Polynomial Transfer Research Articles

Analog synthesizer of orthogonal signals

A new adaptive architecture, an analog synthesizer of orthogonal signals which is able to produce a plurality of mutually orthogonal signals as functions of time, such as Legendre polynomials, cosine basis of functions, smoothed-cosine basis, etc., is proposed and analyzed. A method for the creation of analog devices with Chebyshev polynomials and Legendre polynomial transfer functions is proposed. A proof-of-concept breadboard version of the analog synthesizer is described. The results of smoothed-cosine basis synthesis, as well as characteristics of output functions, are then presented. The device is characterized by a very fast and stable process of signals synthesis.

Blind identification of MIMO-FIR systems: A generalized linear prediction approach

We address the blind identification of multiple input multiple output (MIMO) polynomial transfer functions with the number of inputs strictly less than the number of outputs. According to previous results, the knowledge of the second-order statistics of the output series allows to reduce the convolutive mixture to an instantaneous mixture of the inputs under mild assumptions. Our approach consists of second-order deconvolution via the so-called linear prediction approach with the possibility of further instantaneous mixture separation. Contrary to the standard linear prediction, we propose a statistically optimal least-squares identification procedure. The latter reveals a simple sub-optimal approach characterized by high estimation accuracy as well as robustness to the model order choice. The statistical inference about the asymptotic performance of these methods is validated by computer simulations.

A noniterative polynomial 2-D calibration method implemented in a microcontroller

Output signal handling of a smart sensor usually involves a calibration facility to correct for input-output nonidealities which comprise offset, gain errors, nonlinearity errors, and cross sensitivity. In this paper, a calibration method is presented which features a progressive improvement in the sensor calibrated transfer toward the desired transfer as the user proceeds from one calibration point to the next. The method is based on a set of mathematical formulas whereby a calibration coefficient is calculated at a selected calibration point and used to calculate a first correction of the sensor transfer curve. Further improvement in the sensor transfer is obtained by repeating the process for a second calibration point using the transfer resulting from the first calibration, without the need to review the calibration already carried out at the first point. The process can be repeated until the desired error reduction is obtained. Step by step, the calibration method builds up a polynomial transfer correction. Simulation results are shown to demonstrate the performance of the method, A software implementation of the method for two-dimensional (2-D) calibration in an 8-bit microcontroller is presented. The microcontroller with calibration program can be incorporated in a compact smart sensor system.

Loudspeaker parameter measurement in reverberant environments

Loudspeaker driver small-signal parameter measurement is required to design and quantitatively measure loudspeaker enclosures. Such measurements are conventionally performed in an anechoic chamber with the intent of eliminating the acoustic response of the room from influencing the measurement. A novel procedure is proposed to perform the measurements in arbitrary untreated enclosures using a standard sound card. Aoshima’s optimal time stretched pulse [Aoshima, J. Acoust. Soc. Am. (1981)] [Suzuki et al., J. Acoust. Soc. Am. (1995)] is used to excite the loudspeaker. The response is captured by a high-quality flat response microphone. Knowledge of the minimum first reflection time of the enclosure is used to filter out the effects of room response without affecting the measurement of loudspeaker response. A sliding window is used to remove reverberent effects similar to time delay spectrometry (TDS) [Heyser, J. Audio Eng. Soc. (1967)]. The filtered signal is fit against a complex rational polynomial transfer function, inspired by the method described by Leach et al. [IEEE Trans. ASSP Dec. (1979)] and Jain et al. [IEEE Trans. ASSP Feb. (1983)]. The fit is performed by a nonlinear least-squares method written in AMPL [Fourer et al., 1993], a language for mathematical programming.

Multivariable system identification via continued-fraction approximation

This paper presents theory for multivariable system identification using matrix fraction descriptions and the matrix continued fraction description approach which, in turn, yields a lattice-type order-recursive structure. Once the matrix continued-fraction expansion has been determined, it is straightforward to obtain solutions to both the left and right coprime factorizations of transfer function estimates and, in addition, a solution to problems of state estimation (observer design) and pole-assignment control. An important and attractive technical property is that calculation of transfer functions on the form of right and left coprime factorizations; calculation of state variable observers,and regulators all can be made using causal polynomial transfer functions defined by means of matrix sequences of the continued-fraction expansion applied in causal and stable forward-order and backward-order recursions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Tutorial on the digraph approach to multivariable control : J. Jantzen, pp 711–714

Abstract The digraph approach is a graphical approach to analysis and design of control systems. Its objective is to reveal generic properties of state space models. The approach is based on a mapping of the state space equations into a directed graph and system properties are expressed as properties of the graph. The structural concepts of path, cycle , and cycle family underlie familiar concepts like controllability characteristic polynomial transfer functions and pole assignability. the digraph approach is easy to learn because it is inherently graphical.

Tutorial on the Digraph Approach to Multivariable Control

The digraph approach is a graphical approach to analysis and design of control systems. Its objective is to reveal generic properties of state space models. The approach is based on a mapping of the state space equations into a directed graph and system properties are expressed as properties of the graph. The structural concepts of path, cycle, and cycle family underlie familiar concepts like controllability characteristic polynomial transfer functions and pole assignability. the digraph approach is easy to learn because it is inherently graphical.

A chebyshev rational function with low Q‐factors

AbstractThe presented rational function is a modification of a recently published Chebyshev rational function defined by means of some orthogonal polynomials. the necessary conditions providing for the lowest pole Q‐factors for a given ripple are found. the function is a ratio of two similar Chebyshev polynomial transfer functions with multiple poles.The selectivity of the function can be increased by using the Chebyshev rational characteristic function instead of the characteristic polynomials. the minimum number of active elements in the cascade connection is obtained with third‐order elliptic characteristic functions.The function is compared with the Cauer (elliptic) and MCPER filters. the distinctive features of the presented function are small Q‐factors of the poles, almost ideal dynamic range, simple design and poles with the same multiplicity m, where m designates the number of cascaded blocks.

Simulation and modeling-the macro modeling of phase locked loops for the SPICE simulator

A modeling approach that uses no semiconductor junctions and no regenerative feedback is described. The voltage-controlled oscillator is implemented by means of a circuit producing essentially infinite ringing, and the comparator characteristics are obtained by means of an odd-order polynomial transfer function. The design of the comparator, phase detector, low-pass filter, and error amplifier are also discussed. Performance of the complete macro-model is such that 8000-point simulations, corresponding to some 150 cycles of local oscillator, can be performed in about 15 min on a platform such as the Sun 3/60, or Apollo DN4500.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Frequency Domain Techniques for Modeling Distribution or Transmission Networks Using Capacitor Switching Induced Transients

This paper presents two frequency domain methods to model and identify the equivalent impedance transfer function of a distribution or transmission system using recorded voltage and current transients due to capacitor switching. The first frequency domain method discussed was the fast Fourier transform method of identifying the frequency components of the voltage and current, then using the voltage and current spectra to find the impedance transfer function. The second method first obtains estimates of the sample auto and cross-correlation functions. These functions are then used, via the FFT, to find the respective power spectra of the signals, and finally to find the impedance transfer function. The FFT method was found to produce accurate results when the frequency components signal to noise ratios are large. The power spectral identification method and the coherence function added a measure to reject the frequency components that are contaminated with noise, but this method suffers from spurious cross correlations between the voltage and current. Once the impedance transfer function has been identified, several techniques were investigated that fit a parametric model to the function. Two polynomial transfer functions were fit to the real and imaginary parts of the impedance. These polynomials resulted in accurate representations of the impedance function of the system. The second parametric method discussed used the poles and zeroes identified from the impedance transfer function to develop an equivalent circuit representation of the system. This equivalent circuit did not fit the estimated impedance/frequency characteristics as well as the polynomial transfer function.

Confidence limits of identified frequency responses of multivariable systems

When identifying rational polynomial transfer functions from input-output data, estimates of the coefficients and their covariances can be obtained. However, the estimated value of any given coefficient depends considerably on the chosen model order, leading to segmented confidence regions when the model order is not known. Fortunately the frequency responses of the transfer function models are little affected by using too high a model order. These are more likely to have a consistent estimate with a single surrounding confidence region, In this paper, the confidence limits established for the frequency response of single-input-single-output systems by Edmunds are extended to multivariable systems. A statistical lest of the method is then carried out using an illustrative example.

A simple method for the estimation of rational distributed lag models

In an earlier paper we suggested a method for the identification and estimation of linear transfer function models. The method was claimed to be especially suitable for polynomial transfer function models. In this paper we shall consider the case of rational transfer function models (distributed lag models) in more detail. A simple method for the estimation of the parameters of multiple input rational distributed lag models is suggested. The method is based on simple linear identities that the parameters always fulfill. The asymptotic distribution of the proposed estimator is derived. Two illustrative examples of the use of the new method are given.

A new method for digital all-pass filter design

A new method for the design of digital all-pass filters using Chebyshev criterion is introduced. It is based on a phase approximation algorithm for polynomial transfer functions. The algorithm exploits a scheme of iteratively linearizing the nonlinear constraints in a nonlinear programming and converges theoretically. The design method has the advantage that it finds the best uniform phase approximation to an arbitrarily specified phase response without any initial guess of the solution. Design examples of orders up to 80, obtained on an IBM-PC/XT personal computer, are given to show the practicability of the method.

Radiometric correction of SAR images A new correction algorithm

Abstract Spatial variations of the backscatter coefficient result from synthetic aperture radar imaging systems and their platforms. An adaptive algorithm has been developed to correct the multiplicative variation of the backscatter in the longitudinal (parallel to flight line) and lateral (perpendicular to flight line) directions. The coefficient of variation along the parallel and lateral profiles, which consist of the means of the pixels along these lines perpendicular to the respective profiles, is constant. This implies that the radiometric variations are multiplicative. Because standard correction methods such as polynomial transfer functions give unsatisfactory results, an adaptive correction algorithm was developed to correct these images. The algorithm produces a transfer function by means of a filtering window which moves along the profile in what is essentially a moving mean procedure. However, the length of the window is automatically adjusted as a function of the variation of the profile. The adjustment is based on a calculation of the probability that values to be included or excluded from the window belong to the included population. The effects of the successive correction steps were monitored using several thematic test sites. The resultant images provide both increased quality and quantity of data without any degradation of the statistical properties of the data.

Asymptotical inversion of non-singular digital symmetrical polynomial transfer functions

A new asymptotical method for the inversion of discrete systems with non-singular symmetrical polynomial transfer functions is given. For both, the one- and two-dimensional cases, we place the method in a general framework and show that it can not be applied to general non-singular transfer functions.

Generalized state space realizations of non-proper rational transfer functions

In this paper we propose an algebraic realization theory for polynomial transfer functions based on classical module theory. Realizations are given by linear systems whose generalized state space has a natural module structure over the ring of formal power series K [ z −1 ]. Existence and uniqueness of minimal realizations are proved and their properties are described. The dimension of the minimal realization is called generalized degree of the polynomial transfer function and it is proved to be equal to the rank of an associated generalized Hankel matrix. Finally, a correspondence between realizations and fractional representations over K [ z −1 ]of polynomial matrices is established and some properties of such representations are investigated.

AR and ARMA identification algorithms of Levinson type: An innovations approach

Fast recursive-in-time identification procedures for both AR and ARMA processes (e.g., Chandrasekhar, square root algorithms,... ) have been available for a few years. These algorithms realize the desired transfer functions in the classical polynomial or rational form. On the other hand, the synthesis of polynomial and rational transfer functions in lattice and ladder form has fostered great interest in network theory by virtue of its pleasant properties. This type of synthesis is strongly related to the theory of orthogonal polynomials on the unit circle. An identification procedure with the realization of the desired whitening filter in lattice form was available for AR processes. We give here a simple approach for obtaining such algorithms, investigating furthermore the connections between the so obtained algorithms and the classical ones (least squares procedure). In the same way, we obtain identification procedures with realization of the desired filter in lattice and ladder form for ARMA processes, together with the connection with the classical extended least squares procedure. The method is based upon a fairly general Levinson orthogonalization lemma in Hilbert space, involving innovation techniques. We extend the method to various other estimation problems. The algorithms we obtain are fast (even somewhat faster than the previous fast ones) and seem to be well conditioned.

Parametric time domain analysis of the multiple input/scalar output problem: The source identification problem

A multiple input/scalar output stationary time series identification problem is considered from a parametric model time domain point of view. Particular emphasis is on the source identification problem. Closed form formula estimates of the individual source power contributions are expressed in terms of sample correlations that are obtained from the observed input and output time series and from parametric models fitted to that data. The estimates of the noise power contributions are asymptotically jointly normally distributed. The mean values and covariance matrix of those estimates yield confidence interval estimates of the individual and joint power contributions. The motivation for developing a rational polynomial transfer function or ARMA model of the multi-input scalar output plus additive noise situation is given. A two correlated input/single output version of this model is considered for a Monte Carlo simulation study. Parametric ARMA and approximate AR models are fitted to the simulated data. The asymptotic normality, and the distribution of the mean and covariances of the source power contribution computed from the ARMA and AR models are appraised. Several facets of the relative performance of windowed periodogram and AR model spectral analysis are examined for the multiple input/scalar output identification problem. The points that are emphasized are that conventional windowed periodogram spectral analysis is subjective, not particularly satisfactory for the sharp spectral peak situation that is commonly encountered in vibration data analysis and very likely not as good as “objective” Akaike criterion order AR modelled spectral analysis.

Recurrent Continuant Synthesis of Maximally Flat Time Delay Filters

A new synthesis technique is applied to the synthesis of maximally flat time delay filters. Only the low pass case is considered as the other types may be derived from this by simple frequency transformations. It is shown that the recurrent determinant of a polynomial transfer function with a constant term other than unity can only be obtained in a modified form, being multiplied by the constant term.

The simulation of wave filters having polynomial transfer functions on an analogue computer

The simulation of wave filters having polynomial transfer functions on an analogue computer