Advanced algorithms for harmonic retrieval
諧波恢復的先進算法
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(a1f2aac8f4a94602b3f23abfebd9cd9e).html 

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Abstract
Harmonic retrieval has been one of the fundamental problems across a diverse range of
important fields, from source localization, radar and sonar signal processing, channel
modeling in wireless communication systems, to medical areas. Generally speaking,
harmonic retrieval algorithms can be divided into parametric and nonparametric
approaches. Parametric based algorithms assume that the received signal follows a
specific model. These kinds of estimators provide a superior estimation performance
in terms of accuracy, computational efficiency, resolution and/or identifiability, but
will fail when the assumption of the data model is incorrect. Nonparametric based
algorithms, however, can generate the estimated spectrum without any assumption
of a data model. In this thesis, several algorithms are devised and analyzed in terms
of the harmonic retrieval problem, from 1dimensional (1D) to RD where R ≥ 2.
Synonyms of harmonic retrieval include frequency estimation, parameter estimation
and spectrum/spectral analysis.
The commonly used algorithms for 1D harmonic retrieval of multiple sinusoids
can be divided into two groups: subspace based and iterative approaches. The subspace
based technique is usually computationally efficient but the accuracy is not very
satisfying. Iterative algorithms, however, can achieve a high level of accuracy when
the signal to noise ratio (SNR) is sufficiently high, but will be costly when the data
size is large. This thesis introduces a new approach, the principalsingularvector
utilization for model analysis (PUMA), which combines the advantages of both subspace
and iterative techniques, for the frequency estimation of complexvalued and
realvalued sinusoids. The main idea is to use a matrix without repeated elements to
represent the observed signal and apply the weighted least squares (WLS) on its signal
subspace to retrieve parameters accurately. It is proved that for small error conditions,
the frequency estimate is approximately unbiased. Furthermore, its variance equals to CramérRao lower bound (CRLB) for the single complex cisoid problem.
Although the 1D problem is the most common, the RD harmonic retrieval with
R ≥ 2 has recently been receiving increased interests. Conventional RD parameter
estimation techniques include the maximum likelihood (ML) based and subspace approaches.
The ML based methods are only feasible when the problem is 2D due to
their extremely high computational requirements, while the accuracy and/or computational
efficiency of subspace methodology remains unsatisfactory. Furthermore, the
subspace approaches can be divided into matrix based and tensor based ones. The
latter one usually give a more accurate subspace estimation with a higher complexity.
In this thesis, the tensorPUMA (TPUMA) approach for RD parameter estimation
of a summation of F ≥ 2 sinusoids in additive white Gaussian noise is devised.
Firstly, with the use of an iterative procedure, the sinusoidal parameters at one dimension
are accurately estimated from its signal subspace, which is retrieved using
a decomposition method. The parameters in the remaining dimensions are then
solved such that the RD harmonics are automatically pairedup. Algorithm modification
for a single RD tone is made and it is shown that the frequency estimates
are asymptotically unbiased while their variances approach CRLB at sufficiently high
SNR conditions. It is also shown that the TPUMA algorithms are computationally
simpler than conventional parameter estimators.
In the special case of RD harmonic retrieval, the single cisoid in additive white
Gaussian noise is studied, and two proposals are then developed. By exploiting the
correlation of the data samples, R singletone sequences, which contain the RD
frequencies, are constructed, and accurate frequency estimation is attained from each
constructed sequence. The two devised estimators are proved to be asymptotically
unbiased while their variances achieve CRLB when the SNR and/or data length tend
towards infinity, and also shown to be more computationally attractive than the
conventional schemes.
Sometimes identical frequencies exist along each dimension. When this occurs, the
TPUMA scheme cannot work. To overcome this problem, a tensor eigenvectorbased
(TEV) estimator is proposed. Our underlying idea is to utilize the tensorial structure
of the received data and then apply a higherorder singular value decomposition
(HOSVD) to obtain the tensorbased signal subspace. It is decomposed into a set of singletone tensors from which singletone vectors can be constructed by another
HOSVD later on. In doing so, the RD multiple sinusoids are converted to a set
of singletone sequences whose frequencies are individually estimated according to
structure least squares (SLS). The bias and variance of the subspacebased method
are also analyzed and it is proved that the frequency estimates are approximately
unbiased. It is shown that the proposed algorithm can provide superior estimation
performance.
Recently, an iterative adaptive approach (IAA), which employs the WLS technique
to update the whole spectrum, is proposed for highresolution spectral estimation.
However, the IAA algorithm is computationally expensive for 2D spectrum imaging.
To address this issue, a new fast implementation for IAA utilizing subspace technique
is devised. The main idea is to utilize the vectors of the subspaces to retrieve the
spectrum. It is observed that the whole spectrum can be written as a summation
of several subspectra, and each subspectrum can be updated in an iterated 1D
format based on the dominant singular vectors instead of 2D format based on the
original data samples. Fast computation is thus achieved. Furthermore, the number of
signals, F, can be known or unknown for this procedure, which makes this algorithm
superior to the algorithms where information of F is required. It is shown that the
proposed fast IAA can achieve a performance similar to IAA with less computational
complexity.
 Harmonic analysis