ON ACCUMULATED SPECTROGRAMS

Transcription

1 ON ACCUMULATED SPECTROGRAMS Abstract. We consider the problem of optimizing the concentration of the spectrogram of a function within a given set and give asymptotics for the timefrequency profile of the corresponding solutions. The main result shows that these solutions are organized in such a way that they almost tile the target time-frequency spectrum. This is made precise by means of a detailed analysis of the accumulated spectrogram, which is the sum of the spectrogram of the most significant solutions to the time-frequency concentration problem. We also present an application in signal processing concerning the construction of an approximation of a time-frequency filter in a robust fashion, from information on the spectrogram of a few of its eigenmodes.. Introduction and results.. The time-frequency localization problem. The spectrogram of a function f L 2 (R d ) is defined, by means of a window function g L 2 (R d ), g 2 =, as the square of the absolute value of the short-time Fourier transform: () V g f(z) = f(t)g(t x)e 2πiξt dt, z = (x, ξ) R d R d. R d The number V g f(x, ξ) quantifies the importance of the frequency ξ on f near x. Hence, the spectrogram V g f 2 gives a picture of the distribution of the timefrequency content of f. The Gaussian is an usual choice for the window g, because it provides optimal resolution in both time and frequency. The uncertainty principle in Fourier analysis, in its several versions, sets a limit to the possible simultaneous concentration of a function and its Fourier transform. In terms of the spectrogram, the uncertainty principle can be roughly recasted as follows: if a function f is concentrated inside a region R 2d, then the area of must be at least (see for example the recent survey [27]). For example, if f L 2 (R d ) has norm and V gf(z) 2 dz ε, then it follows that 2 d ( ε) 2 (see [4, Theorem 3.3.3]). Besides the basic restrictions on its measure, not much is known about the possible shapes that such a set can assume. In this article we investigate this question in the following way. We consider the problem of maximizing the concentration of the spectrogram of a function within and give asymptotics for the solutions of that problem, relating the shapes of corresponding spectrograms to. To be precise, let R 2d be a compact set and consider the problem of finding functions whose spectrogram (with respect to a certain window function g) is K. G. was supported in part by the project P26273-N25 and the National Research Network S06 SISE of the Austrian Science Fund (FWF). J. L. R. gratefully acknowledges support by the project M586-N25 of the Austrian Science Fund (FWF).

2 2 optimally concentrated within (2) Problem: maximize V g f(z) 2 dz, with f 2 =. The advantage of studying the time-frequency concentration problem using L 2 - norms is that the solution is given by the spectral analysis of a certain linear operator. The time-frequency localization operator with symbol is defined as [5, 6, 25, 4, 9] (3) H f(t) = V g f(x, ξ)g(t x)e 2πiξt dxdξ, t R d. It can be shown that if is compact, then H is self-adjoint and compact, and hence it can be diagonalized as (4) H f = k λ k () f, h k h k, f L 2 (R d ), with { h k : k } and orthonormal subset of L 2 (R d ) and λ k () 0. (The functions h k and the eigenvalues λ k() depend on the choice of the window g, but we do not make this dependence explicit in the notation.) The reason why H is useful to study (2) is that H f, f = V g f(x, ξ) g( x)e 2πiξ, f dxdξ = V g f(x, ξ) 2 dxdξ. Consequently, h, the first eigenfunction of H, solves (2): λ = H h, h = Vg h (z) { } 2 dz = max V g f(z) 2 dz : f 2 =. If the set is small, we expect λ = Vg h 2 to be small because, as a consequence of the uncertainty principle, no spectrogram fits inside. On the other hand, if is big we expect (2) to have a number of approximate solutions, since a number of spectrograms fit inside. This intuition is made precise by studying of the eigenvalues of H. From the min-max lemma for self-adjoint operators: (5) λ k = V g h k (z) { } 2 dz = max V g f(z) 2 dz : f 2 =, f h,..., h k. Hence, the profile of the eigenvalues of shows how many orthogonal functions have a spectrogram well-concentrated within. The standard asymptotics for the eigenvalues of H involve dilating a fixed set and read as follows. Proposition.. Let g L 2 (R d ) have norm and let R 2d be a compact set. Then for each δ (0, ) (6) # {k : λ k (R ) > δ} R, as R +. Proposition. was proved with additional assumptions on the boundary of by Ramanathan and Topiwala [26] and in the present generality by Feichtinger and Nowak [2]. For sets with smooth boundary, more refined asymptotics are available [5, 26, 6, 2, 8] (see also Section 3). These results parallel the fundamental results

3 ON ACCUMULATED SPECTROGRAMS 3 (a) A domain with the shape of a star and area 23. (b) A plot of the eigenvalues illustrating illustrating Proposition.. Note that the star domain has considerable perimeter in relation to its area Figure. The eigenvalues of a time-frequency localization operator with gaussian window. for Fourier multipliers (ideal low-pass filters) [9, 8, 20, 2]. See Figure for a numerical example for Proposition...2. Accumulation of spectrograms. The asymptotics in (6) imply that, after sufficiently dilating, the L 2 -concentration V gh k (z) 2 dz is close to for k n and decays for large k. The purpose of this article is to refine the description of the time-frequency localization of the eigenfunctions h k by showing that the corresponding spectrograms approximately tile the set. More precisely, we consider the following function, called the accumulated spectrogram: A ρ (z) := V g h k (z) 2, z R 2d, k= where A := is the smallest integer greater than or equal to. The goal of this article is to prove estimates that show that ρ looks approximately like (the characteristic function of ). Since 0 V g h k (z) 2, this means that the spectrograms V g h 2,..., V g h A 2 are approximately a partition of unity on. Indeed, numerical experiments show that ρ resembles a bump function on plus a certain tail around its boundary (see Figure 2). The size of this tail does grow when grows but at a smaller rate compared to. Our main results will validate these observations. To argue that the tail in ρ is asymptotically smaller than we consider dilations of a fixed set and rescale ρ R by a factor of R. We then have the following result. Theorem.2. Let g L 2 (R d ) have norm and let R 2d be compact. Then ρ R (R ), in L (R 2d ), as R +.

4 4 (a) The accumulated spectrogram plotted over the domain. (b) A cross-section of the domain and the accumulated spectrogram. Figure 2. A domain with the form of a star. Under mild regularity assumptions we give a more quantitative estimate, that explains the tail in ρ as the smearing effect of the window g. To quantify the statement in Theorem.2 we assume that g satisfies the following time-frequency concentration condition: (7) g 2 M := R 2d z V g g(z) 2 dz < +. We let M (R d ) denote the class of all L 2 (R d ) functions satisfying the previous condition. It follows easily that the Schwartz class is contained in M (R d ). (The class M is closely related to the modulation space M 2 /2 (Rd ) - see []). Theorem.3. Let g M (R d ) be normalized in L 2 and let R 2d be a compact set with finite perimeter. Then for all p + ρ ( Θ) ( ) p p d p + 4 g M, where Θ := V g g 2, is the perimeter of, and for p = we interpret: p d p =. The technical meaning of the term set of finite perimeter is clarified in Section 3. A compact set R 2d with smooth boundary is of finite perimeter and its perimeter is the 2d dimensional measure of its topological border. Theorem.3 implies the following estimate (for ) (8) ρ C, where C is a constant that depends on the window g (see Corollary 5.). To interpret (8) note that ρ = A = + O() and =. Hence, in analogy with the asymptotics in (6), the difference between ρ and is much smaller than these two functions, when the area of is much bigger than its perimeter.

5 ON ACCUMULATED SPECTROGRAMS 5 (a) A domain of area 23, the spectrograms of the eigenfunctions #, 7 and 7, and the accumulated spectrogram. The window is a Fourier invariant Gaussian. The shading of each eigenfunction is relative to its own scale of values. (b) The same experiment with a Gaussian window g that is more concentrated in frequency than the one used in (a) (and consequently less concentrated in time) Figure 3. The first eigenfunctions of a time-frequency localization operator with different windows, showing two different (almost) tilings on the same domain..3. Strong orthogonality. We stress some consequences of the results. First, the estimate in (8) is a strong almost-orthogonality statement about the eigenfunctions h,..., h A. Indeed, according to (5), the set {V g (h ),..., V g (h A )} is characterized as an orthogonal subset of L 2 (R 2d ), consisting of functions that are maximally concentrated within among all functions of the form V g f, with f L 2 (R d ). The orthogonality of the functions V g (h k ) could in principle be due to cancellations. The fact that ρ = A V g h k 2 approximately looks like implies that the essential supports of the functions V g (h ),..., V g (h A ) are asymptotically disjoint, which is a strong form of orthogonality. In addition, these essential supports almost exhaust the set. We expect this observation to became a useful technical tool in time-frequency analysis. Second, we point out that the shape of the spectrogram of each individual eigenfunction is difficult to describe, since it depends in a intricate manner on the underlying windows g (see Figure 3). Nevertheless, (8) means the spectrograms of the whole family {h,..., h A } asymptotically tile the set ; the particular way in which this happens depends on g..4. Ginibre s law. Before going into the technical work, it is instructive to discuss a case where all the objects can be computed explicitly. Let g(t) := 2e πt 2, t R be the one-dimensional Gaussian, normalized in L 2. Let D := { (x, ξ) R 2 : x 2 + ξ 2 } be the unit disk. In [5], Daubechies calculated the eigenfunctions and eigenvalues of the time-frequency localization operator with window g and domain the disk

6 6 R D. For all R > 0, the eigenfunctions of H R D are the Hermite functions: (9) h k+ (t) = ( ) k (2 k k! π) 2 e t2 /2 dk dt k e t2, k 0. It is a remarkable fact, due to the symmetries of g and D, that the functions h k do not depend on R. Writing z := x + iξ C, the short-time Fourier transform of the Hermite functions with respect to g is ( ) π V g h k (z) = e πixξ k 2 z k e π z 2 /2, k 0. k! Hence the corresponding spectrograms are V g h k (z) 2 = πk k! z 2k e π z 2, k 0. The accumulated spectrogram corresponding to R D is Theorem.2 says that ρ R D (Rz) = R 2 2π k=0 ρ R D (z) = R 2 2π k=0 π k k! z 2k e π z 2. π k k! R2k z 2k e π Rz 2 D (z), in L (C, dxdξ) as n +. This formula is one of Ginibre s limit laws [3]. (Of course, in this case more precise asymptotics are possible.) Notice the following universality property in Theorem.2: the limit function does not depend on the window g. (For an overview of the concept of universality in several contexts see [7])..5. Technical overview. As commonly done in the literature, we transfer the problem to the range of the short-time Fourier transform V g L 2 (R d ). This is a reproducing kernel subspace of L 2 (R 2d ) and the time-frequency localization operator H translates into a Toeplitz operator on V g L 2 (R d ) (multiplication followed by projection). Our proof begins with the observation that ρ is the diagonal of the integral kernel of the orthogonal projection operator P n from V g L 2 (R d ) onto the linear span of the functions {V g h,..., V g h n }. We then asymptotically compare this operator to the Toeplitz operator, which has an explicit kernel on which we can do direct estimates. The relation between the Toeplitz operator and P n can be understood as a threshold on its eigenvalues. We then resort to the rich existing information on these [9, 8, 20, 2, 5, 26, 6, 2, 8]..6. Organization. Section 2 introduces some basic tools from analysis in phasespace, including the interpretation of time-frequency localization operators as Toeplitz operators on a certain reproducing kernel space. In Section 3 we recall and slightly refine estimates on the profile of the eigenvalues of time-frequency localization operators. Section 4 develops a number of technical estimates that are then used in Section 5 to prove the main results. Finally, Section 6 briefly presents an application to the field of signal processing.

7 ON ACCUMULATED SPECTROGRAMS 7 2. Phase-space tools We now introduce the basic phase-space tools related to the short-time Fourier transform. 2.. The range of the short-time Fourier transform. Throughout the article we let g L 2 (R d ) be a unit norm function. The short-time Fourier transform with window g - cf. () - then defines an isometry V g : L 2 (R d ) L 2 (R 2d ). (This is a consequence of Plancherel s theorem, see for example [4, Chapter ]). The adjoint of V g is Vg : L 2 (R 2d ) L 2 (R d ) Vg f(t) = F (x, ξ)g(t x)e 2πiξt dxdξ, t R d. R d R d Let H := V g L 2 (R d ) L 2 (R 2d ) be the (closed) range of the V g. The orthogonal projector P H : L 2 (R 2d ) H is given by P H = V g Vg. Explicitly, it is an integral operator (0) P H f(z) = f(z )K(z, z )dz, z = (x, ξ) R 2d, R 2d with integral kernel () K(z, z ) = V g g(z z )e 2πiξx, z = (x, ξ), z = (x, ξ ) R 2d. Using this description of P H it follows that H is a reproducing kernel Hilbert subspace of L 2 (R 2d ). This means that each function f H is continuous and satisfies: f(z) = f(z )K(z, z )dz, for all z R 2d. The function K is called the reproducing kernel of H. Since K is the integral kernel of an orthogonal projection, it satisfies K(z, z ) = K(z, z) and (2) K(z, z ) = K(z, z )K(z, z )dz, z, z R 2d. R 2d From now on, we use the notation Θ(z) := V g g(z) 2, z R 2d. Then Θ L (R 2d ), Θ =, Θ(z) = Θ( z) and (3) K(z, z ) 2 = Θ(z z ), z, z R 2d. Throughout the article we keep the notation we have just introduced Toeplitz operators. For a compact set R 2d, the (Gabor-Toeplitz) operator M : H H is It is explicitly given by M f := P H ( f), M f(z) = (f H). f(z )K(z, z )dz, where K is the reproducing kernel of H (cf. ()). We now note some properties of Toeplitz operators. These are well-known but we lack a precise reference.

8 8 Lemma 2.. Let g L 2 (R d ) have unit norm and let R 2d be a compact set. Then the M : L 2 (R d ) L 2 (R d ) is trace-class and it satisfies (4) 0 M I. In addition, the traces of M and M 2 are given by (5) trace(m ) = K(z, z)dz =, (6) trace(m) 2 = Θ(z z )dzdz. Proof. For f H we compute M f, f = P H ( f), f = f, f = f(z) 2 dz. This gives (4), and in particular shows that M is bounded and positive. Let M : L 2 (R 2d ) L 2 (R 2d ) be the inflated operator M (f) := P H ( P (f)). Hence, with respect to the decomposition L 2 (R 2d ) = H H, M is given by (7) M = [ M The advantage of working with M is that it is defined on L 2 (R 2d ) and consequently its spectral properties can be easily related to its integral kernel. Using (0) we get the following formula for M M f(z) = f(z ) (z )K(z, z )K(z, z )dz dz, f L 2 (R 2d ). R 2d R 2d That is, M has integral kernel K M (z, z ) := (z )K(z, z )K(z, z )dz. Using (3) we compute K M (z, z)dz = (z )Θ(z z )dzdz = Θ =. R 2d R 2d R 2d Since M is positive, so is M by (7). Hence, the previous calculation shows that M is trace-class and trace(m ) = (see for example [28, Theorems 2.2 and 2.4]). Using (7) we deduce that M is trace class and trace(m ) = trace(m ) =. For the formula in (6), we use again the fact that M is positive to get trace(m) 2 = trace(m 2 ) = K M 2 2 = K M (z, z )K M (z, z)dzdz R 2d R 2d = (w) (w ) K(z, w)k(w, z )K(z, w )K(w, z)dzdz dwdw. R 2d R 2d R 2d R 2d Using (2) and (3) it follows that trace(m) 2 = (w) (w )K(w, w)k(w, w)dwdw R 2d R 2d = (w) (w )Θ(w w )dwdw, R 2d R 2d as desired. ].

9 ON ACCUMULATED SPECTROGRAMS Time-frequency localization. The time-frequency localization operator H : L 2 (R d ) L 2 (R d ) from (3) can be written using V ϕ and V ϕ as Therefore, H f = V g ( V g f), f L 2 (R d ). (V g H V g )f = P H ( f), f H. This says that H and the Toeplitz operator M are related by (8) Vg M V g = H. As a consequence, M and H enjoy the same spectral properties. This, together with Lemma 2., justifies the diagonalization of H discussed in the Introduction (cf. (4)). With that notation, we deduce that M can be diagonalized as M f = k λ k () f, ϕ k ϕ k, f H, where ϕ k = V gh k. In addition, the accumulated spectrogram can be written as A ρ (z) := ϕ k (z) 2, z R 2d. k= 2.4. Lieb s uncertainty principle. We quote the following inequality for the short-time Fourier transform due to Lieb [22] (a proof can also be found in [4, Theorem 3.3.2]). Proposition 2.2. Let f, g L 2 (R d ) and 2 p, then ( ) d 2 p V g f L p (R d ) f 2 g 2, p where for p = we interpret: ( 2 p ) d p =. 3. Profile of the eigenvalues Let g L 2 (R d ) have unit norm and consider the time-frequency localization operator H associated with a compact set. The fundamental asymptotics on the profile of the eigenvalues {λ k () : k } are given by Proposition., due to Ramanathan and Topiwala in a weaker form and to Feichtinger and Nowak in full generality [26, 2]. For sets with smooth boundary, Ramanathan and Topiwala [26] quantified the order of convergence in (6). Later De Marie, Feichtinger and Nowak obtained fine asymptotics for families of sets where certain geometric quantities are kept uniform [8] - the dilations of a set with smooth boundary being one such an example. Moreover their results are also applicable to the hyperbolic setting [8]. We will use the following variation of the result in [26]. Recall that a set E R d is said to have finite perimeter if its characteristic function E is of bounded variation (see for example [0, Chapter 5]). In this case, its perimeter is E := Var( E ). The number E is the 2d Hausdorff measure of the measure-theoretic boundary of E, which for regular domains coincides with the topological boundary (see [0, Chapter 5]).

10 0 Proposition 3.. Let R 2d be a compact set with finite perimeter and assume that g M (R d ). Then for each δ (0, ), where C δ = max{ δ, δ }. # {k : λ k () > δ} C δ g 2 M, The proof of Proposition 3. is a variation of the argument in [26]. Before giving its proof, we point out that [8] provides an asymptotic converse inequality, where, for certain classes of sets and small δ, the error # {k : λ k () > δ} is also shown to be bounded from below by. While Ramanathan and Topiwala use the co-area formula we resort to the following lemma, that we will use again in the article. Lemma 3.2. Let f BV(R d ) and ϕ L (R d ) be such that ϕ =. Then f ϕ f Var(f) x 2 ϕ(x) dx. R d Here, Var(f) denotes the total variation of f. Proof. Assume first that f C (R d ) and estimate f ϕ f = f(y)ϕ(x y) dy f(x) dx R d R d = (f(y) f(x))ϕ(x y) dy dx R d R d = (f)(t(x y) + y), x y ϕ(x y)dtdy dx R d R d 0 (f)(t(x y) + y) 2 x y 2 ϕ(x y) dxdydt = R d R d R d 0 R d 0 (f)(tx + y) 2 x 2 ϕ(x) dxdydt = (f) 2 R d x 2 ϕ(x) dx. For a general bounded variation function f, there exists a sequence {f k : k } BV (R d ) C (R d ) such that f k f in L and (f k ) 2 Var(f) (see [0, Theorem 5.2.2]]. The desired estimate now follows by approximation. We now quote the standard trace-norm estimate normally used to derive lower bounds on eigenvalues of Toeplitz operators. This estimate is ubiquitous in the literature - see for example [20, Th. ], [26, Th. 2] or [2, Remark (iii)]. For lack of an exact quotable reference we sketch a proof following [2]. Lemma 3.3. Let g L 2 (R d ) be of norm and let R 2d be compact. Then { } # {k : λ k () > δ} max δ, Θ(z z )dzdz δ.

11 ON ACCUMULATED SPECTROGRAMS Proof. Let F : [0, ] R be the function { t, if t δ, F (t) := t, if t > δ. and note that F (t) max { estimate δ, δ } (t t 2 ), for t [0, ]. Since 0 M we # {k : λ k () > δ} = F (M ) F (M ) { } max δ, trace(m M 2 δ ). The formulas in Lemma 2. complete the proof. We now obtain Proposition 3.. Proof of Proposition 3.. Recall that Θ(z) = Θ( z), let f := and note that Θ(z z )dzdz = (f Θ)(z ) f(z )dz f Θ f. We now combine Lemmas 3.2 and 3.3 and the desired conclusion follows. (In [26] the authors use the slightly stronger hypothesis: Θ(z)dz z >r r p, for some p > ; the focus there is on orders of convergence). 4. Some technical estimates We now derive a number of lemmas. In the next section we combine them in several ways to obtain the main results. Recall that the ongoing assumption is that g L 2 (R d ) has unit norm and we use the notation from Section 2. The following lemma is one of the key observations in our proof. Lemma 4.. Let R 2d be a compact set. Then the following formula holds ( Θ)(z) = k λ k () ϕ k (z) 2, z R 2d. Proof. Let K(z, z ) = V g g(z z )e 2πiz 2z be the reproducing kernel of H. Hence for all f H, f(z) = K(z, z )f(z ) dz = f, K z, z R 2d, R 2d where K z := K(z, ) H. Let us now compute M K z, K z = K z, K z = (z )K z (z )K z (z ) dz R 2d = (z )Θ(z z ) dz = ( Θ)(z). R 2d Let { } φ k H be an orthonormal basis of ker(m k ) (eventually empty). Hence { } ϕ { } k φ k k is an orthonormal basis of H and therefore the reproducing kernel k

12 = p d p ( + 2E() ). 2 K can be written as K(z, z ) = k ϕ k (z)ϕ k (z ) + k φ k (z)φ k (z ). Using this we compute M K z, K z = M ϕ k (z)ϕ k, ϕ j (z)ϕ j k j ϕ k (z)ϕ j (z) M ϕ k, ϕ j = k,j = k ϕ k (z) 2 λ k (). The lemma follows by equating the two formulae for M K z, K z. We now use the formula in Lemma 4. to bound the error between ρ() and. Lemma 4.2. Let R 2d be a compact set and consider the quantity (9) E() := A Then for all p + ρ ( Θ) p p d p where for p =, p d p =. Proof. Using Lemma 4. we have that k= λ k(). ( ) + 2E(), ρ (z) ( Θ)(z) = k (L k λ k ) ϕ k (z) 2, where L k =, if k A and 0 otherwise. By Lieb s inequality (Proposition 2.2) we have ϕ k 2 p = Vg h 2 k d 2p p p, with the usual interpretation when p =. Using (5) we estimate ρ ( Θ) ( ) ( A p p d p L k λ k = p d p ( λ k ) + ( A = p d p A 2 k λ k k= k>a λ k ) ( ) A = p d p A 2 λ k + λ k + k= k k= ( ) ( ) A A = p d p (A ) + 2( λ k ) p d p + 2( λ k ) k= k= )

13 ON ACCUMULATED SPECTROGRAMS 3 Finally we give estimates for the quantity in (9). Lemma 4.3. Let R 2d be a compact set and consider the number E() defined in (9). Then the following holds. (a) E(R ) 0, as R +. R (b) If has finite perimeter and g M, then (20) 0 E() 2 g M. Proof. First note that from (5) we have that k λ k() = and consequently 0 E(). Let δ (0, ) and define l δ () := min{a, # {k : λ k () > δ}}. It follows that Since A l δ () we estimate Therefore, Since A we get (2) A k= λ k () δ, for k l δ (). λ k () l δ () k= 0 E() ( δ) min λ k () ( δ)l δ (). 0 E() ( δ) l δ() {, # {k : λ } k() > δ}. To prove (a) apply this estimate and Proposition. to R to get 0 lim sup E(R ) ( δ) = δ, R + and then let δ 0 +. Let us prove (b). From Proposition 3. we have, # {k : λ k () > δ} C δ g 2 M, where C δ = max{δ, ( δ) }. Plugging this estimate in (2) gives ( ) E() ( δ) C δ g 2 M = δ + ( δ)c δ g 2 M. Since δ (0, ), we have that ( δ)c δ /δ and therefore ( ) E() ( δ) C δ g 2 M (22) = δ + /δ g 2 M.

14 4 Finally, let δ := g M. Note that we can assume that δ < since otherwise the bound in (20) is trivial because E(). Therefore, we can apply (22) to get the desired conclusion. 5. Proof of the main results We now combine the bounds from Section 4 to get several estimates on ρ(). By combining Lemmas 4.2 and 4.3 we immediately get a proof of Theorem.3. As a consequence we deduce the following estimate for the error when we approximate by ρ. Corollary 5.. Let g M (R d ) and let R 2d be a compact set with finite perimeter. Then ρ (z) (z) dz R + g M + 4 g M. 2d In particular, for (23) ρ (z) (z) dz R 2d where the implicit constant depends on the window g., Proof. For the first part, we simply estimate ρ ρ ( Θ) + ( Θ), and apply Theorem.3 and Lemma 3.2. For the second part, note that ρ = A = + O() and consequently the left-hand side in (23) is. This allows us to assume that. Consequently, the conclusion follows. dominates both and and Finally we prove Theorem.2 that shows that without the finite perimeter assumption, we still have asymptotic convergence for the family of dilations of a single set. To compensate the change in scale we dilate ρ R by a factor of R. Proof of Theorem.2. For R > 0, let Θ R (z) := R 2d Θ(Rz). Note that Let us estimate ( R Θ)(Rz) = ( Θ R )(z), (z R 2d ). ρ R (Rz) (z) dz R 2d ρ R (Rz) ( Θ R )(z) dz + (z) ( Θ R )(z) dz R 2d R 2d = ρ R (Rz) ( R Θ)(Rz) dz + (z) ( Θ R )(z) dz R 2d R 2d = ρ R (z) ( R Θ)(z) dz + (z) ( Θ R )(z) dz. R R 2d R 2d

15 ON ACCUMULATED SPECTROGRAMS 5 Hence, by Lemma 4.2 we get (24) ρ R (Rz) (z) dz E(R ) + R 2d (z) ( Θ R )(z) dz. R 2d By Lemma 4.3 we have that E(R ) 0, as R +. In addition, since Θ =, it follows that Θ R R in L. This completes the proof. 6. Approximate retrieval of time-frequency filters In signal processing, time-frequency localization operators H are also known as time-frequency filters. As opposed to classical time-invariant filters, that multiply the Fourier transform of a signal by a given symbol, time-frequency filters localize signals both in time and frequency and are therefore time-variant. The field of system identification studies the possibility of retrieving an operator from certain measurements, which usually consist in the response of the operator to a set of carefully designed test signals. For time-invariant systems the identification problem is particularly difficult [7, 3, 24, 23]. In the case of a time-frequency filter, it is also important to understand to what extent the operator H can be understood from the measurement of a few of its eigenmodes h,..., h n. In [] the following special case was established (see [] for a discussion on possible applications). Theorem 6.. Let g(t) := 2e πt2, t R, be the one-dimensional Gaussian and let R 2 be compact and simply connected. If one Hermite function - cf. (9) - is an eigenfunction of H, then is a disk. Hence, for a Gaussian window, the information that R 2d is a simply connected domain of a certain area and that one eigenfunction is a Hermite function completely determines. Theorem 6. has however some drawbacks. First, it is non-robust: from the information that the eigenmodes of H look approximately like Hermite functions we cannot conclude the is approximately a disk. Second, it only applies to the restricted situation of a one-dimensional Gaussian window. Both restrictions stem from the one-variable complex analysis techniques on which the proofs in [] rely. While exactly and robustly recovering the fine details of the domain of a timefrequency filter only from measurements of a few of its eigenmodes does not seem to be possible, we can hope to at least recover the coarse shape of the set. We now show how to apply the estimates on accumulated spectrograms to produce an approximation of the domain with bounded average error. Let us consider a timefrequency filter H and suppose that we know and measure the spectrogram of the first eigenmodes V g h 2,..., V g h A 2, A :=. With this information we want to construct an approximation of. The accumulated spectrogram of gives a natural choice: (25) := { z R 2d : ρ (z) > /2 }. We then have the following result.

16 6 Theorem 6.2. Let g M (R d ) and let R 2d be a compact set with finite perimeter and. Let be defined by (25). Then, where denotes the symmetric difference and the implicit constant depends on the window g. Before proving Theorem 6.2 we note the following. In order to construct the approximate set we do not need to measure the phase of the short-time Fourier transforms V g h,... V g h A but only their absolute value. This fact is very valuable in applications (see for example [2]). Proof of Theorem 6.2. Let E δ := { z R 2d : (z) ρ (z) /2 }. Let us note that (26) (R 2d \ E δ ) = (R 2d \ E δ ) Indeed, if z (R 2d \E δ ), then ρ (z) (z) (z) ρ (z) > /2 = /2. Hence z. Second, if z (R 2d \E δ ), then (z) ρ (z) (z) ρ (z) > /2 /2 = 0. Hence z. The equality in (26) simply means that E δ. To finish the proof, we apply Corollary 5. to bound the measure of E δ : E δ 2 ρ L (R 2d ) References. [] L. D. Abreu and M. Dörfler. An inverse problem for localization operators. Inverse Problems, 28, 202. [2] R. M. Balan, B. G. Bodmann, P. G. Casazza, and D. Edidin. Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl., 5: [3] P. A. Bello. Measurement of random time-variant linear channels. IEEE Trans. Inform. Theory, 5(4): , 969. [4] E. Cordero and K. Gröchenig. Time-frequency analysis of localization operators. J. Funct. Anal., 205():07 3, [5] I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, 34(4):605 62, July 988. [6] I. Daubechies. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory, 36(5):96 005, 990. [7] P. Deift. Universality for mathematical and physical systems. In International Congress of Mathematicians. Vol. I, pages Eur. Math. Soc., Zürich, [8] F. DeMari, H. G. Feichtinger, and K. Nowak. Uniform eigenvalue estimates for timefrequency localization operators. J. London Math. Soc., 65(3): , [9] M. Englis. Toeplitz operators and localization operators. Trans. Amer. Math. Soc., 36(2): , [0] L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, 992. [] H. G. Feichtinger. Modulation Spaces: Looking Back and Ahead. Sampl. Theory Signal Image Process., 5(2):09 40, 2006.

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