Gabor orthonormal bases, tiling and periodicity

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space [Formula: see text] from dual pairs of atomic wave packet frames in [Formula: see text] is studied. A class of matrix-valued dual generators from its associated dual pair of atomic wave packets has been obtained. We provide a characterization of matrix-valued dual window functions in terms of orthogonality of wave packet Bessel sequences. A perturbation result with respect to window functions for matrix-valued dual frames is given. It is well known that an orthogonal Parseval Hilbert frame for a Hilbert space turned out be an orthonormal basis for the space, however, this is not true for an orthogonal matrix-valued wave packet Parseval frame for the underlying matrix-valued function space. We give a type of matrix-valued orthonormal basis associated with an orthonormal basis of [Formula: see text].

In practice, the time variable cannot be negative. The space $$L^2({\mathbb {R}}_+)$$ of square integrable functions defined on the right half real line $${\mathbb {R}}_+$$ models the causal signal space. This paper focuses on a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$. The density theorem for Gabor systems in $$L^2(\mathbb {R})$$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $$L^2({\mathbb {R}})$$ in terms of the index set alone-independently of window functions. The space $$L^2({\mathbb {R}}_+)$$ admits no nontrivial Gabor system since $${\mathbb {R}}_+$$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ and the notion of $$\Theta $$-transform matrix. Using the $$\Theta $$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ under the condition that $$\log _ba$$ is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $$L^2({\mathbb {R}}_+)$$ is that $$\log _ba \le 1$$. Simultaneously, we obtain a $$\Theta $$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $$L^2(\mathbb {R}_+)$$.

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L 2(ℝn) under the action of lattice translations and dilations by products of elements drawn from non-commuting sets of matrices A and B. Typically, the members of B are matrices whose eigenvalues have magnitude one, while the members of A are matrices expanding on a proper subspace of ℝn. The theory of these systems generalizes the classical theory of wavelets and provides a simple and flexible framework for the construction of orthonormal bases and related systems that exhibit a number of geometric features of great potential in applications. For example, composite wavelets have the ability to produce “long and narrow” window functions, with various orientations, well-suited to applications in image processing.KeywordsOrthonormal BasisFundamental DomainGabor FrameGabor SystemParseval FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

A low-cost type terahertz time-domain spectroscopic system can be constituted using a multimode laser diode instead of a pulse laser. To suppress noise, a window function is usually used in the Fourier transformation. When this system is used to measure an optically thick sample, the obtained refractive index shows sinusoidal structures. This is caused by ingress of a signal fragment into the measured range and egress of another signal fragment out of the measured range. In addition, a broad positive background component appears in the imaginary part of the refractive index. This is because of the decreased amplitude of the central structure of the signal, which results from the time shift under the window function caused by sample insertion. These false structures can be eliminated by using a modified window function that is shifted with the signal when the sample is inserted.

In this paper we show how to construct a certain class of orthonormal bases in starting from one or more Gabor orthonormal bases in . Each such basis can be obtained acting on a single function with a set of unitary operators which operate as translation and modulation operators in suitable variables. The same procedure is also extended to frames and wavelets. Many examples are discussed.

In this paper we present the Balian-Low theorem for the twosided windowed quaternionic Fourier transform (WQFT), a theorem which expresses the fact that time-frequency concentrations are incompatible with non-redundancy whenever Gabor systems form orthonormal bases or frames. Since uncertainty principles are closely connected with representations of the kernel of the Fourier transform under consideration, we construct a suitable representation for the kernel of our two-sided WQFT which in turn provides suitable Gabor systems. We proceed by deriving several important properties of the WQFT, such as shift and modulation operators, a reconstruction formula, orthogonality relations and a Heisenberg uncertainty principle for the WQFT. Finally, we establish the Balian-Low theorem for Gabor orthonormal bases associated with discrete versions of the kernels of the WQFT and of the right-sided WQFT.

The proposed sound synthesis tool converts a physics-based frequency-domain model of wind turbine trailing edge noise to a time-domain signal while accounting for the appropriate time shift due to the propagation between the moving blades and the fixed observer. A window function that implements cross-fading between consecutive signal grains is proposed and a method to objectively estimate the influence of the synthesis parameters is described. As the synthesis tool is independent of the aerodynamic noise model, it can be readily adapted to auralize other noise sources such as turbulent inflow noise or stall noise.

Background: For the traditional Fourier Transform (FT), it cannot effectively detect the frequency of non-stationary signals with time. Analyzing the local characteristics of time-varying signal by using FT is hard to achieve. Therefore, many time-frequency analysis methods which can meet different needs have been proposed on the basis of the traditional Fourier transform, like the Short Time Fourier Transform (STFT), the widely used Continuous Wavelet Transform (CWT), Wigner-Ville Distribution (WVD) and so on. However, the best time and frequency resolution cannot be achieved at the same time due to the uncertainty criterion. Methods: From the point of view of optimizing time-frequency performance, a new Generalized S Transform (GST) window function optimization method is proposed by combining time frequency aggregation with an improved genetic algorithm in this paper. Results: In the noiseless condition, the Linear Frequency Modulation (LFM), Nonlinear Frequency Modulation (NLFM) and binary Frequency Shift Keying (2FSK) signals are simulated. The simulation results prove that the method can improve the time-frequency concentration and decreasing Rényi entropy effectively. Compared with other four traditional time-frequency analysis methods, the improved GST has more advantages. Conclusion: In this paper, a new method of optimizing the window function in GST based on improved GA is proposed in this paper. Experiments on LFM, NLFM and 2FSK signals show that the proposed method not only has the advantages of high resolution, but also determines the optimal parameters via the time frequency focusing criterion and the Rényi entropy. Compared with the other four kinds of time-frequency analysis methods, the optimized GST based on improved GA in this paper has the best time-frequency focusing.

Gabor analysis is playing an important role in time-frequency analysis for the last 60 years. The fundamental concept of Gabor frames has become a focus also in the finite dimensional setting. Here, we study which sets of discrete time frequency shifts admit a proper choice of window vector, so that they generate an orthonormal Gabor basis in the underlying finite dimensional vector space. We fully characterize such time-frequency shift sets in the case that the vector space dimension is a prime number.

Optimal radar waveform design for target detection has been addressed in prior research literature under various assumptions regarding noise and clutter. A common model of the radar scene in work of this kind is a linear time-invariant (LTI) operator with additive Gaussian noise that acts on the transmitted signal to produce the received signal. This model is intrinsically ill-suited to dynamic scenes or moving radar platforms because it cannot account for Doppler. This paper introduces scene models based on Hilbert-Schmidt class (HS) operators on the space of finite-energy signals. This category of models generalizes the LTI category in the sense that every LTI operator is also a HS operator, but the HS class includes operators that account for frequency shifts as well as time shifts and are thus suitable for modeling radar scenes involving Doppler. Every HS operator is uniquely expressible as a superposition of elementary time and frequency shift operators, thus providing a convenient interpretation of a scene in terms of these physically meaningful operations on the transmitted signal. Application of this perspective to waveform design for target detection in noise and to optimal receiver processing for a given waveform for target detection in clutter and noise are demonstrated.

Frequency shift, due to quartz crystal resonator aging, has been identified as one of the most important quality control problems of quartz crystal products. The problem becomes more significant due to the device miniaturization and high precision standards for telecommunication applications. Since aging induced frequency shift occurs during a long time frame, it is necessary to predict the long-term behavior of the devices based on the short-term data obtained under an accelerated environment. On the other hand, frequency shift is associated with quite large random variation, and thus, a proper probabilistic theory should be used for analyzing test data and for developing a reliable prediction model. Accelerated testing was performed for various types of crystal resonators under elevated temperatures. The frequency shifts of the devices were measured at different testing periods. Markov chain model was used to characterize the frequency shift of the devices. The obtained short-term test results were used for calibrating the probabilistic transition matrix of Markov chain model. The model can then be used for predicting the long-term frequency shift. The time-temperature superposition principle in viscoelasticity was adopted to address the shift in time under different temperatures.

Experimental study and theoretical prediction of aging induced frequency shift of crystal resonators and oscillators

Previous work has shown that time-frequency distributions (TFDs) belonging to Cohen's class can be represented as a sum of weighted spectrograms. This representation offers the means of reducing the computational complexity of TFDs. The windows in the spectrogram representation may either be the eigenfunctions obtained from an eigen decomposition of the kernel or any complete set of orthonormal basis functions. The efficiency of the computation can further be increased by using a set of scaled and shifted functions like wavelets. In this paper, the concept of scaling is considered in discrete-time domain. The scale operator in the frequency domain is formulated and the vectors which correspond to the solutions of this eigenvalue problem in discrete-time are derived. These new eigenvectors are very similar in structure to the eigenvectors obtained from eigensystem decomposition of reduced interference distribution (RID) kernels. The relationship between these two sets of window functions is illustrated and a new efficient way of decomposing time-frequency kernels is introduced. The results are compared to the previous decomposition methods. Finally, some possible applications of these discrete scale functions in obtaining new time-frequency distributions are discussed.

In Section 3.9.3, the point was made that a radar receiver performs a cross-correlation between the received signal and a reference of the transmitted waveform. Furthermore, the received signal may differ from the transmitted signal on account of being time delayed and Doppler shifted. The design of a matched receiver usually entails designing the receiver to match the transmitted waveform and so may no longer match the received signal on account of the time and frequency displacements of the received signal over the transmitted one. Matched reception is tantamount to autocorrelation of the received signal at a particular time shift. In radar, it is most convenient to think of the particular time shift associated with a given range cell. While the autocorrelation function handles shifts in time, it offers no provision for handling shifts in frequency. The ambiguity function describes the response of a matched receiver to a finite duration signal. The use of the ambiguity function in ambiguity analysis considers the receiver to be matched to a signal received from a target at a given time delay (range cell) and frequency. The ambiguity function describes the matched receiver response as a function of any additional time delay and any additional Doppler frequency.