Denominator Coefficients Research Articles

Design of separable-denominator variable 2-D digital filters with guaranteed stability

Stability guarantee is the most important issue in designing variable recursive digital filters since unstable variable filters cannot be applied in signal processing applications. This paper proposes a technique for designing separable-denominator variable two-dimensional (2-D) digital filters having variable magnitude characteristics, and the stability of the designed variable recursive 2-D filters is always guaranteed. The method assumes each coefficient of the variable 2-D recursive digital filter as a polynomial of the spectral parameters that specify different variable magnitude responses. Since the filter coefficients are represented by different polynomials, substitution of different values of the spectral parameters into the polynomials gives different filter coefficients and thus different magnitude responses. To guarantee the stability, we first perform coefficient transformations on the denominator coefficients of the recursive transfer function such that the stability conditions of the recursive filter are always satisfied for arbitrary values of the new coefficients. Then both the denominator and the numerator coefficients are determined as the polynomials of the spectral parameters. The proposed technique is effective in designing separable-denominator variable 2-D digital filters with guaranteed stability.

Realization of a linearized transconductor based on the multitail technique

A new linearization technique of the multitail trans-conductor is presented. In this technique, the parameter conditions for the maximally flat transconductance are easily obtained by comparing the coefficients of denominator and numerator of an output current expanded into a polynomial function using a Taylor expansion. In this paper, linearized transconductors with two, three, and four emitter-coupled pairs are described and discussed. SPICE simulation results show that the total harmonic distortion is less than, 1% for an input signal of 1 Vp–p at 100 kHz. © 2000 Scripta Technica, Electron Comm Jpn Pt 2, 83(12): 42–51, 2000

Adaptive IIR bandpass decimation filter for single sinusoid detection

This paper proposes an adaptive second-order IIR bandpass decimation filter for detecting a single sinusoid with an unknown frequency embedded in noise. The purpose is to realize a filter with high Q (selectivity) and less output amplitude fluctuation after convergence. As the first step, the adaptive second-order IIR bandpass filter with the variable center frequency is outlined. Then, we present a new method that derives the decimation filter function with a small amount of computation. The filter is derived from the above filter with the denominator coefficient made variable, and with the polynomial of z−M as the denominator, where M (integer) is the decimation ratio. A method is shown to design the adaptive signal generator and the coefficient update circuit for a single sinusoid detection adaptive filter based on the proposed method. Lastly, the convergence and the output error characteristics of the proposed adaptive decimation filter are examined by a computer simulation example. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(7): 91–101, 2000

The simultaneous interpolation of antenna radiation patterns in both the spatial and frequency domains using model-based parameter estimation

The Pade rational function fitting model commonly used for model-based parameter estimation (MBPE) in the frequency domain is enhanced to include spatial dependence in the numerator and denominator coefficients. This allows the function to interpolate an antenna radiated electric field pattern in both the frequency and spatial domains simultaneously, such that a single set of coefficients can be used to accurately reconstruct an entire radiation pattern at any frequency in the fitting-model range. A simple procedure is introduced for transforming interpolated electric fields into gain patterns using input impedance versus frequency curves also obtained via MBPE. The utility of this method is demonstrated by applying it to a dipole antenna over a frequency range of 150-950 MHz and using a polynomial representation in /spl theta/ for the coefficient spatial dependence. It is also used to estimate radiation patterns for a three-element Yagi array between the frequencies of 470 and 500 MHz using a binomial representation for the spatial variation that includes terms dependent on /spl theta/ as well as /spl phi/. The use of this method for interpolating radiation patterns has at least two significant advantages; one being large compression ratios for the amount of data that must be stored to accurately reproduce patterns and the other being a significant decrease in the amount of time required for modeling problems with large computational domains.

Searching for Robust Minimal-Order Compensators

A method of designing a family of robust compensators for a single-input/single-output linear system is presented. Each compensator’s transfer function is found by using a genetic-algorithm search for numerator and denominator coefficients. The search minimizes the probabilities of unsatisfactory stability and performance subject to real parameter variations of the plant. As the search progresses, probabilities are estimated by Monte Carlo evaluation. The design procedure employs a sweep from the lowest feasible transfer-function order to higher order, terminating either when design goals have been achieved or when no further improvement in robustness is evident. The method provides a means for estimating the best possible compensation of a given order based on repeated searches.

Robust stability analysis of control systems with parameter uncertainties: An eigenvalue approach

Abstract This paper considers the stability radius problem of control systems whose coefficients of the denominator and the numerator polynomials are affine in a Holder p-norm-bounded uncertain vector δ. It is shown that the stability radius of the closed-loop system can be characterized in terms of the eigenvalues of two sets of frequency independent matrices if the nominal plant is stabilized by the controller. The involved frequency independent matrices can be factorized as Hβ-1 Hγ, where both Hβ and Hγ are Hurwitz-like.

A generalization of filters with monotonic amplitude-frequency response

The polynomial low-pass filters monotonic in the passband amplitude-frequency response are proposed. The characteristics are intermediate between Butterworth and Papoulis filters (with these two classes as limiting) and are obtained starting from the filters of the fifth order. The number of intermediate characteristics increases with the filter order. Decreasing the passband flatness degree, one can obtain an additional attenuation in the stop band. The design tables are given for the filters of the fifth, sixth, and seventh orders, The tables include the filtering functions, transfer function poles, pole frequencies and Q factors, denominator coefficients of transfer functions, and inductor and capacitor values for the transfer impedance realization.

Novel interpretation of the pencil-of-functions approximation/identification method

Jain's pencil-of-functions method based on the linear dependence/independence of a set of functions is revisited. It is shown that no matter what model order is chosen, every estimated denominator coefficient may be regarded as the geometric mean of two values obtained in a least-squares sense.

New method for designing stable recursive variable digital filters

The digital filters with adjustable frequency-domain characteristics are called variable filters. Variable filters are used in many signal processing fields, but the recursive variable filters are extremely difficult to design due to the stability problem. This paper proposes a new method for designing recursive one-dimensional (1-D) variable filters whose stability is always guaranteed. To guarantee the stability, we first perform coefficient substitutions on the denominator coefficients such that for arbitrary real-valued coefficients, the stability condition is always satisfied. Then both the denominator coefficients and the numerator coefficients after substitutions are determined as multi-dimensional (M-D) polynomials of spectral parameters that specify variable magnitude characteristics. In applying the resulting variable filters, substituting different values of the spectral parameters into the M-D polynomials will obtain different filter coefficients and thus different magnitude characteristics. Two examples are given to show the effectiveness of the proposed design technique.

Least-squares approximation of FIR by IIR digital filters

An algorithm is presented for the least-squares approximation of FIR filters by IIR filters. The algorithm is an iterative procedure where each iteration requires the solution of an overdetermined set of linear equations and some digital filtering operations. All calculations are performed with the numerator and denominator coefficients of the transfer functions. A conversion to state-space descriptions is not necessary. Examples show that the approximation error is as small as that of the IIR filters obtained with balanced model reduction. Moreover, the effects of numerical errors are negligible. Thus, our algorithm is applicable even in cases where the FIR filter length is large.

Design of variable 2-D linear phase recursive digital filters with guaranteed stability

This paper proposes a new technique for designing variable two-dimensional (2-D) linear phase recursive digital filters, the stability of which is always guaranteed. The method finds each variable filter coefficient as a multidimensional (M-D) polynomial of a few parameters. The parameters specify different frequency responses, thus they are called the spectral parameters. In applying the resulting variable filters, substituting different spectral parameter values into the M-D polynomials will obtain different filter coefficients and, thus, different frequency responses. To guarantee the stability, we first perform denominator coefficient transformations such that they satisfy the stability conditions. Then, both denominator and numerator coefficients are determined as M-D polynomials.

Structural Properties of Maximally Robust Controllers

This work contains results on properties of integrating feedback controllers which give maximal stability robustness against real coefficient uncertainty in the numerator and denominator coefficients of transfer functions of classes of linear discretetime scalar plants. The integrator imposes simply computed upper bounds on the size of the smallest destabilising plant numerator and denominator uncertainties. The ability for these bounds to be achieved for a given nominal plant depends on the existence of solutions of particular sign properties to interpolation problems. The first main result of this paper is to use these sign constraints to infer properties of the roots of corresponding controller polynomials. In particular it is shown how the robustness bound on denominator uncertainty cannot be achieved by a controller with any strictly unstable poles. The second main result concerns cancellation of nominal minimum phase plant zeros and its effect on robustness.

OPTIMAL 2-D SEPARABLE-DENOMINATOR APPROXIMANTS FOR 2-D TRANSFER FUNCTIONS

ABSTRACT This paper is concerned with the optimal approximation of a general 2-D transfer function by a separable-denominator one with stability preservation. To preserve the stability, the two one-variable denominator polynomials of the reduced model are both represented in their bilinear Routh continued-fraction expansions. The bilinear Routh γ parameters of the two one-variable denominator polynomials and the coefficients of the two-variable numerator polynomial are then determined such that a frequency-domain L 2-norm is minimized. The main advantage of searching bilinear Routh γ parameters instead of denominator coefficients is that the stability constraints on the new decision parameters are simple bounds. To facilitate using a gradient-based algorithm, an effective numerical algorithm is also provided for computing the performance index and its gradients with respect to the decision variables.

Air temperature transfer function of a convection oven

The effect on air temperature of variations in heater power for a forced convection oven were modelled. The transfer functions for five operating ranges, from 140 °C to 236 °C, were determined experimentally. The oven power control was excited using a pseudo-random binary signal and the resulting effect on the air temperature was measured. From this information, transfer functions in the ARMAX form were determined. It was found that while the denominator coefficients of the system do not vary for different ranges of air temperature, the numerator coefficients of the system do vary significantly.

Design of recursive 1-D variable filters with guaranteed stability

The digital filters with adjustable frequency-domain characteristics are called variable filters. Variable filters are used in many signal processing fields, but the recursive variable filters are extremely difficult to design due to the stability problem. This paper proposes a new method for designing recursive one-dimensional (1-D) variable filters whose stability is guaranteed. The method finds the coefficients of the transfer function of a variable digital filter as the multidimensional (M-D) polynomials of a few variables. The variables specify different frequency-domain characteristics, thus, we call the variables the spectral parameters. In applying the resulting variable filters, substituting different values of the spectral parameters into the M-D polynomials will obtain different filter coefficients and, thus, obtain different frequency-domain characteristics. To guarantee the stability, we first perform coefficient substitutions on the denominator coefficients such that they satisfy the stability conditions. Then both denominator and numerator coefficients are determined as M-D polynomials. In determining the M-D polynomials, we also propose an efficient least-squares approximation method that requires only solving simultaneous linear equations. Two examples are given to show the effectiveness of the proposed variable filter design technique.

A structured matrix approach for spatial-domain approximation of 2-D IIR filters

This work addresses least-squares (LS) approximation of a prescribed spatial response by a quarter-plane 2-D IIR filter. Using structured matrix representation it is shown that the spatial domain error vector between the desired and estimated responses is linearly related to the numerator coefficients and nonlinearly related to the denominator coefficients. The numerator and denominator estimation problems are theoretically decoupled without affecting the optimality properties. The decoupled denominator criterion possesses a quasi-quadratic form and its optimization is computationally efficient requiring very few iterations, The numerator is estimated linearly in a single step. Effectiveness of the algorithm is demonstrated with several examples.

A technique for the design of 2-D recursive digital filters with guaranteed stability

In the present technique, the denominator and numerator in the filter are designed separately and recursively. First, the denominator coefficients, derived from the stable matrix of the Roesser model (the Fornasini-Marchesini second model), are found by minimizing a performance index through the alternating variable method. The numerator coefficients are then determined analytically by solving linear simultaneous equations. The above process will be repeated until there is negligible change in the objective function. Finally, a numerical example is given to illustrate the utility of the proposed technique.

Precise insensitive current-mode third-order lowpass Butterworth characteristics

The realisation of third-order lowpass Butterworth characteristics in the current mode using second-generation current conveyor (CC II) elements is presented. With nonideal CC IIs the denominator coefficients are altered slightly, which can be precisely compensated by suitable design. The network parameters are practically active-insensitive.

Least-squares Padé reduction: A nonuniqueness property

It is shown that least-squares Pade reduced order models are not unique for a given order. Depending on which reduced denominator coefficient is chosen to be unity, the method is seen to minimise an error index over different subsets of system time moments when calculating this denominator.

Efficient time domain synthesis of pipelined recursive filters

An efficient technique is developed for the synthesis of pipelined recursive filters directly from the time domain specifications. Based on the modified least-squares approximation, the pipelinable denominator form that contains only the powers of z/sup -R/ is designed inherently by expressing the design criterion in terms of only the nonzero denominator coefficients and a final design is derived. Due to the lack of the conventional transformation overheads, the multiplications needed are fewer in the resulting filters. Moreover, we introduce a new kind of companion matrix that characterizes the stable properties of pipelined recursive filters. Using this result, we prove that the proposed direct synthesis method can always guarantee the stability.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>