The Frequency Response Function of a Linear Time-Varying System

Kamen, Edward W.; Sills, J.A.

doi:10.1016/s1474-6670(17)49134-8

Cited by 4 publications

(4 citation statements)

References 3 publications

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“…Although there is no simple classification of nonstationary noise [43], there are multiple different models (see, for example, Refs. [42,[44][45][46][47][48]). We concern ourselves with models that decompose nonstationary noise in a similar manner to stationary noise in Wold's theorem, this time into a family of nondeterministic stationary processes, or a family of deterministic nonstationary processes.…”

Section: B Validity Of Stationarity Assumptionmentioning

confidence: 99%

Issues of mismodeling gravitational-wave data for parameter estimation

Edy

Nuttall

2021

Phys. Rev. D

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Bayesian inference is used to extract unknown parameters from gravitational-wave signals. Detector noise is typically modeled as stationary, although data from the LIGO and Virgo detectors is not stationary. We demonstrate that the posterior of estimated waveform parameters is no longer valid under the assumption of stationarity. We show that while the posterior is unbiased, the errors will be under-or overestimated compared to the true posterior. A formalism was developed to measure the effect of the mismodeling, and found the effect of any form of nonstationarity has an effect on the results, but are not significant in certain circumstances. We demonstrate the effect of short-duration Gaussian noise bursts and persistent oscillatory modulation of the noise on binary-black-hole-like signals. In the case of short signals, nonstationarity in the data does not have a large effect on the parameter estimation, but the errors from nonstationary data containing signals lasting tens of seconds or longer will be several times worse than if the noise was stationary. Accounting for this limiting factor in parameter sensitivity could be very important for achieving accurate astronomical results. This methodology for handling the nonstationarity will also be invaluable for analysis of waveforms that last minutes or longer, such as those we expect to see with the Einstein Telescope.

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Section: B Validity Of Stationarity Assumptionmentioning

confidence: 99%

Issues of mismodeling gravitational-wave data for parameter estimation

Edy

Nuttall

2021

Phys. Rev. D

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“…The rings A[z], A z −1 , and A z −1 are called skew rings due to the noncommutative multiplication defined in Equation (6). Skew polynomial rings were first introduced and studied by Oystein Ore in his 1933 paper [9].…”

Section: The Vit Transformmentioning

confidence: 99%

“…It is known that it is not possible to express the z-transform of the output response as the product of H Zadeh (z, k) with the z-transform of the input. Moreover, as discussed in [6], when the system or filter is finite-dimensional, this transform is seldom expressible as a polynomial fraction in z with time-varying coefficients. These two limitations were circumvented in [7] by defining the transfer function to be the formal power series Eng 2021, 2…”

Section: Introductionmentioning

confidence: 99%

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Kamen

2021

Eng

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A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

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“…It is known that it is not possible to express the -transform of the output response as the product of ℎ ( , ) with the -transform of the input. Also, as discussed in [10], when the system or filter is finite-dimensional, this transform is seldom expressible as a polynomial fraction in with time-varying coefficients. These two limitations were circumvented in [9] by defining the transfer function to be the formal power series ( , ) = ∑ ℎ( + , )…”

Section: Introductionmentioning

confidence: 99%

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Kamen¹

2021

Preprint

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A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z^(-1) which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z^(- i) by time functions which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of z^(i) at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

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The Frequency Response Function of a Linear Time-Varying System

Cited by 4 publications

References 3 publications

Issues of mismodeling gravitational-wave data for parameter estimation

Issues of mismodeling gravitational-wave data for parameter estimation

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

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