Comments on "On the construction of a digital transfer function from its real part on unit circle"

This paper studies the generation of stable transfer functions for which the real or imaginary part takes prescribed values at discrete uniformly spaced points on the unit circle. Formulas bounding the error between a particular interpolating function and any function consistent with the data are presented, which have the desirable property that the error goes to zero exponentially fast with the number of interpolating points. The paper also examines generation of stable minimum phase transfer functions for which the magnitude takes prescribed values at uniformly spaced points on the unit circle, and presents error bounds for this problem. Connection with the discrete Hilbert transform is made. The effect of uncertainty in the original data is also examined.

Expressions are derived for the Cramer-Rao bounds (CRBs) of the parameters of an exponential model with one set of poles and multiple sets of amplitude coefficients. The poles of this model may lie anywhere in the complex plane. The CRB for pole angle is log-symmetric about the unit circle with minimum on the unit circle, while the CRB for pole magnitude has its minimum inside the unit circle for finite-length datasets and is not symmetric. Simulation results show that error-standard-deviation ellipses about poles turn out to be circular. The CRBs for estimates of the amplitude coefficients of real and imaginary parts are also available. >

In this paper, we present a directly design method of minimum phase FIR filters with complex coefficients using successive projections (SP) method. In general, it is necessary to restrict the zeros of transfer function inside the unit circle or on the unit circle to directly obtain the transfer function of minimum phase FIR filters. We use Rouche's theorem for coefficient update algorithm in SP method to restrict the zeros of the transfer function inside the unit circle and on the unit circle. Therefore, the transfer function of minimum phase FIR filter is obtained without factorization by using the proposed method. Moreover, the proposed method is also able to obtain the transfer function of real coefficients by considering the imaginary part of complex coefficients as the zero. The usefulness of the proposed method is verified through some examples.

The unstable burning of a bipropellant rocket combustion chamber is investigated and a study made of the requirements for an automatic closed loop control circuit to stabilize the motor. The bipropellant combustion chamber equations developed by Dr. L. Crocco(1) are utilised as the analytical description of the rocket motor burning phenomena. Equations similar to those developed by Dr. H. S. Taiga(2) are used for the oxidizer and fuel supply systems and the two closed loop stabilizing circuits. The stability or instability of the system is demonstrated by the use of a special plotting diagram in the complex plane suggested by M. Satche as a means of handling systems with time lag, and developed for this use by H. S. Tsien. This involves separating the transfer function into two parts. In the complex plane the first portion of the transfer function, the exponential variable containing the time lag, plots as a unit circle as the complex variable p is made to take a contour enclosing the positive half of the p—plane. If the remaining portion of the transfer function intersects this unit circle, the rocket motor can be unstable for large reduced time lag; if it does not intersect the unit circle, the system is generally stable, although the roots of the exponential coefficient in the positive half of the complex plane must be investigated. This latter requirement can be conveniently accomplished by the aid of a Nyquist Diagram. The equations for the feedback circuit are developed and the oxidizer and fuel transfer function requirements are determined. Two cases of stable combustion and two cases of unstable combustion are analyzed. One unstable case is stabilized by the addition of a feedback circuit.

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

We consider the problem of reconstructing a real sparse coefficient θ of a transfer function under the orthogonal rational functions from a limit number of linear measurements. Given m randomly selected samples of Φθ, where Φ is a sample matrix constructed by orthogonal rational functions at samples in the unit circle, we show that l 1 minimization can recover the coefficient θ from the real part model or the imaginary part model for the real and imaginary part of Φ have the similar structure of the orthogonal matrix.

Resume.-Solitary wave problem is transformed into the one of holomorphic function Q(') in a unit circle [ C [ < 1, whose real and imaginary parts D, .,. satisfy a certain nonlinear "surface condition" on the circumference of unit circle.This is almost identical formulation as that of deep water due to Levi-Civita.Then the highest wave is determined by numerical iteration procedure, whose one cycle consisting of two steps: the first the derivation of .,. from II by surface condition and the second the determination of 11, the harmonic conjugate function of r, Fourier series with 12 components being used as an available function for adjustment.Further improvement of approximation has been undertaken to satisfy the surface condition within a percent error.

The starting point is the autocorrelation function of the signal. We assume the latter to be the sum of a random component (noise) plus d trigonometric functions with incommensurate frequencies. The time is descretized and the signal has a finite length N. The autocorrelation is thus itself a set of N random numbers Ci, i = 0,…(N - 1); - periodic boundary conditions are assumed. Considered as Fourier coefficients, the Ci define a random measure dμ with support on the unit. One can then build by a recurrence relation the family of polynomials orthogonal with respect to this measure. These Szego polynomials are simply related to the transfer functions of a family of Wierner-Levinson filters. The n complex random zeros of the random orthogonal polynomial of degree n are the focus point of the method, (i) all the zeros are within the unit circle [1]. When n and N increase, they all have their mean distance D to the unit circle going to zero [2]. (ii) d pairs of complex conjugate zeros can be associated to the d trigonometric components of the signal [3]. In the mean their distance (D) to the unit circle goes to zeros like n –2 [4]. Asymptotically, the modulus of their phases are equal to the frequencies of the d trigonometric components [2]. (iii) the (n – 2d) remaining zeros can be associated to the noise. The bulk of these “noisy zeros” is further away from the unit circle [3]. In the mean, the distance to the latter is decreasing like n – 1 [4]. (iv) if only noise is present the mean positions of the n zeros are regularly spaced on an inner circle [5]. This exactly regular spacing or crystallization is the signature of the noise. It is universal with respect to the probability law of the noise and to the length of the signal, provided lnN.

In this work we consider structures for simultaneous multiplication by a small set of two pairwise coefficients where the coefficients are the real and imaginary part of a limited number of points uniformly spread on the unit circle. Hence, each such multiplier forms half of a complex multiplier suitable for twiddle factor multiplication in FFT architectures. Based on trigonometric identities we propose a multiplier for a unit circle resolution of 32 points. Also, we revisit an earlier proposed multiplier for 16 points and show that the complexity can be reduced by using minimum adder constant multipliers compared with the earlier proposed CSD-based multipliers.

This is the third installment in a series of papers concerning the Bounded Real Lemma for infinite-dimensional discrete-time linear input/state/output systems. In this setting, under appropriate conditions, the lemma characterizes when the transfer function associated with the system has contractive values on the unit circle, expressed in terms of a linear matrix inequality, often referred to as the Kalman– Yakubovich–Popov (KYP) inequality. Whereas the first two installments focussed on causal systems with the transfer functions extending to an analytic function on the disk, in the present paper the system is still causal but the state operator is allowed to have nontrivial dichotomy (the unit circle is not contained in its spectrum), implying that the transfer function is analytic in a neighborhood of zero and on a neighborhood of the unit circle rather than on the unit disk. More generally, we consider bicausal systems, for which the transfer function need not be analytic in a neighborhood of zero. For both types of systems, by a variation on Willems’ storage-function approach, we prove variations on the standard and strict Bounded Real Lemma. We also specialize the results to nonstationary discrete-time systems with a dichotomy, thereby recovering a Bounded Real Lemma due to Ben-Artzi–Gohberg–Kaashoek for such systems.

How to count the number of zeros that a polynomial has on the unit circle?

A new method is presented to obtain degree-one or real degree-two transfer scattering matrices of two-port lossless lattice filters through the use of complex Richard's function extractions for the minimum degree cascade synthesis of real, stable, single-input, single-output ARMA (n, m) filters from the transfer function or the input reflection coefficient. The method relies on a four-step Richard's function extraction where two steps are used for reducing the degree of the transfer function and two for obtaining real lattices. We treat the cases where the zeros of transmission are inside and outside the unit circle but not on the unit circle.

A topology for implementing a second-order digital recursive filter using complex arithmetic and having a transfer function with real coefficients is presented. The proposed structure is analyzed using state variables, separating the contributions into a real part and an imaginary part, and combining them at the output. The results give a low sensitivity and low roundoff noise digital filter topology. Accumulators are used to reduce the coefficient sensitivity of the filter structure. The input signal, although real, is applied to both the real and imaginary parts of the filter. It is shown that one of the realizations of the proposed filter could be a switched-capacitors one. >

A cascade synthesis for autoregressive moving-average (ARMA) digital filters, which uses the minimum number of delay elements (the degree of the transfer function in z), is described. The resulting structure results in the very convenient lattice from when the zeros of the transmission are not on the unit circle. The theory rests on conversion to scattering parameters and the use of a Richards' function, valid for complex zeros of rational functions with complex coefficients. Results of zeros of transmission on the unit circle are obtained by a parallel combination of lattices. By a suitable transformation, the sections are made computable while preserving the cascade form.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The denominator polynomial of a given causal stable z-domain transfer function is modified so that the magnitude of the frequency response remains the same. This simple modification permits an infinite number of decompositions of the modified denominator into a mirror-image polynomial (MIP) and an anti-mirror-image polynomial (AMIP). Two types of Discrete Reactance Functions (DRF) are constructed. From these DRFs, continued fraction expansions (CFE) are considered and some properties are obtained. These properties indicate whether the original denominator polynomial has all its roots within the unit circle (is minimum phase) or not.