Tracking properties of a gradient-based second-order adaptive IIRnotch filter with constrained poles and zeros

Gradient-type adaptive IIR notch filters have many attractive merits for various real-life applications since they require a small number of computations and yet demonstrate practical performance. However, it is generally quite difficult to assess their performance analytically. Their tracking properties, in particular, have not yet been investigated. In this paper, the tracking performance of a plain gradient (PG) algorithm is analyzed in detail for a second-order adaptive IIR notch filter with constrained poles and zeros, which takes a linear chirp signal as its input. First, two sets of difference equations for the frequency tracking error and mean square error (MSE) are established in the sense of convergence in the mean and convergence in the mean square, respectively. Closed-form expressions for the asymptotic tracking error and MSE are then derived from these difference equations. An optimum step-size parameter for the algorithm is also evaluated based on the minimization of the asymptotic tracking error or the tracking MSE. It is discovered that the asymptotic tracking error may be driven to zero for a positive chirp rate by selecting a proper step size, which is an interesting property for a real-valued adaptive filtering algorithm. Extensive simulations are performed to support the analytical findings     

Analysis of second-order gratings

The results of an analysis of second-order gratings used as distributed Bragg reflectors in surface-emitting lasers are presented. The gratings provide reflection, output coupling, and power transmission to other gain segments for purposes of injection locking. The analysis determines these quantities for arbitrary-shaped grating teeth and includes the presence of a substrate reflector to reduce the radiated power in that direction. The reflector is shown to be effective, but only if it can be precisely positioned. Examples illustrating variations in dimensions, tooth shapes and heights, waveguide loss, and detuning are included. Second-order, square gratings, whose conventional coupling coefficient is identically zero, exhibit substantial reflectivity which is primarily produced by the radiation reaction. The analysis also forms the basis for calculating the far-field grating coupled radiation patterns     

Selectivity measures of second-order networks

The utility of gain-slope definitions in assessing the selectivity performance of various second-order transfer functions is investigated and their suitability for describing notch functions alone demonstrated.     

Statistics of second-order PMD depolarization

The second-order polarization mode dispersion (PMD) components associated with depolarization of the principal states can cause significant system impairments. We discuss the probability densities of these components and find excellent agreement between extensive measurements, simulations, and theoretical predictions     

Asymptotic performance of second-order algorithms

This paper re-examines the asymptotic performance analysis of second-order methods for parameter estimation in a general context. It provides a unifying framework to investigate the asymptotic performance of second-order methods under the stochastic model assumption in which both the waveforms and noise signals are possibly temporally correlated, possibly non-Gaussian, real, or complex (possibly noncircular) random processes. Thanks to a functional approach and a matrix-valued reformulated central limit theorem about the sample covariance matrix, the conditions under which the asymptotic covariance of a parameter estimator are dependent or independent of the distribution of the signal involved are specified. Finally, we demonstrate the application of our general results to direction of arrival (DOA) estimation, identification of finite impulse response models, sinusoidal frequency estimation for mixed spectra time series, and frequency estimation of sinusoidal signal with very lowpass envelope     

Characterisation of second-order discrete-time filters

Novel but very simple expressions characterising second-order discrete-time filters are presented. The parameters used are the centre angular frequency and the 3 dB bandwidth. The sensitivity analysis points out that Q (=?0/?b) has been misused as the measure related to sensitivity performance of discrete-time filters. Actually the sensitivity is approximately proportional to 1/?0?b for narrowband filters.     

Weakly constrained codes

The author reports on the performance of a new class of constrained codes, called weakly constrained codes. These codes do not strictly guarantee the imposed channel constraints, but rather generate codewords that violate, with a given (small) probability, the prescribed constraint. Weakly constrained codes are specifically of interest when it is desirable that the code rate R=p/q is very high, requiring codewords of length q>100     

Novel second-order switched-capacitor interpolator

A novel switched-capacitor interpolator circuit is proposed for the realisation of arbitrary second-order interpolating functions employing a reduced number of switching waveforms and yielding low capacitance spread and total capacitor area.<>     

A second-order microwave differentiator

A novel approach consisting of discrete signal processing (DSP) technique and optimization method is developed to implement a second-order differentiator at microwave frequencies. To utilize the transfer function of a second-order differentiator developed in the DSP study, we formulate the chain scattering parameters of transmission lines in the Z domain. In particular, it is shown that shunt stubs combining with nonuniform serial lines lead to the realization of a second-order microwave differentiator. Some examples are presented to illustrate the validity of this approach.     

Switched-capacitor second-order noise-shaping coder

The rapid development of digital processing in the fields of instrumentation, control and communications has generated a good deal of research into simple, easily integrable, high-resolution A/D converters. Given the significant promise such converters offer for future application, a switched-capacitor version of a second-order noise-shaping coder intended for high-resolution conversion has been realised. This coder has been used successfully to produce a PCM signal.     

Zero-loss second-order ring modulator

It is shown that zero conversion loss may be obtained for the frequency change from ?q to 2?p ± ?q using an ideal ring modulator switched at a frequency ?p and having a reactive idler circuit at the normal output terminals. The frequency 2 ?p ± ?q is obtained at the normal input terminals, so that advantages are obtained with regard to earthing arrangements.     

Experimental signal/noise-ratio comparison of the second-order phase-locked loop and the second-order frequency-locked loop

The signal/noise ratio performance of a novel f.m. threshold extension demodulator, the frequency-locked loop, is experimentally compared with that of the phase-locked-loop demodulator and the conventional f.m. discriminator. Experimental results indicate that the f.l.l. represents a significant improvement over the p.l.l. and the conventional discriminator for f.m. systems subject to large carrier deviations.     

On the second-order approximation of PMD

A second-order polarization mode dispersion (PMD) approximation based upon the pulse-width distortion has been studied. It shows that a complete second-order approximation should include the second derivative of the PR-ID vector as well as the first derivative of the PMD vector. Second-order pulse distortions are explicitly expressed including a `first-order' term involving principal states of polarization (PSP) of the pulse and a second-order term involving the beating between fiber chromatic dispersion and effective PMD chromatic dispersion. An analytical result is derived for the probability of second-order PR-ID power penalty. It shows that the mean PMD of the fiber should be restricted to 26 ps and 18 ps, respectively for an optical link with zero and 850 ps/nm chromatic dispersion, in order to maintain a one dB second-order PMD power penalty with a probability below 10^-6 at a data rate of 10 Gb/s. The analysis also indicates that a second-order PMD compensator can be used as a dynamic chromatic dispersion compensator     

Above-threshold analysis of second-order circular-grating DFBlasers

The behavior of second-order circular-grating DFB lasers in the above-threshold regime are analyzed. The formulation uses the standing wave method for treating lasers with second-order gratings. The effects of spontaneous emission and the variation of carrier and photon densities in the radial direction are included in a self-consistent fashion in the model. In particular, we examine the output power emitted from the surface of the laser, the near-field, and the far-field pattern of the laser as a function of the injected current     

General nonlinear analysis of second-order oscillators

A general nonlinear analysis of second-order oscillators is presented. The oscillator equation is given in perturbation form, involving a small perturbation parameter, and its periodic solution derived, if any exists, which is calculated using an asymptotic method. Analytical relations between design constraints and circuit parameters are deduced and applied     

Analysis of constrained active networks

A simplified procedure for computing the transfer functions of an active filter involving single or multiple operational amplifiers is presented. Illustrative examples are given.     

Planar three-dimensional constrained lenses

A new design for a beamforming lens is presented. Both faces are planar arrays of radiating elements interconnected by transmission lines whose length varies as a function of radius. While the front face elements are regularly spaced, the back face elements are displaced radially from their corresponding front face elements, the amount of displacement also being a function of radius. We show that such a lens is capable of forming low sidelobe beams over an angular sector 36 beamwidths across in all planes ofphiby switching between clusters of only seven feed elements. Because both faces are planar, construction of lightweight lenses for multibeam antennas should be feasible.     

The constrained signal detector

The problem of detecting a material-of-interest in a hyperspectral image is considered. Knowledge of the background materials in the image is assumed. It is also assumed that the stochastic noise in the system has a Gaussian distribution with a known covariance matrix. Using these assumptions, along with the requirement that the material abundances in the pixel must sum to one, a filter called the constrained signal detector (CSD) is derived. The CSD is a variation of the generalized likelihood ratio test (GLRT). Where the GLRT uses maximum-likelihood estimates (MLEs) of the noise in the received signal, the CSD uses constrained least squares (CLS) noise estimates. It will be shown that the CSD is actually a scaling of the CLS target abundance estimate which has been derived elsewhere. However, the CSD computes that estimate much more efficiently then existing methods do. It is proved that the CSD outperforms the orthogonal subspace projection (OSP) detector and that the CSD is the optimal detector when there is only one background material present.     

Iterative second-order derivative-based deinterlacing algorithm

In this paper, we introduce an iterative second-order derivative-based intra-field deinterlacing algorithm (ISDD). The proposed ISDD is composed of two steps: (1) preprocessing by a fast second-order derivative-based deinterlacing algorithm (FSDD), (2) iteratively refining the interpolated pixel by minimizing a mass second-order function. The function includes three components: the derivative difference, directional derivative, and isophote smoothing. The derivative difference is used to detect any intensity discontinuity in an image; the directional derivative is a supplement to the derivative difference to avoid oversmoothing; and isophote smoothing can reduce artifacts. The proposed ISDD can preserve image features and reduce the artifacts caused by conventional deinterlacing methods. Compared with existing deinterlacing algorithms, the proposed algorithm improves the peak signal-to-noise ratio while maintaining high efficiency.