FRAMES AND TIME-FREQUENCY ANALYSIS

Transcription

1 FRAMES AND TIME-FREQUENCY ANALYSIS LECTURE 5: MODULATION SPACES AND APPLICATIONS Christopher Heil Georgia Tech heil

2 READING For background on Banach spaces, Hilbert spaces, and operator theory: Chapters 1 2 in C. Heil, A Basis Theory Primer, Birkhäuser, Boston, For background on the Fourier transform: Chapter 9 in C. Heil, A Basis Theory Primer, Birkhäuser, Boston, Today s lecture is based upon: K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, C. Heil, Integral operators, pseudodifferential operators, and Gabor frames, in: Advances in Gabor Analysis, Birkhäuser, Boston, 2003, pp For further reading: C. Heil, History and evolution of the Density Theorem for Gabor frames, J. Fourier Anal. Appl., 13 (2007), pp C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and their Applications, Allied Publishers, New Delhi (2003), pp

3 REVIEW Translation: (T a f)(t) = f(t a), a R. Modulation: (M b f)(t) = e 2πibt f(t), b R. (Regular or Lattice) Gabor (Gah-bor) System: G(g, a, b) = {M bn T ak g} = {e 2πibnt g(t ak)}. Analysis map: C g f = { } f, M bn T ak g. STFT: V g f(x, ξ) = f, M ξ T x g = f(t) g(t x)e 2πiξt dt, (x, ξ) R 2. Orthogonality relations: Vg1 f 1, V g2 f 2 = f1, f 2 g 1, g 2. Inversion: f = 1 γ, g V g f(x, ξ) M ξ T x γ dξ dx (weakly). Modulation space M p (R): f M p = V φ f p = ( V φ f p ) 1/p. 3

4 ADJOINT OF THE STFT Given 1 p, let p be its dual index, defined by 1 p + 1 p = 1. The STFT is (by definition) an isometric map of M p (R) into L p (R 2 ): V φ : M p (R) L p (R), f M p = V φ f p. Therefore it has an adjoint that is a bounded linear map V φ : L p (R 2 ) M p (R). If F L p (R 2 ) is given, then V φ F is a uniquely defined element of Mp (R). Given F L p (R 2 ) and f M p (R), we have Vφ F, f = F, V φ f (definition of the adjoint) = F(x, ξ) V φ f(x, ξ)dx dξ = F(x, ξ) M ξ T x φ, f dx dξ = F(x, ξ) M ξ T x φ, f. Therefore V φ F = F(x, ξ) M ξ T x φ, where we interpret the integral weakly. 4

5 V φ F = F(x, ξ) M ξ T x φ. Corollary 1. If f M p (R) then V φ V φf = f. Proof. V φ V φ f = = V φ f(x, ξ) M ξ T x φ 1 φ, φ V φ f(x, ξ) M ξ T x φ = f (by Inversion). Corollary 1 and most other results, can be formulated in terms of other windows. A first step is to extend to windows in S(R), but then once all the tools are in place, we can extend to any window in M 1 (R). In this sense the Feichtinger algebra M 1 (R) is the correct space of windows for time-frequency analysis. We will concentrate on the Gaussian window for convenience. Corollary 1 does not imply that V φ is invertible. It is injective, but it does not map onto L p (R 2 ). Even so, from f = V φ V φf we obtain the reproducing formula V φ f = V φ V φ V φ f. This will be very useful when we combine it with the next (seemingly technical but very important) result. 5

6 Theorem 2. If F L p (R 2 ), then V φ V φ F F V φφ. Proof. V φ V φ F(u, η) = V φ F, M η T u φ = F, V φ (M η T u φ) = = = F(x, ξ) V φ (M η T u φ)(x, ξ) dx dξ F(x, ξ) M η T u φ, M ξ T x φ dx dξ F(x, ξ) M ξ η T x u φ, φ dx dξ F(x, ξ) V φ φ(u x, η ξ) dx dξ = ( ) F V φ φ (u, η). Remark: Since φ S(R), we know that V φ φ S(R 2 ). In fact, V φ φ is a two-dimensional Gaussian. When going to general windows, the important fact is that if g, γ M 1 (R), then V γ g W(C, l 1 ), so we have both local and global control of V γ g. 6

7 AMALGAM PROPERTIES OF THE STFT V φ V φ F F V φφ. Corollary 3. If f M p (R), then F = V φ f L p (R 2 ) satisfies V φ f = V φ V φ V φf V φ f V φ φ. Theorem 4 (Inclusions for Amalgams). If p 1 p 2 and q 1 q 2, then W(L p 1, l q 1 ) W(L p 2, l q 2 ). Theorem 5 (Convolution for Amalgams). For appropriate indices, W(L p 1, l q 1 ) W(L p 2, l q 2 ) W(L p 1 L p 2, l q 1 l q 2 ). Consequently, L p W(L, l 1 ) = W(L p, l p ) W(L, l 1 ) W(L 1, l p ) W(L, l 1 ) W(L 1 L, l p l 1 ) W(L, l p ). 7

8 V φ V φ F F V φφ L p W(L, l 1 ) W(L, l p ) Corollary 6. If f M p (R), then V φ f W(L, l p ). Proof. We have V φ f = V φ V φ V φf V φ f V φ φ L p W(L, l 1 ) W(L, l p ). Amalgam spaces are solid, i.e., g W(L p, l q ) and h g = h W(L p, l q ). Therefore V φ f W(L, l p ). The mapping in Corollary 6 is bounded; a slightly more careful proof shows that V φ f W(L,l p ) C f M p V φ φ W(L,l 1 ). Not every space is solid! Spaces defined in terms of the Fourier transform, like the Wiener algebra A(R) or the Sobolev spaces H s (R), are not solid. The modulation spaces M p (R) are not solid either. 8

9 GABOR FRAMES IN THE MODULATION SPACES f M p (R) = V φ f W(L, l p ) The analysis map is a sampling of the STFT: { { } C g f = f, M bn T ak g } = V g f(ak, bn). Samples are a local property of the STFT, while what l p space they belong to is a global property. We have a relation between the samples and the STFT because the STFT belongs to an amalgam space! Theorem 7. Fix a, b > 0. Analysis (using the Gaussian window) is a bounded map of M p (R) into l p (Z 2 ): C φ : M p (R) l p (Z 2 ) is bounded. Proof. Let Q kn = [ak, a(k + 1)] [bn, b(n + 1)]. Then C φ f p l p = { f, M bn T ak φ } p = V φ f(ak, bn) p p sup V φ f(x, ξ) p (x,ξ) Q kn = V φ f p W(L,l p ) C f p M p V φ φ p W(L,l 1 ). 9

10 Restated explicitly, there exists a constant B > 0 ( B = C V φ φ p W(L,l )) such that 1 f, M bn T ak φ p B f p M, all f M p (R). p In essence, G(φ, a, b) is a Bessel sequence for M p (R). The same is true if we replace φ by any other window g M 1 (R). Since synthesis is the adjoint of analysis, we obtain the following. Corollary 8. Let R φ : l p (Z 2 ) M p (R) be the adjoint of C φ : M p (R) l p (Z 2 ). Then R φ is bounded. Assignment 1. Prove that if c = {c kn } has only finitely many nonzero components, then R φ c = c kn M bn T ak φ. Here (and elsewhere) we are sweeping under the rug some technical issues concerning p =. Sometimes this involves interpreting series or integrals in a weak or weak sense. 10

11 Theorem 9. If 1 p < and c l p (Z 2 ), then R φ c = c kn M bn T ak φ, (1) with (unconditional) convergence in M p -norm. Proof. Fix c l p (Z 2 ), and let G = V φ φ and χ = χ [0,a] [0,b]. Since G W(L, l 1 ), we have 0 G a jl T (aj,bl) χ, j,l Z where a = {a jl } is summable. Aside. To show that a series n=1 converges, we must show that the partial sums are Cauchy. A difference of two partial sums has the form N M N =. n=1 n=1 n=m+1 We want this < ε when M, N are large enough. For unconditional convergence, we want < ε whenever F is a finite subset of the index set and min(f) is large enough. n F 11

12 Let F be a finite subset of Z 2, and let c F be c restricted to F. By Assignment 1, R φ c F = c kn M bn T ak φ, so to estimate R φ c F M p = V φ (R φ c F ) L p we first compute that V φ (R φ c F ) = (k,n) F c kn V φ (M bn T ak φ) (k,n) F (k,n) F (k,n) F (k,n) F (k,n) F c kn V φ ( ak, bn) c kn T (ak,bn) G ( ) c kn T (ak,bn) a jl T (aj,bl) χ j,l Z c kn c kn a jl T (aj+ak,bl+bn) χ j,l Z a j k,l n T (aj,bl) χ j,l Z = j,l Z ( ) c kn a j k,l n T (aj,bl) χ (k,n) F = j,l Z( cf a ) jl T (aj,bl) χ. 12

13 We ve shown: (k,n) F c kn V φ (M bn T ak φ) j,l Z( cf a ) jl T (aj,bl) χ. Therefore R φ c F M p = V φ (R φ c F ) L p = = (k,n) F (k,n) F c kn M bn T ak φ M p c kn V φ (M bn T ak φ) L p ( cf a ) jl T (aj,bl) χ Lp j,l Z = C c F a l p (it s a step function) C c F l p a l 1 < ε for all large enough tails F. This shows that the series R φ c = c kn M bn T ak φ (2) converges unconditionally in M p -norm, and repeating the above work with F = Z 2 shows that R φ c M p B c l p, where B = C a l 1. 13

14 Finally, repeating Assignment 1 with F = Z 2 shows that (2) really is the adjoint of the analysis map C φ. Remark: For general windows, the only important point is to have V γ g W(L, l 1 ), which is ensured if g, γ M 1 (R). Summary: If g, γ belong to M 1 (R), then C g : M p (R) l p (Z 2 ) and R γ : l p (Z 2 ) M p (R) are bounded, and therefore S g,γ f = R γ C g f = is a bounded mapping of M p (R) into itself. f, M bn T ak g M bn T ak γ, f M p (R), What is S g,γ? If G(g, a, b) is a Gabor frame for L 2 (R) and γ = g is the dual window, then S g,γ = I on the space L 2 (R). 14

15 What all this work shows is that if we start with a Gabor frame for L 2 (R) whose window g belongs to the correct window class M 1 (R) ( W(C, l 1 )), then it is a Gabor frame for every modulation space M p (R), not just L 2 (R). Corollary 10. Assume g M 1 (R) is such that G(g, a, b) is a Gabor frame for L 2 (R), and let g be the dual window. Assume g M 1 (R) (this turns out to be automatic). Then there exist constants A, B > 0 such that for each 1 p < we have A f p M f, M p bn T ak g p B f p M, f M p (R), p and f = f, M bn T ak g M bn T ak g = f, M bn T ak g M bn T ak g, f M p (R). (3) Proof. The upper inequality in (3) is a restatement of the boundedness of C g : M p (R) l p (Z 2 ). A close examination of the preceding results shows that the bound (which is the operator norm of C g ) does not depend on p. Fix f S(R), which is dense in M p (R). Then R g C g f = f, M bn T ak g M bn T ak g (convergence in M p -norm). 15

16 R g C g f = f, M bn T ak g M bn T ak g (convergence in M p -norm). Because G(g, a, b) is a frame for L 2 (R), we have f = f, M bn T ak g M bn T ak g (convergence in L 2 -norm). This implies (after some work) that R g C g f = f. Using the boundedness of R g and C g and the density of the Schwartz space, this extends to all f M p (R). Finally, f M p = R g C g f M p R g g Cf M p, which gives the lower inequality in (3). Remark 1: For p =, we still obtain the norm equivalence, but the series converge weak instead of in norm. Remark 2: These results extend to general mixed-norm, weighted modulation spaces M p,q w (R). Remark 3: A deep result of Gröchenig and Leinert shows that if g M 1 (R) and G(g, a, b) is frame for L 2 (R), then g belongs to M 1 (R). 16

17 Corollary 11. Assume g M 1 (R) is such that G(g, a, b) is a tight Gabor frame for L 2 (R). Then f M p (R) if and only if C g f l p (Z 2 ), and f = 1 A f, M bn T ak g M bn T ak g, f M p (R), with unconditional convergence of this series in M p -norm. Further results: Although Gabor frames do not yield unique representations and hence are not Schauder bases for the modulation spaces, there is a remarkable related construction of Wilson bases that share the same norm equivalence and series representation theorems as Gabor frames with unique representations. Consequently Wilson bases are unconditional Schauder bases for the modulation spaces. As a consequence, the norm equivalence implies an actual isomorphism between M p and l p : M p (R) = l p (Z 2 ) (in the sense of topological isomorphism). This implies additional results. In particular, the modulation spaces are ordered identically to the l p spaces: 1 p q = M p (R) M q (R). 17

18 AN APPLICATION TO PSEUDODIFFERENTIAL OPERATORS Definition 12. (a) Given k L 2 (R 2 ), the integral operator A k : L 2 (R) L 2 (R) is A k f(x) = k(x, y) f(y) dy, f L 2 (R). (Analogous to matrix-vector multiplication: (Au) j = j a iju i.) (b) Given a symbol function σ L 2 (R 2 ), the Weyl transform of σ is L σ : L 2 (R) L 2 (R) given by L σ f(x) = = ( ( x + y ) σ 2, ξ e 2πi(x y)ξ f(y) dy dξ ( x + y ) σ 2, ξ e 2πi(x y)ξ dξ ) f(y) dy. (c) Given a symbol function τ L 2 (R 2 ), the Kohn-Nirenberg transform of τ is K τ : L 2 (R) L 2 (R) defined by K τ f(x) = τ(x, ξ) ˆf(ξ) e 2πixξ dξ = τ(x, ξ) e 2πi(x y)ξ f(y) dy dξ. L σ and K τ are pseudodifferential operators. 18

19 Contrast the following formulas. Identity: f(x) = ( f ) (x) = f(ξ) e 2πiξx dξ Filtering (convolution): (f g)(x) = (f g) (x) = ( f ĝ ) (x) = f(ξ) ĝ(ξ) e 2πiξx dξ (f with frequency amplitudes adjusted, like a signal through an equalizer.) Time-varying filtering (Kohn-Nirenberg transform): K τ f(x) = ˆf(ξ) τ(x, ξ) e 2πixξ dξ (f with frequency amplitudes adjusted continuously with time.) (f after transmission through a wireless channel; superposition of time delays and doppler shifts.) 19

20 To convert from Weyl to integral operator: ( x + y ) k(x, y) = σ 2, ξ e 2πi(x y)ξ dξ This is a composition of a change of variable and a partial Fourier transform, and M p is invariant under both of these. Therefore σ M p (R 2 ) k M p (R 2 ) τ M p (R 2 ). As far as modulation properties go, we can work either with integral operators or pseudodifferential operators. We ll focus on integral operators, but the application (and generalization) we have in mind is based on pseudodifferential operators. 20

21 INTEGRAL OPERATORS ON L 2 (R 2 ) If k L 2 (R 2 ), then the corresponding integral operator A k is bounded on L 2 (R), because A k f 2 2 = A k f(x) 2 dx = ( = f 2 2 k(x, y) f(y) dy 2 dx ) ( k(x, y) 2 dy k(x, y) 2 dy dx ) f(y) 2 dy dx = f 2 2 k 2 2. Thus, A k f 2 k 2 f 2. Taking the supremum over all unit vectors f, we see that A k is bounded and its operator norm satisfies A k = sup f 2 =1 A k f 2 k 2. 21

22 More is true: A k is a compact operator when k is square-integrable. To prove this, let {e n } n N be an ONB for L 2 (R). Set e mn (x, y) = (e m ē n )(x, y) = e m (x) e n (y). Then {e mn } m,n N is an ONB for L 2 (R 2 ). Since k L 2 (R 2 ), we have k = k, e mn e mn, m=1 n=1 with convergence in L 2 -norm. Make an approximation to k: N N k N = k, e mn e mn. m=1 n=1 Let A N = A kn be the corresponding integral operator: N N A N f(x) = k N (x, y) f(y) dy = k, e mn e mn (x, y) f(y) dy m=1 n=1 N N = k, e mn e m (x) e n (y)f(y) dy m=1 n=1 = N N k, e mn f, e n e m (x). m=1 n=1 22

23 A N f = N N k, e mn f, e n e m m=1 n=1 span{e 1,..., e N }. Hence A N has finite-dimensional range, it is a linear, continuous, finite-rank operator. All such operators are compact (the image of the unit ball has compact closure). The integral operator A does not have finite-dimensional range, but A k A N = A k kn k k N 2 0. It is the limit (in operator norm) of compact operators, and this implies that it is compact. In general A k is not self-adjoint: A k is self-adjoint k(x, y) = k(y, x), but on the other hand the composition A k A k is both compact and self-adjoint. 23

24 The (most basic) version of the Spectral Theorem says that A k A k has an ONB of eigenvectors and corresponding nonnegative eigenvalues µ n 0. The singular values of A k are s n = µ n. If A k happens to be self-adjoint, then s n = λ n, where λ n are the eigenvalues of A k. Theorem 13. If k L 2 (R 2 ), then the singular numbers of A k satisfy s 2 n = k 2 2 <. (Therefore we say that A k is a Hilbert Schmidt operator.) n=1 We say that A k is trace-class if the singular values are summable, i.e., if s n <. n=1 24

25 We can give a very simple sufficient condition to be trace-class. Theorem 14. If k M 1 (R 2 ), then A k is trace-class. Proof. Choose g M 1 (R) so that G(g, a, b) is a Parseval frame for L 2 (R). Enumerate this Gabor system as G(g, a, b) = {e n } n N, and let e mn = e n ē m. Then {e mn } m,n N is a Parseval frame for every modulation space M p! In particular, k = k, e mn e mn m=1 n=1 m=1 n=1 with convergence in M 1 -norm, and k, e mn <. Via linearity and absolute convergence, A k = k, e mn A emn. (4) m=1 n=1 As before, A emn has one-dimensional range. It has a single nonzero singular value, and that singular value is g 2 2. As a consequence A e mn is a trace-class operator. The space of trace-class operators is a Banach space, so an absolutely convergence series of trace-class operators is trace-class. Therefore equation (4) converges absolutely in trace-class norm, so A k is traceclass. 25

26 More refined analysis using weighted modulation spaces gives, with a simple proof, an improvement to known results: If k Ms 2 (= intersection of weighted L2 space and a Sobolev space) then A k is trace-class. Further, the invariance of the modulation spaces gives us equivalent theorems for pseudodifferential operators instead of integral operators. 26

27 Thank You! 27