Frequency Response Error Research Articles

Minimax Design of Low-Complexity Allpass Variable Fractional-Delay Digital Filters

This paper proposes noniterative and iterative linear programming (LP) procedures for designing low-complexity allpass variable fractional-delay (VFD) digital filters in the minimax sense. Expressing each coefficient of an allpass VFD filter as a polynomial in the VFD parameter p, we show that the frequency response error of an allpass VFD filter can be written as a pure imaginary part divided by its denominator. Thus, the minimax design can be approximately formulated as an LP problem through neglecting the denominator, which leads to a noniterative minimax design. To improve the minimax design accuracy, we propose an iterative LP procedure that utilizes the denominator from the preceding iteration as a known. The iterative LP minimization is repeated until it converges to the minimax solution. Moreover, we also present a two-stage algorithm for optimizing the optimal variable range p ∈ [pMin, pMax] of the VFD parameter p and successively reducing the subfilter orders. Design examples are given to show that both noniterative and iterative LP methods can achieve much better minimax designs (smaller peak errors) than the existing iterative weighted-least-squares (WLS) approaches, which aim to minimize the peak errors of VFD response and variable phase response, respectively. Moreover, the resulting allpass VFD filters have lower complexities than those from the iterative WLS approaches.

Optimizing Capacitance Ratio Assignment for Low-Sensitivity SC Filter Implementation

Accurate capacitance matching is essential for switched-capacitor filters implementation because the filter coefficients depend upon the capacitance ratios. The use of identical unit capacitors in parallel to form larger capacitances and careful layout design can provide, in many cases, an accuracy of 0.1%. Unfortunately, this technique can be directly applied only if the filter coefficients are rational numbers. In general, coefficient approximations are required, leading to frequency response errors. In this paper, a new design method, using a genetic algorithm, is proposed to find the optimum capacitance ratio approximations by rational numbers which minimize the total number of unit capacitors for a given acceptable frequency response error, in order to save die area.

Using a Genetic Algorithm to Optimize Capacitance Ratio Approximations in SC Filters

Accurate capacitance matching is one of the main design issues in switched-capacitor (SC) filters. Using identical unit capacitors in parallel to implement each filter capacitor, combined with a careful layout design, can provide an accuracy of 0.1% in the filter coefficients. The disadvantage of this technique is the fact that it can be directly applied only if the coefficients can be expressed as ratios of integer numbers. As a result, coefficient approximations are required, leading to frequency response errors. In this paper, a genetic algorithm (GA) is used to find the optimum capacitance ratio approximations by rational numbers that minimize the total number of unit capacitors for a given error tolerance in the frequency response. Design examples in 0.35 μm CMOS are presented and simulated to illustrate the proposed approach and verify its effectiveness.

A New Criterion for the Design of Variable Fractional-Delay FIR Digital Filters

Conventionally, variable fractional-delay finite-impulse-response digital filters are generally designed by minimizing the root-mean-square error of variable frequency response. In this paper, a new criterion concerning the minimization of the root-mean-square error of variable group-delay response is proposed. However, minimization is a highly nonlinear problem, so an iterative method is proposed in this paper to overcome it. To further reduce the maximum absolute group-delay error in the least squares design, an iterative weighting-updated technique is also proposed, which constitutes the outer loop of the overall iterative process while the iteration stated earlier makes up the inner loop. Several design examples will be presented to demonstrate the effectiveness of the proposed method.

Generalized WLS Method for Designing All-Pass Variable Fractional-Delay Digital Filters

This paper presents an optimal weighted least squares (WLS) method for designing low-complexity all-pass variable fractional-delay (VFD) digital filters. Instead of using a fixed range for the VFD parameter p and same-order constant-coefficient filters (subfilters), both the VFD parameter range p isin [p Min,p Max] and subfilter orders are optimized such that a low-complexity all-pass VFD filter can be achieved for the LS design. To suppress the peak errors of variable frequency response, weighting functions are adopted and optimized such that the boundary peak errors can be further reduced but without noticeably increasing the total error energy (integral of squared error) of variable frequency response. After optimizing the variable range of the VFD parameter, weighting functions, and subfilter orders, an all-pass VFD filter can be designed by using a generalized noniterative WLS method, which yields a closed-form solution. Design examples are given to illustrate that utilizing different-order subfilters, along with the optimal range and optimal weighting functions, can yield an all-pass VFD filter with significantly reduced complexity and design errors as compared with existing ones.

Room mode bass absorption through combined diaphragmatic and Helmholtz resonance techniques

An acoustical bass absorber which reduces peak and dip frequency response errors caused by interference from naturally occurring axial standing waves in rectangular rooms. The apparatus uses two forms of simple harmonic resonance: pistonic diaphragm resonance and Helmholtz cavity resonance. The pistonic diaphragm resonance is achieved by attaching a rigid planar membrane to metal springs. The Helmholtz cavity resonance is achieved by constructing an enclosed chamber attached to an open cylindrical tube. Coupling these two dissipation devices leads to several-fold improvement in absorption and total room mode attenuation.

An iterative method for the design of variable fractional-order FIR differintegrators

In the paper, a new method is proposed for the design of variable fractional-order (VFO) FIR differintegrators. Comparing with the existing methods, the elements of relevant matrices can be determined just by the given specification, which makes the method easier. An iterative technique is also incorporated to adjust the weighting function, such that the peak absolute error of variable frequency response can be reduced drastically. Several design examples, including a VFO differintegrator, two VFO differentiators and a VFO integrator, are presented to demonstrate the effectiveness and flexibility of the proposed method.

Model Reduction Via Limited Frequency Interval Gramians

An improved frequency domain interval Gramian-based model reduction scheme for discrete time systems is presented. It is first shown that two of the main results presented in the model reduction method of are incorrect. Improved methods which overcomes these shortcomings are then presented. Improved methods not only yields stable reduced-order models but also have easily computable frequency response error bounds. The method is further extended to 2-D separable denominator system approximation. The simulation results show the effectiveness of the proposed scheme.

An approximation algorithm to improve capacitance matching in the design of SC filters

A critical issue in the design of switched-capacitor (SC) filters is the capacitance matching, because the filter coefficients depend on capacitance ratios. The most successful design method to achieve an accurate capacitance matching employs a parallel arrangement of identical unit capacitors to implement each filter capacitor. However, this procedure can be directly applied only if the filter coefficients can be written as rational numbers, since each capacitor is implemented as an integer number of unit capacitors in parallel. This paper presents a systematic procedure, with low computational effort, to approximate the filter coefficients by integer ratios causing acceptable errors in the filter frequency response, whereas keeping the total number of unit capacitors small, in order to save die area. This procedure was applied in the design of a sixth-order SC band-pass filter, which has been fabricated in a 0.35 μm CMOS technology. The fabricated filter occupies an area of 0.913 mm2, exhibits low sensitivity to fabrication process deviations and has an output dynamic range of 79.2 dB.

A broadband method to measure magnetic spectra of thin films

Conventional single-port strip-line techniques to measure permeability of thin films are based on the quasistationary approximation. In this paper, wave phenomena in the measurement cell are taken into account. Particularly, the dielectric response of the sample may be accounted for and removed by the consideration of the geometry of the cell and the sample. An additional measurement with a reference sample with known constitutive parameters allows the directivity error and the frequency response error to be removed from the experimental reflectivity data; accounting for the field inhomogeneity inside the strip cell is also introduced. A proper selection of the reference sample is shown to be critical for the measurement accuracy. The measurement results may also be affected by the demagnetization of the sample. With these corrections introduced, the operating frequency range of the technique is extended up to 8GHz. The measurement results are validated by the data obtained with either standard materials or by conventional techniques.

Constraints on a synthetic-noise source for Johnson noise thermometry

Conventional Johnson noise thermometers based on switching correlators have conflicting matching requirements for the sensing resistance. To mitigate distortion effects in the correlator, the RT products of the two sensors must be the same, and to mitigate frequency-response errors in nominally identical input circuits, the two sensing resistances should be the same. A noise thermometer using synthetic noise for the primary reference signal overcomes this conflict because the output voltage and output resistance are independent. This paper presents the rationale and design constraints for a noise thermometer using a synthetic-noise source based on Josephson junctions. The quantized voltage noise source developed at NIST produces tunable waveforms with a spectral density composed of a comb of frequencies of equal amplitude and random phase. In addition to the conventional noise-power and impedance constraints, it has additional constraints relating to the number of tones and the tone spacing.

Fractional Delay Filter Design Based on Truncated Lagrange Interpolation

A new design method for fractional delay filters based on truncating the impulse response of the Lagrange interpolation filter is presented. The truncated Lagrange fractional delay filter introduces a wider approximation bandwidth than the Lagrange filter. However, because of truncation, a ripple caused by the Gibbs phenomenon appears in the filter's frequency response. Proper choices of filter order and prototype filter order allow adjusting the overshoot to a desired level and simultaneously reducing the overall frequency-response error. The design of the proposed filter is computationally efficient, because it is based on polynomial formulas, which have common terms for all coefficients.

Optimal model reduction of non-causal systems via least squares frequency fitting

A new model reduction method, based on frequency fitting, is proposed for single-input discrete-time singular (non-causal) systems. The reduced-order models are obtained by minimising the least square frequency response error between the original system and the reduced-order model. Finally, the method is illustrated by a numerical example and is compared with other related techniques.

Correlation and Error Metrics for Plant Identification of On-Orbit Space Structures

In plant identification and modeling efforts, metrics are needed to quantitatively compare measured data to numeric models. Quantitative metrics broadly fall into five categories: performance metrics, correlation metrics, error metrics, data quality metrics, and model interpretation metrics. This paper describes the utility of frequency response function correlation and error metrics for plant identification, particularly for complex space structures. Three correlation metrics and three error metrics are discussed. Multiple-input, multiple-output extensions to the correlation metrics are presented and their application is demonstrated on data collected from spacecraftrepresentative structures.

Approximate LMS tuning of continuous time filters: convergence and sensitivity analysis

The quality factor of continuous time filters may be tuned using a master/slave arrangement in which an analogue implementation of the LMS algorithm is used to tune the master filter. Two such schemes are analysed, and their sensitivities to the effects of parasitic capacitances are determined theoretically and then compared. The two schemes are shown to have similar sensitivities to parasitically induced errors in the magnitude frequency response of the master filter, whereas only one of them is shown to be immune to phase errors in the master filter. The theoretical results are verified using a behavioural simulation. Approximate hardware costs of the two tuning schemes are also compared.

Frequency response and systematic errors in STI measurements

The paper examines two fundamental problems when measuring and verifying the STI of sound systems. These relate to the overall accuracy of the measurement platform in use and the fact that STI does not correctly account for system frequency response and spectral aberrations under certain conditions. As has previously been reported by the author, STI appears to significantly overestimate the intelligibility of limited bandwith sound systems under reverberant, high S/N ratio conditions. Further evidence and word score comparisons for a number of different frequency response conditions, measured under several reverberant situations are presented. The results indicate that errors of up to 30 comparisons with low-reverberant field and anechoic conditions are also made. It is also shown that the measurement platform itself and sound source characteristics can affect the resultant error. Tests, carried out under highly controlled conditions, are reported that enable the causes of the potential errors to be established. The implications of reported limitations in STI and STIPa, when assessing the potential intelligibility of a sound system or acoustics of a space, are discussed.

Design and parallel implementation of FIR digital filters with simultaneously variable magnitude and non-integer phase-delay

Variable fractional-delay digital filters are useful in various signal processing applications. To perform both fractional-delay filtering and signal frequency selecting, variable fractional-delay filters must also have variable magnitude characteristics. This paper proposes a method for designing variable finite-impulse response (FIR) filters with both variable magnitude and variable noninteger phase-delay. First, the coefficients of a variable FIR filter are expressed as different 2-variable polynomials of a pair of spectral parameters; one is for varying magnitude response, and the other is for varying noninteger phase-delay. Then the optimal coefficients of the 2-variable polynomials are found by minimizing the total weighted squared error of the variable frequency response. Since the coefficients of the obtained variable FIR filter are the polynomials of the two spectral parameters, we can yield variable magnitude and variable noninteger phase-delay simultaneously or independently by substituting different spectral parameter values to the 2-variable polynomial coefficients. Finally, we show that the resulting variable FIR filter can be implemented in a parallel form, which is suitable for high-speed signal processing.

Assessment of computational model updating procedures with regard to model validation

In this paper two different computational model updating (CMU) methods are compared. Both procedures make use of the inverse sensitivity approach. The residuals of the first method are formed by eigenvalue and mode shape differences whereas the residuals of the second method include eigenvalue and frequency response errors. Computational model updating of physical mass, stiffness and geometric parameters is possible with the first technique. The second method additionally allows updating of viscous modal damping parameters. This second method uses directly measured frequency response data taken only at the resonance peaks. Since both methods allow to handle incomplete test data vectors, that means that the number of measured degrees of freedom (dof) is much less than the dof no. of the mathematical model, they are able to deal with large order aerospace finite element (FE) models. Both procedures were used to investigate the prediction capability of an updated analytical model of a benchmark test structure with respect to the test data of a modified structure. The results of both techniques are compared, advantages and disadvantages are pointed out and an assessment with respect to the operational validity is given.

Frequency and Magnitude Response Design Approximation Problems for Nonlinear-phase Nonrecursive Digital Filters

In this article special (possibly constrained) problems of linear and nonlinear complex approximation are studied with respect to the existence and uniqueness of solutions and the convergence of the approximation errors, where the errors are measured by an arbitrary L p and l p norm respectively. The problems arise in connection with the frequency and magnitude response approximation at the design of nonrecursive digital filters in the frequency domain. Two main results of the article concern the completeness of the functions exp ( m ik y ), k = 0,1,2,… , with respect to a certain space of continuous functions. These results imply that, under usual assumptions and with increasing number of approximating functions exp ( m ik y ), the errors in the frequency and magnitude response approximation problems converge to zero when the design region is not the total interval [0, ~ ] (in case of real coefficients) and not [ m ~ , ~ ] (in case of complex coefficients) which is given for the majority of filter design problems, but that the frequency response errors may not converge to zero when the design region equals [0, ~ ] or [ m ~ , ~ ] respectively.

On an instantaneous frequency estimator with FIR filters having maximally flat frequency response error magnitude

In this paper an instantaneous frequency estimator (IFE) of a discrete-time base band complex signal is considered. The IFE is built around one-band, maximally flat linear-phase FIR filters, which are used for differentiating and delaying Cartesian components of the complex signal. One of the key features of the estimator is that it avoids problems related to the ambiguity of the instantaneous phase waveform. The quality of the estimator is tested. A closed-form formula for the static characteristic of the IFE is derived and expressed as a function of the frequency responses of the filters used. Two representative test signals: a full band complex linear frequency modulated (LFM) chirp and a three-component complex synthetic signal are used to demonstrate the characteristic features of the estimator. If the chirp is sufficiently long in comparison with the length of the filters, the instantaneous frequency (IF) estimation errors are comparable to those obtained by using the static characteristic. For this case, the IF estimation error plots for the practical versus ideal IFE are presented and a design chart showing the dependence of the IF estimation error magnitude on the input signal bandwidth and the FIR filters’ length is given. This chart can be exploited in, e.g., FM-telemetry applications, where the IF carries a very slowly changing telemetric message. The three-component signal chosen allows demonstration of the ability of the estimator to track the IF which extends beyond the signal spectral range, permitting measurement even beyond the Nyquist frequency. Finally, the power of the proposed IFE to measure the stability of highly precise frequency oscillators is shown.