Cascade realization of 2-D separable in denominator filters

After selecting the most suitable model for the class of two-dimensional quarter-plane-causal, recursive, and separable in denominator (CRSD) filters, a procedure for realizing a cascade system of such filters is developed. The realization utilizes the individual CRSD models constituting the cascade system. An example is given to further illustrate the technique.     

Design of multistage separable two-dimensional digital filters with frequency domain specifications

The separable decomposition method for 2-D filter design is examined in the continuous frequency domain. In the general case, improved accuracy and better insight can be obtained in comparison with the discrete counterparts as examined in the literature. Substantial simplifications both for the off-line computational load and the subsequent implementation are possible for frequency response functions being constant on their effective domain. This is achieved through a piecewise linear approximation of the domain boundary which reduces the original eigenvalue problem for operators with 2-D kernel into a solution of decoupled simple boundary-value problems in each direction. The latter are solvable through a simple Newton-Raphson procedure combined with a shooting method. This results into 1-D constituent filters with piecewise sinusoidal frequency responses. Completely analytical results are obtained for fan and directional filter design. Error estimates are given and examples illustrate the applicability of the results.     

Approximation of multivariable functions by M-D transfer functions with separable denominator

The approximation of a given function by a rational function has been considered extensively by mathematicians. A particular result has been stated by Walsh that the best approximation of a given analytical function is one which interpolates the given function at several properly chosen points. In this paper, transfer functions of multidimensional digital filters with separable denominator are used for the approximation of given multivariate functions. It is shown that the result of Walsh can be generalized in a straightforward manner. By an example it is illustrated how the new result can be applied to, e.g., the order reduction of a higher-order system. In the conclusion we state the usefulness and the limitation of the result.     

Design of two-dimensional IIR digital filters by using a novel hybrid optimization algorithm

This paper proposes a hybrid optimization algorithm named as BBO–PSO, which is a combination of biogeography-based optimization (BBO) and particle swarm optimization (PSO). In BBO–PSO, the whole population will be split into several subgroups and BBO is employed for local search in each subgroup independently to achieve the different local optima while PSO is employed for global search based on the local optima to achieve the global optimum. The test results on the benchmark functions show that BBO–PSO has powerful search ability with great robustness. Furthermore, the proposed algorithm is applied to the design of the 2-D IIR digital filters and the simulation results show that it outperforms the existing methods on this problem.     

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.     

Realization of IIR LDM digital filters

A method is presented of realizing an infinite impulse response (IIR) digital filter (DF) using linear delta modulation (LDM) as a simple analog/digital (A/D) converter. This method makes the realization of IIR digital filters much simpler than that of conventional ones because it does not require hardware multipliers or a pulse code modulation (PCM) A/D converter. Compared to the finite impulse response (FIR) LDMDF, this IIR LDMDF requires significantly less computation time     

Design of IIR digital notch filters

A method is presented for the design of notch filters with specified notch frequency ₀ and 3-dB rejection bandwidthB _t, using a first-order real all-pass filter, wherein the only coefficient is used to control the notch frequency. To control the bandwidth, use is made of a new amplitude change function (ACF), and it is shown that given notch filter specifications can be exactly met thereby. Also, using the ACF, it is shown that stability of the second-order notch filter designs can be improved along with the noise gain.     

Design of IIR notch filters with different passband gains

A method is proposed for the design of infinite-duration impulse response (IIR) notch filters with different passband gains using the standard lowpass IIR digital-filter design methods. This method satisfies the notch-frequency specification exactly and realises a rejection bandwidth, defined by the frequencies at which the gain is 3 dB below the lower of the two passband gains, which is smaller than that specified. Illustrative examples are given     

On the implementation of nonseparable two-dimensional Haar wavelettransforms

We introduce a transformation to localize the equations defining the successive levels of the Mallat (1989) pyramid for two-dimensional (2-D) Haar wavelets. We propose a methodology for implementing these wavelet transforms in parallel architectures like systolic arrays. More specifically, we show that there is a perfect match between the wavelet algorithms and the multiphase multirate array (MPRA) architectures     

Multidimensional IIR filters and robust rational interpolation

It is well-known that IIR filters can have a much lower order than FIR filters with the same performance. On the downside is that the implementation of an IIR filter is an iterative procedure while that of an FIR filter is a one-shot computation. But in higher dimensions IIR filters are definitely more attractive. We offer a technique where the filter’s performance specifications, stability constraints, its convergence speed and a protection against possible adverse effects of perturbations are all included in the design from the start. The technique only needs an off-the-shelf LP solver because the filter is obtained as a Chebyshev center of a convex polytope. The method deals with general non-causal non-separable filters.     

Designing IIR filters with a given 3-dB point

Often in infinite impulse response (IIR) filter design, our critical design parameter is the cutoff frequency at which the filter's power decays to half (-3 dB) the nominal passband value. This article presents techniques that aid in the design of discrete-time Chebyshev and elliptic filters given a 3-dB attenuation frequency point. These techniques place Chebyshev and elliptic filters on the same footing as Butterworth filters, which traditionally have been designed for a given 3-dB point. The result is that it is easy to replace a Butterworth design with either a Chebyshev or an elliptic filter of the same order and obtain a steeper rolloff at the expense of some ripple in the passband and/or stopband of the filter.     

A design of linear-phased IIR Nyquist filters

This paper discusses a new method of designing linear-phased IIR Nyquist filters with zero intersymbol interference. The filters designed by this method possess linear-phase characteristics and are lower in order than other Nyquist filters designed by existing methods. Expressions are derived for zero-phased IIR Nyquist filters and efficient design methods are examined for them. The opted design method is based on an iteration process, and in each iteration step a modified version of the Remez exchange algorithm is used. In addition, the implementation of the designed zero-phased IIR filters is considered. Finally, the proposed design method is demonstrated through various design examples     

Design of multiplierless elliptic IIR filters with a smallquantization error

We present a new technique for the design of multiplierless IIR elliptic filters. The multiplierless filter has all multiplication constants implemented with a small number of shifters and adders. The proposed technique is based on sensitivity analysis. An analytical expression for amplitude response sensitivity is derived for the filter structures consisting of two allpass subfilters in parallel. It is shown that the amplitude response sensitivity to some constant x can be expressed as a product of the filter reflectance function and the phase sensitivity of the allpass section that implements the constant. The closed-form expressions for the phase sensitivities of the first- and second-order allpass sections are also developed. It is shown in the paper that the (n+1)/2 most sensitive constants can be directly controlled by the transfer function parameters if the transfer function is derived by the bilinear transformation from an elliptic minimal Q-factors (EMQF) analog prototype. This way, (n+1)/2 multiplication constants can be implemented without quantization, leaving the filter characteristic strictly elliptic. This is achieved for a class of low-noise allpass sections and for the wave lattice digital filter as well. The quantization of the remaining (n-1)/2 less-sensitive constants is performed using the phase-tolerance scheme and phase-sensitivity functions. The proposed design technique is straight-forward and, consequently, very fast. The application is demonstrated on the examples of narrowband, wideband, and halfband filters     

Canonic ladder realizations of IIR digital filters

Two new canonic realizations of IIR digital filters age presented. The realizations are derived employing modified continued fraction expansion procedures. Proposed methods also provide alternate derivations to the digital lattice and ladder structures based on two-pair extractions and orthogonal polynomial expansions.     

Exact realisation of 2-dimensional digital filters by separable filters

A scheme for the exact realisation of 2-dimensional digital filters by using separable filters is proposed. The desired impulse response is obtained by a process of successive corrections. The performance criterion in terms of computing speed is compared with that of another existing technique.     

Design of three-dimensional adaptive IIR notch filters

This paper investigates a realization of a three-dimensional (3-D) adaptive notch filter. The procedures are mainly divided into two parts: frequency-detecting and sinusoidal interference removal. The detections are based on adaptive line enhancer on infinite impulse response (IIR) lattice structure. In the interference removal part, a non-separable version of a 3-D notch filter is effectively applied. The magnitude response of a 3-D adaptive IIR notch filter is illustrated. At the end of the paper, the implementation of an IIR notch filter on a 3-D image is also conducted in order to show how to remove a sinusoidal interference superimposed on a 3-D image.     

Adaptive signal processing using FIR and IIR filters

A method is presented for designing optimal adaptive digital filters. The derivation is based upon filtering a desired signal which has been corrupted by a "noise" like signal so the least-square error between the filtered output and the desired output is minimized. The filter is contrasted with both the Levinson and Widrow filters. The greatest utility of the derived filters are in digital signal processing applications where real-time or stability are critical constraints.     

Two-channel IIR QMF banks with approximately linear-phase analysisand synthesis filters

Perfect linear-phase two-channel QMF banks require the use of finite impulse response (FIR) analysis and synthesis filters. Although they are less expensive and yield superior stopband characteristics, perfect linear phase cannot be achieved with stable infinite impulse response (IIR) filters. Thus, IIR designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank. However, the analysis and synthesis filters of such an IIR filter bank are not linear phase. In this paper, a computationally simple method to obtain IIR analysis and synthesis filters that possess negligible phase distortion is presented. The method is based on first applying the balanced reduction procedure to obtain nearly allpass IIR polyphase components and then approximating these with perfect allpass IIR polyphase components. The resulting IIR designs already have only negligible phase distortion. However, if required, further improvement may be achieved through optimization of the filter parameters. For this purpose, a suitable objective function is presented. Bounds for the magnitude and phase errors of the designs are also derived. Design examples indicate that the derived IIR filter banks are more efficient in terms of computational complexity than the FIR prototypes and perfect reconstruction FIR filter banks. Although the PR FIR filter banks when implemented with the one-multiplier lattice structure and IIR filter banks are comparable in terms of computational complexity, the former is very sensitive to coefficient quantization effects     

Optimization-based design and implementation of multidimensional zero-phase IIR filters

This paper considers multidimensional infinite-impulse response (IIR) filters that are iteratively implemented. The focus is on zero-phase filters with symmetric polynomials in the numerator and denominator of the multivariable transfer function. A rigorous optimization-based design of the filter is considered. Transfer function magnitude specifications, convergence speed requirements for the iterative implementation, and spatial decay of the filter impulse response (which defines the boundary condition influence in the spatial domain of the filtered signal) are all formulated as optimization constraints. When the denominator of the zero-phase IIR filter is strictly positive, these frequency domain specifications can be cast as a linear program and then efficiently solved. The method is illustrated with two two-dimensional IIR filter design examples.