The Experts below are selected from a list of 288 Experts worldwide ranked by ideXlab platform
Ullrich J. Monich - One of the best experts on this subject based on the ideXlab platform.
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Non-Existence of Convolution Sum System Representations
IEEE Transactions on Signal Processing, 2019Co-Authors: Holger Boche, Ullrich J. Monich, Bernd MeinerzhagenAbstract:Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper, we ask a far more general question: is it at all possible to define a generalized convolution sum with natural properties that works for all absolutely integrable continuous signals that Vanish at Infinity and all stable LTI systems? We prove that the answer is “no.” Further, for certain subspaces, we give a sufficient and necessary condition for uniform convergence. Finally, we discuss the implications of our results on the effectiveness of window functions in the convolution sum.
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ICASSP - Analytic Properties of Downsampling for Bandlimited Signals
ICASSP 2019 - 2019 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2019Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper we study downsampling for bandlimited signals. Downsampling in the discrete-time domain corresponds to a removal of samples. For any downsampled signal that was created from a bandlimited signal with finite energy, we can always compute a bandlimited continuous-time signal such that the samples of this signal, taken at Nyquist rate, are equal to the downsampled discrete-time signal. However, as we show, this is no longer true for the space of bounded bandlimited signals that Vanish at Infinity. We explicitly construct a signal in this space, which after downsampling does not have a bounded bandlimited interpolation. This shows that downsampling in this signal space is an operation that can lead out of the set of discrete-time signals for which we have a one-to-one correspondence with continuous-time signals.
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Distributional Behavior of Convolution Sum System Representations
IEEE Transactions on Signal Processing, 2018Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper, we study the validity of the usual convolution sum sampling representation of linear time-invariant (LTI) systems. We consider continuous input signals with finite energy that are absolutely integrable and Vanish at Infinity. Even for these benign signals, the convolution sum does not always converge. There exist LTI systems and signals such that the convolution sum diverges even in a distributional sense. This result shows that the practice of multiplying a signal with a Dirac comb and convolving subsequently with the impulse response of the LTI system is not valid for this signal space. We further fully characterize the LTI systems for which we have convergence for all signals in the space, and establish a connection between the pointwise, uniform, and distributional convergence. In particular, we show that the convolution sum converges in a distributional sense if and only it converges in a classical pointwise sense. Hence, for this signal space, nothing can be gained by treating the convergence in a distributional sense.
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Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals
Problems of Information Transmission, 2013Co-Authors: Holger Boche, Ullrich J. MonichAbstract:The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty $ of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for $\mathcal{B}_\pi ^\infty $ , there is a more general definition of the Hilbert transform, which is based on the abstract H ^1-BMO(ℝ) duality. It was recently shown in [1] that, in addition to this abstract definition, there exists an explicit formula for calculating the Hilbert transform. Based on this formula we study properties of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty $ of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace $\mathcal{B}_{\pi ,0}^\infty $ of bounded bandlimited signals that Vanish at Infinity. By studying the properties of the Hilbert transform, we continue the work [2].
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EUSIPCO - Complete characterization of bandlimited signals with bounded Hilbert transform
2012Co-Authors: Holger Boche, Ullrich J. MonichAbstract:For the space of bounded bandlimited signals a definition of the Hilbert transform by the usual Hilbert transform integral is not possible, because the integral diverges for certain bounded bandlimited signals. There are other ways to define the Hilbert transform meaningfully. Recently, it was shown that, for bounded bandlimited signals, a simple formula can be used to calculate the Hilbert transform. However, the Hilbert transform of a bounded bandlimited signal is not necessarily bounded again. In this paper, we completely characterize the bounded bandlimited signals that have a bounded Hilbert transform by giving a necessary and sufficient condition for the boundedness. Further, we use this condition to prove that there exist bounded bandlimited signals that even Vanish at Infinity, the Hilbert transform of which is unbounded.
Holger Boche - One of the best experts on this subject based on the ideXlab platform.
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Non-Existence of Convolution Sum System Representations
IEEE Transactions on Signal Processing, 2019Co-Authors: Holger Boche, Ullrich J. Monich, Bernd MeinerzhagenAbstract:Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper, we ask a far more general question: is it at all possible to define a generalized convolution sum with natural properties that works for all absolutely integrable continuous signals that Vanish at Infinity and all stable LTI systems? We prove that the answer is “no.” Further, for certain subspaces, we give a sufficient and necessary condition for uniform convergence. Finally, we discuss the implications of our results on the effectiveness of window functions in the convolution sum.
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ICASSP - Analytic Properties of Downsampling for Bandlimited Signals
ICASSP 2019 - 2019 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2019Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper we study downsampling for bandlimited signals. Downsampling in the discrete-time domain corresponds to a removal of samples. For any downsampled signal that was created from a bandlimited signal with finite energy, we can always compute a bandlimited continuous-time signal such that the samples of this signal, taken at Nyquist rate, are equal to the downsampled discrete-time signal. However, as we show, this is no longer true for the space of bounded bandlimited signals that Vanish at Infinity. We explicitly construct a signal in this space, which after downsampling does not have a bounded bandlimited interpolation. This shows that downsampling in this signal space is an operation that can lead out of the set of discrete-time signals for which we have a one-to-one correspondence with continuous-time signals.
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Distributional Behavior of Convolution Sum System Representations
IEEE Transactions on Signal Processing, 2018Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper, we study the validity of the usual convolution sum sampling representation of linear time-invariant (LTI) systems. We consider continuous input signals with finite energy that are absolutely integrable and Vanish at Infinity. Even for these benign signals, the convolution sum does not always converge. There exist LTI systems and signals such that the convolution sum diverges even in a distributional sense. This result shows that the practice of multiplying a signal with a Dirac comb and convolving subsequently with the impulse response of the LTI system is not valid for this signal space. We further fully characterize the LTI systems for which we have convergence for all signals in the space, and establish a connection between the pointwise, uniform, and distributional convergence. In particular, we show that the convolution sum converges in a distributional sense if and only it converges in a classical pointwise sense. Hence, for this signal space, nothing can be gained by treating the convergence in a distributional sense.
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Characterization of the peak value behavior of the Hilbert transform of bounded bandlimited signals
Problems of Information Transmission, 2013Co-Authors: Holger Boche, Ullrich J. MonichAbstract:The peak value of a signal is a characteristic that has to be controlled in many applications. In this paper we analyze the peak value of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty $ of bounded bandlimited signals. It is known that for this space the Hilbert transform cannot be calculated by the common principal value integral, because there are signals for which it diverges everywhere. Although the classical definition fails for $\mathcal{B}_\pi ^\infty $ , there is a more general definition of the Hilbert transform, which is based on the abstract H ^1-BMO(ℝ) duality. It was recently shown in [1] that, in addition to this abstract definition, there exists an explicit formula for calculating the Hilbert transform. Based on this formula we study properties of the Hilbert transform for the space $\mathcal{B}_\pi ^\infty $ of bounded bandlimited signals. We analyze its asymptotic growth behavior, and thereby solve the peak value problem of the Hilbert transform for this space. Further, we obtain results for the growth behavior of the Hilbert transform for the subspace $\mathcal{B}_{\pi ,0}^\infty $ of bounded bandlimited signals that Vanish at Infinity. By studying the properties of the Hilbert transform, we continue the work [2].
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EUSIPCO - Complete characterization of bandlimited signals with bounded Hilbert transform
2012Co-Authors: Holger Boche, Ullrich J. MonichAbstract:For the space of bounded bandlimited signals a definition of the Hilbert transform by the usual Hilbert transform integral is not possible, because the integral diverges for certain bounded bandlimited signals. There are other ways to define the Hilbert transform meaningfully. Recently, it was shown that, for bounded bandlimited signals, a simple formula can be used to calculate the Hilbert transform. However, the Hilbert transform of a bounded bandlimited signal is not necessarily bounded again. In this paper, we completely characterize the bounded bandlimited signals that have a bounded Hilbert transform by giving a necessary and sufficient condition for the boundedness. Further, we use this condition to prove that there exist bounded bandlimited signals that even Vanish at Infinity, the Hilbert transform of which is unbounded.
Liuyang Shao - One of the best experts on this subject based on the ideXlab platform.
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Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential Vanishing at Infinity
Open Mathematics, 2019Co-Authors: Liuyang ShaoAbstract:Abstract The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term $$\begin{array}{} \displaystyle \begin{cases} -\triangle u+V(x)u-l(x)\phi|u|^{3}u=\eta K(x)f(u),~~\mbox{in}~~\mathbb{R}^{3}, \notag\\ -\triangle\phi=l(x)|u|^{5},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{in}~~\mathbb{R}^{3}, \end{cases} \end{array}$$(1.1) where the potential V(x) and K(x) are positive continuous functions that Vanish at Infinity, and l(x) is bounded, nonnegative continuous function. Under some simple assumptions on V, K, l and f, we prove that the problem (1.1) has a non-trivial solution.
Bernd Meinerzhagen - One of the best experts on this subject based on the ideXlab platform.
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Non-Existence of Convolution Sum System Representations
IEEE Transactions on Signal Processing, 2019Co-Authors: Holger Boche, Ullrich J. Monich, Bernd MeinerzhagenAbstract:Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper, we ask a far more general question: is it at all possible to define a generalized convolution sum with natural properties that works for all absolutely integrable continuous signals that Vanish at Infinity and all stable LTI systems? We prove that the answer is “no.” Further, for certain subspaces, we give a sufficient and necessary condition for uniform convergence. Finally, we discuss the implications of our results on the effectiveness of window functions in the convolution sum.
Francisco J. Mendoza-torres - One of the best experts on this subject based on the ideXlab platform.
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A contribution to the Dirichlet-Jordan theorem for non Lebesgue integrable functions
2021Co-Authors: Francisco J. Mendoza-torres, Edgar Torres-teutle, Ugur SengulAbstract:We prove that the convergence involved in the inversion of the Fourier transform of bounded variation functions which Vanish at Infinity is uniform at points of continuity. We do not impose the condition that the functions belong to L1(R):
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A Note on Two Classical Theorems of the Fourier Transform for Bounded Variation Functions
Communications in Mathematics and Applications, 2016Co-Authors: Francisco J. Mendoza-torresAbstract:Employing the Henstock-Kurzweil integral, we make simple proofs of the Riemann-Lebesgue lemma and the Dirichlet-Jordan theorem for functions of bounded variation which Vanish at Infinity.
