Signal recovery from a few linear measurements of its high-order spectra

Bendory, Tamir; Edidin, Dan; Kreymer, Shay

doi:10.1016/j.acha.2021.10.003

“…2) Phase preservation: Spectra of order greater than 2 are not blind to time in the same way but preserve information about temporal structure, up to the discarded time shift. In fact, for a transient signal, whose HOS is non-vanishing almost everywhere (by virtue of having a finite duration), it is possible to uniquely recover a waveform from HOS, up to time shift and, in the case of even orders, sign inversion [2], [47].…”

Section: Appendix a Higher Order Spectra A Main Propertiesmentioning

confidence: 99%

“…7) HOS inversion: For systems whose outputs are both transient and have deterministic HOS, it is not only possible to recover the signal from HOS, but HOS overdetermine the signal, meaning that it is not necessary to estimate HOS over the entire domain to fully recover the signal phase spectrum. The minimum number of HOS samples for recovery is on the order of the number of samples in the original signal [47]. For example, from the bispectrum, one can reconstruct a transient, deterministic signal from the diagonal axis X (ω) X (ω) X * (2ω).…”

Section: Appendix a Higher Order Spectra A Main Propertiesmentioning

confidence: 99%

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A Slice of Trispectrum for Patterns of Modulation

Kovach¹

2022

Preprint

0

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<p>Modulation can be identified in a signal from correlations of power across frequencies. Resolving power over time and frequency using standard techniques of bandpass filtering and squaring creates bias: filtering presupposes a timescale and bandwidth for modulation, while squaring complicates separation from noise. Higher-order spectra (HOS) can be used to estimate power correlations without bias. HOS are not widely familiar because their meaning is thought to be obscure. A natural interpretation of the fourth-order spectrum (trispectrum) as a measure of linear relationships between power across frequencies is explained. This interpretation motivates a new tool, the modulogram, for identifying various types of modulation. The modulogram can be further decomposed into a low-dimensional feature space describing characteristic patterns of modulation. These developments are demonstrated with the blind identification of beta burst in rodent local field potential (LFP) recordings.</p>

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“…2) Phase preservation: Spectra of order greater than 2 are not blind to time in the same way but preserve information about temporal structure, up to the discarded time shift. In fact, for a transient signal, whose HOS is non-vanishing almost everywhere (by virtue of having a finite duration), it is possible to uniquely recover a waveform from HOS, up to time shift and, in the case of even orders, sign inversion [2], [47].…”

Section: Appendix a Higher Order Spectra A Main Propertiesmentioning

confidence: 99%

“…7) HOS inversion: For systems whose outputs are both transient and have deterministic HOS, it is not only possible to recover the signal from HOS, but HOS overdetermine the signal, meaning that it is not necessary to estimate HOS over the entire domain to fully recover the signal phase spectrum. The minimum number of HOS samples for recovery is on the order of the number of samples in the original signal [47]. For example, from the bispectrum, one can reconstruct a transient, deterministic signal from the diagonal axis X (ω) X (ω) X * (2ω).…”

Section: Appendix a Higher Order Spectra A Main Propertiesmentioning

confidence: 99%

A Slice of Trispectrum for Patterns of Modulation

Kovach¹

2022

Preprint

0

View full text Add to dashboard Cite

<p>Modulation can be identified in a signal from correlations of power across frequencies. Resolving power over time and frequency using standard techniques of bandpass filtering and squaring creates bias: filtering presupposes a timescale and bandwidth for modulation, while squaring complicates separation from noise. Higher-order spectra (HOS) can be used to estimate power correlations without bias. HOS are not widely familiar because their meaning is thought to be obscure. A natural interpretation of the fourth-order spectrum (trispectrum) as a measure of linear relationships between power across frequencies is explained. This interpretation motivates a new tool, the modulogram, for identifying various types of modulation. The modulogram can be further decomposed into a low-dimensional feature space describing characteristic patterns of modulation. These developments are demonstrated with the blind identification of beta burst in rodent local field potential (LFP) recordings.</p>

show abstract