Model Selection for Sinusoids in Noise: Statistical Analysis and a New Penalty Term

Nadler, Boaz; Kontorovich, Aryeh

doi:10.1109/tsp.2011.2105482

Cited by 43 publications

(69 citation statements)

References 40 publications

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“…The 3k unidentified parameters are estimated by the grid search, i.e., all possible values of frequency and phase (amplitude can be estimated using the assumed frequency and phase by using this relation;â j = t x(t) sin(ω j t+φ) t (x(t) sin(ω j t+φ)) 2 [140] are tested and the one maximizing the likelihood function (66) is selected as the best estimate.…”

Section: Preliminariesmentioning

confidence: 99%

A unified approach to sparse signal processing

Marvasti

Amini

Haddadi

et al. 2012

EURASIP J. Adv. Signal Process.

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A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, component analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing, and rate of innovation. The redundancy introduced by channel coding in finite and real Galois fields is then related to over-sampling with similar reconstruction algorithms. The error locator polynomial (ELP) and iterative methods are shown to work quite effectively for both sampling and coding applications. The methods of Prony, Pisarenko, and MUltiple SIgnal Classification (MUSIC) are next shown to be targeted at analyzing signals with sparse frequency domain representations. Specifically, the relations of the approach of Prony to an annihilating filter in rate of innovation and ELP in coding are emphasized; the Pisarenko and MUSIC methods are further improvements of the Prony method under noisy environments. The iterative methods developed for sampling and coding applications are shown to be powerful tools in spectral estimation. Such narrowband spectral estimation is then related to multi-source location and direction of arrival estimation in array processing. Sparsity in unobservable source signals is also shown to facilitate source separation in sparse component analysis; the algorithms developed in this area such as linear programming and matching pursuit are also widely used in compressed sensing. Finally, the multipath channel estimation problem is shown to have a sparse formulation; algorithms similar to sampling and coding are used to estimate typical multicarrier communication channels.

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Section: Preliminariesmentioning

confidence: 99%

A unified approach to sparse signal processing

Marvasti

Amini

Haddadi

et al. 2012

EURASIP J. Adv. Signal Process.

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“…The implication of such a conclusion is that if one pursues an approach based on the generalized likelihood ratio test (GLRT) between ML estimates of models of different orders (see [20]) they may find a penalty term that does not exactly satisfy the EDC consistency requirements, but indeed results in consistent model selection with interestingly good performance for finite-sample cases. Based on this, [20] provides a new penalty term based on extreme value theory results on the excursion probability of Gaussian random fields. Such random fields are associated with the GLRT for the sinusoid model selection problem when the observations are scalar.…”

Section: Introductionmentioning

confidence: 99%

“…Such random fields are associated with the GLRT for the sinusoid model selection problem when the observations are scalar. More precisely, [20] shows that for such assumptions, the distribution of the GLRT is related to the distribution of maxima of a two-dimensional Gaussian random field since two parameters, frequency and phase, are needed to be estimated for each successive model order. In the case of complex-valued vector observations that we deal with in this paper, we show that the GLRT distribution is related to the distribution of maxima of a one dimensional (only frequency dimension) chi-square random field.…”

Section: Introductionmentioning

confidence: 99%

“…Our treatment of the model selection for complex-valued sinusoidal components considers multiple sensors (vector observations) within vector noise with an unknown correlation matrix while most of the literature deals with problems involving a single sensor (and usually real-valued observations) [12], [15], [17], [20], [26], [27]. References that consider multi-sensor observations often assume spatially white noise [28], [29].…”

Section: Introductionmentioning

confidence: 99%

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Model Order Selection for Complex Sinusoids in the Presence of Unknown Correlated Gaussian Noise

Talebi

Pratt

2015

IEEE Trans. Signal Process.

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We consider the problem of detecting and estimating the amplitudes and frequencies of an unknown number of complex sinusoids based on noisy observations from an unstructured array. In parametric detection problems like this, information theoretic criteria such as minimum description length (MDL) and Akaike information criterion (AIC) have previously been used for joint detection and estimation. In our paper, model selection based on extreme value theory (EVT), which has previously been used for enumerating real sinusoidal components from one-dimensional observations, is generalized to the case of multidimensional complex observations in the presence of noise with an unknown spatial correlation matrix. Unlike the previous work, the likelihood ratios considered in the mutlidimensional case cannot be addressed using Gaussian random fields. Instead, chi-square random fields associated with the generalized likelihood ratio test are encountered and EVT is used to analyze the model order overestimation probability for a general class of likelihood penalty terms including MDL and AIC, and a novel likelihood penalty term derived based on EVT. Since the exact EVT penalty term involves a Lambert-W function, an approximate penalty term is also derived that is more tractable. We provide threshold signal-to-noise ratios (SNRs) and show that the model order underestimation probability is asymptotically vanishing for EVT and MDL. We also show that MDL and EVT are asymptotically consistent while AIC is not, and that with finite samples, the detection performance of EVT outperforms MDL and AIC. Finally, the accuracy of the derived threshold SNRs is also demonstrated.Index Terms-Akaike information criterion, detection of sinusoids, extreme value theory, generalized likelihood ratio test, maximum likelihood estimation, minimum description length, model order selection.

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“…The tonality detection itself is a common problem in many different technical aspects. From radio or echolocation (radar or sonar) [7] through detection of birds in noisy environment [8] to perceptual audio coding [9]. This short list of examples shows how diverse a domain of tones detection is.…”

Section: Introductionmentioning

confidence: 99%

Detection of multi-tones in white and colored noise with application to asthmatic wheezes detection

Wisniewski

Zieliński

2012

2012 International Conference on Signals and Electronic Systems (ICSES)

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The signal tonality detection is a common problem in many different technical fields, e.g. radar, sonar, perceptual audio encoders, birds recognition etc. Wheezes recognition at asthma patients is another example, which we are interested in, where the multi-tone signal is to be detected in colored noise of normal breath. The spectrum of the typical breath decreases roughly exponentially with frequency and this situation can cause serious problems for tonality descriptors designed for the white noise background. Therefore, the aim of this paper is to examine the impact of spectrum shape of background noise (its colority) on the detection of multi-tonal sinusoidal signal lying in a specified range of frequency.

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Model Selection for Sinusoids in Noise: Statistical Analysis and a New Penalty Term

Cited by 43 publications

References 40 publications

A unified approach to sparse signal processing

A unified approach to sparse signal processing

Model Order Selection for Complex Sinusoids in the Presence of Unknown Correlated Gaussian Noise

Detection of multi-tones in white and colored noise with application to asthmatic wheezes detection

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