The theory of stationary point processes

Beutler, Frederick J.; Leneman, O.

doi:10.1007/bf02392816

Cited by 73 publications

(29 citation statements)

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“…The analysis on continuous time random sampling can be traced back to Beutler and Leneman's [5]- [8] publications on the theory of stationary point process and random sampling of random process in the late 1960s. [9] extended the theory to address discrete time additive random sampling.…”

Section: Discrete Time Jittered Random Sampling Theorymentioning

confidence: 99%

Jittered random sampling with a successive approximation ADC

Luo

Zhu

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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This paper proposes a randomly jittered temporal sampling scheme with a successive approximation register (SAR) ADC. The sampling time points are jittered around a uniform sampling clock. A control logic is implemented on a traditional SAR ADC to force it to terminate the sample conversion before reaching the full precision at the jittered time points. As a result, variable word length data samples are produced by the SAR converter. Based on a discrete jittered random sampling theory, this paper analyzes the impact of the random jitters and the resulting randomized quantization noise for a class of sparse or compressible signals. A reconstruction algorithm called Successive Sine Matching Pursuit (SSMP) is proposed to recover spectrally sparse signals when sampled by the proposed SAR ADC at a sub-Nyquist rate.

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Section: Discrete Time Jittered Random Sampling Theorymentioning

confidence: 99%

Jittered random sampling with a successive approximation ADC

Luo

Zhu

2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…In this letter, we demonstrate a new φ-OTDR system inspired by the concept of non-uniform sampling [10][11][12]. Thanks to its antialiasing characteristic in frequency domain, the proposed method, named sub-Nyquist additive random sampling (sNARS) technique, is capable of sampling the wide-band sparse frequency signals with sub-Nyquist sampling rate [8,12].…”

Section: Xxxxmentioning

confidence: 99%

“…Unlike the uniform sampling case, the interval Δtn=tn-tn-1 of additive random sampling (ARS) is an independent identically distributed (IID) random variable with probability density function (PDF) p(t). Shapiro [10] and Beutler [11] established that the analytic expression of Ws(f) of ARS is …”

Section: Xxxxmentioning

confidence: 99%

“…As for rail track monitoring, most of the long-range waves decay rapidly because only several vibration modes with certain frequencies are supported [7], owing to the fact that the vibration waves degenerate to sparse signal [8] over 10 kHz to 60 kHz. It should be noted that sparse signal is also used widely in many fields, such as metal damage detection [9].In this letter, we demonstrate a new φ-OTDR system inspired by the concept of non-uniform sampling [10][11][12]. Thanks to its antialiasing characteristic in frequency domain, the proposed method, named sub-Nyquist additive random sampling (sNARS) technique, is capable of sampling the wide-band sparse frequency signals with sub-Nyquist sampling rate [8,12].…”

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confidence: 99%

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Distributed fiber sparse-wideband vibration sensing by sub-Nyquist additive random sampling

et al. 2018

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The round trip time of the light pulse limits the maximum detectable vibration frequency response range of phasesensitive optical time domain reflectometry (φ-OTDR). Unlike the uniform laser pulse interval in conventional φ-OTDR, we randomly modulate the pulse interval, so that an equivalent sub-Nyquist additive random sampling (sNARS) is realized for every sensing point of the long interrogation fiber. For an φ-OTDR system with 10 km sensing length, the sNARS method is optimized by theoretical analysis and Monte Carlo simulation, and the experimental results verify that a wide-band spars signal can be identified and reconstructed. Such a method can broaden the vibration frequency response range of φ-OTDR, which is of great significance in sparse-widebandfrequency vibration signal detection, such as rail track monitoring and metal defect detection. XXXXDistributed vibration sensing by phase-sensitive optical time domain reflectometry (φ-OTDR) has been intensively studied for its potential applications in fields like intrusion detection, oil and gas pipeline monitoring, high speed rail or urban rail transit monitoring, and so on [1][2][3][4][5]. It is important for the sensing distance and the frequency response range to meet the requirements of practical applications. Up to now, ultra-long φ-OTDR with 175 km sensing range has been achieved [5], but its vibration frequency response range is limited to less than 285 Hz. In order to broaden the frequency response range, two kinds of approaches have been proposed: the combination of interferometer with φ-OTDR [2], and the frequency multiplexing method [3,4].Taking the φ-OTDR based high-speed rail track monitoring system as an example, the length of sensing fiber should be larger than 50 km or even 100 km, which corresponds to a maximum detectable frequency of 1 kHz or 500 Hz, respectively. However, in terms of the wave propagation along a railway track, the frequency of traininduced vibration is up to 80 kHz [6]. It is difficult for the conventional φ-OTDR system to detect such a broadband signal.In some cases, the vibration waves are sparse in frequency domain, which gives a chance for φ-OTDR system to handle with low pulse repetition rate. As for rail track monitoring, most of the long-range waves decay rapidly because only several vibration modes with certain frequencies are supported [7], owing to the fact that the vibration waves degenerate to sparse signal [8] over 10 kHz to 60 kHz. It should be noted that sparse signal is also used widely in many fields, such as metal damage detection [9].In this letter, we demonstrate a new φ-OTDR system inspired by the concept of non-uniform sampling [10][11][12]. Thanks to its antialiasing characteristic in frequency domain, the proposed method, named sub-Nyquist additive random sampling (sNARS) technique, is capable of sampling the wide-band sparse frequency signals with sub-Nyquist sampling rate [8,12]. The sNARS method is studied theoretically and verified with Monte Carlo simulation, by which the key parameters for φ-O...

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“…In order to preserve stationarity, we introduce randomness (i.e., jitter) into the sampling times. These so-called stationary point random processes were investigated in [22]. One special case is when the sampling times are , where are independent random variables with some distribution function Let be its characteristic function.…”

Section: B Nonuniform Samplingmentioning

confidence: 99%

Generalized sampling theorems in multiresolution subspaces

Djbkovic

Vaidyanathan²

1997

IEEE Trans. Signal Process.

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Abstract-It is well known that under very mild conditions on the scaling function, multiresolution subspaces are reproducing kernel Hilbert spaces (RKHS's). This allows for the development of a sampling theory. In this paper, we extend the existing sampling theory for wavelet subspaces in several directions. We consider periodically nonuniform sampling, sampling of a function and its derivatives, oversampling, multiband sampling, and reconstruction from local averages. All these problems are treated in a unified way using the perfect reconstruction (PR) filter bank theory. We give conditions for stable reconstructions in each of these cases. Sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported, except in trivial cases. This restriction no longer applies for the generalizations we study here, due to the existence of FIR PR banks. In fact, with nonuniform sampling, oversampling, and reconstruction from local averages, we can guarantee compactly supported synthesizing functions. Moreover, local averaging schemes have additional nice properties (robustness to the input noise and compression capabilities). We also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis. After this, we extend the sampling theory to random processes. We require autocorrelation functions to belong to some subspace related to wavelet subspaces. It turns out that we cannot recover random processes themselves (unless they are bandlimited) but only their power spectral density functions. We consider both uniform and nonuniform sampling.

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The theory of stationary point processes

Cited by 73 publications

References 0 publications

Jittered random sampling with a successive approximation ADC

Jittered random sampling with a successive approximation ADC

Distributed fiber sparse-wideband vibration sensing by sub-Nyquist additive random sampling

Generalized sampling theorems in multiresolution subspaces

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