Chapter 2: Paper Review
2.6 The Issue of Application to the Finding of UBE
In this study, we will consider the use of robust mean estimation in signal detection.
Generally, the energy-based signal activity detection approach is robust to noise and may cost down the non-coherent detection within a communication receiver. Zeng et al. [31] showed the benefits of using the maximum eigenvalue as a result of energy representation on large sample size. Recently, compressive sampling (CS) [32] is an emerging research topic aiming at restoring a signal in an undersampled condition using special vector bases with prior knowledge of the signal. In addition to CS, eigen-analysis is also a popular technique to consider spanning a signal with sparse eigenvectors in which the prior knowledge of needing signal to be normally distributed is released. We will not only consider the combination of energy detection and sparse data sampling, but also fuse the demand of practical signal processing. For instance, measuring signal in a time-varying environment usually results in representing the measured signal as the output of combining quantities by an additive mixture model, as suggested and outlined in the manual published by JCGM [33].
Moreover, the combined quantities are usually resulted from the propagations of multi-source signals with different pdfs so that the representation for the pdf of the output random variable is not tractable.
Unexpectedly, the pdf of the maximum eigenvalue is too complex and inconvenient for computation [34] so that Ma and Zarowski [35] have tried to use the upper bound of the maximum eigenvalue, i.e., Dembo’s bound, for an efficient signal representation. In the study, we are interested in using more accurate mean estimation to improve the finding of upper bound of eigenvalues (UBE) from sparse observed samples.
Since the environmental noise is usually time-varying or color, the traditional white-noise assumption is not realistic so that the mean value of noise can not always be regarded as zero. Hence this study proposes a new algorithm to evaluate the mean value in terms of noise combined with signal.
Let x , i n: 1 i n≤ ≤ , represent the ranked random samples generated from the output of Eq.(2-23). In this study, we plan to estimate the mean value of a QSAW signal by a new quantile-based maximum likelihood estimator (QMLE) using only the
quasi-symmetric quantiles (QSQ), i.e., the minimum sample, x , and the maximum 1:n sample, x . We will compare the performances of the QMLE and sample mean on n n: mean estimation as well as on UBE finding.
There are two parts in our task: one is the QMLE mean estimation aiming at reducing the uncertainty of the estimated correlation matrix and another is the improved upper bound of eigenvalues finding. Conventionally, the mean value of a signal is estimated by sample mean which is UMVUE derived basing on the assumption of normally distributed observations. Although sample mean is a good mean estimator, there still exist some biased estimators that outperform it [23]. In mean estimation for quasi-normal signals, the non-parametric order statistics method was applied to overcome the mismatch between normal and quasi-normal data. In the study, we are interested in the special case of quantile application to mean estimation using the QSQ. The QSQ are determined by the maximum percentage of the observed samples covering the original population, i.e., the coverage which is the cumulative probability calculated between the two endpoints of range. There are good evidences to show that the symmetric property of QSQ is more efficient if they occupy either a very large or very small percentage of the population [36]. Lastly, the task of UBE finding is attractive because the maximum eigenvalue is an important cue of signal activity detection for fading channels with unknown dispersion [31] in multiple-input multiple-output (MIMO) systems [37]. Taparugssanagorn and Ylitalo [38] further indicated the upper bound of MIMO channel capacity being affected by the distribution of the maximum eigenvalue, which was evaluated by the covariance of short-term phase noise. Zhang and Ovaska [39] extended the eigenanalysis to singular value decomposition based on signal-to-noise ratio for the analog-to-digital converter, but their method is not realistic for the cyclostationary detection in spectrum reuse application. Wu et al. [40] proved that the well-trained eigenvector feature of vehicle sound signature was capable of vehicle recognition. UBE acts as the maximum eigenvalue owing to the fact that this representation has been well discussed for the case of deterministic covariance matrix with Hermitian, symmetric positive-definite, or Toeplitz property, Park and Lee [41] improved it by using the technique of series expansion. They proposed the following equations to find a better upper bound of maximum eigenvalue than the classical Dembo’s bound:
(m1) (m1) (m-1)-dimensional vector, and a is a scalar. Up till now, there are seldom studies devoting to the uncertainty analysis for the estimation of correlation matrix on sparse data condition. This study proposes the refreshing change-solution against the issue. It avoids the well-known heavy resampling and computation of the bootstrapping method [42] for small sample size. The main uncertainties of additive model result from the propagation of each source signal. In the reasoning for uncertainty of propagation, Denguir-Rekik et al.[43] fused the multiple marginal effects based on the multi-criteria for aggregated decision making. Ferrero and Salicone [ 44 ] addressed the issue of utilizing the random-fuzzy variable to fit the propagation of distribution.