There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Clock synchronization is a critical component in the operation of wireless sensor networks (WSNs), as it provides a common time frame to different nodes. It supports functions such as fusing voice and video data from different sensor nodes, time-based channel sharing, and coordinated sleep wake-up node scheduling mechanisms. Early studies on clock synchronization for WSNs mainly focused on protocol design. However, the clock synchronization problem is inherently related to parameter estimation, and, recently, studies on clock synchronization began to emerge by adopting a statistical signal processing framework. In this article, a survey on the latest advances in the field of clock synchronization of WSNs is provided by following a signal processing viewpoint. This article illustrates that many of the proposed clock synchronization protocols can be interpreted and their performance assessed using common statistical signal processing methods. It is also shown that advanced signal processing techniques enable the derivation of optimal clock synchronization algorithms under challenging scenarios.

Clock Synchronization of Wireless Sensor Networks

This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the “Fast Fourier Transform” as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the “Fast Fourier Transform” as a method of solving the evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the “Fast Fourier Transform” that can also be understood by someone who has an understanding of the topic from the engineering perspective.
The manuscript will also introduce several new algorithms that solve the fast multipoint evaluation problem over certain finite fields and require fewer finite field operations than existing techniques. The document will also demonstrate that these new algorithms can be used to multiply polynomials with finite field coefficients with fewer operations than Schonhage's algorithm in most circumstances.
A third objective of this document is to provide a mathematical perspective of several algorithms which can be used to multiply polynomials of size which is not a power of two. Several improvements to these algorithms will also be discussed.
Finally, the document will describe several applications of the “Fast Fourier Transform” algorithms presented and will introduce improvements in several of these applications. In addition to polynomial multiplication, the applications of polynomial division with remainder, the greatest common divisor, decoding of Reed-Solomon error-correcting codes, and the computation of the coefficients of a discrete Fourier Series will be addressed.

/pdf/fast-fourier-transform-algorithms-with-applications-2vinpci6f6.pdf

Fast fourier transform algorithms with applications

The Karhunen-Loeve transform for a class of signals is proven to be a set of periodic sine functions and this Karhunen-Loeve series expansion can be obtained via an FFT algorithm. This fast algorithm obtained could be useful in data compression and other mean-square signal processing applications.

A fast Karhunen-Loeve transform for a class of random processes

Concentric circular antenna array (CCAA) has interesting features over other array configurations.A uniform arrangement of elements is assumed where the interelement spacing is kept almost half of the wavelength and the array parameters such as the steering matrix and gain are determined.The array performance such as beam power pattern, sidelobe level and beamwidth are discussed in two cases of central element feeding.The two cases are compared showing the reduction in the sidelobe level to more than 20 dB in the case of central element feeding without extra signal processing especially for small- sized arrays that have smaller number of elements and rings.

/pdf/efficient-sidelobe-reduction-technique-for-small-sized-lq2o6ypxob.pdf

Efficient sidelobe reduction technique for small-sized concentric circular arrays

This paper studies the design and implementation of finite-impulse response (FIR) and infinite-impulse response (IIR) variable digital filters (VDFs), whose frequency characteristics can be controlled continuously by some control or tuning parameters. A least squares (LS) approach is proposed for the design of FIR VDFs by expressing the impulse response of the filter as a linear combination of basis functions. It is shown that the optimal LS solution can be obtained by solving a system of linear equations. By choosing the basis functions as piecewise polynomials, VDFs with larger tuning range than that of ordinary polynomial based approach results. The proposed VDF can be efficiently implemented using the familiar Farrow structure. Making use of the FIR VDF so obtained, an Eigensystem Realization Algorithm (ERA)-based model reduction technique is proposed to approximate the FIR VDF by a stable IIR VDF with lower system order. The advantages of the model reduction approach are: 1) it is computational simple which only requires the computation of the singular value decomposition of a Hankel matrix; 2) the IIR VDF obtained is guaranteed to be stable; and 3) the frequency response such as the phase response of the FIR prototype is well preserved. Apart from the above advantages, the proposed IIR VDF does not suffer from undesirable transient response during parameter tuning found in other approaches based on direct tuning of filter parameters. For frequency selective VDFs, about 40% of the multiplications can be saved using the IIR VDFs. The implementation of the proposed FIR VDF using sum-of-powers-of-two (SOPOT) coefficient and the multiplier block (MB) technique are also studied. Results show that about two-third of the additions in implementing the multiplication of the SOPOT coefficients can be saved using the multiplier block, which leads to significant savings in hardware complexity.

/pdf/on-the-design-and-implementation-of-fir-and-iir-digital-49ewj7rp70.pdf

On the design and implementation of FIR and IIR digital filters with variable frequency characteristics

The sequential partial-update least mean square (S-LMS)-based algorithms are efficient methods for reducing the arithmetic complexity in adaptive system identification and other industrial informatics applications. They are also attractive in acoustic applications where long impulse responses are encountered. A limitation of these algorithms is their degraded performances in an impulsive noise environment. This paper proposes new robust counterparts for the S-LMS family based on M-estimation. The proposed sequential least mean M-estimate (S-LMM) family of algorithms employ nonlinearity to improve their robustness to impulsive noise. Another contribution of this paper is the presentation of a convergence performance analysis for the S-LMS/S-LMM family for Gaussian inputs and additive Gaussian or contaminated Gaussian noises. The analysis is important for engineers to understand the behaviors of these algorithms and to select appropriate parameters for practical realizations. The theoretical analyses reveal the advantages of input normalization and the M-estimation in combating impulsive noise. Computer simulations on system identification and joint active noise and acoustic echo cancellations in automobiles with double-talk are conducted to verify the theoretical results and the effectiveness of the proposed algorithms.

/pdf/new-sequential-partial-update-least-mean-m-estimate-4w55u1il7v.pdf

New Sequential Partial-Update Least Mean M-Estimate Algorithms for Robust Adaptive System Identification in Impulsive Noise

The design of a digital uniform circular array frequency invariant beamformer (UCA-FIB) is proposed. After transformation and removal of the frequency dependency of the individual phase modes, the frequency invariant phase modes are linearly combined with the spatial filter coefficients to form the far field pattern of the array. As a result, such a beamformer is a wideband frequency invariant beamformer. For simultaneous beamforming, we propose a UCA-FIB DFT filterbank for concurrent beamforming in different directions. Design examples demonstrate that the proposed UCA-FIB can form deep nulls and sharp peaks with good frequency invariant characteristics, while the UCA-FIB DFT can efficiently divide the array neighboring area into small sectors spatially.

On the design of digital broadband beamformer for uniform circular array with frequency invariant characteristics

This letter proposes a new method for designing finite-impulse response (FIR) filters with variable characteristics. The impulse response of the variable digital filter (VDF) is parameterized as a linear combination of functions in the spectral or tuning parameters. Using the least square objective function, the optimal solution is obtained by solving a system of linear equations. Design results show that this method is simple and effective in designing FIR VDF with good frequency characteristics. Furthermore, by using a piecewise polynomial, instead of an ordinary polynomial, more complicated frequency characteristics, or a larger tuning range can be approximated.

/pdf/a-new-method-for-designing-fir-filters-with-variable-3w5pshjkkq.pdf

A new method for designing FIR filters with variable characteristics

This paper proposes two new methods for designing a class of two-channel perfect reconstruction (PR) finite impulse response (FIR) filterbanks (FBs) and wavelets with K-regularity of high order and studies its multiplier-less implementation. It is based on the two-channel structural PR FB proposed by Phoong et al (1995). The basic principle is to represent the K-regularity condition as a set of linear equality constraints in the design variables so that the least square and minimax design problems can be solved, respectively, as a quadratic programming problem with linear equality constraints (QPLC) and a semidefinite programming (SDP) problem. We also demonstrate that it is always possible to realize such FBs with sum-of-powers-of-two (SOPOT) coefficients while preserving the regularity constraints using Bernstein polynomials. However, this implementation usually requires long coefficient wordlength and another direct-form implementation, which can realize multiplier-less wavelets with K-regularity condition up to fifth order, is proposed. Several design examples are given to demonstrate the effectiveness of the proposed methods.

https://hub.hku.hk/bitstream/10722/42960/1/90027.pdf

S.C. Chan

Papers

On the design and implementation of FIR and IIR digital filters with variable frequency characteristics

New Sequential Partial-Update Least Mean M-Estimate Algorithms for Robust Adaptive System Identification in Impulsive Noise

On the design of digital broadband beamformer for uniform circular array with frequency invariant characteristics

A new method for designing FIR filters with variable characteristics

New design and realization techniques for a class of perfect reconstruction two-channel FIR filterbanks and wavelets bases