Spectral analysis of sinusoidal signals from multiple channels
多通道正弦信號的譜分析
Student thesis: Doctoral Thesis
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Award date  2 Oct 2013 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(9597dae05fc64c96b71101ac55c87ad1).html 

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Abstract
Spectral analysis of sinusoidal signals is a classical but still open problem in statistical
signal processing, finding its applications in a wide range of areas. This problem
consists of two parts  sinusoidal model order detection and parameter estimation.
During the recent decades, the problem of analyzing the sinusoidal signals from multiple
channels, which are contaminated by different undesired harmonics, has attracted
considerable attention. Given the corresponding observations, the goal is to determine
the unknown orders and parameters of the sinusoidal signals in the multiple channels,
after which the signal parametrization is complete. This problem is of great research
value not just because it is interesting and practical, but also there are two main
significant advantages compared with singlechannel modeling:
• The multichannel setup means more observed data and the parameter estimation
refinement of the common mode sinusoidal components is expected, which
makes it feasible to extract the common information in a more accurate way.
• In a multisource scenario, if the sources are overlapping in one dimension, a
singlechannel setup will not be able to resolve the sources. On the other hand,
this issue can be alleviated with the multichannel setup by considering joint
higher dimensional modeling.
In addition, decimation technique is utilized in the parameter estimation of oversampled
multiple complex sinusoids for the sake of lower computational complexity
and higher estimation resolution, where the decimative signals belong to a special
form of multichannel sinusoidal signals with the same amplitudes and frequencies.
And accurate parameter estimation for dualchannel sinusoidal signal is extensively
useful in the electronic measurement. Such applications are also the motivations of the research on spectral analysis of multichannel sinusoidal signals.
In this thesis, the multichannel sinusoidal modeling consists of four parts, that is,
oversampling parameter estimation for multiple sinusoids; accurate dualchannel sinewave
parameter estimation; parametric modeling for damped sinusoids from multiple
channels; and spatialtemporal modeling for harmonic signal from microphone array.
In the oversampling parameter estimation for multiple complex sinusoids, the parameters
of continuoustime frequencies are of interest. In signal processing, oversampling
technique is the process of sampling a signal with a sampling rate significantly
higher than the Nyquist frequency of the signal being sampled. Oversampling is
utilized to obtain more data in a fixed duration, and is expected to improve the estimation
accuracy. Nevertheless, two problems occur in spectral estimation, that is the
problems of smaller frequency separation and higher computational complexity. To
alleviate these problems, the oversampling weighted least squares frequency estimator
with decimation is proposed.
For the problem of sinusoidal parameter estimation at two channels, the parameters
of common frequency, amplitudes, initial phases and possibly DC offsets, are
of interest. Under the assumption of white Gaussian noise, an iterative linear leastsquares
algorithm for accurate frequency estimation is devised. The remaining parameters
are then determined according to linear leastsquares fit with the use of the
frequency estimate. The parameter of phasedifference is another key quantity. To
estimate it, two algorithms have been proposed. The first one utilizes the maximum
likelihood criterion to find the initial phases of dualchannel outputs, respectively, and
the phasedifference estimate is then given by their difference. Algorithm extension
to unknown frequency and/or noise powers is also studied. The development of the
second method is based on the weighted linear prediction approach with a properly
chosen sampling frequency.
On parametric modeling of damped sinusoidal signals from multiple channels,
it is aimed at addressing the issues of their model order detection and parameter
estimation from a new and complete viewpoint via performing the parametric modeling
with joint model selection and parameter estimation. It consists of three parts.
Firstly, we extend the subspacebased automatic model order selection method to the multichannel scenario, and detect the number of the distinct sinusoidal poles in
the multiple channels with the multichannel model order estimator. Secondly, we
extend the iterative quadratic maximum likelihood approach to the current problem,
that is, parameter estimation for the sinusoidal poles from multiple channels, which
is referred to as the multichannel iterative quadratic maximum likelihood estimator.
Thirdly, sinusoidal model selection, or matching the estimated poles to their
corresponding channels, is realized based on a sequence of hypothesis tests. At each
test, we compute the significance of the maximum correlation between the estimation
residual and a sinusoidal function, whose statistical property is derived from the extreme
value theory about the distribution of the maximum of stochastic fields. We
refer this scheme to as extreme value theory selector.
The problem of spatialtemporal modeling for harmonic signal from microphone
array is solved from two aspects. Firstly, we propose to estimate the fundamental
frequency and direction of arrival (DOA) in two stages. At first, the multichannel
optimally weighted harmonic multiple signal classification estimator is devised, and
the estimation of fundamental frequency is conducted. Then we make use of the
spatialtemporal multiple signal classification estimator to estimate the DOAs with
the estimated fundamental frequencies. Although the twostage method is more computationally
efficient, it cannot resolve the sources with overlapping frequencies or
DOAs. To overcome this problem, in the second part, we perform DOA and fundamental
frequency estimation in a joint way. In practice, there also occur the problems
of order detection and detection of missing harmonics of each source. To take this issue
into account in harmonic modeling, we propose to perform joint estimation based
on optimal filtering method and with the maximum harmonic model, and then the
model selection is accomplished according to the powers of the respective harmonic
components of each source, and the maximum a posteriori criterion.
 Statistical methods, Spectral theory (Mathematics), Signal processing