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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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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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System Representations for the Paley–Wiener Space $$\mathcal {PW}_{\pi }^2$$ PW
Journal of Fourier Analysis and Applications, 2018Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space $$\mathcal {PW}_{\pi }^2$$ PW π 2 , i.e., the set of bandlimited functions with finite $$L^2$$ L 2 -norm, by Convolution Sums. It is possible to use either, the Convolution Sum where the time variable is in the argument of the bandlimited impulse response, or the Convolution Sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the $$L^2$$ L 2 -norm in our case, is important. While it is well-known that both Convolution Sums converge uniformly on the whole real axis, the $$L^2$$ L 2 -norm of the second Convolution Sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.
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System Representations for the Paley–Wiener Space \(\mathcal {PW}_{\pi }^2\)
Journal of Fourier Analysis and Applications, 2016Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by Convolution Sums. It is possible to use either, the Convolution Sum where the time variable is in the argument of the bandlimited impulse response, or the Convolution Sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both Convolution Sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second Convolution Sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.
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Distributional System Representations on Bandlimited Signals
IEEE Transactions on Signal Processing, 2010Co-Authors: Ullrich J. Monich, Holger BocheAbstract:In this paper we analyze the distributional convergence behavior of time-domain Convolution type system representations on the Paley-Wiener space PWπ1. Two Convolution integrals as well as the discrete counterpart, the Convolution Sum, are treated. It is shown that there exist stable linear time-invariant (LTI) systems for which the Convolution integral representation does not exist because the integral is divergent, even if the convergence is interpreted in a distributional sense. Furthermore, we completely characterize all stable LTI systems for which a Convolution representation is possible by giving a necessary and sufficient condition for convergence. The classical and the distributional convergence behavior are compared, and differences between the convergence of the Convolution integral and the Convolution Sum are discussed. Finally, the results are illustrated by numerical examples.
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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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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System Representations for the Paley–Wiener Space $$\mathcal {PW}_{\pi }^2$$ PW
Journal of Fourier Analysis and Applications, 2018Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space $$\mathcal {PW}_{\pi }^2$$ PW π 2 , i.e., the set of bandlimited functions with finite $$L^2$$ L 2 -norm, by Convolution Sums. It is possible to use either, the Convolution Sum where the time variable is in the argument of the bandlimited impulse response, or the Convolution Sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the $$L^2$$ L 2 -norm in our case, is important. While it is well-known that both Convolution Sums converge uniformly on the whole real axis, the $$L^2$$ L 2 -norm of the second Convolution Sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.
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System Representations for the Paley–Wiener Space \(\mathcal {PW}_{\pi }^2\)
Journal of Fourier Analysis and Applications, 2016Co-Authors: Holger Boche, Ullrich J. MonichAbstract:In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by Convolution Sums. It is possible to use either, the Convolution Sum where the time variable is in the argument of the bandlimited impulse response, or the Convolution Sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both Convolution Sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second Convolution Sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.
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Distributional System Representations on Bandlimited Signals
IEEE Transactions on Signal Processing, 2010Co-Authors: Ullrich J. Monich, Holger BocheAbstract:In this paper we analyze the distributional convergence behavior of time-domain Convolution type system representations on the Paley-Wiener space PWπ1. Two Convolution integrals as well as the discrete counterpart, the Convolution Sum, are treated. It is shown that there exist stable linear time-invariant (LTI) systems for which the Convolution integral representation does not exist because the integral is divergent, even if the convergence is interpreted in a distributional sense. Furthermore, we completely characterize all stable LTI systems for which a Convolution representation is possible by giving a necessary and sufficient condition for convergence. The classical and the distributional convergence behavior are compared, and differences between the convergence of the Convolution integral and the Convolution Sum are discussed. Finally, the results are illustrated by numerical examples.
Ebénézer Ntienjem - One of the best experts on this subject based on the ideXlab platform.
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Evaluation of Convolution Sums entailing mixed Divisor Functions for a Class of Levels.
arXiv: Number Theory, 2019Co-Authors: Ebénézer NtienjemAbstract:Let $0< n,\alpha,\beta\in\mathbb{N}$ be such that $\gcd{(\alpha,\beta)}=1$. We carry out the evaluation of the Convolution Sums $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {\alpha\,k+\beta\,l=n} } }{\Sum}\sigma(k)\sigma_{3}(l)$ and $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {\alpha\,k+\beta\,l=n} } }{\Sum}\sigma_{3}(k)\sigma(l)$ for all levels $\alpha\beta\in\mathbb{N}$, by using in particular modular forms. We next apply Convolution Sums belonging to this class of levels to determine formulae for the number of representations of a positive integer $n$ by the quadratic forms in twelve variables $\underset{i=1}{\overset{12}{\Sum}}x_{i}^{2}$ when the level $\alpha\beta\equiv 0\pmod{4}$, and $\underset{i=1}{\overset{6}{\Sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)$ when the level $\alpha\beta\equiv 0\pmod{3}$. Our approach is then illustrated by explicitly evaluating the Convolution Sum for $\alpha\beta=3$, $4$, $6$, $7$, $8$, $9$, $12$, $14$, $15$, $16$, $18$, $20$, $21$, $27$, $32$. These Convolution Sums are then applied to determine explicit formulae for the number of representations of a positive integer $n$ by quadratic forms in twelve variables.
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Evaluation of the Convolution Sum involving the Sum of divisors function for 22, 44 and 52
Open Mathematics, 2017Co-Authors: Ebénézer NtienjemAbstract:Abstract The Convolution Sum, $ \begin{array}{} \Sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the Convolution Sums for αβ = 22. We then use these Convolution Sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).
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Elementary Evaluation of Convolution Sums involving primitive Dirichlet Characters for a Class of positive Integers
arXiv: Number Theory, 2016Co-Authors: Ebénézer NtienjemAbstract:We extend the results obtained by E. Ntienjem to all positive integers. Let $\EuFrak{N}$ be the subset of $\mathbb{N}$ consisting of $\,2^{\nu}\mho$, where $\nu$ is in $\{0,1,2,3\}$ and $\mho$ is a squarefree finite product of distinct odd primes. We discuss the evaluation of the Convolution Sum, $\underset{\substack{ {(l,m)\in\mathbb{N}^{2}} {\alpha\,l+\beta\,m=n} } }{\Sum}\sigma(l)\sigma(m)$, when $\alpha\beta$ is in $\mathbb{N}\setminus\EuFrak{N}$. The evaluation of Convolution Sums belonging to this class is achieved by applying modular forms and primitive Dirichlet characters. In addition, we revisit the evaluation of the Convolution Sums for $\alpha\beta=9$, $16$, $18$, $25$, $36$. If $\alpha\beta\equiv 0 \pmod{4}$, we determine natural numbers $a,b$ and use the evaluated Convolution Sums together with other known Convolution Sums to carry out the number of representations of $n$ by the octonary quadratic forms $a\,(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})+ b\,(x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2})$. Similarly, if $\alpha\beta\equiv 0 \pmod{3}$, we compute natural numbers $c,d$ and make use of the evaluated Convolution Sums together with other known Convolution Sums to determine the number of representations of $n$ by the octonary quadratic forms $c\,(\,x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} + x_{3}^{2} + x_{3}x_{4} + x_{4}^{2}\,) + d\,(\,x_{5}^{2} + x_{5}x_{6} + x_{6}^{2} + x_{7}^{2} + x_{7}x_{8} + x_{8}^{2}\,)$. We illustrate our method with the explicit examples $\alpha\beta = 3^{2}\cdot 5$, $\alpha\beta = 2^{4}\cdot 3$, $\alpha\beta = 2\cdot 5^{2}$ and $\alpha\beta = 2^{6}$, .
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The Convolution Sum $\Sum_{al+bm=n} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (1,14), (2,7), (1,7)$
arXiv: Number Theory, 2016Co-Authors: Ayșe Alaca, Şaban Alaca, Ebénézer NtienjemAbstract:We evaluate the Convolution Sum $\displaystyle W_{a,b}(n):= \Sum_{al+bm=n} \hspace{-3mm} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known Sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of $n$ by the octonary quadratic form $x_1^2 + x_2^2 +x_3^2 + x_4^2 + 7(x_5^2 + x_6^2 + x_7^2 + x_8^2)$. Finally we compare our evaluations of the Sums $W_{1,7}(n)$ and $W_{1,14}(n)$ with the evaluations of Lemire and Williams [10] and Royer [13] to express the modular forms $\Delta_{4,7}(z)$, $\Delta_{4,14, 1}(z)$ and $\Delta_{4,14, 2}(z)$ (given in [10, 13]) as linear combinations of eta quotients.
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the Convolution Sum Sum_ al bm n sigma l sigma m for a b 1 28 4 7 1 14 2 7 1 7
arXiv: Number Theory, 2016Co-Authors: Ayșe Alaca, Şaban Alaca, Ebénézer NtienjemAbstract:We evaluate the Convolution Sum $\displaystyle W_{a,b}(n):= \Sum_{al+bm=n} \hspace{-3mm} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known Sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of $n$ by the octonary quadratic form $x_1^2 + x_2^2 +x_3^2 + x_4^2 + 7(x_5^2 + x_6^2 + x_7^2 + x_8^2)$. Finally we compare our evaluations of the Sums $W_{1,7}(n)$ and $W_{1,14}(n)$ with the evaluations of Lemire and Williams [10] and Royer [13] to express the modular forms $\Delta_{4,7}(z)$, $\Delta_{4,14, 1}(z)$ and $\Delta_{4,14, 2}(z)$ (given in [10, 13]) as linear combinations of eta quotients.
Nankun Hong - One of the best experts on this subject based on the ideXlab platform.
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Ramanujan’s Convolution Sum twisted by Dirichlet characters
International Journal of Number Theory, 2019Co-Authors: Zafer Selcuk Aygin, Nankun HongAbstract:We find formulas for Convolutions of Sum of divisor functions twisted by the Dirichlet character [Formula: see text], which are analogous to Ramanujan’s formula for Convolution of usual Sum of divisor functions. We use the theory of modular forms to prove our results.
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Ramanujan’s Convolution Sum twisted by Dirichlet characters
International Journal of Number Theory, 2019Co-Authors: Zafer Selcuk Aygin, Nankun HongAbstract:We find formulas for Convolutions of Sum of divisor functions twisted by the Dirichlet character −4 ∗, which are analogous to Ramanujan’s formula for Convolution of usual Sum of divisor functions. ...
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ramanujan s Convolution Sum twisted by dirichlet characters
International Journal of Number Theory, 2019Co-Authors: Zafer Selcuk Aygin, Nankun HongAbstract:We find formulas for Convolutions of Sum of divisor functions twisted by the Dirichlet character −4 ∗, which are analogous to Ramanujan’s formula for Convolution of usual Sum of divisor functions. ...
Kenneth S. Williams - One of the best experts on this subject based on the ideXlab platform.
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Some arithmetic Convolution identities
The Ramanujan Journal, 2017Co-Authors: Kenneth S. WilliamsAbstract:Let n be a positive integer. Let $$\delta _3(n)$$ δ 3 ( n ) denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic Convolution Sum $$\begin{aligned} D_3(n):=\Sum _{j=1}^{n-1}\delta _3(j)\delta _3(n-j) \end{aligned}$$ D 3 ( n ) : = ∑ j = 1 n - 1 δ 3 ( j ) δ 3 ( n - j ) satisfies the relation $$\begin{aligned} 3D_3(n)+\delta _3(n)={\Sum _{\mathop {_{d \mid n}}\limits _{3 \not \mid d}}}d. \end{aligned}$$ 3 D 3 ( n ) + δ 3 ( n ) = ∑ d ∣ n 3 ∤ d d . In this paper, we use a result about binary quadratic forms to prove a general arithmetic Convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new Convolution formulas.
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The Convolution Sum Σm
Pacific Journal of Mathematics, 2006Co-Authors: Kenneth S. WilliamsAbstract:The Convolution Sum Σ m
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the Convolution Sum σm n 8 σ m σ n 8m
Pacific Journal of Mathematics, 2006Co-Authors: Kenneth S. WilliamsAbstract:The Convolution Sum Σ m
quadratic form x 2 1 + x 2 2 + x 2 3 + x 2 4 +2x 2 5 + 2x 2 6 + 2x 2 7 , + 2x 2 8 . -
the Convolution Sum Sum limits_ m n 9 sigma m sigma n 9m
International Journal of Number Theory, 2005Co-Authors: Kenneth S. WilliamsAbstract:The evaluation of the Sum ∑m