions. M. FAULHUBER
Proposition 2.5.
With the notation of Definition 2.4 we get that any free symplectic matrix S = (cid:18) A BC D (cid:19) can be factored as (2.8) S = V − DB − M B − JV − B − A . A proof is given in [6] or [15]. It makes use of well-known factorization results and propertiesof symplectic matrices.2.3.
Generating Functions.
Following [6] we will point out connections between quadratic formsin ( x, x ′ ) and free symplectic matrices. The motivation comes from Hamiltonian mechanics . Wewant to describe the motion of a particle depending on two variables usually called position ( x )and momentum ( p ) which depend on time ( t ) and are coupled by Hamilton’s equations (2.9) ˙ x ( t ) = ∂∂p H ( x ( t ) , p ( t ))˙ p ( t ) = − ∂∂x H ( x ( t ) , p ( t )) . Here, H ( x ( t ) , p ( t )) is the Hamiltonian or Hamilton function . For more details on Hamiltonianmechanics see [2].Given two different positions x and x ′ of a particle we want to know the initial and finalmomentum p and p ′ assuming that the motion is linear, meaning we have the linear system( x, p ) = S ( x ′ , p ′ ). This is equivalent to(2.10) x = Ax ′ + Bp ′ p = Cx ′ + Dp ′ . In order to solve this system of equations for ( p, p ′ ), clearly B has to be invertible.In the case of time-frequency analysis the proper way to use and interpret Hamiltonian me-chanics is by replacing position by time and momentum by frequency. The following propositionis again formulated in the context of time-frequency analysis. Proposition 2.6.
Let S = (cid:18) A BC D (cid:19) ∈ Sp (2 d, R ) be a free symplectic matrix. Let P, Q be d × d symmetric matrices and let L be a d × d invertible matrix.(i) Then we have (2.11) ( x, ω ) = S ( x ′ , ω ′ ) ⇐⇒ ( ω = ∂ x W ( x, x ′ ) ,ω ′ = − ∂ x ′ W ( x, x ′ ) where W is the quadratic form (2.12) W ( x, x ′ ) = 12 DB − x − B − x · x ′ + 12 B − Ax ′ where DB − and B − A are symmetric.(ii) To every quadratic form (2.13) W ( x, x ′ ) = 12 P x − Lx · x ′ + 12 Qx ′ we can associate the free symplectic matrix (2.14) S W = (cid:18) L − Q L − P L − Q − L T P L − (cid:19) . We call the quadratic form in (2.13) the generating function of S W in (2.14) Remark.
Note the connection between the generating function and the factorization of a freesymplectic matrix.
Theorem 2.7.
For every S ∈ Sp (2 d, R ) there exist two (non-unique) free symplectic matrices S W and S W ′ such that S = S W S W ′ . ABOR FRAME SETS OF INVARIANCE 5
Corollary 2.8.
The set of all matrices (2.15) { V P , M L , J } generates the symplectic group Sp (2 d, R ) . The Metaplectic Group
The second way to perform a deformation of a Gabor frames is to perturb the window. We willdescribe this process by letting some unitary operator act on the window. In fact, we will onlydeal with some special operators called metaplectic.The metaplectic group and its elements, the metaplectic operators, are widely used in quantummechanics and in time-frequency analysis. There is a close connection to the symplectic groupand this interplay might be used to solve problems in quantum mechanics once the solution forthe corresponding classical problem is known [6]. In time-frequency analysis this property canbe used to deform Gabor frames without destroying their frame property and even keeping theoptimal frame bounds [8], [16].Again, since there is not much difference between formulating the results for d = 1 and d > d = 1 later on. The resultscan be found in [6] in full detail.3.1. The Group
M p (2 d, R ) .Definition 3.1. The metaplectic group
M p (2 d, R ) is the connected two-fold cover of the symplecticgroup Sp (2 d, R ). Equivalently, we can define M p (2 d, R ) by saying that the sequence(3.1) 0 → Z → M p (2 d, R ) → Sp (2 d, R ) → Remark.
A sequence(3.2) A → A → · · · → A n → A n +1 of morphisms is called exact , if the image of each morphism is equal to the kernel of the nextmorphism(3.3) im ( A k − → A k ) = ker ( A k → A k +1 ) , k = 1 , . . . , n. We want to use a more constructive approach to define the metaplectic group.3.2.
Metaplectic Operators and the Quadratic Fourier Transform.
The metaplectic groupis a group of unitary operators on L ( R d ) [5], [6], [16], [21]. Let ψ ∈ S ( R d ) be a function in theSchwartz space. Following de Gosson [6] we define the following operators. • The modified Fourier transform b J defined by(3.4) b Jψ ( t ) = i − d/ Z R d ψ ( t ′ ) e − πi t · t ′ dt ′ . • The linear “chirps”(3.5) d V − P ψ ( t ) = e πi P t · t ψ ( t )with P being a real, symmetric d × d matrix. • The rescaling operator(3.6) [ M L,n ψ ( t ) = i n p | det ( L ) | ψ ( Lt ) , where L is invertible and n is an integer corresponding to a choice of arg ( det ( L )), to bemore precise(3.7) nπ ≡ arg ( det ( L )) mod 2 π. The class modulo 4 of the integer n appearing in the definition of the rescaling operator (3.6) iscalled “Maslov index” [6], [9].As in the section on the symplectic group we will associate quadratic forms to metaplecticoperators and we will also see the interplay between the symplectic and the metaplectic group. M. FAULHUBER
Definition 3.2.
Let S W be the free symplectic matrix(3.8) S W = (cid:18) L − Q L − P L − Q − L T P L − (cid:19) associated to the quadratic form W ( t, t ′ ) = P t − Lt · t ′ + Qt ′ (compare proposition 2.6equations (2.13) and (2.14)). Let the operators b J , d V − P and [ M L,n be defined as in (3.4), (3.5) and(3.6) respectively. We call the operator(3.9) [ S W,n = d V − P [ M L,n b J d V − Q the quadratic Fourier transform associated to the free symplectic matrix S W .For ψ ∈ S ( R d ) we have the explicit formula(3.10) [ S W,n ψ ( t ) = i n − d p | det ( L ) | Z R d ψ ( t ′ ) e πi W ( t,t ′ ) dt ′ , where W ( t, t ′ ) is again the quadratic form as defined in (2.13) and Definition 3.2. Remark.
Although, all statements in this section were formulated for the Schwartz space S ( R d ),they also hold for Feichtinger’s algebra S ( R d ) as well as for the Hilbert space L ( R d ) [8]. Remark.
We will frequently drop one or both of the indices W and n and will write S instead of S W and b S or d S W instead of [ S W,n . When the context allows, we will also use other indices thanthe ones mentioned.
Remark.
As can be seen by formula (3.9) a quadratic Fourier transform is a manipulation of a(suitable) function by a chirp, a modified Fourier transform, a dilation and another chirp. This isthe exact same way in which the fractional Fourier transform is described in [1] with an additionaldilation in between the modified Fourier transform and the second chirp. Hence, the quadraticFourier transform is an extension of the fractional Fourier transform in the sense that the directionsin the time-frequency plane are scaled by some factor depending on the angle. For more detailson the fractional Fourier transform see also [9].
Proposition 3.3.
The operators [ S W,n extend to unitary operators on L ( R d ) and the inverse isgiven by (3.11) [ S W,n − = \ S W ∗ ,n ∗ , where W ∗ ( t, t ′ ) = − W ( t ′ , t ) and n ∗ = d − n . The fact that [ S W,n is a unitary operator is clear since, d V − P , [ M L,n and b J are unitary. Obviously,we have(3.12) d V − P − = c V P , [ M L,n − = \ M L − , − n and the inverse of the modified Fourier transform is given by(3.13) b J − ψ ( t ) = i d/ Z R d ψ ( t ′ ) e πit · t ′ dt ′ We note that(3.14) b J − \ M L − , − n = \ M − L T ,d − n b J and hence,(3.15) [ S W,n − = c V Q b J − \ M L − , − n c V P = \ S W ∗ ,n ∗ . Definition 3.4.
The subgroup of U ( L ( R d )) generated by the quadratic Fourier transforms [ S W,n is called the metaplectic group and is denoted by
M p (2 d, R ). Its elements are called metaplecticoperators. ABOR FRAME SETS OF INVARIANCE 7
Remark.
To each quadratic form W ( t, t ′ ) we can actually associate not one but two metaplecticoperators as, due to (3.7), [ S W,n and \ S W,n +2 = − [ S W,n are equally good choices. This reflects thefact that the metaplectic operators are elements of the two-fold cover of the symplectic group.
Theorem 3.5.
For every b S ∈ M p (2 d, R ) there exist two quadratic Fourier transforms [ S W,n and \ S W ′ ,n ′ such that b S = [ S W,n \ S W ′ ,n ′ . The factorization in Theorem 3.5 is not unique as the identity operator can always be writtenas [ S W,n \ S W ∗ ,n ∗ . Corollary 3.6.
The set of all operators (3.16) { d V − P , [ M L,n , b J } generates the metaplectic group. Without further preparation we introduce the natural projection of the metaplectic group
M p (2 d, R ) onto the symplectic group Sp (2 d, R ), which we will denote by π Mp . For the details werefer to [6]. Theorem 3.7.
The mapping (3.17) π Mp : M p (2 d, R ) −→ Sp (2 d, R ) [ S W,n S W which to a quadratic Fourier transform associates a free symplectic matrix with generating function W , is a surjective group homomorphism. Hence, (3.18) π Mp (cid:16) b S b S ′ (cid:17) = π Mp (cid:16) b S (cid:17) π Mp (cid:16) b S ′ (cid:17) . and the kernel of π Mp is given by (3.19) ker ( π Mp ) = {− I, + I } . hence, π Mp : M p (2 d, R ) Sp (2 d, R ) is a two-fold covering of the symplectic group. Definition 3.8.
The mapping π Mp in Theorem 3.7 is called the natural projection of M p (2 d, R )onto Sp (2 d, R ). Remark.
The natural projections of the metaplectic generator elements are the symplectic gener-ator elements.(3.20) π Mp (cid:16) c V P (cid:17) = V P , π Mp (cid:16) [ M L,n (cid:17) = M L , π Mp (cid:16) b J (cid:17) = J. Gabor Frame Sets of Invariance
We prepared the machinery of the symplectic and metaplectic group to the extend we needit in order to be able to deform Gabor frames without destroying their frame property. We areinterested in Gabor frame deformations which leave the frame bounds invariant.
Definition 4.1.
Assume G ( g, Λ) and G ( g ′ , Λ ′ ) are Gabor frames with the same optimal framebounds A and B . We write(4.1) G ( g, Λ) ∼ = G ( g ′ , Λ ′ ) . Theorem 4.2.
Let G ( g, Λ) be a Gabor frame with optimal frame bounds A and B . Let b S ∈ M p (2 d, R ) with projection π Mp ( b S ) = S ∈ Sp (2 d, R ) . Then G ( b Sg, S Λ) is also a Gabor frame andhas the same optimal frame bounds A and B . A full proof is given in [8].
Remark.
For any window a phase factor c ∈ C with | c | = 1 is negligible in the sense that G ( g, Λ)and G ( c g, Λ) have the same frame bounds as can directly be seen from equation (1.6).
M. FAULHUBER
Theorem 4.2 is a particular case of the notion of
Hamiltonian deformation of Gabor frames (see [8]). It tells us under which conditions the frame property as well as the optimal framebounds are kept when a Gabor frame suffers some disturbances. This is a very special case, asin general neither the optimal frame bounds nor the frame property might be kept under somegeneral deformation of the frame. However, there are cases when the frame property might bekept without keeping the optimal frame bounds (see [13], [18]). This is usually done by eitherdeforming the window and fixing the lattice or the other way round. By Theorem 4.2 we know thatthese approaches are equivalent as long as we stick to symplectic and metaplectic deformations.What we will see in the following sections is that it is possible to keep both, the frame propertyand the optimal frame bounds under certain lattice deformations, without changing the window.This is due to the fact that generalized Hermite functions, including the generalized Gaussians,are eigenfunctions with eigenvalues of modulus 1 of certain metaplectic operators. Hence, thecorresponding symplectic matrix will deform the lattice, while the window can remain unchanged.4.1.
Lattice Rotations and the Standard Gaussian.
From this point on, we will only considerthe 1-dimensional case. The most popular 1-dimensional window function is probably the standardGaussian g ( t ) = 2 / e − πt . Although, Gabor frames with Gaussian window have been studiedintensively, we still want to explore and exploit the Gabor family G ( g , Λ) with vol (Λ) < S τ = (cid:18) cos τ sin τ − sin τ cos τ (cid:19) and the corresponding deformation of the window is given by the action of the quadratic Fouriertransform on the window g . To derive a formula for the resulting window we use Proposition 2.6and Equation (3.10).(4.3) c S τ g ( t ) = i n ( τ ) − s | sin( τ ) | Z R e πi W τ ( t,t ′ ) g ( t ′ ) dt ′ , where n ( τ ) ∈ { , , , } depends on τ and the choice of arg (cid:16)p sin( τ ) (cid:17) and where(4.4) W τ ( t, t ′ ) = 12 sin( τ ) (cid:0) ( t + t ′ ) cos( τ ) − tt ′ ) (cid:1) . This manipulation is meaningful whenever τ = kπ , k ∈ Z . Performing the calculations, we get(4.5) c S τ g ( t ) = 2 / i n ( τ ) e − i τ e − πt = c g ( t ) , with | c | = 1. Hence, we have the result(4.6) G ( g , Λ) ∼ = G ( g , S τ Λ) , which means that the frame bounds of a Gabor frame with window g stay invariant under rotationof the lattice.Although, the ambiguity function of the standard Gaussian g = 2 / e − πt is well-known and,though, it is an easy exercise to compute it, we will still do the calculations, as the procedure willbe used extensively in a somewhat more general form in the rest of this work. For the definitionand an interpretation of the ambiguity function see Appendix A. ABOR FRAME SETS OF INVARIANCE 9 (4.7) Ag ( x, ω ) = Z R g (cid:16) t + x (cid:17) g (cid:16) t − x (cid:17) e − πiωt dt = Z R / e − π ( t + x/ / e − π ( t − x/ e − πiωt dt = √ Z R e − π ( t + x / e − πiωt dt = √ e − π x Z R √ e − πt e − πi ω √ t dt = e − π x − / F g (cid:16) ω/ √ (cid:17) = e − π x − / g (cid:16) ω/ √ (cid:17) = e − π x e − π ( w/ √ = e − π ( x + ω )Here, F denotes the Fourier transform which is given by(4.8) F f ( ω ) = Z R d f ( t ) e − πiω · t dt. What we used in the calculations above are a change of variables and the Fourier invariance ofthe standard Gaussian, F ( g ) = g . For the latter argument see [15], [16].4.2. Elliptic Deformations and Dilated Gaussians.
In Section 4.1 we saw that using thestandard Gaussian window the Gabor frame bounds stay invariant under a rotation of the lattice.We will now generalize this result using ideas from Hamiltonian mechanics. For an introductionto Hamiltonian mechanics we refer to [2]. The rotation matrix(4.9) S τ = (cid:18) cos τ sin τ − sin τ cos τ (cid:19) determines the flow of the harmonic oscillator with mass m = 1 and resonance Ω = 1. TheHamiltonian of this problem is given by(4.10) H ( x, ω, τ ) = ω x ddτ λ = J (cid:18) ∂∂x H ∂∂ω H (cid:19) = Jλ, where λ = ( x, ω ) T and both, x and ω depend on τ .Written in its most general form the Hamiltonian of the harmonic oscillator is given by(4.12) H ( x, ω, τ ) = ω m + m Ω x , where m is the mass of the particle and Ω is the resonance. The trajectories will be ellipses instandard position with semi-axis ratio m Ω.Assume, we are given the Gabor frame G ( g , Λ) with standard Gaussian window and arbitrarylattice Λ, vol (Λ) <
1. Any dilation of the lattice by a symplectic matrix M √ m can be compensatedby a metaplectic dilation of the window such that the frame bounds remain unchanged, so(4.13) G ( g , Λ) ∼ = G ( \ M √ m g , M √ m Λ) . We compute that the dilated window is of the form(4.14) \ M √ m g ( t ) = c (2 m ) / e − πmt = g m ( t ) , where | c | = 1. Next, we compute the ambiguity function Ag m .(4.15) Ag m ( x, ω ) = √ m Z R e − πm ( t + x/ e − πm ( t − x/ e − πiωt dt = √ m e − πmx / Z R e − πm t e − πiωt dt = e − π (cid:16) mx + ω m (cid:17) . Hence, any level set of Ag m will be an ellipse in standard position with semi-axis ratio m . It iskind of self-evident to examine the behavior of G ( g m , M √ m Λ) under deformations induced by theharmonic oscillator given by (4.12) with Ω = 1. The flow is then determined by the symplecticmatrix(4.16) S τ,m = (cid:18) cos τ m sin τ − m sin τ cos τ (cid:19) . The corresponding metaplectic operator [ S τ,m is determined by(4.17) [ S τ,m g m ( t ) = i n ( τ ) − r m | sin τ | Z R e πi W τ,m ( t,t ′ ) g m ( t ′ ) dt ′ , where n ( τ ) ∈ { , , , } depends on the parameter τ as well as on the choice of arg (cid:16) √ sin τ (cid:17) andwhere(4.18) W τ,m ( t, t ′ ) = m τ (cid:0) ( t + t ′ ) cos τ − tt ′ ) (cid:1) is the generating function of S τ,m . The verification of the upcoming formulas (4.19) and (4.21)can be found in appendix B. We have(4.19) [ S τ,m g m ( t ) = c g m ( t ) , with | c | = 1. Hence, equation (4.19) implies that(4.20) G ( g m , Λ) ∼ = G ( g m , S τ,m Λ) , where S τ,m is defined as in (4.16), as well as(4.21) Ag m ( x, ω ) = A (cid:16) [ S τ,m g m (cid:17) ( x, ω ) , where [ S τ,m is defined as given by equation (4.17). Summing up the results we get the followingtheorem. Theorem 4.3.
Let g m ( t ) = c (2 m ) / e − πmt with | c | = 1 and let Λ ⊂ R be a lattice with vol (Λ) < Let (4.22) S τ,m = (cid:18) cos τ m sin τ − m sin τ cos τ (cid:19) . be the deformation matrix acting on the lattice, then (4.23) d S τ,m g m ( t ) = i n ( τ,m ) − r m | sin τ | Z R e πi W τ,m ( t,t ′ ) g m ( t ′ ) dt ′ = c g m ( t ) , with | c | = 1 hence, (4.24) G ( g m , Λ) ∼ = G ( g m , S τ,m Λ) . Furthermore, for the ambiguity function A (cid:16) d S τ,m g m (cid:17) we have (4.25) A (cid:16) d S τ,m g m (cid:17) ( x, ω ) = Ag m ( x, ω ) . ABOR FRAME SETS OF INVARIANCE 11
Setting m = 1, Theorem 4.3 implies that the Gabor frame bounds of a Gabor frame withstandard Gaussian window stay invariant under a rotation of the lattice and hence, theorem 4.3is a generalization of a result given in [8].Also, in Theorem 4.3 the symplectic geometry of the lattice remains unchanged, whereas theEuclidean geometry of the lattice will change in general. In the case where m = 1, meaning thatwe only rotate the lattice, the symplectic as well as the Euclidean geometric properties are kept.In order to derive Theorem 4.3 we used a very geometric approach and a clear picture in mindabout the flow induced by the harmonic oscillator. The crucial ingredient for theorem 4.3 towork is that we could easily and explicitly calculate the eigenfunctions of the metaplectic operatorinvolved. We note that similar approaches have already been made in [4], characterizing the(dilated) Hermite functions as eigenfunctions of certain localization operators. The geometricapproach to phase space has also been used in [10] to construct frames consisting of eigenfunctionsof localization operators. We only stated theorem 4.3 for the dilated Gaussian window, but theresult holds for all dilated Hermite functions since they are eigenfunctions of the quadratic Fouriertransform and have eigenvalues of modulus 1.Let H T H = √ (cid:18) ± ± (cid:19) , such that H generates some version of a rotated hexagonallattice of volume 1. Taking the standard Gaussian g as window function, any rotated versionof the hexagonal lattice gives the same frame bounds, hence, we may choose one representativeamong all versions of the rotated hexagonal lattice and denote its generator matrix by H . Let δ > G ( g , √ δ H Z ) is a Gabor frame. Then, using theorem4.3, we gain a result closely related to the problem of optimal pulse shape design for LOFDM [24].The matrix S = √ √ √ ! generates a lattice which is a 45 degrees rotated version of theinteger lattice. If we want(4.26) G (cid:18) g , √ δ H Z (cid:19) ∼ = G (cid:18) g, √ δ S Z (cid:19) for some g ∈ L ( R ), then(4.27) Ag ( x, ω ) = e − π (cid:16) √ x + ω √ (cid:17) . So, the quadratic form in the exponent of the ambiguity function describes an ellipse in standardposition with semi-axis ratio √
3. Furthermore, we know that there are uncountably many otherlattice arrangements which together with g lead to the same frame bounds, namely(4.28) G (cid:18) g, √ δ S Z (cid:19) ∼ = G (cid:18) g, √ δ S τ,m S Z (cid:19) . This result carries over to and extends the results given in [24]. We will also discuss this result inExample A.4.3.
Modular Deformations of Gabor Frames.
In this section we will deal with discretedeformations of Gabor frames. In particular, the objects of interest are taken from the modulargroup which we define as follows.
Definition 4.4.
The modular group Sp (2 , Z ) consists of all 2-dimensional symplectic matriceswith integer entries. Remark.
Usually the modular group Γ is defined as the group of linear fractional transformationson the upper half of the complex plane which have the form(4.29) z az + bcz + d , with a, b, c, d ∈ Z and ad − bc = 1. For more details on the modular group see [23]. Consider the integer lattice Z . The action of the modular group leaves Z invariant, i.e. B Z = Z for B ∈ Sp (2 , Z ). In other words, B is just another choice for a basis of Z . Inparticular, any B ∈ Sp (2 , Z ) provides a basis for Z . Taking any symplectic matrix S ∈ Sp (2 , R )and any basis B ∈ Sp (2 , Z ) for Z this implies that(4.30) S Z = S B Z . We stay with the integer lattice for the beginning. Let G ( g, √ δ Z ), δ > B = (cid:18) a bc d (cid:19) ∈ Sp (2 , Z ) . The corresponding metaplectic operator is given by(4.32) b B g ( t ) = i − s | b | Z R e πiW ( t,t ′ ) g ( t ′ ) dt ′ , where W ( t, t ′ ) = db t − b tt ′ + ab t ′ and b = 0. In general b Bg will differ from g by more thanjust a phase factor as we apply a chirp a modified Fourier transform a dilation and again a chirp,but the lattice Λ I,δ = √ δ Z remains invariant under a modular deformation. Hence,(4.33) G ( g, Λ I,δ ) ∼ = G ( b B g, Λ I,δ ) . This result can be extended in an obvious way. Let S ∈ Sp (2 , R ) and let b S ∈ M p (2 , R ) be thecorresponding metaplectic operator, then,(4.34) G ( b Sg, S Λ I,δ ) ∼ = G ( b S b B g, S Λ I,δ ) . Therefore, given any lattice Λ = S Λ I,δ there are countably many possible windows which lead tothe same Gabor frame bounds. We sum up the results in the following theorem.
Theorem 4.5.
Let S ∈ Sp (2 , R ) , B ∈ Sp (2 , Z ) and let b S and b B be the corresponding metaplecticoperators. Let Λ I,δ = √ δ Z with δ > and let g be a window function. Then (4.35) G ( b Sg, S Λ I,δ ) ∼ = G ( b S b B g, S Λ I,δ ) . Remark.
Whereas the deformations in the previous section have been derived from a continuous,compact group, the deformations in this section are derived from a discrete, non-compact group.Continuous deformation groups will in general change the lattice, whereas the window might stayinvariant under the corresponding deformation. Discrete deformation groups will in general changethe window, whereas the lattice might stay invariant under the corresponding deformation.4.4.
Examples for Generalized Gaussians.
We will now illustrate our intuitive geometricapproach by example. We will use different generalized Gaussians and different lattices.
Example A.
We start with an example inspired by [24]. Let(4.36) Λ H = 1 √ δ s √ (cid:18) cos( π/
6) cos( π/ − sin( π/
6) sin( π/ (cid:19) be a version of the hexagonal lattice of density δ >
1. Since, Λ H is radial symmetric, we choose thestandard Gaussian g as window function and have the Gabor system G ( g , Λ H ) which is a Gaborframe. We apply the dilation matrix M − / on the lattice and the rescaling operator \ M − / onthe window. By theorem 4.3 we know that(4.37) G ( g , Λ H ) ∼ = G ( \ M − / g , M − / Λ H ) . Furthermore, we compute(4.38) M − / Λ H = 1 √ δ (cid:18) cos( π/
4) sin( π/ − sin( π/
4) cos( π/ (cid:19) Z = 1 √ δ S π Z , which is a 45 degrees rotated version of the integer lattice scaled to have density δ > ABOR FRAME SETS OF INVARIANCE 13 (a) τ = π/
12 (b) τ = 0(c) τ = − π/
12 (d) τ = − π/ Figure 1.
Illustration of the action of S τ,m on the lattice and of [ S τ,m on theambiguity function. The small ellipses illustrate the ambiguity functionscentered at lattice points. The ellipses centered at the origin indicate flow linesof the harmonic oscillator.The ambiguity function of g is given by(4.39) Ag ( x, ω ) = e − π ( x + ω ) and the ambiguity function of \ M − / g = g √ is given by(4.40) Ag √ ( x, ω ) = e − π (cid:16) x √ + √ ω (cid:17) . Applying the matrix(4.41) S τ, √ = cos τ √ τ − sin τ √ cos τ ! , derived from the flow of the harmonic oscillator with mass m = √ on the lattice will leave theframe bounds unchanged and we have(4.42) G ( g , Λ H ) ∼ = G (cid:18) g √ , √ d S π Z (cid:19) ∼ = G (cid:18) g √ , √ d S τ, √ S π Z (cid:19) . The deformation process is illustrated in the Figure 1.
Example B.
Let Λ
I,δ = √ δ Z be the scaled integer lattice of density δ > g ( t ) =2 / e − πt be the standard Gaussian. The standard symplectic form J belongs to the modulargroup Sp (2 , Z ). Hence, J Λ I,δ = Λ
I,δ and b Jg = c g with | c | = 1. In this case neither the changeof basis, nor the metaplectic operation have an effect on the Gabor frame. This is due to the factthat we could interpret the change of basis as a rotation of the time-frequency plane. Using thequadratic representation of g , the ambiguity function, we see that a rotation does not have aneffect since it reads Ag ( x, ω ) = e − π ( x + ω ) . Let us now consider the case of g √ and Λ = √ δ S π Z which is a 90 degrees rotated versioncompared to the window in Example A which can also be interpreted as the deformation of thewindow under a change of basis using J as basis. We know the ambiguity function of Ag √ ( x, ω ) = e − π (cid:16) √ x + ω √ (cid:17) . We rotate our lattice by the matrix S − π and apply the corresponding operator d S − π on the window. Hence, the ambiguity function of the new window becomes(4.43) A d S − π g √ ( x, ω ) = Ag √ (cid:16) S − − π λ (cid:17) = Ag (cid:16) M − / S − − π λ (cid:17) = e − π h (cid:16) S − π M / (cid:17) − λ, (cid:16) S − π M / (cid:17) − λ i = e − π (cid:16) M − / S π λ (cid:17) T · (cid:16) M − / S π λ (cid:17) = e − π λ T S Tπ M T − / M − / S π λ = e − π λ T S − π M − / S π λ = e − π √ ( x + xω + ω ) . Therefore, the level lines of the ambiguity function will be ellipses rotated by 45 degrees withsemi-axis ratio equal to √ δ >
1. The action of the metaplectic operator can be interpreted in a very natural andgeometric way as can be seen by the calculations above. (a) A [ S − π g √ (b) A [ S − π g √ Figure 2.
Contour plots of the ambiguity functions of two possible generalizedGaussians which lead to the same frame bounds for the scaled integer lattice ofdensity δ > √ δ S π Z in the opposite direction by S π . Then, wewould have had √ δ J Z as our lattice, which leads basically to the same lattice, but with anotherchoice of basis. Hence, the window as well as the ambiguity function would have changed. In thiscase, we can interpret the deformation as a rotation, a shearing or a choice of a different basis.The interpretation is left to the reader, but the ambiguity function will be(4.44) A d S − π − g √ ( x, ω ) = A d S − π g / √ ( x, ω ) = e − π √ ( x − xω + ω ) . ABOR FRAME SETS OF INVARIANCE 15
The calculations are left to the reader, but the geometric understanding of the lattice deformationis sufficient to understand the deformation of the ambiguity function. The level lines describe thesame ellipses as before only rotated by 90 degrees (see Figure 2).
Remark.
Instead of the ambiguity function, which is commonly used in time-frequency analysis,we could as well have used the Wigner distribution (see appendix A). The Hamiltonian formulationof the harmonic oscillator describes the behaviour of a system in classical mechanics which canalso be formulated as a problem in quantum mechanics. The Hermite functions are eigenstatesof the propagator which corresponds to the flow of the classical problem. Hence, the deformationof the lattice corresponds to a problem in classical mechanics and the deformation of the windowcorresponds to the same problem formulated in terms of quantum mechanics. Therefore, usingthe Wigner distribution, all results can also be interpreted for problems in classical and quantummechanics. A.
The Ambiguity Function
Definition A.1.
The ambiguity function of a function f ∈ L ( R d ) is given by(A.1) Af ( x, ω ) = Z R d f (cid:16) t + x (cid:17) f (cid:16) t − x (cid:17) e − πiω · t dt. In a similar way we define the cross-ambiguity function of two functions f, g ∈ L ( R d )(A.2) A g f ( x, ω ) = Z R d f (cid:16) t + x (cid:17) g (cid:16) t − x (cid:17) e − πiω · t dt. The cross-ambiguity function and hence, just as well the ambiguity function are closely relatedto the short-time Fourier transform (A.3) V g f ( x, ω ) = h f, π ( λ ) g i = Z R d f ( t ) g ( t − x ) e − πiω · t dt. In fact, they only differ by a phase factor and we have(A.4) A g f ( x, ω ) = e πix · ω V g f ( x, ω ) . The difference in the phase factor is due to the fact that the translation and modulation operatorsdo not commute.(A.5) M ω T x = e πix · ω T x M ω , therefore, we have(A.6) V g f ( x, ω ) = h f, M ω T x g i = h T − x/ M − ω f, T x/ g i = e − πix · ω h M − ω T − x/ f, T x/ g i = e − πix · ω h M − ω/ T − x/ f, M ω/ T x/ g i = e − πix · ω h π ( − λ/ f, π ( λ/ g i = e − πix · ω A g f ( x, ω ) . The ambiguity function is somehow a more symmetric time-frequency representation of a signalthan the short-time Fourier transform. In addition, the ambiguity function of a dilated Gaussianis real-valued as we have seen in (4.15). Furthermore, the ambiguity function already determinesa function up to a phase factor [16]. The usual interpretation of the ambiguity function is thatit tells how much a function is spread in time and frequency, similar to the interpretation of the
Wigner distribution in physics. The Wigner distribution is given by(A.7) W f ( x, ω ) = Z R d f (cid:18) x + t (cid:19) f (cid:18) x − t (cid:19) e − πiω · t dt. It is related to the ambiguity function by the symplectic Fourier transform(A.8) W f ( λ ) = F ( Af )( Jλ ) , where F f ( ω ) = R R d f ( t ) e − πiω · t dt is the Fourier transform on L ( R d ).B. Proof of Theorem 4.3
We will now prove equations (4.23) and (4.25) of theorem 4.3. Since the ambiguity functionalready determines a function up to a factor of modulus 1, equation (4.23) and (4.25) are equivalent.First, we note that(B.1) S τ,m M √ m = (cid:18) cos τ √ m sin τ √ m −√ m sin τ √ m cos τ (cid:19) = M √ m S τ , where S τ = S τ, is a rotation by − τ . This means, that imposing the elliptic flow S τ,m on thedilated lattice is the same as rotating the lattice by the corresponding angle followed by the samedilation. The second ingredient we need in order to perform the proof is the covariance principle(B.2) π ( λ ) b S = b Sπ (cid:0) S − λ (cid:1) , where π ( λ ) = M ω T x is the time-frequency shift as defined in (1.1). This implies that(B.3) A (cid:16)b Sf (cid:17) ( λ ) = Af ( S − λ ) . This is a classical result [15] which is also used in [24] and a similar result for the STFT is givenin [16]. Hence, using the fact that(B.4) Ag ( λ ) = Ag ( x, ω ) = e − π ( x + ω ) = e − π h λ, λ i we compute(B.5) A (cid:16) [ S τ,m g m (cid:17) ( x, ω ) = A (cid:16) [ S τ,m \ M √ m g (cid:17) ( x, ω )= Ag (cid:16)(cid:0) S τ,m M √ m (cid:1) − λ (cid:17) = e − π h ( S τ,m M √ m ) − λ, ( S τ,m M √ m ) − λ i = e − π h ( M √ m S τ ) − λ, ( M √ m S τ ) − λ i = e − π h S − τ M − √ m λ, S − τ M − √ m λ i = e − π h M − √ m λ, M − √ m λ i = e − π (cid:16) mx + ω m (cid:17) = Ag m ( x, ω ) . This proves equation (4.25) and hence, the proof of theorem 4.3 is complete.
Acknowledgements.
The author wishes to thank Maurice de Gosson and Karlheinz Gr¨ochenig formany fruitful discussions on the topic. The author wishes to thank the reviewers for helpfulcomments to improve this work. The author was supported by the Austrian Science Fund (FWF):[P26273-N25].
References [1] Luis B. Almeida. The fractional Fourier transform and time-frequency representations.
Signal Processing, IEEETransactions on , 42(11):3084–3091, 1994.[2] Vladimir I. Arnold.
Mathematical methods of classical mechanics . Number 60 in Graduate texts in mathemat-ics. Springer-Verlag, New York, 1989.[3] Ole Christensen.
An Introduction to Frames and Riesz Bases . Applied and Numerical Harmonic Analysis.Birk¨auser, Boston, MA, 2003.[4] Ingrid Daubechies. Time-frequency localization operators: a geometric phase space approach.
InformationTheory, IEEE Transactions on , 34(4):605–612, 1988.[5] Maurice A. de Gosson. Metaplectic representation, Conley–Zehnder index, and Weyl calculus on phase space.
Reviews in Mathematical Physics , 19(10):1149–1188, November 2007.[6] Maurice A. de Gosson.
Symplectic Methods in Harmonic Analysis and in Mathematical Physics , volume 7 of
Pseudo-Differential Operators. Theory and Applications . Birkh¨auser/Springer Basel AG, Basel, 2011.
ABOR FRAME SETS OF INVARIANCE 17 [7] Maurice A. de Gosson. The symplectic egg in classical and quantum mechanics.
American Journal of Physics ,81(5):328–337, 2013.[8] Maurice A. de Gosson. Hamiltonian deformations of Gabor frames: First steps.
Applied and ComputationalHarmonic Analysis , 2014.[9] Maurice A. de Gosson and Franz Luef. Metaplectic Group, Symplectic Cayley Transform, and FractionalFourier Transforms.
Journal of Mathematical Analysis and Applications , 416(2):947–968, 2014.[10] Monika D¨orfler and Jos´e Luis Romero. Frames adapted to a phase-space cover.
Constructive Approximation ,39(3):445–484, 2014.[11] Hans G. Feichtinger. On a new Segal algebra.
Monatshefte f¨ur Mathematik , 92(4):269–289, 1981.[12] Hans G. Feichtinger and Karlheinz Gr¨ochenig. Gabor Frames and Time-Frequency Analysis of Distributions.
Journal of Functional Analysis , 146(2):464–495, 1997.[13] Hans G. Feichtinger and Norbert Kaiblinger. Varying the time-frequency lattice of Gabor frames.
Trans. Am.Math. Soc. , 356(5):2001–2023, 2004.[14] Hans G. Feichtinger and Franz Luef. Gabor analysis and time-frequency methods.
Encyclopedia of Applied andComputational Mathematics , 2012.[15] Gerald B. Folland.
Harmonic analysis in phase space . Number 122 in Annals of Mathematics Studies. PrincetonUniversity Press, 1989.[16] Karlheinz Gr¨ochenig.
Foundations of Time-Frequency Analysis . Applied and Numerical Harmonic Analysis.Birkh¨auser, Boston, MA, 2001.[17] Karlheinz Gr¨ochenig. The Mystery of Gabor Frames.
Journal of Fourier Analysis and Applications , 20(4):865–895, 2014.[18] Karlheinz Gr¨ochenig, Joaquim Ortega-Cerd`a, and Jos´e Luis Romero. Deformation of Gabor systems.
Advancesin Mathematics , 277(0):388–425, 2015.[19] Christopher Heil.
Harmonic Analysis and Applications . Applied and Numerical Harmonic Analysis. Birkh¨auser,Boston, MA, 1 edition, 2006.[20] Yurii Lyubarskii. Frames in the Bargmann space of entire functions. In
Entire and Subharmonic Functions ,page 167. American Mathematical Society, Providence, RI, 1992.[21] Hans Reiter. Metaplectic groups and Segal algebras.
Lecture notes in mathematics , 1382, 1989.[22] Kristian Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space.
American Math-ematical Society. Bulletin. New Series , 26(2):322–328, 1992.[23] Elias Stein and Rami Shakarchi.
Complex Analysis . Princeton University Press, Princeton, NJ, 2003.[24] Thomas Strohmer and Scott Beaver. Optimal OFDM design for time-frequency dispersive channels.
Commu-nications, IEEE Transactions , 51(7):1111–1122, July 2003.
Markus Faulhuber: NuHAG, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
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