Tapering Promotes Propriety for Fourier Transforms of Real-Valued Time Series

Walden, A. T.; Leong, Z.

doi:10.1109/tsp.2018.2858185

Cited by 4 publications

(7 citation statements)

References 21 publications

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“…Discrete Fourier transform (DFT) [42]: In [43], a time series is approximately presented with discrete Fourier transform in time domain and frequency domain. In specific, the mapping from time domain space to frequency domain space is based on spectral analysis.…”

Section: Domain Transformmentioning

confidence: 99%

An edge-cloud collaboration architecture for pattern anomaly detection of time series in wireless sensor networks

Gao

Yang

Chen

et al. 2021

Complex Intell. Syst.

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With large deployment of wireless sensor networks, anomaly detection for sensor data is becoming increasingly important in various fields. As a vital data form of sensor data, time series has three main types of anomaly: point anomaly, pattern anomaly, and sequence anomaly. In production environments, the analysis of pattern anomaly is the most rewarding one. However, the traditional processing model cloud computing is crippled in front of large amount of widely distributed data. This paper presents an edge-cloud collaboration architecture for pattern anomaly detection of time series. A task migration algorithm is developed to alleviate the problem of backlogged detection tasks at edge node. Besides, the detection tasks related to long-term correlation and short-term correlation in time series are allocated to cloud and edge node, respectively. A multi-dimensional feature representation scheme is devised to conduct efficient dimension reduction. Two key components of the feature representation trend identification and feature point extraction are elaborated. Based on the result of feature representation, pattern anomaly detection is performed with an improved kernel density estimation method. Finally, extensive experiments are conducted with synthetic data sets and real-world data sets.

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Section: Domain Transformmentioning

confidence: 99%

An edge-cloud collaboration architecture for pattern anomaly detection of time series in wireless sensor networks

Gao

Yang

Chen

et al. 2021

Complex Intell. Syst.

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“…To ensure that x(t) is real-valued, the Hermitian symmetry condition, x * (t, ω) = x(t, −ω), must be satisfied so that we can express (14) in terms of TFRs associated with the positive angular frequencies only, to yield…”

Section: Multivariate Complex Tfr With Ergodicity In the Momentsmentioning

confidence: 99%

“…Owing to the linearity property of the Fourier transform in (14), it follows that if the TFRs are multivariate complex Gaussian distributed, that is, x(t, ω) ∼ CN (m(ω), R(ω)), then their time-domain counterpart, x(t), is also multivariate Gaussian distributed, since a linear function of Gaussian random variables is also Gaussian distributed. The signal, x(t), is therefore distributed according to…”

Section: A Class Of Nonstationary Signals With Time-invariant Spectra...mentioning

confidence: 99%

“…From (39), consider the statistical expectation of the spectral expansion of x(t) as in (14), to yield…”

Section: A Harmonic Signalmentioning

confidence: 99%

“…While it was first noted in [1], [2] that wide-sense stationary (WSS) random signals exhibit circularly distributed Fourier transforms, the existence of noncircularly distributed counterparts was not considered until the seminal work in [3]. Thereafter, a class of general complex time-frequency representations (TFRs) was proposed in [4], which has motivated various developments of complex-valued spectral analysis techniques for modelling real-valued nonstationary random signals [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. These methods employ augmented complex statistics [15], which provide a complete second-order statistical description of a general complex random variable, x ∈ C, by incorporating the standard Hermitian variance, E |x| 2 ∈ R, which parametrizes the scale of a pdf, in conjunction with the pseudovariance, E x 2 ∈ C, which parametrizes the eccentricity and angle of an elliptical pdf in the complex plane [16].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Probabilistic Spectral Analysis of Multivariate Real-Valued Nonstationary Signals

Scalzo¹,

Stanković²,

Mandic³

2020

Preprint

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A class of multivariate spectral representations for real-valued nonstationary random variables is introduced, which is characterised by a general complex Gaussian distribution. In this way, the temporal signal properties -harmonicity, wide-sense stationarity and cyclostationarity -are designated respectively by the mean, Hermitian variance and pseudo-variance of the associated time-frequency representation (TFR). For rigour, the estimators of the TFR distribution parameters are derived within a maximum likelihood framework and are shown to be statistically consistent, owing to the statistical identifiability of the proposed distribution parametrization. By virtue of the assumed probabilistic model, a generalised likelihood ratio test (GLRT) for nonstationarity detection is also proposed. Intuitive examples demonstrate the utility of the derived probabilistic framework for spectral analysis in low-SNR environments.

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On impropriety for a large-sized discrete fourier transform of a real-valued stationary process

2022

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Tapering Promotes Propriety for Fourier Transforms of Real-Valued Time Series

Cited by 4 publications

References 21 publications

An edge-cloud collaboration architecture for pattern anomaly detection of time series in wireless sensor networks

An edge-cloud collaboration architecture for pattern anomaly detection of time series in wireless sensor networks

A Probabilistic Spectral Analysis of Multivariate Real-Valued Nonstationary Signals

On impropriety for a large-sized discrete fourier transform of a real-valued stationary process

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