Sequences with minimal time–frequency uncertainty

Parhizkar, Reza; Barbotin, Yann; Vetterli, Martin

doi:10.1016/j.acha.2014.07.001

Cited by 29 publications

(17 citation statements)

References 24 publications

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“…By comparison with CRLB in frequency domain, it is clear that they have the same expression. Therefore, taking the uncertainty principle of time-bandwidth product into the equation of estimation variance of TDE, this paper illustrates that Gaussian pulse is just the only waveform to make CRLB be the maximum value [6]. In section 3, simulations firstly verifies uncertainty principle of time-bandwidth product of Gaussian pulse, then shows time-bandwidth product of cosine rising and falling pulse.…”

Section: Introductionmentioning

confidence: 90%

The maximum of CRLB of time delay estimation

Wang

Xia

et al. 2014

2014 8th International Conference on Signal Processing and Communication Systems (ICSPCS)

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Cramer-Rao lower bound (CRLB) is very significant to evaluate parameter's estimation results. This paper proposes to study the maximum of CRLB of time delay estimation (TDE) under certain signal to noise ratio, when we only need to get a reasonable range of estimation variance to test the validity of an algorithm without knowing the exact formula of signal, or to design a waveform which is difficult to be located. First, we deduce the estimation variance of time delay using the principle of least squares. Then, through comparing to CRLB in frequency domain, we know that they are same to express the lower variance bound for unbiased TDE. Finally, with time-frequency uncertainty principle, we could know that Gaussian envelope is the waveform to maximize CRLB of TDE under certain signal to noise ratio. Simulations also illustrate that Gaussian envelope is close to the maximum of CRLB of time delay.

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Section: Introductionmentioning

confidence: 90%

The maximum of CRLB of time delay estimation

Wang

Xia

et al. 2014

2014 8th International Conference on Signal Processing and Communication Systems (ICSPCS)

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“…, is indirectly calculated from the pulse arrival time, the maximum resolution of time and frequency of the time-frequency spectrum obtained from PATFTM is no longer bound by Heisenberg's uncertainty principle [22] but depends upon the measuring precision of the pulse arrival time .…”

Section: (D) Plane Transformation Of Instantaneous Frequency Withmentioning

confidence: 99%

A Fault Feature Extraction Method for Rolling Bearing Based on Pulse Adaptive Time-Frequency Transform

Yao

Tang

Zhao

2016

Shock and Vibration

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Shock pulse method is a widely used technique for condition monitoring of rolling bearing. However, it may cause erroneous diagnosis in the presence of strong background noise or other shock sources. Aiming at overcoming the shortcoming, a pulse adaptive time-frequency transform method is proposed to extract the fault features of the damaged rolling bearing. The method arranges the rolling bearing shock pulses extracted by shock pulse method in the order of time and takes the reciprocal of the time interval between the pulse at any moment and the other pulse as all instantaneous frequency components in the moment. And then it visually displays the changing rule of each instantaneous frequency after plane transformation of the instantaneous frequency components, realizes the time-frequency transform of shock pulse sequence through time-frequency domain amplitude relevancy processing, and highlights the fault feature frequencies by effective instantaneous frequency extraction, so as to extract the fault features of the damaged rolling bearing. The results of simulation and application show that the proposed method can suppress the noises well, highlight the fault feature frequencies, and avoid erroneous diagnosis, so it is an effective fault feature extraction method for the rolling bearing with high time-frequency resolution.

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“…The detailed proof is provided in [15]. This is a fundamental result showing that although the Heisenberg uncertainty principle provides a lower bound for the timefrequency spread of the signal, it cannot be achieved for any given frequency (or equivalently time) spread.…”

Section: Theorem 1 If Xn Is Maximally Compact For a Givenmentioning

confidence: 99%

“…The proof is provided in [15]. Lemmas 1 and 2 greatly reduce the complexity of the problem, and from here on we only consider-without loss of generalityreal, positive sequences x, with µn(x) = 0 and x 2 = 1.…”

Section: Lemmamentioning

confidence: 99%

Sequences with minimal time-frequency spreads

Parhizkar

Barbotin

Vetterli

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

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For a given time or frequency spread, one can always find continuoustime signals, which achieve the Heisenberg uncertainty principle bound. This is known, however, not to be the case for discrete-time sequences; only widely spread sequences asymptotically achieve this bound. We provide a constructive method for designing sequences that are maximally compact in time for a given frequency spread. By formulating the problem as a semidefinite program, we show that maximally compact sequences do not achieve the classic Heisenberg bound. We further provide analytic lower bounds on the time-frequency spread of such signals.

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Sequences with minimal time–frequency uncertainty

Cited by 29 publications

References 24 publications

The maximum of CRLB of time delay estimation

The maximum of CRLB of time delay estimation

A Fault Feature Extraction Method for Rolling Bearing Based on Pulse Adaptive Time-Frequency Transform

Sequences with minimal time-frequency spreads

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