H �� Filtering for Markovian Switching Genetic Regulatory Networks with Time-Delays and?Stochastic?Disturbances

This paper is concerned with the H _∞ filtering problem for a class of nonlinear Markovian switching genetic regulatory networks (GRNs) with time-delays, intrinsic fluctuation and extrinsic noise. The delays, which exist in both the translation process and feedback regulation process, are not dependent on the system model. The intrinsic fluctuation is described as a state-dependent stochastic process, while the extrinsic noise is modeled as an arbitrary signal with bounded energy, and no exact statistics about the noise are required to be known. The aim of the problem addressed is to design a Markovian jump linear filter to estimate the true concentrations of mRNA and protein through available measurement outputs. By resorting to the Lyapunov functional method and some stochastic analysis tools, it is shown that if a set of linear matrix inequalities (LMIs) is feasible, then the desired linear filter exists. The designed filter ensures asymptotic mean-square stability of the filtering error system and two prescribed L ₂-induced gains from the noise signals to the estimation errors. Finally, an illustrative example is given to demonstrate the effectiveness of the approach proposed.     

A new method for designing multiplierless two-channel filterbank using shifted-Chebyshev polynomials

This paper presents an efficient design method for a digital multiplierless two-channel filterbank using the shifted-Chebyshev polynomials and common sub-expression elimination (CSE) algorithm for reducing hardware requirements such as adders and multipliers. For designing a two-channel filterbank, the design problem is constructed as minimization of integral mean square error between the desired and designed response of a prototype filter in the passband and stopband. For controlling the performance in passband and stopband, two parameters (K_P, and K_S) are used, whose optimum values are determined by swam optimization techniques such as differential evolution algorithm, artificial bee colony optimization, particle swarm optimizations, cuckoo search algorithm and hybrid method using a fitness function, constructed by perfect reconstruction condition of a filterbank. The number of polynomials used for approximation depends upon the order of a prototype filter. A new hybrid CSE is proposed for further reduction of hardware requirement. A comparative study of various CSE techniques such as horizontal, vertical and proposed hybrid CSE is also made. Numerical examples illustrate the effectiveness of the proposed algorithm in the reduction of adders with comparisons accomplished using existing methods. It has been found that almost 43% adder gain can be achieved when a filter is designed with N = 32 and wordlength (WL) as 12 using proposed methodology.     

A class of predictive analog filters for sensor signal processingand control instrumentation

A class of analog continuous-time filters is introduced, having predictive properties for specified narrow-band signal models, such as low-order polynomials or sinusoids. Such filters are designed by using model transfer functions designed in the discrete-time domain. Z-to-s-domain mapping is done using the inverse bilinear transformation. The analog filters are implemented with active-RC structures, using the state-variable structure for biquads and a single-op-amp structure for real poles and zeros. The application examples include a filter for zero-crossing detectors, polynomial predictors for sensor signal smoothing, and an optimized sixth-order ramp-tracking filter for anti-aliasing and anti-imaging in digital signal processor (DSP) systems where high selectivity is required     

Signal limitations of the current-mode filters employing current conveyors

It is well known that second-generation current conveyors (CCII) are widely used for the realization of second-order current-mode universal filters. A filter with high-quality factor (Q) and gain constant (K) suffers from various signal swing restrictions especially at its angular resonance frequency (ω₀). This is due to the terminal voltages of the CCIIs limited by the power supply voltages and maximum allowable terminal currents of the CCIIs. In this paper, signal limitations of the CCII-based current-mode filters are investigated in detail. A filter example is given to exhibit the signal limitations of a universal current-mode filter. The time-domain and frequency-domain results of the proposed filter are also given to verify the theoretical analysis.     

A new closed form method for design of variable bandwidth linear phase FIR filter using different polynomials

In this paper, a new method for the design of variable bandwidth linear-phase finite impulse response (FIR) filters using different polynomials such as shifted Chebyshev polynomials, Bernstein polynomials and shifted Legendre polynomials is proposed. For this purpose, the transfer function of a variable bandwidth filter, which is a linear combination of fixed-coefficient linear-phase filters and the above polynomials are separately exploited as tuning parameters to control bandwidth of the filter. In order to determine the filter coefficients, mean squared difference between the desired variable bandwidth filter and the practical filter is minimized by differentiating it with respect to its coefficients leading to a system of linear equations. The matrix elements can be expressed in form of Toeplitz-plus-Hankel matrix, which reduces the computational complexity. Several examples are included to demonstrate effectiveness of the proposed method in terms of passband error (e_p), stopband error (e_s) and stopband attenuation (A_s).     

On the Efficient Generation of Prime-Order Elliptic Curves

We consider the generation of prime-order elliptic curves (ECs) over a prime field \mathbbF_p\mathbb{F}_{p} using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones. These polynomials are uniquely determined by the CM discriminant D. In this paper, we consider a variant of the CM method for constructing elliptic curves (ECs) of prime order using Weber polynomials. In attempting to construct prime-order ECs using Weber polynomials, two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that D≡3mod8), which gives Weber polynomials with degree three times larger than the degree of their corresponding Hilbert polynomials (a fact that could affect efficiency). The second difficulty is that these Weber polynomials do not have roots in \mathbbF_p\mathbb{F}_{p} .     

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.     

New current-mode current-controlled universal filter with single input and three outputs

In this study, a new current-mode current-controlled universal filter with single input and three outputs is presented. The proposed circuit uses single-output current controlled conveyors (CCCIIs) and can simultaneously realize lowpass, bandpass and highpass filter functions all at high impedance outputs. Realization of notch and allpass responses does not require additional active elements. The circuit enjoys independent current-control of the parameters ω₀ and ω₀/Q without disturbing the gains of the lowpass, bandpass and highpass filters. Both its active and passive sensitivities are low.     

Zero-Phase Filter Bank and Wavelet Code r Matrices: Properties,Triangular Decompositions,and a Fast Algorithm

The class of digital filter banks (DFB's) and wavelets composed of zero-phase filters is particularly useful for image processing because of the desirable filter responses and the possibility of using the McClellan transformation for two-dimensional design. In this paper unimodular polyphase matrices with the identity matrix for Smith canonical form are introduced, and then decomposed to a product of unit upper and lower triangular or block-triangular matrices which define ladder structures. A fundamental approach to obtaining suitable unimodular matrices for one and two dimensions is to focus on the shift (translation) operators, as is done in the harmonic analysis discipline. Several matrix shift operators of different dimensions are introduced and their properties and applications are presented, most notable of which is that the McClellan transformation can be effected by a simple substitution of a 2 × 2 circulant matrix for the polynomial variable, w = (z + z ^-1)/2. Unimodular matrix groups and pertinent subgroups are identified, and these are observed to be subgroups of the special linear group over polynomials, SL(k[w]) . A class of coiflet-like wavelets containing the well-known wavelet, based on the Burt and Adelson filter, is decomposed by these methods and is seen to require only 3/2 multiplications/sample if a scaling property, introduced herein, is satisfied. Making use of certain paraunitary wavelets, coiflets, that are closely comparable to the zero-phase wavelets of this class, it is seen that, in these cases, the zero-phase ladder algorithm is twice as fast as the paraunitary lattice algorithm.     

Design and performance evaluation of narrowband rectangular optical filter based on stimulated Brillouin scattering in fiber

We propose a rectangular optical filter based on stimulated Brillouin scattering (SBS) in fiber with tunable bandwidth from 50 MHz to 4 GHz at 15-MHz tuning resolution. The steep-edged rectangular shape of the filter is precisely controlled utilizing digital feedback compensation of the multi-tone pump light. The passband ripple is \(\sim \)1 dB by nonlinearity management of the pump light and using the fiber with a single Brillouin peak. The filter selectivity is improved to more than 40 dB by using pump-splitting dual-stage configuration. We analyze the noise performance of the proposed SBS filter and demonstrate a sub-band extraction of a multi-band orthogonal frequency division multiplexing (OFDM) signal. Furthermore, we validate the amplification performance with different gains for OFDM signal, which shows the potential capability of the filter in the fields of optical signal processing.     

A Practical Parallel Architecture for Stacks Filters

Stack filters belong to the class of non-linear filters and include the well-known median filter, weighted median filters, order statistic filters and weighted order statistic filters. Any stack filter can be implemented by using the parallel threshold decomposition architecture which allows implementing their non-linear processing by means of a collection of identical binary filters (Boolean logic circuits). Although it is conceptually simple and useful to study the filter properties, this architecture is not practical for direct hardware implementation because as many as (M – 1) binary filters are required for a M-valued input signal and M is large in many applications.In this paper we introduce a new parallel architecture for stack filter implementations. The complexity is now proportional to the window width L of the filter, instead of to M. In most applications L is much smaller than M which translates into efficient hardware implementations. The attractive characteristic of ease of design exhibited by the threshold decomposition architecture is kept. In fact, for a given stack filter both in the conventional implementation and in the proposed one, the same binary filter is required. The key concept supporting the new architecture is a modified decomposition scheme which generates L binary signals for a multi-valued input. As an application example, a complex WOS filter is designed and prototyped in an FPGA.     

A Constructive Approach to Stabilizability and Stabilization of a Class of nD Systems

This paper presents a constructive approach to the problem of output feedback stabilizability and stabilization of a class of linear multidimensional (nD, n>2) systems, whose varieties of the ideals generated by the reduced minors are infinite with respect to not more than two variables. The main idea of the proposed approach is to decompose the variety of an nD system in this class into a union of several varieties, each of which is defined by polynomials in just two variables. The new method can be considered as a combination of Gröbner bases and existing results on two-dimensional (2D) digital filter stability tests and on stabilizability and stabilization of 2D systems. An example is illustrated.     

An improved LMS algorithm for mode demultiplexing in frequency domain

In order to solve the problem that the traditional frequency domain least mean square(FD-LMS) algorithm will lose efficacy with the increase of differential mode group delay(DMGD) when the algorithm is used for demultiplexing of the 6×6 mode division multiplexing(MDM) system, an improved FD-LMS demultiplexing algorithm is proposed. By improving the error signal calculation method, the convergence performance of the output signal of the equalization filter is improved, and the steady-state error of the algorithm is reduced. Besides, the equalization performance of the traditional FD-LMS algorithm is compared with the improved FD-LMS algorithm. Simulation results show that the improved FD-LMS algorithm has great advantage over the traditional FD-LMS algorithm in demultiplexing performance on the premise that the computation complexity does not significantly increase. The optical signal to noise ratio(OSNR) penalty of the improved FD-LMS algorithm is 2.6 dB lower than that of traditional FD-LMS algorithm at a transmission distance of 80 km with DMGD is 50 ps/km.     

Frequency estimation of distorted non-stationary signals using complex H∞ filter

Mechanical vibration signals are always composed of harmonics of different order. A novel estimator is proposed for estimating the frequency of sinusoidal signals from measurements corrupted by White Gaussian noise with zero mean. Also low frequency sinusoidal signal is considered along with third and fifth order harmonics in presence of noise for estimating amplitudes and phases of different harmonics. The proposed estimator known as complex H_∞ filter is applied to a noisy sinusoidal signal model. State space modeling with two and three state variables is used for estimation of frequency in presence of white noise. Various comparisons in terms of simulation results for time varying frequency reveal that the proposed adaptive filter has significant improvement in noise rejection and estimation accuracy. Comparison in performance between two and three states modeling is presented in terms of mean square error (MSE) under different SNR conditions .The computer simulations clearly indicate that two states modeling based on Hilbert transform performs better than three states modeling under high noisy condition. Frequency estimation performance of the proposed filter is also being compared with extended complex Kalman filter (ECKF) under same noisy conditions through simulations.     

A fundamental theorem of algebra, spectral factorization, and stability of two-dimensional systems

In his doctoral dissertation in 1797, Gauss proved the fundamental theorem of algebra, which states that any one-dimensional (1-D) polynomial of degree n with complex coefficients can be factored into a product of n polynomials of degree 1. Since then, it has been an open problem to factorize a two-dimensional (2-D) polynomial into a product of basic polynomials. Particularly for the last three decades, this problem has become more important in a wide range of signal and image processing such as 2-D filter design and 2-D wavelet analysis. In this paper, a fundamental theorem of algebra for 2-D polynomials is presented. In applications such as 2-D signal and image processing, it is often necessary to find a 2-D spectral factor from a given 2-D autocorrelation function. In this paper, a 2-D spectral factorization method is presented through cepstral analysis. In addition, some algorithms are proposed to factorize a 2-D spectral factor finely. These are applied to deriving stability criteria of 2-D filters and nonseparable 2-D wavelets and to solving partial difference equations and partial differential equations.     

Voltage step-up filters not using transformers

A network of resistors and capacitors is to serve as a filter for an arbitrary signal voltagef(t). It consists of twin parts connected in parallel across the input terminals. One part is a low pass ladder filter; the rungs of the ladder are capacitors, and the other branches are resistors. The twin part is connected to the input terminals in theopposite direction. The output signal is the voltageg(t) measured between the end nodes of the ladders. Kirchhoff's current law gives a differential equation relatingf and (g−f). IfF is the Fourier transform off, then the transform of the differential equation is an algebraic equation relating the transformsF and (G−F). This equation shows that ⋎G/F⋎>1. Then by Parseval's theorem the norm off is greater than the norm ofg. Thus there is a stepup; the norm of the output always exceeds the norm of the input. Such a filter is here termed anRCformer. This paper is dedicated to Abraham Charnes, mathematician (1917–1992)     

Realization of Tunable Pole-Q Current-Mode OTA-C Universal Filter

A realization of a current-mode operational transconductance amplifier-capacitor (OTA-C) universal filter with tunable pole-Q is proposed. A biquadratic band-reject function is used as the initial synthesis function based on three integrator blocks. Consequently, the proposed filter uses a total of three multiple-output OTAs and three grounded capacitors. Five types of transfer functions, namely, low-pass, high-pass, band-pass, band-reject, and all-pass responses, can be obtained without changing the circuit topology. The pole-Q (Q ₀) and the pole-frequency (ω ₀) parameters are independently tuned. The Q ₀ and ω ₀ parameters are electronically tuned by adjusting the transconductance gains of the OTAs. Furthermore, Q ₀ can be tuned by varying the capacitor manually without affecting ω ₀. SPICE simulation results of the proposed filter are presented.     

Simulation of an on-board hierarchical multistage digital FDM demultiplexer for mobile scpc satellite communications

This paper is focused on the finite-precision time-domain simulation of the hierarchical multistage method detailed in References 1 and 2 for demultiplexing of an FDM signal being composed of L = 32 slot signals. This approach to FDM demultiplexing is based on the processing of complex signals by linear-phase FIR filters, where at any stage of processing the respective signals are always oversampled by a factor of two. The simulation results fully confirm the system performance predicted in Reference 1 by modelling the distortions (spectral foldover and quantization noise) inherent in the system: assuming an ideal FDM signal at the input port of the analogue anti-aliasing bandpass filter in front of the analogue-to-digital converter, a minimum signal-to-distortion ratio (S/D) of 30 dB is achieved at all L output ports of the demultiplexer with {w, w_F, w_i} = {11,12,14}, as anticipated in Reference 1. The signal word length w apply to the A/D conversion, w_F to requantization between the filter stages, and w_i to filter internal signal processing.     

Envelope-constrained H2 FIR filter design

In this paper, we consider an envelope-constrained (EC)H ₂ optimal finite impulse response (FIR) filtering problem. Our aim is to design a filter such that theH ₂ norm of the filtering error transfer function is minimized subject to the constraint that the filter output with a given input to the signal system is contained or bounded by a prescribed envelope. The filter design problem is formulated as a standard optimization problem with linear matrix inequality (LMI) constraints. Furthermore, by relaxing theH ₂ norm constraint, we propose a robust ECFIR filter design algorithm based on the LMI approach.     

Statistically optimal null filter based on instantaneous matched processing

A novel approach is proposed for solving the problem of enhancement/suppression of narrowband signals of short-record length based on combining the maximum signal-to-noise ratio (SNR _o) and the least-squares (LS) optimization criteria. This two-fold optimization is implemented by scaling the output of an instantaneous matched filter used for the maximization of theSNR _o, over a variable-time observation interval, with the locally generated function (t) whose gain is optimized through the LS procedure. The intrinsic property of the proposed statistically optimal null filter (SONF) is its ability to track rapidly, leading to a more practical processing of short duration signals (transients). The theoretical analysis and simulation studies show that the SONF, based on this proposed two-fold optimization procedure, is closely related to the Kalman filter. On the other hand, the design of the SONF does not require the solution of a nonlinear equation of the Ricatti type that is necessary in finding the gains of the Kalman filter. Consequently, the proposed algorithm may be considered as an alternate approach to Kalman filtering. The paper also presents some simulation results illustrating the application of the proposed SONF.This work was partially supported by Micronet, a Candidan Network of Centre of Excellence, by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by Fonds Pour la Formation de Chercheurs et L'Aide a la Recherche (FCAR) of Province of Quebec, Canada.