Prescribed Frequency Band Research Articles

An Experimental Analysis of an Active Elastic Metamaterial

Abstract Elastic and acoustic metamaterials provide an approach to creating materials which possess properties which are not found in natural or conventional materials. This notably includes dynamic effective material properties, i.e. density and modulus, which can become negative in prescribed frequency bands. Elastic and acoustic metamaterials provide alternative solutions to existing problems in acoustics and provide a route to realizing novel new concepts. Novel concepts associated with metamaterials include acoustic invisibility and sub-wavelength imaging. Elastic metamaterials with negative effective parameters can be realized using arrays of local resonators. However the performance of these devices suffers due to a fixed and usually narrow frequency band in which the effective parameters are negative. Active metamaterials provide an alternative approach in which control forces applied to a transmission medium emulate the desired response. Such devices have the benefit of being able to realize a response which is both tunable and not possible with passive components alone. This paper focuses on the development of an experimental demonstration of active elastic metamaterials with negative mass or stiffness across desired frequency bands. The active elastic metamaterial is a one-dimensional structure composed of a series of constrained lumped masses to which control forces are applied through a feedback control. The dynamic effective parameters are extracted from the measured data using a least squares approach to fit the data to an effective system. Comparison of the measured effective parameters to the simulated results verifies the effective response of the active elastic metamaterial in experimentation.

Non-uniform PCB traces with prescribed frequency bands for improved crosstalk immunity

In this work, a new design approach for realizing printed circuit board (PCB) trace configuration with continuous impedance perturbation to reduce crosstalk in PCB interconnects is investigated. The purpose is to reduce far-end crosstalk without incorporating additional PCB components in order to modify the reactive couplings between the adjacent traces, however at the expense of the available frequency band. This is achieved by replacing a uniform victim trace on a PCB with a continuously varying-impedance trace that exhibits a lowpass filter frequency characteristic. A simplified design approach of the proposed trace impedance profile which is governed by a truncated fourier series is presented. The proposed trace shows better crosstalk reduction results compared to conventional intervening guard trace schemes. Several configurations were designed, implemented, and tested to demonstrate the advantages of the underlying principle.

Performance and stability analysis of active elastic metamaterials with a tunable double negative response

Metamaterials can possess material parameters which do not exist in conventional materials. Consequently metamaterials provide a novel way to control wave propagation within a structure. These characteristics are achieved by designing a material with a particular sub-wavelength structure which leads to negative constitutive parameters in the long wavelength limit. For elastic materials these parameters are the density and modulus. In a previous study, a theoretical design for a novel active elastic metamaterial (AEM) was proposed. In this material control forces are applied to an array of resonant units in order to achieve a simultaneously negative effective density and modulus over a prescribed frequency band. This design potentially overcomes some of the restrictions imposed by previous passive designs. In this paper a new design of AEM is proposed which compensates for actuator dynamics and provides a basis for practical implementation. This design is shown to have a stable response with a tunable double negative frequency band.

Optimization of multiple-layer microperforated panels by simulated annealing

Sound absorption by microperforated panels (MPP) has received increasing attention the past years as an alternative to conventional porous absorbers in applications with special cleanliness and health requirements. The absorption curve of an MPP depends on four parameters: the holes diameter, the panel thickness, the perforation ratio, and the thickness of the air cavity between the panel and an impervious wall. It is possible to find a proper combination of these parameters that provides an MPP absorbing in one octave band or two, within the frequency range of interest for noise control applications. However, when a wider absorption frequency band is required, it is necessary to design multiple-layer MPP (ML-MPP). The design of an N-layers MPP depends on 4 N parameters. Consequently, the tuning of an optimal ML-MPP by exhaustive search within a prescribed frequency band becomes impractical. Therefore, simulated annealing is proposed in this paper as a tool to solve the optimization problem of finding the best combination of the constitutive parameters of an ML-MPP providing the maximum average absorption within a prescribed frequency band.

SYNTHESIS OF MULTI-STEP COPLANAR WAVEGUIDE-TO-MICROSTRIP TRANSITION

A synthesis procedure is developed in this paper for the design of N-step coplanar waveguide-to-microstrip transitions. An equivalent circuit approach is adopted to model the structure in terms of N cascaded ABCD matrices relative to the N coplanar waveguide sections forming the transition. A constrained optimization problem is formulated as the minimum flnding of a proper functional to accurately determine the transition dimensions by imposing an upper bound to the return loss within a prescribed frequency band. An iterative N- step procedure is developed to flnd the optimization problem solution. Numerical results on millimeter-wave transition conflgurations are provided to demonstrate the efiectiveness of the proposed synthesis method. A back-to-back transition prototype with N = 3 sections is then fabricated and characterized in terms of measured S-parameters to experimentally demonstrate a return loss better than 10dB in the frequency range from 1GHz up to 40GHz.

On the multi-frequency obstacle reconstruction via the linear sampling method

This paper investigates the possibility of multi-frequency reconstruction of sound-soft and penetrable obstacles via the linear sampling method involving either far-field or near-field observations of the scattered field. On establishing a suitable approximate solution to the linear sampling equation and making an assumption of continuous frequency sweep, two possible choices for a cumulative multi-frequency indicator function of the scatterer's support are proposed. The first alternative, termed the ‘serial’ indicator, is taken as a natural extension of its monochromatic companion in the sense that its computation entails space-frequency (as opposed to space) L2-norm of a solution to the linear sampling equation. Under a set of assumptions that include experimental observations down to zero frequency and compact frequency support of the wavelet used to illuminate the obstacle, this indicator function is further related to its time-domain counterpart. As a second possibility, the so-called parallel indicator is alternatively proposed as an L2-norm, in the frequency domain, of the monochromatic indicator function. On the basis of a perturbation analysis which demonstrates that the monochromatic solution of the linear sampling equation behaves as O(|k2 − k2∗|−m), m ⩾ 1 in the neighborhood of an isolated eigenvalue, k2∗, of the associated interior (Dirichlet or transmission) problem, it is found that the ‘serial’ indicator is unable to distinguish the interior from the exterior of a scatterer in situations when the prescribed frequency band traverses at least one such eigenvalue. In contrast the ‘parallel’ indicator is, due to its particular structure, shown to be insensitive to the presence of pertinent interior eigenvalues (unknown beforehand and typically belonging to a countable set), and thus to be robust in a generic scattering configuration. A set of numerical results, including both ‘fine’ and ‘coarse’ frequency sampling, is included to illustrate the performance of the competing (multi-frequency) indicator functions, demonstrating behavior that is consistent with the theoretical results.

Sound absorption of single- and double-layer microperforated insertion units.

Sound absorption by microperforated panels (MPPs) has received increasing attention recently as an alternative in applications with special cleanliness and health requirements. Since the hole diameter-to-panel thickness ratio must be close to unity, MPPs result limp, and therefore difficult to handle for practical applications. Microperforated insertion units (MIUs) were proposed recently to solve this trouble. They combine a millimetric panel with a micrometric mesh to provide an absorption curve similar to that of an MPP. The disadvantage of an MIU, as compared with an MPP, is that its performance depends on the tuning of seven parameters. Furthermore, when a double-layer absorber is required to increase the absorption frequency band, the number of constitutive parameters to be tuned becomes the design rather arduous. Therefore, it could be attractive to know the combination of constitutive parameters, within given variation ranges, which provides the maximum mean absorption for a prescribed frequency band. This is a typical optimization problem. Thus, this paper proposes the application of simulated annealing for optimizing the mean absorption of single- and double-layer MIUs within given variation ranges of their constitutive parameters.

Chebishev Bandpass Spatial Filter Composed of Strip Gratings

A new filter composed of a cascade of strip gratings with a 3rd order Chebishev frequency response that operates within a prescribed frequency band, is presented in this paper. Theoretical development was undertaken to establish equivalence between electromagnetic characteristics of a frequency selective surface (FSS) configuration and equivalent model based on circuit parameters that are used in the realization of Chebishev frequency response. The FSSs replace the impedance inverter elements in circuit implementation of the filter and the spacing between FSSs, which are adjusted to obtain the desired electrical performance, correspond to resonant elements in the circuit implementation. Dielectric slabs similar to the ones that were employed in FSS stages were used as the spacers between the constituent FSS layers to increase the robustness and rigidity of the structure. The fabrication process is described in detail and methods are introduced to account for tolerances in the permittivity and etching process during the design stage. Close agreement between measurement results demonstrates the validity of the design procedure.

Optimization of natural frequencies of bidirectional functionally graded beams

We propose a methodology to optimize the natural frequencies of functionally graded structures by tailoring their material distribution. The element-free Galerkin method is used to analyze the two-dimensional steady-state free and forced vibration of functionally graded beams. To optimize the material composition, the spatial distribution of volume fractions of the material constituents is defined using piecewise bicubic interpolation of volume fraction values that are specified at a finite number of grid points. Subsequently, we use a real-coded genetic algorithm to optimize the volume fraction distribution for three model problems. In the first problem, we seek material distributions that maximize each of the first three natural frequencies of a functionally graded beam. The goal of the second model problem is to minimize the mass of a functionally graded beam while constraining its natural frequencies to lie outside certain prescribed frequency bands. The last problem aims to minimize the mass of a functionally graded beam by simultaneously optimizing its thickness and material distribution such that the fundamental frequency is greater than a prescribed value.

The Wave Annihilation Technique and the Design of Nonreflective Coatings

We develop theory and algorithms for the design of coatings which either eliminate or enhance reflection of waves from surfaces. For steady-state harmonic waves with continuous frequency spectrum that covers an arbitrarily prescribed frequency band, coatings are designed that essentially eliminate reflections of all frequencies within the band. Although we focus on acoustic waves in elastic media, the methods developed here can be adapted to electromagnetism or other phenomena governed by variants of the linear wave equation.To create a nonreflective coating which is to operate in a frequency band $[\Omega_0, \Omega_1]$, we select n frequencies, uniformly distributed in the band, and design an n-layer coating of given thickness that completely eliminates reflections of waves at these frequencies. We show that if n is large, then the reflectivity of the coating designed by our method is small for all frequencies in the band. More precisely, the reflectivity at an arbitrary frequency $\omega \in (\Omega_0, ...

Design of broadband phasing network by real frequencytechnique

The application of the real frequency technique in broadband phasing network design is proposed. The technique uses real frequency load data over any prescribed frequency band for the design of a phasing network. The advantage is that no circuit topology or analytic form of the transfer function is required.

Design criteria for multistage microwave amplifier with match requirements at input and output

An approach to the design of multistage microwave amplifiers in a prescribed frequency band with requirements about the transducer gain flatness and the maximum magnitude of the reflection coefficient at input and output is presented. The interstage equalizers are designed by imposing a suitable constraint on the maximum transducer gain, obtainable directly from the specifications; the input and output equalizer are obtained by imposing only the matching requirement. The method proposed allows a separate design of each network, which can be performed either through direct optimization or by means of a numerical synthesis. >

A photonic variable RF delay line for phased array antennas

It is shown that by applying spatial frequency-dependent phase compensation in an optical heterodyne process a variable RF delay line can be synthesized over a prescribed frequency band. Experimental results which demonstrate the performance of the delay line with regard to both maximum delay and resolution over a broad bandwidth are presented. A spatially integrated optical system is proposed for control of phased array antennas. The integrated system provides mechanical stability essentially eliminates the drift problems associated with freespace optical systems, and can provide high-packing density. The approach uses a class of spatial light modulator known as a deformable mirror device and leads to a steerable arbitrary antenna radiation pattern of the true time-delay type. >

The absolutely stable Nyquist-stable non-linear feedback system design

Abstract A synthesis procedure is developed for a Feedback system consisting of two non-Jinear links : a non-dynamic link v in the plant and a dynamic one—non-linear corrector— consisting of a linear link looped by the link v or by parallel connection of the link — v and a unitary one. Global stability of the system is guaranteed by the absolute stability criterion. It is shown that in such a system the permissible phase shift, of the loop transmission function and then the ultimate achievable feedback in the prescribed frequency band are much larger than those in the conventional absolutely stable single-loop system. The method described is applicable to the band-pass and the low-pass system. Even under the more restricted limitation on the output process (absolutely stable), the benefit in the feedback in the proposed system is remarkable. Examples are included to illustrate the procedure.

Optimum pulse shape for minimum spectral occupancy in FSK signals

The characteristics of a frequency shift keying (FSK) signal giving rise to maximum power within a prescribed frequency band are determined. The baseband pulse shape extends over any integer number of bit periods. An integral equation for the optimum pulse shape is derived and some simple properties of the solution are investigated. The equation has been solved for some values of the product of the prescribed band times the pulse duration and the signal shape is shown. Our results for the particular case of pulses only one bit period long are presented for purpose of comparison with results known in the literature. Plots of out-of-band power for some values of the prescribed band and of pulse duration are reported. Finally a comparison with the well-known TFM, Nyquist, and raised cosine signals in terms of psdf and out-of-band power is exhibited.

Experimental study for the globally-stable Nyquist-stable 2-channel feedback system

In order to check the theoretical results of a previous study (Lurie 1980 a) a 16-order 2-ehannel non-linear feedback system was designed and tested. The average slope of the feedback cut-off was — 20 dB/octave. The system was unconditionally (globally) stable. In the prescribed frequency band the attained feedback was 30 dB greater than the theoretical limit for a single-loop system. A good agreement with the theory of the previous study was achieved It was found that (1) in order for the advantage over ordinary single-loop systems to be appreciable, the high-order linear part must be used ; (2) the large discrepancies in the characteristics of the non-linear links are admissible ; (3) the high-frequency noise components are damped significantly ; (4) output response to a step input is good for both small- and high-level signals to the input. No unexpected troubles were encountered during the experiments and the system was easy to handle.

The prolate filter: An ideal lowpass filter with optimum step-response

An ideal lowpass filter permits no signal transmission outside of a prescribed frequency band centered around the origin. In this paper, it is shown that the ideal lowpass filter which provides the fastest monotonic step-response for a prescribed bandwidth is the prolate filter. The system-function of this circuit is obtained from the autocorrelation of the zero-order, zero degree angular prolate spheroidal wave function. The bandwidth and risetimes of the filter are related through the largest eigenvalue of the same wave function. The computation and realization of the optimum frequency- and time-responses is discussed in detail and is illustrated by a numerical example. It is also shown how the asymptotic forms of the optimum system function degenerate, for very small or very large risetimes, into the well-known Hadamard and Gaussian filter functions, respectively.

Synthesis of Transfer Functions in a Prescribed Frequency Band

A signal-processing system for synthesizing complicated transfer functionisn a prescribed frequency band is described. The system consists of a multiplier followed by a closed loop containing a delay line and phase modulator in series. One of the inputs to the multiplier is the signal to be filtered. The second input to the multiplier is periodic, the period being equal to the loop delay. One cycle of the periodic waveform is identicatlo the real part of the transfer function overa prescribed frequency band of width equatlo the reciprocal of the loop delay. The imaginary componeonf t the transfer functionis the negative of the Hilbert transform of the real part, as in all physically realizable filters. Two applications of the system are discussed. It is shown how a continuously variable delay line and chirp filter can be synthesized using these techniques.

Synthesis of Multiple Antireflection Films over a Prescribed Frequency Band

The synthesis (as opposed to analysis) of multilayer dielectric films as antireflection coatings over any specified frequency band has not been attempted before. The similarity with transmission lines, which can be synthesized as multisection quarter-wave transformers to minimize reflection over any prescribed frequency band, is utilized by applying recently developed synthesis procedures for quarter-wave transformers to the synthesis of multiple antireflection films. [ See O. S. Heavens , Optical Properties of Thin Solid Films ( Academic Press, Inc., New York, 1955); and P. J. Leurgans , J. Opt. Soc. Am.41, 714 ( 1951).] A brief review of network synthesis is given, leading up to the synthesis of quarter-wave transformers and multilayer films. Numerical tables are presented in this paper from which antireflection coatings of up to four layers can be designed by interpolation. The design procedure (synthesis) described herein leads to the best possible antireflection films, but demands precisely controlled materials with certain refractive indices. It is hoped that this paper will help to promote the development of such materials, and will encourage opticists to try experimentally for the optimum performances possible in theory.

Multi-Element Directional Couplers

It is shown that the backward wave in a directional coupler is related to the shape of the function describing the coupling between tansmission lines by the Fourier transform. This facilitates the design of directional couplers for arbitrary directivities over any prescribed frequency band. Tightly coupled directional couplers are analyzed in simple terms, and it is shown that any desired loss ratio, including complete power transfer between lines, may be achieved. The theories are verified using waveguide models operating at 4,000, 24,000, and 48,000 mc, and it is indicated that the work is applicable to many types of electrical and acoustic trnsmission lines.