Augmented Slepians: Bandlimited Functions that Counterbalance Energy in Selected Intervals
Slepian functions provide a solution to the optimization problem of joint time-frequency localization. Here, this concept is extended by using a generalized optimization criterion that favors energy concentration in one interval while penalizing ener…
Authors: Robin Demesmaeker, Maria Giulia Preti, Dimitri Van De Ville
SUBMITTED TO IEEE TRANSA CTIONS ON SIGNAL PR OCESSING 1 Augmented Slepians: Bandlimited Functions that Counterbalance Ener gy in Selected Interv als Robin Demesmaeker , Maria Giulia Preti, Member , IEEE, and Dimitri V an De V ille, Senior Member , IEEE Abstract —Slepian functions pro vide a solution to the opti- mization problem of joint time-fr equency localization. Here, this concept is extended by using a generalized optimization criterion that fa vors energy concentration in one interv al while penalizing energy in another interval, leading to the “augmented” Slepian functions. Mathematical foundations together with examples ar e presented in order to illustrate the most interesting properties that these generalized Slepian functions show . Also the rele vance of this novel energy-concentration criterion is discussed along with some of its applications. Index T erms —prolate spheroidal wav e functions, Slepian func- tions, localization, signal processing I . I N T RO D U C T I O N H EISENBERG’ s uncertainty principle states that the en- ergy of a signal can nev er be strictly localized both in the temporal and the Fourier domain. In a series of seminal papers, Slepian, Pollak, and Landau [1]–[4] study the case where maximal energy concentration on a selected interval is sought for a band-limited function. They sho w that the solution can be found from an inte gral eigen value equation where eigen values indicate ener gy concentration in the selected interv al, and eigenfunctions define a basis that is orthonormal on R , and orthogonal on the selected interv al. The sum of the eigenv alues—which exhibit a striking phase transition between high and low ener gy concentrations— corresponds to the time-bandwidth product (a.k.a. the Shannon number) and characterizes the dimensionality of the linear space of band-limited functions associated to an interval with giv en width. The functions defined by Slepian et al. , called Slepian functions onward, are known as prolate spheroidal wa ve functions and hav e a number of elegant properties and applications, including band-limited extrapolation. They ha ve also been extended for other domains such as their spherical counterparts with applications in geophysics [5]. It is straightforward to extend the Slepian construction for two or more interv als in the temporal domain. In such case, the solution maximizes the energy simultaneously in all intervals. Howe ver , in some applications, it can be useful to be able to specify interv als that are counteracting; i.e., when one wants to obtain functions that are maximally concentrated in one interval, while being minimally concentrated in another one. Therefore, in this paper , we generalize Slepian functions by pursuing band-limited functions that not only maximize ener gy concentration in one interval, but are also penalized by their R. Demesmaek er, M. G. Preti, D. V an De V ille are with the Institute of Bioengineering/Center for Neuroprosthetics, Ecole Polytechnique F ´ ed ´ erale de Lausanne, and the Department of Radiology and Medical Informatics, Univ ersity of Genev a, Switzerland. Manuscript received xxx; revised xxx; accepted xxx; published xxx. energy concentration in another one. W e demonstrate that the solution can still be found from an inte gral eigen value equation where the eigen values indicate the differ ence in energy concentration between both interv als. The eigenspec- trum rev eals two phase transitions with corresponding time- bandwidth products. The eigenfunctions are approximately orthogonal on the selected intervals. The interaction between both interv als makes the solution ef fectiv ely different from combining solutions for the interv als separately . The paper is organized as follows. After a short revie w of 1-D Slepian theory in Sec. II, we introduce the proposed gen- eralization (Sec. III). W e provide the mathematical foundations together with instructiv e 1D e xamples and se veral properties of the “augmented” Slepian functions. T o conclude, we discuss possible applications of this novel view on energy localization in domains such as signal recovery and data analysis. I I . S L E P I A N F U N C T I O N S Slepian and colleagues were the first to propose an ele- gant solution to the problem of finding continuous-domain functions that are band-limited, but with maximal energy concentration in an interv al. W e briefly revie w the Slepian theory , highlighting those aspects that are important for the generalization. Let us start by introducing the Hilbert space of square- integrable functions L 2 ( R ) with associated inner product h f , g i R = Z R f ( t ) ¯ g ( t ) dt, (1) where ¯ · is the complex conjugate. The Fourier transform (and its inv erse) is defined as F ( ω ) = Z R f ( t ) exp( − iω t ) dt, (2) f ( t ) = 1 2 π Z R F ( ω ) exp( iω t ) dω . (3) The Slepian design problem can be formulated as finding the band-limited function g ( t ) that maximizes the energy concentration in an interv al. The temporal interval is chosen [ − T , + T ] and thus centered around the origin with a half width of T . The spectral band-limit is specified as [ − W , + W ] where W indicates the one-sided bandwidth. The following optimization problem for maximizing the energy concentration can then be written: λ = max g ( t ) ∈B W R + T − T | g ( t ) | 2 dt R R | g ( t ) | 2 dt , (4) SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 2 where B W is the space of band-limited functions in [ − W , + W ] . As shown by Slepian et al. , this criterion can be reformulated in the Fourier domain as 1 2 π R + W − W R + W − W G ( ω ) ¯ G ( ω 0 ) = D ( ω 0 − ω ) z }| { 1 2 π Z + T − T e − i ( ω 0 − ω ) t dt dω dω 0 1 2 π R + W − W G ( ω ) ¯ G ( ω ) dω , where the kernel D ( ω ) = 1 2 π Z + T − T e − iω t dt = sin( T ω ) π ω (5) is the scaled Fourier transform of the indicator function of the interval [ − T , + T ] . Therefore, maximizing the energy concen- tration leads to an equiv alent integral eigenv alue equation in the Fourier domain: Z + W − W sin T ( ω − ω 0 ) π ( ω − ω 0 ) G ( ω 0 ) dω 0 = λG ( ω ) . (6) This equation can be written in its canonical form by replacing G ( ω ) = ψ ( ω /W ) and the change of variables ω = W ξ : Z +1 − 1 sin T W ( ξ − ξ 0 ) π ( ξ − ξ 0 ) ψ ( ξ 0 ) dξ 0 = λψ ( ξ ) . (7) Since the kernel in this homogeneous Fredholm equation of the second kind is symmetric positiv e definite, the integral operator is compact and its solutions λ k , ψ k , k ∈ N , are countable where the eigen values λ k are positiv e (and tend to zero), and the real-valued eigenfunctions ψ k , known as prolate spheroidal wav e functions (PSWF), hereafter also referred to as Slepian functions, form an orthogonal basis of L 2 ([ − 1 , 1]) . The PSWF can be extended to b uild an orthogonal basis of L 2 ( R ) by defining Eq. (7) for all ξ ∈ R . This leads to the double orthogonality property h ψ k , ψ l i R = δ k − l , (8) h ψ k , ψ l i [ − 1 , +1] = λ k δ k − l , (9) where δ k is the Kronecker delta. In addition, any PSWF also satisfies the follo wing intriguing Fourier property: ψ k ( t ) = 1 µ k Z +1 − 1 ψ k ( ξ ) e iT W tξ dξ , (10) where µ k ∈ C is a scaling factor up to which the PSWF has the same shape as its Fourier transform in the interval [ − 1 , +1] . This property plays a key role in relating the PSWF to the prolate dif ferential equation that justifies their name and provides an alternative numerical procedure for their computation. An important feature of the Slepian basis is the Shannon number N Shannon , which is giv en by the sum of all eigen values. It can easily be sho wn that this number only depends on the time-bandwidth product 2 T W : N Shannon = ∞ X k =0 λ k = Z + W − W lim ξ 0 → ξ D ( ξ − ξ 0 ) dξ = 2 T W π (11) Since the characteristic spectrum sho ws a step-like behaviour with eigenv alues either close to 1 or 0 separated by a narrow transition band, N Shannon moreov er approximately represents the number of eigenfunctions that are well concentrated within the selected region of interest. Therefore, it is also a measure for the dimension of the subspace spanned by the band-limited functions that are well localized. The Slepian construction can easily be extended for an interval that is not centered at the origin. In such case, the following modified properties hold: Pr oposition 1 (T ranslated temporal interval): The Slepian design for a translated temporal interv al [ − T + P, + T + P ] satisfies the following Fourier property ψ k ( t ) = e iP W t µ k Z +1 − 1 ψ k ( ξ ) e i ( P W + T W t ) ξ dξ , (12) and corresponding integral eigen value equation: ψ k ( ξ ) = 2 π | µ k | 2 T W | {z } =1 /λ k Z +1 − 1 ψ k ( ξ 0 ) e iP W ( ξ − ξ 0 ) sin( T W ( ξ − ξ 0 )) π ( ξ − ξ 0 ) dξ 0 . (13) Proof: See Appendix A. One can ask whether the Slepian design can be further ex- tended to a union of intervals. The answer is af firmativ e from a point-of-view of the construction of the energy-concentration criterion to be optimized. Specifically , let us consider the union of intervals S = N [ n =1 [ − T n + P n , + T n + P n ] , for which criterion to be maximized is λ = max g ( t ) ∈B W R S | g ( t ) | 2 dt R R | g ( t ) | 2 dt , (14) which can be turned into the equi valent integral eigen value equation G k ( ω ) = 1 λ k Z + W − W N X n =1 e iP n ( ω − ω 0 ) sin( T n ( ω − ω 0 )) π ( ω − ω 0 ) | {z } = D ( ω − ω 0 ) G k ( ω 0 ) dω 0 . Unfortunately , the Fourier property (10) no longer holds. Therefore, we need to explicitly define the temporal domain version of these functions as g k ( t ) = 1 2 π Z + W − W G k ( ω ) e itω dω , (15) to which we will refer to as the “Slepian functions” since they do not necessarily correspond to PSWFs. Using the normalization k g k k 2 = 1 and thus k G k k 2 = 2 π (Parsev al identity), we can still easily prove the double orthogonality property in the temporal domain. Pr oposition 2 (Orthogonality of Slepian functions for union of intervals): The Slepian functions g k , k ∈ N , associated SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 3 Fig. 1. Schematic presentation of the concept of selecting two types of intervals. The one in green (“positiv e”, by the plain indicator function) and the one in red (“negativ e”, by the dashed indicator function). The criterion that is maximized will be the difference between the energies in the green and the red interval, respectiv ely , normalized with respect to the energy on the real line. to the union of intervals S , satisfy the following double orthogonality property: h g k , g l i R = δ k − l , (16) h g k , g l i S = λ k δ k − l . (17) The proof is given in Appendix B. I I I . A U G M E N T E D S L E P I A N F U N C T I O N S A. Design The PSWF introduced in the previous section are driv en by maximizing energy concentration in a chosen interv al. This implies that the energy is minimized ev erywhere else since max g ( t ) ∈B W R S | g ( t ) | 2 dt R R | g ( t ) | 2 dt ! = max g ( t ) ∈B W 1 − R S c | g ( t ) | 2 dt R R | g ( t ) | 2 dt ! , where S c = R \S is the complement of S . Here, we propose to introduce explicitly the notion of a second type of interv al. As illustrated in Fig. 1, we want to find the band-limited functions that maximize the energy con- centration in the green interval while minimizing it in the red interval, where both can be chosen by the user . Consequently , the energy in the two intervals will be counterbalanced. This additional freedom in the design leads to what we term as “augmented Slepians”. Mathematically , the criterion to be maximized is defined as λ = max g ( t ) ∈B W R S + | g ( t ) | 2 dt − R S − | g ( t ) | 2 dt R R | g ( t ) | 2 dt ! , (18) where S + and S − are two (disjoint) unions of interv als: S + = N + [ n =1 [ − T + n + P + n , + T + n + P + n ] , S − = N − [ n =1 [ − T − n + P − n , + T − n + P − n ] . By turning the criterion (18) in the Fourier domain, we find the corresponding integral eigen value equation for a generalized kernel: Z W − W D ( ω − ω 0 ) G ( ω 0 ) dω 0 = λG ( ω ) , (19) where D ( ω ) = N + X n =1 e iP + n ω sin( T + n ω ) π ω − N − X n =1 e iP − n ω sin( T − n ω ) π ω . This kernel is no longer positi ve definite, b ut the solutions of the eigen value problem are still countable. This can easily be understood by deri ving an equi valent kernel. First of all, all eigen values are bounded between − 1 and +1 due to Eq. (18). Second, we can of fset the eigen values with +1 by adding the Dirac distribution δ ( ω ) to the kernel, which does not modify the eigenfunctions. This equi valent kernel is now positiv e definite, ensuring the countability of its solutions and therefore the solutions of the augmented Slepian design are also countable. The eigenv alues cluster around three values: +1 for eigen- functions well concentrated in S + , − 1 for eigenfunctions well concentrated in S − , and 0 for eigenfunctions that are neither concentrated in S + nor S − . By conv ention, we will rank the eigen values λ k , k ∈ N , according to decreasing absolute value. W e also introduce the following notation for positive λ > 0 k and negati ve eigen values λ < 0 k , ranked according to decreasing ab- solute value within their subsets. This grouping of eigen values is graphically represented in Fig. 2. W e define the Shannon number N Shannon in the same way as in the original Slepian design, i.e. as the sum over all eigen values: N Shannon = ∞ X k =0 λ k (20) = Z W − W lim ω 0 → ω D ( ω − ω 0 ) dω (21) = 2 W P N + n =1 T + n π − 2 W P N − n =1 T − n π (22) = 2 W P N + n =1 T + n − P N − n =1 T − n π . (23) From Eq. (22) and Eq. (11) it is straightforward to show that N Shannon is equal to the difference of the Shannon numbers N + Shannon and N − Shannon obtained for regular Slepians associated to S + and S − , respectively: N Shannon = N + Shannon − N − Shannon . (24) W e also use the notations λ + k and λ − k to refer to the eigen- values of the regular Slepian constructions for S + and S − , respectiv ely . From Eq. (24), it is clear that N Shannon is equal to 0 if and only if the Shannon numbers of the regular Slepian constructions for both interval unions separately are equal. This can, for a gi ven bandwidth, only happen if the total sizes of both unions are equal. As such, N Shannon is proportional to the size difference between both unions. B. An instructive example In order to get a handle on what kind of results can be expected from the augmented Slepian frame work, Fig. 3 shows SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 4 0 10 20 30 40 k -1 -0.5 0 0.5 1 Eigenvalue Descending magnitude 0 5 10 15 20 k -1 -0.5 0 0.5 1 Eigenvalue Regrouped positive and negative spectra Fig. 2. Graphical representation of the splitting of the positive and negati ve parts of the spectrum. an example where both intervals are put next to each other . All simulations in this and the following sections are based on the numerical method presented in Appendix E. In the top figure, the first three original Slepian functions resulting from the selection of the green interval only are shown. These functions are highly concentrated in the green interval (i.e., λ + k close to 1) and are always e ven or odd around the center of the interv al. The middle figure shows the first three augmented Slepian functions corresponding to positive eigenv alues when the red interval is neg ativ ely selected. It is clear that, while keeping high concentration within the green interval, these functions are not ev en or odd anymore around any point. Indeed, energy concentration is pushed away from the red interval and, therefore, more energy is located on the left side of the green interval. Finally , in the bottom figure, the first three augmented Slepian functions with negati ve eigen values are shown. As expected, they are highly concentrated within the red interv al. Here, asymmetry exists as well since the green interval is no w repulsing signal energy . C. Pr operties The classical PSWFs are known for a number of remark- ably elegant properties. W e now present how these original properties hold for augmented Slepians, as well as properties which are specific to the augmented setting. Pr operty 1 (Equivalence with conventional Slepian func- tions): Augmented Slepians associated with S − = R \S + are equiv alent to con ventional Slepian functions with S = S + . 0 1 3 -0. 2 -0. 1 0 0 .1 0 .2 0 1 3 -0. 2 -0. 1 0 0 .1 0 .2 0 1 3 -0. 2 -0. 1 0 0 .1 0 .2 Fig. 3. Example of augmented Slepians, the one-sided continuous time bandwidth was set to W = 2 π : (top) the first three Slepian functions resulting from the green selection are highly concentrated in the selected interval and are all e ven or odd around its center; (middle) the first three augmented Slepian functions with positiv e eigen values are still highly concentrated in the green interval, but are less concentrated in the red interval than the original Slepian functions; (bottom) the first three augmented Slepian functions with negativ e eigen values are highly concentrated in the red interval. Pr oof: It is straightforward to show that con ventional Slepians are a special case. W e plug S − = R \S + into the SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 5 energy optimization criterion (18) for augmented Slepians: g ( t ) = argmax g ( t ) ∈B W R S + | g ( t ) | 2 dt − R S − | g ( t ) | 2 dt R R | g ( t ) | 2 dt = argmax g ( t ) ∈B W R S + | g ( t ) | 2 dt − R R \S + | g ( t ) | 2 dt R R | g ( t ) | 2 dt ! = argmax g ( t ) ∈B W 2 R S + | g ( t ) | 2 dt − R R | g ( t ) | 2 dt R R | g ( t ) | 2 dt = argmax g ( t ) ∈B W 2 R S + | g ( t ) | 2 dt R R | g ( t ) | 2 dt − 1 = argmax g ( t ) ∈B W R S + | g ( t ) | 2 dt R R | g ( t ) | 2 dt , which reverts to the con ventional criterion for S = S + . Pr operty 2 (Symmetry of solutions): Interchanging the role of the union of interv als S + and S − as positiv e and negati ve selections in the design of augmented Slepians, leads to an equiv alent solution where the signs of the eigen values λ k are in versed, but where the same associated eigenfunctions are found. Pr operty 3 (Orthogonality): The augmented Slepians are double orthogonal ov er R and the union of intervals S + and S − in a generalized way: h g k , g l i R = δ k − l , (25) h g k , g l i S + − h g k , g l i S − = λ k δ k − l . (26) In addition, the augmented Slepians are approximately orthog- onal on the union of intervals S + : k 6 = l : |h g k , g l i S + | ≤ p (1 − λ k )(1 − λ l ) 2 (27) k = l : h g k , g k i S + ≥ λ k , (28) which become tight upper and lower bounds for Slepians well concentrated on S + (i.e., λ k close to 1 ). Notice that for k 6 = l , we have h g k , g l i S + = h g k , g l i S − due to Eq. (26), and thus the orthogonality of these Slepians well concentrated on S + , becomes also strong on S − . Similar results hold for the union of intervals S − : k 6 = l : |h g k , g l i S − | ≤ p (1 + λ k )(1 + λ l ) 2 (29) k = l : h g k , g k i S − ≥ − λ k , (30) which become tight for Slepians well concentrated on S − (i.e., λ k close to − 1 ). Moreov er , using the definition for the angle α between two eigenfunctions: cos α g k ,g l = h g k , g l i || g k || · || g l || , (31) we can show that the following property holds regarding the cosine of the angle between two eigenfunctions for k 6 = l and λ k > 0 and λ l > 0 : | cos α g k ,g l | ≤ 1 2 √ 1 − λ k √ 1 − λ l √ λ k λ l . (32) For eigenfunctions for which λ k < 0 and λ l < 0 , the analogous property is given by: | cos α g k ,g l | ≤ 1 2 √ 1 + λ k √ 1 + λ l √ λ k λ l . (33) The proof is given in Appendix C. In Fig. 4, we show an example comparing the actual inner products and the cosines together with their respectiv e bounds. Pr operty 4 (Interaction parameter): W e introduce the inter- action parameter ∆ + as the difference between the Shannon number of the con ventional Slepian design for S + and the sum of the positiv e eigenv alues of the augmented Slepian spectrum. It turns out to be equal to the interaction ∆ − between the con ventional design for S − and the negati ve part of the augmented Slepian spectrum: ∆ = N + S hannon − ∞ X k =0 λ > 0 k | {z } =∆ + = N − S hannon + ∞ X k =0 λ < 0 k | {z } =∆ − . (34) The proof is given in Appendix D. A visual interpretation of these parameters is sho wn in Fig. 5. These values can be used to quantify how adding a negati ve region to the concentration problem makes it more difficult to achiev e high (generalized) energy concentration in the original interval when this region is placed close to the positiv e region. Howe ver , a wide spacing of the regions does not highly influence the achiev able energy concentration in the positiv ely selected region. I V . D I S C U S S I O N Now that the theoretical framework and properties of the augmented Slepians have been introduced, we will discuss in more details some of their features, including their importance for practical applications. A. Interplay between two types of intervals The main adv antage of the proposed design is the possibility to specify two types of intervals that play a different role in the optimization criterion. Consequently , while a single basis is obtained, Slepian functions that are well-localized in one versus the other type of interv al are associated with different eigen values; i.e., positive and negati ve ones, respectiv ely . One question then is whether a similar result could have been obtained by combining two con ventional Slepian bases. The answer is no because such a dual construction would not have led to orthogonality properties on the interv als. In particular , as shown by Property 3, Slepians well concentrated on S + (i.e., λ k close to 1 ) are (approximately) orthogonal on both S + and S − taken separately . Consequently , inner products taken on either of the different interv als between a signal and the augmented Slepians can be considered independent. In addition to the orthogonality and independence proper- ties, when the two types of interv als are close enough, an effect of interaction can be observed on the eigen value spectrum as quantified by ∆ of Property 4. This parameter can be interpreted as the difference in energy concentration between SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 6 0 14 0 14 0 14 0 14 k 0 14 0 14 0 14 0 14 0 14 0 14 0 l k 0 14 0 14 0 14 0 14 k 0 14 0 14 0 0.5 1 0 14 -1 0 1 eigenvalue l k 1 3 Fig. 4. The augmented Slepians are approximately orthogonal on the subsets S + and S − , respectively . Comparison between the actual measures (top) and their bounds (bottom): inner product (left) and cosine angle (right), respectively . The selected intervals are shown on the top left and the one-sided bandwidth is 5% of the Nyquist frequency . Fig. 5. Eigen value spectra for augmented and regular Slepian designs on the positively and negati vely selected intervals, respectively . The interaction parameter ∆ corresponds to each of the shaded surface areas. con ventional Slepians for S + , and the augmented Slepians that come with positi ve eigen values and thus are more concentrated in S + than S − . This phenomenon is illustrated in Figs. 6 and 7. In particular , Fig. 6 shows how bringing the interv als closer together shifts the positi ve (resp. negati ve) part of the spec- trum downward (resp. upward). Also the first two augmented Slepian functions are shown on the insets; in (a), the functions resemble more con ventional Slepians (i.e., ev en and odd with respect to the center of the green interval, so no preference for a certain side), in (b)-(d), the (anti-)symmetry gets lost as the intervals move closer together which is indicati ve for the interaction. Fig. 7 shows the e volution of the interaction parameter ∆ normalized by the bandwidth as a function of the spacing between the two intervals of interest. As expected, ∆ decreases with increased spacing. The results for 3 different bandwidths are shown and, although the relationship between ∆ and W is clearly not linear since the curves do not coincide, they roughly have the same shape. On a side note, if the bandwidth is infinite, the spectrum will always show a perfect step-like shape in both the con ventional and the augmented Slepian frameworks irrespectively of the spacing between the regions of interest. Therefore, in this extreme case, ∆ will always be equal to 0 . Fig. 6. Eigenv alue spectrum of augmented Slepian design for different spacings between two equally sized intervals that are positively and negativ ely weighted, respectively . The distance between the two intervals is changed and specified as the percentage of the interval size. The insets show the corre- sponding first two eigenfunctions associated with the two largest eigenv alues. The clear distinction in the eigen value spectrum between functions having high concentration in either of the intervals can be exploited when considering the bandlimited reconstruc- tion of a signal on distinct intervals, for instance. W e provide an example of how the augmented Slepian design can be used to reconstruct a signal on two intervals independently , SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 7 Fig. 7. Interaction parameter ∆ normalised by the bandwidth W as a function of the spacing between selected and penalised intervals for 3 different bandwidths. As expected, the interaction parameter decreases as the intervals are spaced more largely . Increasing bandwidth lowers the normalised interaction parameter, but the behavior as function of spacing remains similar . though being linked by the same decomposition. W e start from measurements in the Fourier domain, which is suggestiv e of measurements taken in k -space as in magnetic resonance imaging. Let us consider the signal f ( t ) shown by the black full line in Figure 8. Assume now that we want to reconstruct the signal in the interv als S + = [1 , 4] and S − = [4 , 7] , independently , and that data can only be acquired within the bandwidth [ − W , W ] where W = 1 . 5 π . W e can then use the augmented Slepian design with band-limit W to reconstruct the signals as follows: f > 0 rec ( t ) = N + Shannon X n =0 G > 0 n , F R g > 0 n ( t ) (35) f < 0 rec ( t ) = N − Shannon X n =0 G < 0 n , F R g < 0 n ( t ) , (36) where F is the Fourier transform of f . Figure 8 shows the original signal and the reconstructed signals on both intervals, as well as the reconstructed signal f rec using the original Slepian design on the union of S + and S − with the total number of eigenfunctions used for the augmented design (i.e., N Shannon = N + Shannon + N − Shannon ). The reconstructions f > 0 rec and f < 0 rec are well localized within their respecti ve interv als. By construction, these reconstructions are orthogonal and thus explain separate parts of the measured energy . The sum of both reconstructions is very close to the one using con ven- tional Slepians on the combined intervals, but it approximates better the ground truth at the boundary between S + and S − , which is unknown to the con ventional design. This example illustrates how the proposed design can be beneficial when additional prior information is av ailable to tailor band-limited reconstructions. B. Alternative way to penalize energy concentration In less known work, Gilbert and Slepian [6] have proposed a generalization of the Slepian functions that maximizes the 1 4 7 0 0.5 1 1 4 7 0 0.5 1 1 4 7 0 0.5 1 Fig. 8. Bandlimited reconstruction of a function in two intervals indepen- dently . The original signal is shown together with the reconstructions based on the augmented Slepian design and on the original Slepian construction for the union of both intervals. ratio λ = max g ( t ) ∈B W R S + | g ( t ) | 2 dt R S − | g ( t ) | 2 dt ! , (37) which re verts to the original Slepian design in case S − = R . In this theory , the ratio between the energy concentrations in S + and S − is optimized as opposed to their difference in our augmented Slepian design. Although the resulting functions hav e the nice property of being orthogonal on both intervals separately , the eigen value spectrum does in general not show the striking phase transi- tions visible with the augmented Slepian framew ork. Moreov er , Gilbert and Slepian reported that a corresponding differential equation (i.e., the lucky accident [7]) could only be found for special cases such as in the original Slepian design. The reason behind is that the differential operator needs to commute with the characteristic function [8], [9]. Therefore, it might not be possible to find such operator for the augmented Slepian design either , except in some particular choices of the intervals. C. Applications and extensions Giv en its fundamental, but at the same time practical objec- tiv es, the original Slepian framew ork has found a wide range of applications, ranging from signal processing (filtering and multitaper spectral analysis [10], e xtrapolation and compressed SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 8 sensing [11]–[13], compression [14]) to geophysics [15]–[17], ultrawideband communications (to describe radiation patterns of antennas [18] or pulse designs [19], [20]), and magnetic resonance imaging (for extrapolation [21], speeding up data acquisition within a predefined region-of-interest [22]). Many of these applications ha ve been built upon extensions of the original framew ork to higher-dimensional spaces [4], to the sphere [5], [23], [24], or more recently to graphs [25]–[27]. Other generalizations hav e been proposed for a weighted criterion to optimize steerable filters [28], for the quaternionic Fourier transform [29], or for matrix-v alued functions [30]. Even a Fast Slepian T ransform has been proposed as an alternativ e to the Fast Fourier T ransform for time-limited signals [31]. The proposed design of augmented Slepians can probably be made useful in many of these applications. In extrapolation, for instance, conv entional Slepians are used to compute inner products on an interval where measures are av ailable, to then be used to obtain a bandlimited extrapolation. W ith the proposed design, tw o separated intervals could be specified and lead to two extrapolations, but that would remain or- thogonal thanks to the joint optimization criterion. Howe ver , extrapolation requires the calculation of Slepian functions corresponding to eigenv alues close to zero [11]. At some point, these functions and eigenv alues cannot be calculated precisely enough using the numerical procedure presented in Appendix E to provide good extrapolation results. For the original Slepian design, the commuting differential equation (or “lucky accident”) has been used to provide an alternati ve way to calculate the eigenfunctions more precisely . Further research into more accurate calculation methods, and whether a commuting kernel exists, could be extremely useful to pave the way for application of the augmented Slepian framework. Extending the augmented Slepian design to more- dimensional and more complex spaces should also be possible. For this, writing the optimisation problem using an operator formalism can be instructive: B ( D S + − D S − ) B g = λg , (38) where B is the band-limiting operator and D ( · ) the time- limiting operator . D. Indefinite inner product and Kr ein spaces There is an interesting link between the kernel D of the augmented Slepian design and Krein spaces [32]. In fact, since D is indefinite (i.e., it has both positive and negati ve eigen values), we cannot define a Hilbert space based on it, but it is possible to define a Krein space K , by building on the indefinite inner product over the generalized selection S + ∪ S − : ( x, y ) = Z S + x ( t ) ¯ y ( t ) dt | {z } h x,y i S + − Z S − x ( t ) ¯ y ( t ) dt | {z } −h x,y i S − , (39) which admits a direct orthogonal sum decomposition K = K + ⊕ K − , where K + , h· , ·i S + and K − , − h· , ·i S − are Hilbert spaces, and which has ( x, y ) = 0 for any x ∈ K + , y ∈ K − . This also means that we can define the projection operators that map onto these constituent spaces as K + = P + K and K − = P − K , which can then be combined in an endomor- phism operator on K as J = P + − P − ; i.e., this operator defines a positive semi-definite inner product: h x, y i def = ( x, J y ) (40) = ( x, ( P + − P − ) y ) (41) = Z S + x ( t ) ¯ y ( t ) dt + Z S − x ( t ) ¯ y ( t ) dt (42) = Z S + ∪S − x ( t ) ¯ y ( t ) dt (43) and satisfies the property J 3 = J . It can be readily verified that applying the kernel three times indeed rev erts to a single application. In many application fields such as data analysis and learning tasks, kernels are typically required to be positiv e semi- definite, howe ver , there is also an interest in using non-positive kernels [33], [34] and therefore the augmented Slepians might be useful to guide new designs in this much larger search space. E. Generalized weightings The energy concentrations within the intervals are sub- tracted and thus weighted with coefficients -1 and 1, respec- tiv ely . Obviously these weights could be changed depending on the application; e.g., reciprocally scaled w .r .t. the size of the interv als. One might consider this as a particular case of expressing the energy concentration using weighting functions that vary within each interval; i.e., the criterion could be generalized as λ = max g ( t ) ∈B W R S + w + ( t ) | g ( t ) | 2 dt − R S − w − ( t ) | g ( t ) | 2 dt R R | g ( t ) | 2 dt ! , (44) where w + ( t ) and w − ( t ) are positi ve real-valued functions. T urning this into the Fourier domain, we get the following eigen value equation: Z W − W D ( ω − ω 0 ) G ( ω 0 ) dω 0 = λG ( ω ) , (45) where D ( ω ) = W + ( ω ) − W − ( ω ) with W + and W − the Fourier transformed window functions. As with the original and augmented Slepian framework, this equation is still a homogeneous Fredholm equation of the second kind with a Hermitian kernel. This more general optimisation problem is a special case, as is the original Slepian frame work, of Franks’ variational framew ork where a general optimisation criterion is built with energy constraints in both time and Fourier domain [35]. It can also be seen as a special case of the pseudo-differential operator framework and the corresponding asymptotic theory presented in [36], [37], which can be useful for further SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 9 theoretical study and extension of the concept of augmented Slepians. Most properties outlined here are tightly linked to the specific case of having weights -1 and 1, and, therefore, further research is needed to better understand how these prop- erties can be further generalised for more general weighting functions. A final question that emerges naturally about the extension presented here, is whether or not it is possible to consider more than two types of interv als. Intriguingly , such goal can be reached when the weighting factors are allo wed to be comple x- valued. For instance, we can select weights of 1, exp( i 2 π 3 ) and exp( − i 2 π 3 ) for three different types of intervals, which provides an eigen value spectrum where eigen values are closely located to lines in the complex plane with angles 0, 2 π 3 and − 2 π 3 , respectively . V . C O N C L U S I O N W e presented an extension of the Slepian design that leads to band-limited functions that simultaneously maximize and minimize energy concentration in different types of intervals. W e showed the mathematical background of these “aug- mented” Slepian functions, together with their main properties and how they can be practically obtained. The eigenv alue spectrum exhibits some essential features such as two phase transitions—one for each type of interv al. The degree of “interaction” between both intervals is also embedded in the eigen value spectrum. Just as regular Slepian functions, their augmented v ariants are orthogonal over the whole domain, in a generalized way over the selected intervals, and approximately (within giv en bounds) ov er the intervals of each type. Gi ven the broad impact of Slepian functions, we expect this work can find various applications. A P P E N D I X A P RO O F O F P R O P O S I T I ON 1 Pr oof: Using the notation σ = T W , we postulate the following variant of the Fourier property for the case of a shifted interval ψ k ( t ) = e iP W t µ k Z +1 − 1 ψ k ( ξ ) e i ( P W + σ t ) ξ dξ , (46) into which we plug the complex conjugate ψ k ( ξ ) = e − iP W ξ ¯ µ k Z +1 − 1 ψ k ( ξ 0 ) e − i ( P W + σ ξ ) ξ 0 dξ 0 . (47) which leads to ψ k ( t ) = e iP W t | µ k | 2 Z +1 − 1 ψ k ( ξ 0 ) e − iP W ξ 0 Z +1 − 1 e iσ ( t − ξ 0 ) ξ dξ dξ 0 . (48) W ith the change of variable w = σ ξ , we obtain: ψ k ( t ) = e iP W t | µ k | 2 σ Z +1 − 1 ψ k ( ξ 0 ) e − iP W ξ 0 Z + σ − σ e i ( t − ξ 0 ) w dw dξ 0 . Using the in verse Fourier transform of the window function [ − σ, + σ ] , we further obtain ψ k ( t ) = 2 π | µ k | 2 σ Z +1 − 1 ψ k ( ξ 0 ) e iP W ( t − ξ 0 ) sin( σ ( t − ξ 0 )) π ( t − ξ 0 ) | {z } = D ( t − ξ 0 ) dξ 0 , (49) which is the integral equation that we would obtain by expressing the maximal energy concentration in the shifted interval; i.e., the kernel D can be directly related to its Fourier transform. In addition, we identified the relationship λ k = | µ k | 2 σ / (2 π ) , which is the same as for the conv entional PSWF . A P P E N D I X B P RO O F O F P R O P O S I T I ON 2 Pr oof: The first property is tri vial giv en that the Slepian functions g k are eigenfunctions and thus orthogonal and nor- malized such that k g k k 2 = 1 . The second property can be deriv ed as follows: h g k , g l i S = Z S g k ( t ) ¯ g l ( t ) dt = Z S g k ( t ) 1 2 π Z + W − W ¯ G l ( ω ) e − iω t dω dt = 1 2 π Z + W − W ¯ G l ( ω ) Z S g k ( t ) e − iω t dtdω = 1 2 π Z + W − W ¯ G l ( ω ) Z + W − W G k ( ω 0 ) D ( ω 0 − ω ) dω 0 dω = 1 2 π Z + W − W G k ( ω 0 ) Z + W − W ¯ G l ( ω ) D ( ω 0 − ω ) dω dω 0 = λ l 2 π Z + W − W G k ( ω ) ¯ G l ( ω ) dω = λ l δ k − l . A P P E N D I X C P RO O F O F P R O P E RT Y 3 Pr oof: W e start from the following energy-concentration property which follo ws from the definition of the augmented Slepian functions: h g k , g k i S + − h g k , g k i S − = λ k h g k , g k i . (50) Since h g k , g k i S − ≥ 0 and h g k , g k i = 1 , this means that h g k , g k i S + ≥ λ k . (51) When k 6 = l , Eq. 25 can be rewritten as the sum of its parts: h g k , g l i S + + h g k , g l i S − + h g k , g l i S ∗ = 0 , (52) where S ∗ is the full domain minus S + and S − . Using Eq. (26), it follows that h g k , g l i S + = − 1 2 h g k , g l i S ∗ . (53) Applying Cauchy-Schwartz to h g k , g l i S ∗ then shows |h g k , g l i S ∗ | ≤ q h g k , g k i S ∗ q h g l , g l i S ∗ , (54) SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 10 where the right-hand terms can be rewritten as q h g k , g k i S ∗ = q h g k , g k i − h g k , g k i S + − h g k , g k i S − ≤ q 1 − h g k , g k i S + ≤ p 1 − λ k , using (51) . (55) This further simplifies Eq. 54 into |h g k , g l i S ∗ | ≤ p 1 − λ k p 1 − λ l (56) and using this in Eq. 53, the following bound can be found: |h g k , g l i S + | ≤ 1 2 p 1 − λ k p 1 − λ l . (57) In order to rule out the ef fect of the signal magnitude inside the region of interest, the geometrical definition of the inner product is used in Eq. 57: | g k | S + | g l | S + | cos α g k ,g l | ≤ 1 2 p 1 − λ k p 1 − λ l . (58) Using λ k = | g k | 2 S + − | g k | 2 S − , it can be concluded that λ k ≤ | g k | 2 S + . For all λ k > 0 it is then also true that √ λ k ≤ | g k | S + and since λ k ≤ 1 and | g k | S + ≤ 1 this leads to: 1 | g k | S + ≤ 1 √ λ k if λ k > 0 . (59) Using this inequality in Eq. 58, the final bound on the approximate orthogonality on the positiv ely selected region of interest for the eigenfunctions for which λ > 0 becomes: | cos α g k ,g l | ≤ 1 2 √ 1 − λ k √ 1 − λ l √ λ k λ l if λ k,l > 0 (60) A fully analogous deriv ation leads to a bound on the approximate orthogonality on the positiv ely selected region of interest for the eigenfunctions for which λ < 0 : | cos α g k ,g l | ≤ 1 2 √ 1 + λ k √ 1 + λ l √ λ k λ l if λ k,l < 0 . (61) A P P E N D I X D P RO O F O F P R O P E RT Y 4 Pr oof: Using the identity ∞ X k =0 λ k = ∞ X k =0 λ + k − ∞ X k =0 λ − k , (62) which follows from Eq. 24, and the fact that P ∞ k =0 λ k can be written as the sum of its positiv e and negativ e parts: ∞ X k =0 λ k = ∞ X k =0 λ > 0 k + ∞ X k =0 λ < 0 k , (63) the difference ∆ + − ∆ − becomes = ∞ X k =0 λ + k − ∞ X k =0 λ > 0 k − ( ∞ X k =0 λ − k + ∞ X k =0 λ < 0 k ) (64) = ∞ X k =0 λ + k − ∞ X k =0 λ − k − ∞ X k =0 λ > 0 k − ∞ X k =0 λ < 0 k (65) = ( ∞ X k =0 λ + k − ∞ X k =0 λ − k ) − ( ∞ X k =0 λ > 0 k + ∞ X k =0 λ < 0 k ) (66) = ( ∞ X k =0 λ + k − ∞ X k =0 λ − k ) − ∞ X k =0 λ k (67) = ∞ X k =0 λ k − ∞ X k =0 λ k (68) = 0 (69) This finishes the proof that ∆ + and ∆ − are equal. A P P E N D I X E N U M E R I C A L M E T H O D While the theoretical developments in this work were in the continuous domain, all examples were simulated numerically and thus in the discrete domain. The original Slepian opti- mization criterion in discrete time can be written as a Rayleigh quotient: λ = v H Cv v H v , (70) where C = F H W S + F W is the concentration matrix with F W the unitary Discrete Fourier Transform matrix limited to the selected frequenc y band (bandwidth W ) and S + is the selection matrix (i.e., diagonal matrix with 1 on the selected region and 0 elsewhere). The discrete prolate spheroidal se- quences are then the eigenv ectors of the concentration matrix C multiplied by F W . This discrete sequence will con verge to the continuous-domain solution when the sampling step decreases and the overall support increases. If the original selection matrix S + is substituted by a generalized selection matrix S = S + − S − where S + and S − are the selection matrices of the selected and penalized regions respectively , the optimization criterion becomes the generalized optimization criterion that is the topic of this Paper: λ = v H Cv v H v with C = F H W SF W . (71) Since all Fourier modes, except the constant mode, come in pairs with the same eigenv alue/frequency , taking an ev en bandwidth W would mean that one of the pairs is split and therefore the truncated DFT matrix would be ambiguous. Therefore, in this Paper all simulations are done using odd values for the bandwidth. The matrix F W can be formed by taking the eigen vectors of the Laplacian of a ring graph with N nodes. In order to approximate the continuous case with a discrete time simulation, two steps are needed. First, the continuous signals are sampled at sampling frequency f s . If the sampled signal is interpreted as a discrete signal, but still with infinite SUBMITTED TO IEEE TRANSACTIONS ON SIGNAL PR OCESSING 11 T ABLE I L I ST O F PAR A M E TE R S F O R T HE N U M E RI C A L S I M UL ATI O N U S E D I N D I FFE R E N T F I GU R E S O F T H IS PA PE R , ∆ P I N DI C A T E S A V A R IA B L E S H I FT F O R F I G . 6 - 7 . Fig. f s [ H z ] N P + n [ s ] T + n [ s ] P − n [ s ] T − n [ s ] α [%] 2 100 4096 2.0 1.0 3.5 0.5 5 3 100 4096 0.5 0.5 2.0 1.0 2 4 100 4096 0.5 0.5 2.0 1.0 5 5 100 4096 2.0 1.0 3.5 0.5 4 6 100 4096 1.5 0.5 2 . 5 + ∆ P 0.5 1 7 100 4096 1.5 0.5 2 . 5 + ∆ P 0.5 1,2,3 8 100 4096 2.5 1.5 5.5 1.5 1.5 length, the corresponding frequency domain is limited to the interval [ − f s 2 , f s 2 ] , though still continuous. Since it is not possible to run simulations on a signal of infinite length, a finite support of N samples is taken, corresponding to a time duration of N f s . The Discrete Fourier T ransform of the resulting discrete time signal of finite length corresponds to the frequencies − f s 2 + k f s N with k = 0 ..N − 1 . (72) Here f s is always chosen to be 100Hz and N = 4096 . If the one-sided continuous time bandwidth is chosen to be F = α 2 π f s 2 with 0 < α < 1 , the corresponding indices k to be kept are giv en by: k ∈ N : − f s 2 + k f s N ≤ α f s 2 . (73) This rev erts to taking into account only the first 1 + 2 α N 2 columns of the DFT matrix. T aking for example α = 5% yields the following values for k : k ∈ N : − 50 + k 100 4096 ≤ 2 . 5 = { 1946 .. 2150 } . (74) This means that the first 2150 − 1946 + 1 = 205 = 1 + 2 0 . 05 4096 2 columns of the DFT matrix will be kept in the calculations. A summary of all parameter values used for the Figures in this work is giv en in T able I. R E F E R E N C E S [1] D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty - i, ” The Bell System T echnical Journal , vol. 40, no. 1, pp. 43–63, 1961. [2] H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertaintyii, ” Bell System T echnical Journal , vol. 40, no. 1, pp. 65–84, 1961. [3] H. J. Landau and H. O. 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Robin Demesmaeker obtained his B.Sc. in Elec- trical Engineering from Ghent University , Belgium, in 2015 and his M.Sc. in Electrical and Electronic Engineering from the Ecole Polytechnique F ´ ed ´ erale de Lausanne (EPFL), Switzerland. During his M.Sc. he joined the Medical Image Processing Labora- tory (MIP:lab) as a semester project student un- der supervision of Prof. Dimitri V an De V ille and Dr . Maria Giulia Preti. His main field of interest is the application of information technologies and signal processing in biomedical applications. Maria Giulia Preti has joined Prof. V an De V ille group as a post-doc in 2013. Her current research aims at understanding the connections between brain functionality and brain microscopic anatomy by using advanced techniques of Magnetic Resonance Imaging. She receiv ed her Ph.D. in Bioengineering at Politecnico di Milano (Milan, Italy) in 2013, after her M.Sc. (2009) and B.Sc. (2007) in Biomedical Engineering, as well at Politecnico di Milano. Dur- ing her Ph.D., mentored by Prof. Giuseppe Baselli, she focused on advanced techniques of brain mag- netic resonance imaging, in particular she dev eloped a method of groupwise fMRI-guided tractography , that revealed to be useful in the in-vivo investi- gation of the pathophysiological changes across the ev olution of Alzheimer’s disease. For this project, she had been collaborating full-time with the hospital Fondazione Don Gnocchi in Milan (Magnetic Resonance Laboratory). In 2011, she was awarded a Progetto Rocca fellowship from MIT -Italy and spent a visiting research period at the MIT and Harvard Medical School (Boston, USA), under the supervision of Prof. Nikos Makris. Dimitri V an De Ville (M’02,SM’12) receiv ed the M.S. degree in engineering and computer sciences and the Ph.D. degree from Ghent University , Bel- gium, in 1998, and 2002, respectively . After a post- doctoral stay (2002-2005) at the lab of Prof. Michael Unser at the Ecole Polytechnique F ´ ed ´ erale de Lau- sanne (EPFL), Switzerland, he became responsible for the Signal Processing Unit at the Uni versity Hospital of Geneva, Switzerland, as part of the Centre d’Imagerie Biom ´ edicale (CIBM). In 2009, he received a Swiss National Science Foundation professorship and since 2015 became Professor of Bioengineering at the EPFL and the Univ ersity of Genev a, Switzerland. His research interests include wav elets, sparsity , pattern recognition, and their applications in computational neuroimaging. He was a recipient of the Pfizer Research A ward 2012, the N ARSAD Independent In vestigator A ward 2014, and the Leenaards Founda- tion A ward 2016. Dr . V an De V ille served as an Associate Editor for the IEEE TRANS- A CTIONS ON IMAGE PR OCESSING from 2006 to 2009 and the IEEE SIGN AL PROCESSING LETTERS from 2004 to 2006, as well as Guest Editor for several special issues. He was the Chair of the Bio Imaging and Signal Processing (BISP) TC of the IEEE Signal Processing Society (2012- 2013) and is the Founding Chair of the EURASIP Biomedical Image & Signal Analytics SA T . He is Co-Chair of the biennial W av elets & Sparsity series conferences, together with V . Goyal and M. Papadakis.
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