Detection of sinusoids in unknown coloured noise using ratios of AR spectrum estimates

Abstract

This paper is an introductory exposition of the application of the ratio of the 2pth order autoregressive (AR) power spectrum estimate to the p t h order AR power spectrum estimate to the problem of detecting sinusoids of unknown frequency in additive coloured noise with unknown power spectral density. The resulting threshold test is independent of a priori knowledge of both the sinusoid's parameters and the PSD of the additive noise. Simulations are presented for the Gaussian case, focussing primarily on the SNR thresholding behaviour of the scheme. These simulations illustrate the power of the technique when highly resonant background noise is present, such as is often the case in rotating machinery, for example.Significant threshold extension, and slightly superior subthreshold performance compared to the energy detector is observed in these cases. The paper also examines the application of the technique to the problem of identifying tones in diesel engine vibration data.

A Study of Periodograms Standardized Using Training Datasets and Application to Exoplanet Detection

“…Techniques reducing the influence of signal peaks under the alternative are proposed in [39], [40]. Different approaches, related to standardization (2), can be found in [41]- [44]. When following Generalized Likelihood Ratio (GLR) approaches for detecting multiple sinusoids in unknown number, the GLR must be combined with model selection procedures.…”

“…the phase of z p obtained from the real and imaginary parts of (38) [96]. With these notations, it is easy to show that Consequently, the expression of the {γ k } in (37) becomes γ k = N 4S E (ν k ) Ns q=1 α 2 q κ 2 q +2α q κ q Ns ℓ=q+1 α ℓ κ ℓ cos(θ q − θ ℓ ) (41) and the non centrality parameters {λ k } of (6) follow from (34),with κ q and θ q given by (39) and (40). Note that if all signal frequencies {f p } fall on the Fourier frequency grid, the crossed term in (41) vanish owing to the orthogonality of the Fejér kernels centered at different signal frequencies.…”

“…q κ 2 q +2α q κ q Ns ℓ=q+1 α ℓ κ ℓ cos(θ q − θ ℓ ) (41) and the non centrality parameters {λ k } of (6) follow from (34),with κ q and θ q given by ( 39) and (40).…”

Asymptotic statistical properties of AR spectral estimators for processes with mixed spectra

“…The use of spectral families, specifically the AR and MV families, as demonstrated in [14]- [16], represents a relatively new and as yet not popular approach to characterizing the spectral information associated with a mixed process. The fact that the results in this work apply to a wide range of model orders allows one to use it to develop statistically robust sinusoid detection methods for the case of arbitrary noise of unknown color, such as was done in [17]. Such methods could be notably improved if one could obtain the joint spectral statistics of family members, as well as the behavior of the mean and variance of the family as .…”

Using hydrodynamical simulations of stellar atmospheres for periodogram standardization: Application to exoplanet detection

“…local SNR [9], modified periodogram smoothers [10,11], robust M-estimators [12,13]) or parametric methods (e.g. iterative Yule-Walker [14], ratio of autoregressive (AR) spectral estimates [15], balanced model truncation [16]). Even if some of these estimators are asymptotically unbiased, the unavoidable injection of estimation noise in the denominator of the standardized periodogram makes the statistical characterization of the test statistics difficult.…”