Lattice size estimates for Gabor decompositions

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.

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

We obtain a sufficient condition of the weak localizability of a principal submodule in the module of entire functions of exponential type and polynomial growth on the real line. Applications to the problem of the (weak) spectral synthesis in the Schwartz space C∞ (a; b) are discussed.

Orthogonal time frequency space (OTFS) modulations are robust to time-varying frequency-selective fading channels. OTFS modulations operate in the delay-Doppler domain whereas two-dimensional (2D) orthogonal frequency division multiplexing (OFDM) modulations operate in the time-frequency domain. For 2D OFDM modulations in time-varying frequency-selective fading channels, we investigate the pilot pattern design problem, which minimizes the mean square error (MSE) of channel estimation. The MSE lower bound (LB) is theoretically derived to provide the design criterion. Based on the criterion, we show that the LB achieving design can be found by exhaustive 2D pilot location search. Exhaustive 2D search has high computational complexity. To reduce the complexity, sufficient conditions on the existence of LB achieving design are provided to decouple the 2D problem into two one-dimensional (1D) problems. For the 1D problem, we propose the LB achieving design when the number of possible pilot locations is divisible by the number of pilots. Simulation results illustrate that our proposed LB achieving design is superior to the random pilot pattern design and the bilinear interpolation channel estimation method.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Let E′ be the dual of a nuclear Frechet space E (of which the Schwartz space ℒ(ℝd) of tempered distributions is one). We give a simple sufficient condition for the strong continuity of a weakly continuous E′-valued stochastic process and as applications examine the Kolmogorov-Hahn-Delporte criteria.

We utilize the concept of an abstract Wiener space to prove a converse to a theorem of Minlos, thereby obtaining necessary and sufficient conditions for a Hilbert subspace of $\mathcal {S}’({{\text {R}}^d})$ to support a given Gaussian process over $\mathcal {S}({{\text {R}}^d})$.

We present new necessary and sufficient conditions for a function on $\partial \Omega \times S^1$ to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set $\overline \Omega \subset \mathbb {R}^2$. The approach is based on an explicit Hilbert transform associated with traces on the boundary of A-analytic functions in the sense of Bukhgeim.

For the non-band-limited functionΨ, a sufficient condition is presented under which\(\{ \sqrt {s_j } \psi (s_j \cdot - kb)\} \) is a frame for L2(R). The stability of these frames is studied. For the wavelets frequently used in signal processing, some concrete results are given.

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.

We have recently shown that click evoked auditory brainstem responses (ABRs) can be efficiently processed using a novelty detection paradigm. Here, ABRs as a large-scale reflection of a stimulus locked neuronal group synchronization at the brainstem level are detected as novel instance-novel as compared to the spontaneous activity which does not exhibit a regular stimulus locked synchronization. In this paper we propose for the first time Gabor frame operators as an efficient feature extraction technique for ABR single sweep sequences that is in line with this paradigm. In particular, we use this decomposition technique to derive the Gabor frame phase stability (GFPS) of sweep sequences of click and chirp evoked ABRs. We show that the GFPS of chirp evoked ABRs provides a stable discrimination of the spontaneous activity from stimulations above the hearing threshold with a small number of sweeps, even at low stimulation intensities. It is concluded that the GFPS analysis represents a robust feature extraction method for ABR single sweep sequences. Further studies are necessary to evaluate the value of the presented approach for clinical applications.

Invertibility of the Gabor frame operator on the Wiener amalgam space

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

Banach frames in coorbit spaces consisting of elements which are invariant under symmetry groups