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## Multidimensional Analytic Signals and the Bedrosian Identity (2012)

Citations: | 1 - 1 self |

### Citations

1367 |
Theory of Communication
- Gabor
- 1946
(Show Context)
Citation Context ...Secondary 44A15. 1 Introduction Signals are carrier of information. In many applications, feature information of a signal usually persists in the time-frequency domain [9]. The analytic signal method =-=[10]-=- is a classical way of defining without ambiguity the local amplitude and frequency of one-dimensional signals. Among others, it has been proven useful in meteorological and atmospheric applications, ... |

682 |
The empirical mode decomposition and the Hilbert spectrum from nonlinear and non-stationary time series analysis
- Huang, Shen, et al.
- 1998
(Show Context)
Citation Context ...pplications, ocean engineering, structural science, and imaging processing, [13, 15, 16, 20, 21]. Especially, it motivates the widely used empirical mode decomposition and the Hilbert-Huang transform =-=[7, 13, 14, 15]-=-. The analytic signal method makes use of the Hilbert transform H defined for functions f ∈ L2(R) as (Hf)(x) := p.v. 1 π ∫ R f(y) x− ydy := limε→0+ 1 π ∫ |y−x|≥ε f(y) x− ydy, x ∈ R. (1.1) Let H1(R) be... |

491 |
Harmonic Analysis
- Stein
- 1993
(Show Context)
Citation Context ..., is defined for f ∈ L2(Rd) as (Hjf)(x) := p.v. 1 π ∫ Rd f(y) xj − yj dy := limε→0+ 1 π ∫ |yj−xj |≥ε f(y) xj − yj dy, x ∈ R d. It is known that Hj is a bounded linear operator from L 2(Rd) to L2(Rd), =-=[26]-=-. Moreover, there holds (Hjf )̂ (ξ) = −i sgn (ξj)f(ξ), ξ ∈ Rd, j ∈ Nd. (2.1) The purpose of the this section is to justify the using of linear combinations of compositions of the partial Hilbert trans... |

253 |
The Importance of Phase in Signals
- Oppenheim, Lim
- 1981
(Show Context)
Citation Context ...litude and frequency of one-dimensional signals. Among others, it has been proven useful in meteorological and atmospheric applications, ocean engineering, structural science, and imaging processing, =-=[13, 15, 16, 20, 21]-=-. Especially, it motivates the widely used empirical mode decomposition and the Hilbert-Huang transform [7, 13, 14, 15]. The analytic signal method makes use of the Hilbert transform H defined for fun... |

99 |
A new view of nonlinear water waves: the Hilbert Spectrum,
- Huang, Shen, et al.
- 1999
(Show Context)
Citation Context ...litude and frequency of one-dimensional signals. Among others, it has been proven useful in meteorological and atmospheric applications, ocean engineering, structural science, and imaging processing, =-=[13, 15, 16, 20, 21]-=-. Especially, it motivates the widely used empirical mode decomposition and the Hilbert-Huang transform [7, 13, 14, 15]. The analytic signal method makes use of the Hilbert transform H defined for fun... |

60 |
A product theorem for Hilbert transforms. In:
- Bedrosian
- 1963
(Show Context)
Citation Context ...> 0, φ0 ∈ R. This implies that the analytic signal method is the only one that satisfies some reasonable physical requirements [28, 29]. Finally, the Hilbert transform satisfies the Bedrosian theorem =-=[1]-=-: If f, g ∈ L2(R) satisfy either supp f̂ ⊆ R+, supp ĝ ⊆ R+ or supp f̂ ⊆ [−a, a], supp ĝ ⊆ (−∞,−a] ∪ [a,∞) for some positive number a, then there holds the Bedrosian identity [H(fg)](x) = f(x)(Hg)(x)... |

60 |
Computing the discrete-time “analytic” signal via FFT
- Marple
- 1999
(Show Context)
Citation Context ...ical soundness of the analytic signal method. Secondly, the multiplier definition (1.3) makes it possible to develop fast algorithms making use of the lighting fast FFT to compute the analytic signal =-=[18]-=-. In addition to the above two properties, it was shown in [28, 29] that the Hilbert transform is the only continuous and homogeneous operator L such that L(cos(ω0t+ φ0)) = sin(ω0t+ φ0), for all ω0 > ... |

58 |
Time-Frequency Analysis: Theory and Applications
- Cohen
- 1995
(Show Context)
Citation Context ... Classification: Primary 65R10, Secondary 44A15. 1 Introduction Signals are carrier of information. In many applications, feature information of a signal usually persists in the time-frequency domain =-=[9]-=-. The analytic signal method [10] is a classical way of defining without ambiguity the local amplitude and frequency of one-dimensional signals. Among others, it has been proven useful in meteorologic... |

49 |
The zeros of certain integral functions
- Titchmarsh
- 1926
(Show Context)
Citation Context ...y (ϕ ∗ ψ)(x) := ∫ Rd ϕ(x− t)ψ(t)dt, x ∈ Rd. 9 It is well-known that suppϕ ∗ ψ ⊆ suppϕ + suppψ. In the case that both ϕ,ψ are compactly supported, we have the celebrated Titchmarsh convolution theorem =-=[17, 27]-=-. For each subset A ⊆ Rd, we denote by convA the convex hull of A in Rd. Lemma 3.5 If both ϕ,ψ ∈ L2(Rd) are compactly supported then conv supp (ϕ ∗ ψ) = conv suppϕ+ conv suppψ. Theorem 3.6 Let ν be an... |

48 |
Fourier Analysis and Applications
- Gasquet, Witomski
- 1999
(Show Context)
Citation Context ...defined as ϕ̂(ξ) := ∫ Rd ϕ(x)e−i(x,ξ)dx, ξ ∈ Rd, which is again a function in S(Rd). The Fourier transform can be extended to the space S ′(Rd) of temperate distributions on Rd by a duality principle =-=[11]-=-. The Hilbert transform has an equivalent definition via the Fourier multiplier −i sgn , where sgn (ξ) takes value −1, 0, 1 for ξ < 0, ξ = 0 and ξ > 0, respectively. Specifically, we have for all f ∈ ... |

32 |
On the quadrature approximation to the hilbert transform of modulated signals,”
- Nuttall
- 1966
(Show Context)
Citation Context ... Hilbert transform of a product of functions, helps understand the instantaneous amplitude and frequency of signals, and provides a method of constructing basic signals in the time-frequency analysis =-=[1, 3, 4, 9, 19, 21, 23, 25, 36]-=-. It has attracted 2 much interest from the mathematical community [6, 8, 22, 24, 30, 31, 33, 34, 35, 36]. Here, we mention an observation in [30]. It states that the Hilbert transform is essentially ... |

32 |
On the Analytical Signal, the TeagerKaiser Energy Algorithm, and other Methods for Defining Amplitude and Frequency”
- Vakman
- 1996
(Show Context)
Citation Context ...tiplier definition (1.3) makes it possible to develop fast algorithms making use of the lighting fast FFT to compute the analytic signal [18]. In addition to the above two properties, it was shown in =-=[28, 29]-=- that the Hilbert transform is the only continuous and homogeneous operator L such that L(cos(ω0t+ φ0)) = sin(ω0t+ φ0), for all ω0 > 0, φ0 ∈ R. This implies that the analytic signal method is the only... |

31 |
Multidimensional complex signals with single-orthant spectra,
- Hahn
- 1992
(Show Context)
Citation Context ... essentially the only operator that satisfies the Bedrosian identity. We conclude that the analytic signal (1.2) is justified by the above mathematical properties. This paper is motivated by the need =-=[5, 12, 32]-=- of defining multidimensional analytic signals for the timefrequency analysis of multidimensional signals. Naturally, we are inclined to define the analytic signal of f ∈ L2(Rd) through a fixed operat... |

24 |
2006: A B-spline approach for empirical mode decompositions
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(Show Context)
Citation Context ...pplications, ocean engineering, structural science, and imaging processing, [13, 15, 16, 20, 21]. Especially, it motivates the widely used empirical mode decomposition and the Hilbert-Huang transform =-=[7, 13, 14, 15]-=-. The analytic signal method makes use of the Hilbert transform H defined for functions f ∈ L2(R) as (Hf)(x) := p.v. 1 π ∫ R f(y) x− ydy := limε→0+ 1 π ∫ |y−x|≥ε f(y) x− ydy, x ∈ R. (1.1) Let H1(R) be... |

22 | Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case
- Bülow, Sommer
- 2001
(Show Context)
Citation Context ... essentially the only operator that satisfies the Bedrosian identity. We conclude that the analytic signal (1.2) is justified by the above mathematical properties. This paper is motivated by the need =-=[5, 12, 32]-=- of defining multidimensional analytic signals for the timefrequency analysis of multidimensional signals. Naturally, we are inclined to define the analytic signal of f ∈ L2(Rd) through a fixed operat... |

20 |
Analytic unit quadrature signals with nonlinear phase,”
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Citation Context ...uency of signals, and provides a method of constructing basic signals in the time-frequency analysis [1, 3, 4, 9, 19, 21, 23, 25, 36]. It has attracted 2 much interest from the mathematical community =-=[6, 8, 22, 24, 30, 31, 33, 34, 35, 36]-=-. Here, we mention an observation in [30]. It states that the Hilbert transform is essentially the only operator that satisfies the Bedrosian identity. We conclude that the analytic signal (1.2) is ju... |

16 | Trigonometric Fourier Series and Their Conjugates, Mathematics and its Applications, 372 - Zhizhiashvili - 1996 |

13 |
Attoh-Okine: The Hilbert Huang Transform in Engineering (Boca
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- 2005
(Show Context)
Citation Context ...litude and frequency of one-dimensional signals. Among others, it has been proven useful in meteorological and atmospheric applications, ocean engineering, structural science, and imaging processing, =-=[13, 15, 16, 20, 21]-=-. Especially, it motivates the widely used empirical mode decomposition and the Hilbert-Huang transform [7, 13, 14, 15]. The analytic signal method makes use of the Hilbert transform H defined for fun... |

12 |
S.: Two-dimensional empirical mode decomposition by finite elements
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Citation Context ... essentially the only operator that satisfies the Bedrosian identity. We conclude that the analytic signal (1.2) is justified by the above mathematical properties. This paper is motivated by the need =-=[5, 12, 32]-=- of defining multidimensional analytic signals for the timefrequency analysis of multidimensional signals. Naturally, we are inclined to define the analytic signal of f ∈ L2(Rd) through a fixed operat... |

11 |
An extension to the Hilbert transform product theorem
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Citation Context ... Hilbert transform of a product of functions, helps understand the instantaneous amplitude and frequency of signals, and provides a method of constructing basic signals in the time-frequency analysis =-=[1, 3, 4, 9, 19, 21, 23, 25, 36]-=-. It has attracted 2 much interest from the mathematical community [6, 8, 22, 24, 30, 31, 33, 34, 35, 36]. Here, we mention an observation in [30]. It states that the Hilbert transform is essentially ... |

10 | The Bedrosian identity for the Hilbert transform of product functions
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Citation Context ...uency of signals, and provides a method of constructing basic signals in the time-frequency analysis [1, 3, 4, 9, 19, 21, 23, 25, 36]. It has attracted 2 much interest from the mathematical community =-=[6, 8, 22, 24, 30, 31, 33, 34, 35, 36]-=-. Here, we mention an observation in [30]. It states that the Hilbert transform is essentially the only operator that satisfies the Bedrosian identity. We conclude that the analytic signal (1.2) is ju... |

10 |
The Bedrosian Identity and Homogeneous Semi-Convolution Equations, accepted to appear
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Citation Context ... Hilbert transform of a product of functions, helps understand the instantaneous amplitude and frequency of signals, and provides a method of constructing basic signals in the time-frequency analysis =-=[1, 3, 4, 9, 19, 21, 23, 25, 36]-=-. It has attracted 2 much interest from the mathematical community [6, 8, 22, 24, 30, 31, 33, 34, 35, 36]. Here, we mention an observation in [30]. It states that the Hilbert transform is essentially ... |

8 |
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Citation Context ...tiplier definition (1.3) makes it possible to develop fast algorithms making use of the lighting fast FFT to compute the analytic signal [18]. In addition to the above two properties, it was shown in =-=[28, 29]-=- that the Hilbert transform is the only continuous and homogeneous operator L such that L(cos(ω0t+ φ0)) = sin(ω0t+ φ0), for all ω0 > 0, φ0 ∈ R. This implies that the analytic signal method is the only... |

6 | Orthonormal bases with nonlinear phases,”
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6 | Fourier spectrum characterization of Hardy spaces and applications, accepted to appear
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Citation Context ...uency of signals, and provides a method of constructing basic signals in the time-frequency analysis [1, 3, 4, 9, 19, 21, 23, 25, 36]. It has attracted 2 much interest from the mathematical community =-=[6, 8, 22, 24, 30, 31, 33, 34, 35, 36]-=-. Here, we mention an observation in [30]. It states that the Hilbert transform is essentially the only operator that satisfies the Bedrosian identity. We conclude that the analytic signal (1.2) is ju... |

5 |
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5 |
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2 |
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Citation Context ...y (ϕ ∗ ψ)(x) := ∫ Rd ϕ(x− t)ψ(t)dt, x ∈ Rd. 9 It is well-known that suppϕ ∗ ψ ⊆ suppϕ + suppψ. In the case that both ϕ,ψ are compactly supported, we have the celebrated Titchmarsh convolution theorem =-=[17, 27]-=-. For each subset A ⊆ Rd, we denote by convA the convex hull of A in Rd. Lemma 3.5 If both ϕ,ψ ∈ L2(Rd) are compactly supported then conv supp (ϕ ∗ ψ) = conv suppϕ+ conv suppψ. Theorem 3.6 Let ν be an... |

2 |
Charaterizing the Hilbert transform by the Bedrosian theorem
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2 |
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2 |
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1 |
An efficient algorithm for the computation of the multidimensional discrete Fourier transform, Multidimens. Systems Signal Process
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Citation Context ... L2(Rd). (2.5) In other words, operators (2.2) are given by a very simple Fourier multiplier. Therefore, efficient numerical algorithms based on fast multidimensional discrete Fourier transforms (see =-=[2]-=- and the references cited therein) can be developed for the computation of the multidimensional analytic signal (2.3). 2.3 Physical Requirements Following [28, 29], we ask what continuous linear opera... |

1 |
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