The Mystery of Gabor Frames

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as hat{g}(xi )= prod _{j=1}^n (1+2pi idelta _jxi )^{-1} , e^{-c xi ^2} for delta _1,ldots ,delta _nin mathbb {R}, c >0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling’s sampling theorem for the Paley–Wiener space of entire functions, we prove that every separated set with lower Beurling density >1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of g with respect to a rectangular lattice alpha mathbb {Z}times beta mathbb {Z} forms a frame, if and only if alpha beta <1. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets “without inequalities” in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann–Fock space.

Motivated by the human auditory system, a new signal transform is presented which models the way humans hear. Cochlear processing acts like a constant bandwidth bank of filters in the low frequency range but is of proportional bandwidth at higher frequencies. This new transform, which we call the composite wavelet transform, is better able to model this process than standard signal processing techniques such as the short‐time Fourier transform (STFT) and the continuous wavelet transform (CWT). The composite wavelet transform in fact provides a signal analysis tool that is able to examine signals with competing signal structures whereas the STFT and the CWT do not. In order to insure stable signal recovery, the theory of frames is examined. An overview of the theory of frames will be given and wavelet and Gabor frames will be discussed. Recent work on frames for the composite wavelet transform will also be presented. Finally, some applications of frames, especially to speech signal processing will be considered.

Time-frequency partitions and characterizations of modulation spaces with localization operators

We study time-frequency localization operators of the form A a φ 1 , φ 2 A_a^{\varphi _1\!,\varphi _2} , where a a is the symbol of the operator and φ 1 , φ 2 \varphi _1 , \varphi _2 are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) , the Schatten class of order p p , is that a a belongs to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) and the window functions to the modulation space M 1 M^1 . Here we prove a partial converse: if A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) for every pair of window functions φ 1 , φ 2 ∈ S ( R 2 d ) \varphi _1,\varphi _2\in \mathcal {S}(\mathbb {R}^{2d}) with a uniform norm estimate, then the corresponding symbol a a must belong to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p = ∞ p=\infty and p = 2 p=2 , we recapture earlier results, which were obtained by different methods.

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θa transform in , and using Θa‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.

The main issue in Chapter 11 was to state necessary and/or sufficient conditions for a Gabor system \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb{Z}}\) in \(L^{2}(\mathbb{R})\) to form a frame. We will now take the next step and consider Gabor frames that are convenient to apply in practice. From the general frame theory in Chapter 5 and Chapter 6 we know that frames are particularly useful when the frame decomposition takes a simple form, which is the case if either the frame is tight or we have access to a convenient dual frame.

Gabor Frames and Contact Structures: Signal Encoding and Decoding in the Primary Visual Cortex

Signal processing by alternate dual Gabor frames

The material presented so far naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. In this final chapter we make connections to abstract harmonic analysis and show how we can gain insight about frames via the theory for group representations. More precisely, we show how the orthogonality relations for square-integrable group representations lead to series expansions of the elements in the underlying Hilbert space; on a concrete level, this gives an alternative approach to Gabor systems and wavelet systems. Feichtinger and Gröchenig proved that the group-theoretic setup even allows us to obtain series expansions in a large scale of Banach spaces, a result which leads Gröchenig to define frames in Banach spaces. By removing some of the conditions we obtain p-frames, first studied separately by Aldroubi, Sun, and Tang.

In the long-term service, the wharf structure can be damaged by ship impact, wave load, and even earthquake, which will affect the safe production and smooth operation of the port. Based on the theory of structural dynamic response analysis and wavelet packet analysis principle, this paper established the damage identification index of the wharf frame structure. Combining with the finite element method and the dynamic response theory of the wharf frame structure, it set up a finite element analysis model of the dynamic response of the wharf frame structure under the action of multiple loads, with the impact load of the ship as the dynamic load under the non-damaged state and the different damaged states. In addition, the characteristic response point location was drawn up. Furthermore, the transient dynamic analysis and damage index analysis of the frame structure in the non-damaged and damaged state were conducted respectively. In addition, the model test and numerical simulation results were combined to compare and analyze the identification of damage indicators, so as to verify the identification effect of the established damage identification indicators on the structural damage, which lays a foundation for the next step of structural damage identification.

A calculation of higher-order relativistic contributions to the Zeeman effect in the $2^{3}P$, $2^{3}S$, $3^{3}P$, $4^{3}P$, and $5^{3}P$ states of helium is given. The contributions to the $g$ factors ${g}_{S}^{\ensuremath{'}}$, ${g}_{L}^{\ensuremath{'}}$, and ${g}_{X}$, have been calculated to order ${\ensuremath{\alpha}}^{2}$ and $\frac{m}{M}$ where $m$ is the electron mass and $M$ is the helium mass. Relativistic contributions are obtained from the matrix elements of a reduced Breit Hamiltonian with a series of numerical wave functions using a Hylleraas basis. The comparison of the theoretical values with the experimental determinations of the $g$ factors is given for the $2^{3}P$ and $2^{3}S$ states. The value of ${g}_{S}^{\ensuremath{'}}$ calculated for $2^{3}P$ helium is within two standard deviations of the experimental error of the measurement of Lewis et al. A recent precise measurement of the $g$ factor in $2^{3}S_{1}$ helium by Brossel et al. is three standard deviations of the experimental error from the value calculated here and from the result of Perl and Hughes. These higher-order contributions are necessary for the determination of the $2^{3}P$ fine-structure intervals from the Zeeman effect, and hence to test the theory of fine structure for helium and to determine precisely the fine-structure constant $\ensuremath{\alpha}$ from the helium fine structure. The relativistic contributions to ${g}_{S}^{\ensuremath{'}}$ have been calculated for the helium isoelectronic series from Li II through Ne IX for the $2^{3}P_{1}$, $3^{3}P_{1}$, $4^{3}P_{1}$, and $5^{3}P_{1}$ states.

We analytically derive the change in exciton fine structure splitting (FSS) under external stresses in self-assembled InAs/GaAs quantum dots, using the Bir–Pikus model. We find that the FSS change is mainly due to the strain-induced valence band mixing and valence-conduction band coupling. The exciton polarization angle under strain is determined by the argument of the electron-hole off-diagonal exchange integrals. The theory agrees well with the empirical pseudopotential calculations.

BackgroundThe autobiographical Implicit Association Test (aIAT) is a novel application of the implicit association concept for detecting life events. It has been used to reveal concealed knowledge in clinical and forensic settings, including detecting drug use. In this study, we aimed to explore the functionality of the aIAT to identify drug use in real-world settings.MethodsThe study used mixed methodology with known groups of drug users and nonusers. Recreational cocaine users (n = 23) and non-users (n = 23) were recruited through ethnographic methodology and assessed using a bespoke brief aIAT for cocaine use. An identical aIAT test for heroin detection was also administered to a sub-sample of 10 cocaine users and 13 nonusers. The accuracy of the cocaine aIAT was measured through ROC analysis. Paradoxical aIAT results were explored by integrating craving, consumption measures and life-story interviews into the analysis.ResultsWhilst the two brief aIATs showed good concurrent validity for cocaine users by accurately detecting drug using status for 18 of the 23 users (78.3%), the test falsely reported 61% cocaine users in the non-user comparison group. The average D-scores were 0.257±0.246 for the cocaine users and 0.134±0.367 for the non-users, showing no discriminatory power (t(44) = 1.339, p = 0.187; AUC = 0.605, p = 0.223). Results were independent from craving and recent cocaine use. The comparison group’s cocaine and heroin aIAT scores correlated significantly (r(13) = 0.776, p = 0.002) whilst an accurate absence of such relationship was evidenced in the cocaine using sample (r(10) = 0.061, p = 0.866). Triangulation with life-story interviews suggests that in the absence of an autobiographical event, this test may measure an alternative cognitive construct linked to the Self-concept.ConclusionThe aIAT is a variant of an attitude measure and can be better rationalized if propositional thinking is implied to explain outcomes. The Relational Frame and Social Knowledge Structure theories can perhaps provide a more plausible theoretical background. Further work is required to clarify which factors underlie this testing technique’s functioning. Reappraisal is advised before further forensic use of the instrument to ensure that general associations not related to autobiographical memory do not confound results.

Using a Doppler-tuned fast ion-beam--laser technique, with co- and counterpropagating laser beams, the intercombination $1s2s{}^{1}{S}_{0}$--$1s2p{}^{3}{P}_{1,F}$ transitions in ${}^{14}{\mathrm{N}}^{5+}$ and ${}^{15}{\mathrm{N}}^{5+}$ have been measured to 0.7 ppm. This precision is equivalent to 20 ppm of the two-electron Lamb shift and is more than two orders of magnitude more precise than current relativistic and QED theoretical results at this $Z$. A comparison is made with theory for the hyperfine structure and isotope shift. Additionally, using a single copropagating laser beam, the differences between the $2{}^{1}{S}_{0}$--$2{}^{3}{P}_{1,F=1}$ and $2{}^{1}{S}_{0}$--$2{}^{3}{P}_{0}$ intervals in ${}^{14}{\mathrm{N}}^{5+}$ and between the $2{}^{1}{S}_{0}$--$2{}^{3}{P}_{1,F=3/2}$ and $2{}^{1}{S}_{0}$--$2{}^{3}{P}_{0}$ intervals in ${}^{15}{\mathrm{N}}^{5+}$ have also been measured. From the former we obtain an improved result for the ${}^{14}{\mathrm{N}}^{5+}$ $2{}^{3}{P}_{1}$--$2{}^{3}{P}_{0}$ fine structure.