Nonlinear Transformation of Differential Equations into Phase Space

Cohen, Leon; Galleani, Lorenzo

doi:10.1155/s1110865704402224

Cited by 13 publications

(18 citation statements)

References 37 publications

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“…We consider the deterministic and random case separately. The obtained results extend the single-input single-output (SISO) case considered in [11], [12].…”

Section: Transformation To the Time-frequency Domainsupporting

confidence: 72%

“…To transform the dynamical system (1) to the time-frequency domain, we need the differential properties (10) (11) where , are arbitrary vector signals, the element-wise operators and are defined as (14) To prove this result, we simply consider that, for the th entry of , it holds [11] (15)…”

Section: A Deterministic Casementioning

confidence: 99%

“…The operators for the Wigner distribution are given in [12], whereas the smoothed Wigner with Gaussian kernel and the Rihaczek distributions are considered in [11]. We now derive the operator for the Choi-Williams distribution.…”

Section: A Deterministic Casementioning

confidence: 99%

“…The transformation generates a time-frequency MIMO dynamical system, whose input is the time-frequency distribution of the input and whose output is the time-frequency distribution of the output . The transformation takes into account both deterministic and random MIMO systems, and it represents an extension of the results derived in [11], [12].…”

Section: Introductionmentioning

confidence: 99%

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Time-Frequency Representation of MIMO Dynamical Systems

Galleani

2013

IEEE Trans. Signal Process.

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Many deterministic and random physical signals can be modeled as the output of a multi-input-multi-output (MIMO) dynamical system. Since physical signals are typically nonstationary, their frequency content changes with time. To understand this time variation, we transform the MIMO system to the time-frequency domain. The result is a time-frequency MIMO dynamical system, whose input and output are the time-frequency spectra of the original input and output signals in the time domain. The time-frequency system reveals the spectral mechanisms involved in the generation of nonstationary signals. We apply our method to the case of a MIMO system with two vibrational modes and a nonstationary noise at the input. We obtain the time-frequency spectrum of the output, which shows how the spectrum of the modes changes with time. This result cannot be achieved with classical spectral techniques, because they require the input random process to be wide sense stationary.

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“…We consider the deterministic and random case separately. The obtained results extend the single-input single-output (SISO) case considered in [11], [12].…”

Section: Transformation To the Time-frequency Domainsupporting

confidence: 72%

Section: A Deterministic Casementioning

confidence: 99%

Section: A Deterministic Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Time-Frequency Representation of MIMO Dynamical Systems

Galleani

2013

IEEE Trans. Signal Process.

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“…The dynamical system (1) can be actually transformed to any distribution of the Cohen class [14]. The obtained time-frequency dynamical system is defined, in general, by a partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

Response of Dynamical Systems to Nonstationary Inputs

Galleani

2012

IEEE Trans. Signal Process.

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We obtain the time-frequency response of the output of a dynamical system when the input belongs to a class of common nonstationary signals, namely, an impulse, a linear chirp, a causal sinusoid, and a short duration sinusoid. The obtained results clarify how the system processes the time-varying frequencies of the input signal to generate the time-frequency spectrum of the output. All analytic results are exact. The solution is obtained by developing a method which can be used to evaluate the output of a dynamical system for complex combinations of nonstationary inputs. We show numerical examples which prove that the response of a system to a nonstationary input is made by a series of events occurring in the joint time-frequency domain.

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