Chebyshev Window Research Articles

Anti-image FIR filters for large interpolation factors

In this paper, the design of FIR filters using novel window families is introduced. These window functions are having high side-lobe-fall-off-rate (SLFOR). The designed filters can be used as anti-imaging filters for large interpolation factors used in multirate signal processing. The performance of the proposed filters is compared with similar filters designed with Kaiser window, Dolph–Chebyshev window and optimum filters designed using Parks–McClellan method. It is observed that, filters designed with high SLFOR combinational window families are providing better image suppression with large interpolation factor. These filters provide smaller aliasing and imaging errors at the cost of smaller initial stopband attenuation.

A hybrid method for the design of oversampled uniform DFT filter banks

Subband adaptive filters have been proposed to speed up the convergence and to lower the computational complexity of time domain adaptive filters. However, subband processing causes signal degradations due to aliasing effects and amplitude distortions. This problem is unavoidable due to further filtering operations in subbands. In this paper, the problem of aliasing effect and amplitude distortion is studied. The prototype filter design problem is formulated as a multi-criteria optimization problem and all the Pareto optima are sought. Since the problem is highly nonlinear and nonsmooth, a new hybrid optimization method is proposed. Different prototype filters are used and their performances are compared. Moreover, the effect of the number of subbands, the oversampling factors and the length of prototype filter are also studied. We find that prototype filters designed via Kaiser or Dolph–Chebyshev window provide the best overall performance. Also, there is a critical oversampling factor beyond which the improvement in performance is not justified. Finally, if the length of the prototype filter increases with the number of subbands, an increase in the subband level will not deteriorate the performance.

Acoustic beamforming of a parametric speaker comprising ultrasonic transducers

A directional audible sound can be generated in air by means of the nonlinear interaction between intense amplitude modulated (AM) ultrasonic waves, which attracts much attention in the audio industry. In the case of sound reproduction by a parametric speaker comprising an array of ultrasonic transducers, a novel algorithm with Chebyshev window has been proposed to control the sidelobe level of the beam pattern by utilizing the acoustic nonlinearity and array signal processing technique. Furthermore, the beamforming approach is extended to the design of a broadband beamformer. In this paper, a single sideband modulation (SSB) system is developed for transmitting audio signal due to its efficient use of both power and bandwidth. By adding different weightings to the carrier and sideband frequencies, a constant beamwidth beamforming is achieved and confirmed in the simulation experiments.

Improved measurement of grain–wall contact forces in granular beds using wavelength scanning interferometry

We describe a wholefield optical technique based on a wavelength scanning Fizeau interferometer for measuring the contact forces between a granular bed and a transparent substrate. The substrate material is polymethyl methacrylate (PMMA), aluminium-coated on the internal surface, and changes to the interference pattern formed by reflection from the two surfaces allow the displacement field induced by the contacting grains to be visualized. Quantitative displacement data are obtained by phase shifting the interference pattern using a tunable laser. In order to avoid miscalibration errors associated with the non-linearities of the laser source, a Newton–Raphson iteration scheme is employed to search for the correct PZT voltage required for the linear phase-shifting steps. A new 31-frame phase-shifting algorithm based on the Chebyshev window function was designed to deal with the problems of residual miscalibration and higher harmonics created by multiple reflections within the substrate. The resulting noise in the measurements is below 1 nm, and the repeatability of the load–displacement relationship was found to be approximately 10 nm.

Impact of the Digital Filter as a Weak Constraint in the Preoperational 4DVAR Assimilation System of Météo-France

In this paper, a weak constraint formulation of the digital filter based on the Dolph–Chebyshev window is introduced in a preoperational version of the 4DVAR analysis of Météo-France. The constraint is imposed only on the analysis increments to damp spurious fast oscillations associated with gravity–inertia waves. In the incremental formulation of 4DVAR, the analysis increments are obtained from a global model at a uniform low resolution with a simplified set of physical parameterizations, while the high-resolution forecast is obtained with a model that uses a variable-resolution grid having a higher resolution over France and the complete set of physical parameterizations. Both models have the same vertical resolution. In a set of preliminary experiments using the same background field and the same set of observations, it is shown that the weak constraint imposed only on the low-resolution increments manages to control efficiently the emergence of fast oscillations in the resulting high-resolution forecast while maintaining a closer fit to the observations than is possible if the digital filter initialization is applied externally on the final analysis increments. It is also shown that this weak constraint does not add any significant computer cost to the 4DVAR analysis. Finally, 4DVAR has been cycled over a period of 2 weeks and the results show that, compared to 3DVAR, the initial dynamical imbalances are significantly less in 4DVAR even if no constraint is imposed at all. However, it has been noted that the innovation statistics show a positive impact when a constraint is applied.

Maximally Robust Input Preconditioning for Residual Vibration Suppression Using Low-Pass FIR Digital Filters

A method for suppressing residual vibrations in flexible systems is presented and experimentally demonstrated. The proposed method is based on the preconditioning of the inputs to the system using low-pass Finite Impulse Response (FIR) digital filters. Provided that the cutoff frequency of FIR filters is selected lower than the lowest expected natural frequency of the system and their stop-band is maximized, we show that these filters can be designed to exhibit maximally robust behavior with respect to changes of the system natural frequencies. To perform the proper design of FIR filters for robust vibration suppression, this paper introduces a series of dimensionless performance indexes and the Delay-Error-Order (DEO) curves that represent graphically the delay time introduced by the filter as a function of the remaining residual vibrations, and the filter order. Several classes of FIR filters such as: a) Parks-McClellan; b) Window-based methods (using Chebyshev window); and c) Constrained Least Squares method, are shown to present maximally robust behavior, almost identical to the theoretically predicted. Parallel, they demonstrate excellent vibration suppression while they introduce the minimum possible delay. Further advantages offered by the proposed method, is that no modeling of the flexible system is required, the method can be used in a variety of systems exhibiting vibrations, it is independent of the guidance function and it is simple to implement in practical applications. The results are experimentally verified on a flexible aluminum beam with a significantly varying mass, attached to the end-effector of a robot manipulator. The beam is rotated, using one joint of the manipulator, from an initial to a final position. It is shown that the preconditioned inputs to the flexible system induce very little amount of residual vibrations compared to the inputs with no preconditioning.

Optimal amplitude shading for arrays of irregularly spaced or non-coplanar elements

The majority of optimal amplitude shading methods for arrays of irregularly spaced or non-coplanar elements rely on numerical optimizations and iterative techniques to compute the desired weighting function because analytic solutions generally do not exist. Optimality is meant here in the Dolph–Chebyshev sense to provide the narrowest mainlobe width for a given sidelobe level. A simpler and more efficient technique to compute the shading weights for arbitrary line array shapes or element spacings is presented and it is shown that it is sufficient to sample the optimal Dolph–Chebyshev window, computed for a uniform line array of equivalent aperture length, at the element position of the nonuniform array. Examples are given for narrow-band plane-wave beamforming with curved arrays in which phase compensation is achieved by projecting the elements on a line tangent to the array. For the same mainlobe width, the resulting peak sidelobe levels are within 3 dB of a 30-dB Dolph–Chebyshev weighted uniform line array of equal aperture length and number of elements. Results are presented for computer simulations and for data collected at sea with the Toroidal Volume Search Sonar by the Coastal Systems Station, Panama City, Florida. [Work sponsored by ONR-NRL (Contract No. N00014-96-1-G913).]

Note the window function with nearly minimum sidelobe energy

It is shown that a previously published class of window functions is complementary to the Chebyshev window function. The impulse response coefficients are expressed in closed form. Chebyshev window functions of the first and second kind are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Earth-tide observations made during the international geophysical year

Results of world-wide earth-tide observations made by the Institute of Geophysics during the IGY are presented. The observations were made at 13 stations distributed around the world with two LaCoste and Romberg tidal gravity meters, each observation lasting 30 to 40 days. The data were analyzed by a method consisting of (a) a power spectrum analysis to determine the general noise background in the spectrum and (b) Fourier scanning of the major tidal lines to determine the amplitudes and phases of the gravity variation at each site as compared with the variation computed for a perfectly rigid earth. A Chebyshev window was used in the Fourier scanning to reduce contamination of the estimates by neighboring lines. Knowledge of the noise levels permits error limits to be assigned to the determination of the amplitude and phase relationships. The results obtained at the various stations differ significantly in relation to the estimated error limits and may be summarized as follows in comparison with theoretical results computed for various earth models. The amplitude ratio of the static tide on a homogeneous incompressible fluid earth to that on a rigid earth would be 1.25. For a semidiurnal tide on model earths of the Bullard I, II, Bullen B, and Gutenberg types, this ratio would have the values 1.161, 1.150, 1.162, and about 1.156, respectively. Our maximum observed value for the largest semidiurnal tide, M2, is 1.249 (Bermuda) and the minimum is 1.153 (New Delhi). The mean for the eight oceanic stations is 1.192, that for the four continental stations is 1.169, and the mean for all stations is 1.184. Thus our continental value is about 1 per cent greater than those computed for the Bullen B or Gutenberg models, but our mean oceanic value is 3 per cent greater. The causes of these variations require further investigation. It is believed that both the oceanic tides and regional geologic differences contribute to the variations. The O1 tide, whose period differs significantly from the solar day, also has a mean amplitude ratio of 1.18, but the P1 and K1 combination (whose period is 24 hours) gives the significantly different ratio 1.14 and is possibly influenced by the daily temperature cycle. The observed average phase lag of the M2 tide is compatible with the astronomical evidence concerning the changing rotation rate of the earth. However, the phases exhibit rather large local variations, the causes of which need further examination.