Design Of IIR Digital Filters Research Articles

A new approach to the design of complex all-pass IIR digital filters

A new method is proposed for designing complex all-pass IIR filters, the all-pass IIR filters with complex coefficients, in this paper. By minimizing the integration of certain square phase error over interested frequencies, an eigenvector of an appropriate real, symmetric and positive-definite matrix is computed to get the filter coefficients. The stability is achieved by specifying properly the desired phase specifications. If an appropriate iterative process is used, equiripple complex all-pass filter design can be obtained. The method is simple and the performance is comparable to the existing methods. Several examples are presented to demonstrate the effectiveness of the approach.

Design of digital all-pass filters using a weighted least squares approach

This paper considers the minimax design of digital all-pass filters via the FIR and IIR approximations. For the FIR digital all-pass filter design, the problem is appropriately formulated so that a recently-developed weighted least squares (WLS) algorithm can be used to efficiently solve the associated minimization problem. We also design FIR digital all-pass filters using the WLS algorithm to have equiripple group delay error. For the IIR digital all-pass filter design, we reformulate the design problem to result in a simple minimization problem. Therefore, a nonlinear minimax algorithm can be employed to obtain the solution. The key operation of the nonlinear minimax algorithm is to find the Chebyshev approximation of a desired function from a linear subspace. This can be efficiently solved by using the WLS algorithm. Several design examples demonstrating the effectiveness of the proposed approaches are also provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two dimensional IIR digital notch filter design

In this paper, the two dimensional (2D) IIR digital notch filter design problem is investigated. First, we develop a simple algebraic method for 2D IIR notch filter design. This approach not only has closed-form transfer function but also satisfies the bounded-input/bounded-output (BIBO) stability condition. Next, we apply the 2D notch filter to eliminate sinusoidal/narrowband interferences superimposed on an image. Simulation results are presented to demonstrate the effectiveness of this approach. >

Corrections on "Two dimensional IIR digital notch filter design"

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Design of IIR digital filters with given magnitude specifications using allpass networks

AbstractThis paper proposes a direct design method in the z‐domain for the IIR digital filter with a parallel structure composed of allpass networks with complex and real coefficients, with the specifications for the bandedge frequencies of the passband and stopband as well as the magnitude in the passband (or stopband).First, a filter composed of two allpass networks with conjugate complex coefficients is considered. Also, the phase difference characteristic between the two networks is considered. By specifying the phase difference error, the determination of the filter coefficients is reduced to the eigenvalue problem of the matrix using Remez' algorithm. The filter coefficients are determined from the eigenvector corresponding to the maximum eigenvalue. Then iterative calculation is executed to realize the equiripple characteristic, and the optimal solution is derived.Next, the relation between the phase difference errors and the magnitude errors is discussed and it is shown that the filter can be designed by the proposed method by specifying the absolute value of the magnitude error. The usefulness of the proposed method is shown by design examples. The filter based on two allpass networks with real coefficients is also considered, and it is shown that the filter can be designed by the same procedure by formulating the procedure for the sum and the difference of the filter coefficients.

Magnitude and Phase Approximations of Linear Phase FIR Digital Filters by Reduced-Order IIR Digital Filters

This paper compares the time and memory requirements of an approximately linear phase IIR digital filter with those of a linear phase FIR digital filter, and shows that, only if a substantial difference between the orders of the filters exists, the former can be considered as an attractive substitute for the latter. Furthermore, it presents a method which makes feasible the design of reduced-order IIR digital filters that resemble the characteristics of a linear plase FIR digital filter. Some design examples are shown, illustrating how the proposed method can be used to design approximately linear phase IIR digital filters that are faster and require less memory than their equivalent linear plase FIR counterparts.

On the design of 2D IIR digital filters using penalty optimization techniques

In this paper an efficient technique is developed for the design of two-dimensional recursive digital filters in the frequency domain. the formulation of the design problem is based upon a rational approximation method in parallel with a suitable interior penalty technique. the structure of the proposed approach permits one to include in the design optimization problem additional constraints related to the stability, linear phase and symmetries properties of the resultant filter. In contrast with previous design methods, the coefficients of the filters are determined directly and it is not required to apply any ‘inverse’ or stabilization technique. Analytical examples illustrate the features and effectiveness of the proposed approach.

A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design

It has been demonstrated by several authors that if a suitable frequency response weighting function is used in the design of a finite impulse response (FIR) filter, the weighted least squares solution is equiripple. The crux of the problem lies in the determination of the necessary least squares frequency response weighting function. A novel iterative algorithm for deriving the least squares frequency response weighting function which will produce a quasi-equiripple design is presented. The algorithm converges very rapidly. It typically produces a design which is only about 1 dB away from the minimax optimum solution in two iterations and converges to within 0.1 dB in six iterations. Convergence speed is independent of the order of the filter. It can be used to design filters with arbitrarily prescribed phase and amplitude response.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of IIR digital filters using complex allpass network based on eigenvalue problem

A new design method is presented for complementary IIR digital filters with given magnitude specifications based on the eigenvalue problem, which are composed of a single complex allpass section.

Design of zero-phase spherically symmetric N-dimensional IIR digital filters

The paper presents a new method for designing zero-phase N-dimensional IIR digital filters satisfying given amplitude design specifications. The method is based on applying the N-dimensional extension of McClellan transformation to the square of a one-dimensional IIR digital filter design recently proposed by Charalambous. The resulting N-dimensional digital filter is of zero-phase and hyperspherically symmetric but unstable. Stabilisation of the filter is achieved by dividing it into a product of 2N filters, each of which is stable if realised through a recursion in the suitable quadrant. The stabilisation procedure uses the properties of the complex cepstrum. Three examples are introduced to show the performance and validity of the design method presented. The designs obtained are also tested in image processing applications.

On the design of 1D IIR digital filters with coefficients of finite word length

AbstractThis paper presents new linear programming techniques for the design of optimal 1D IIR digital filters. the first method is based on a linear minimax criterion and leads to a linear programming approximation problem whose optimal solution is attained by a new fast algorithm. Next, the linear minimax approximation problem is extended and formulated as two linear integer programming problems which permit the design of 1D IIR digital filters with coefficients of finite word length. the first of these methods is formulated as a 0‐1 integer linear programming problem and its optimal solution is attained rapidly by a new algorithm. This method appears to be suitable for the design of high‐order digital filters. the second method is more general and is formulated as a mixed integer linear programming problem. to solve the design problem efficiently, a new integer tree search algorithm is introduced. the feasibility of the proposed algorithms is illustrated with detailed solutions or numerical examples.

A fast linear programming approach to the design of 1D IIR digital filters

AbstractA general method is proposed for the design of optimal 1D IIR digital filters via linear programming. the method is based upon the formulation of an appropriate linear problem which assures satisfactory approximation error for the filter's magnitude frequency response. the linear problem does not require differential correction methods or other iterative techniques for its solution, thus resulting in a computationally efficient algorithm. Analytical examples are presented to illustrate the efficiency of the method for the design of stable 1D recursive digital filters.

N-dimensional symmetries and their applications in digital filters

N-dimensional ( N-D) digital signal processing always results in enormous data processing and numerous parameters to be determined. The use of symmetries existing in the desired frequency specifications largely reduces the independent parameters, and furthermore, reduces the computation in the implementation. N-D digital filter symmetries existing in the frequency domain are developed and are related to numerator and denominator polynomials C( z 1, …, z N ). Constraints for identify magnitude ∥ C( rme jω 1, …, e jω N )∥ symmetries, anti-identity phase θ( ω 1,…, ω N ) symmetries, and conjugate complex frequency response C(e jω 1,…e j ω N symmetries, are developed. The necessary and sufficient conditions for a polynomial C( z 1, …, z N ) to satisfy a certain symmetry constraint are proposed. The newly developed symmetry constraints are used to derive special forms of discrete polynomials C( z 1, …, z N ) possessing these symmetries. The stability constraint is then imposed on the newly derived polynomial forms. These polynomial forms can be used in the design (in the frequency domain) and implementation of FIR and IIR digital filters, and will provide great saving of computation work and simplified hardware architecture.

Complex chebyshev approximation for IIR digital filters with arbitrary log magnitude and phase functions

AbstractThis paper proposes a technique for designing IIR digital filters to have prescribed magnitude and phase responses. The technique is based on an iterative weighted least‐squares approach in the frequency domain.A weight updating procedure is introduced to minimize the complex logarithmic error in the Chebyshev sense. The real and imaginary parts of the complex logarithmic error represents the log magnitude and phase errors, respectively. Both the log magnitude and phase errors of the resulting filter become nearly equiripple. Filter coefficients at each iteration are computed efficiently using FFT and a fast recursive algorithm for a set of linear equations derived from the weighted least‐squares problem. Several design examples are shown to demonstrate the effectiveness of the proposed technique.

On the evaluation of finite precision coefficients in IIR digital filter design

ABSTRACT A simple and effective algorithm is presented for the evaluation of coefficients of reduced word-length in the design of IIR digital filters. The transfer function is expressed as a cascade of second order sections. Each section is considered individually and coefficients of finite precision word-length are derived, using the Pade ' series coefficients in conjunction with a minimum pole displacement approach. The proposed algorithm makes use of two criteria The first criterion preserves a minimum pole displacement, while the second criterion preserves the minimum Pade ' coefficients modification. Once the evaluation of the coefficients has been accomplished the least squares error function is applied in order to check for the total error resulting in the frequency response of the filter. Thus the overall computational effort is reduced substantially. Two examples of one second order section filter and one example of two sections are presented to prove the algorithm' s effectiveness

Simple and computationally efficient design of two-dimensional circularly symmetric IIR digital filters

Abstract A simple and computationally efficient method for the design of a two-dimensional circularly symmetric IIR digital filters is introduced. This method has several advantages. Firstly, the digital filter format is a cascade of first- and second-order filter modules and is suitable for the VLSI modular implementation. Secondly, most of the coefficients can be determined directly from an one-dimensional analogue filter table and the stability of the filter is preserved. Finally, the unknown coefficients are very few in number and can be obtained by an unconstrained optimization method. Three examples are shown to illustrate the design method.

On the design of IIR digital filters with equiripple stopband transmission

Two methods of obtaining low-pass digital filter functions with equiripple loss in the stopband and prescribed poles have been described recently. It is shown that both methods give the same filter functions even though the approaches are different.

Time domain approach to recursive digital filter synthesis

The problem of the design of IIR digital filters, when the analog filter has been synthesized, resides in the approximation of a continuous system by means of a discrete one. We present a synthesis method which discretizes an Input-State-Output (I-S-O) model of the analog filter and is completely performed in the time domain. The basis of the method is the way in which the problem is stated, which points out the fundamental role played by the interpolator and allows an easy implementation. The main advantages of our method, with reference to traditional approaches, are: • - it allows direct working on continuous representations of the analog filter and the discretization does not require symbolic calculations as in approaches based on substitution techniques (for example bilinear transformation); • - it is completely numerical and is very simple from a computational point of view; • - it can be used for single and multiple input-output filters; • - it allows longer sampling periods using suitable interpolators. An example is discussed which allows a comparison with the synthesis method based on bilinear transform and shows, also, that computing time may be saved as the sampling period may be traded off with the interpolator order.

Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications

It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(z)</tex> (for example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(z)</tex> (for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(z)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(z)</tex> have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined.

On the approximation of the magnitude response of two-dimensional IIR digital filters using linear programming

The design of two-dimensional IIR digital filters is approached using linear-programming techniques. The problem is formulated as an approximation problem and filter frequency response characteristics are represented via the square of the magnitude of the filter transfer function. An appropriate approximation error is minimized leading to a linear problem formulation. The latter problem does not require differential correction methods for its solution, thus resulting in a computationally efficient algorithm. Stability conditions and symmetry considerations are easily accounted for. The linear programming approach assures convergence of the solution to a global minimum, among other advantages over nonlinear methods. Several examples illustrate the strength and effectiveness of the methodology.