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     

Cascade lattice IIR adaptive filters

The feedback lattice filter forms, including the two-multiplier form and the normalized form, are examined with respect to their relationships to the feedback direct form filter. Specifically, the transformation matrix between the lattice forms and the direct form is derived; parameter and state relationships between the lattice forms and the direct form are therefore obtained. An IIR filter structure-the cascade lattice IIR structure-is constructed. Based on this structure, three IIR adaptive filtering algorithms in the two-multiplier form can then be developed following the gradient approach, the Steiglitz-McBride approach and the hyperstability approach. Convergence of these algorithms is theoretically analyzed using either the ODE approach or the hyperstability theorem. These algorithms are then simplified into forms computationally as efficient as their corresponding direct form algorithms. Relationships of the simplified algorithms to the direct form algorithms are also studied, which disclose a consistency in algorithm structure regardless of the filter form. Three normalized lattice algorithms can also be derived from the two-multiplier lattice algorithms. Experimental results show much improved performance of the normalized lattice algorithms over the two-multiplier lattice algorithms and the direct form algorithms     

Adiabatic 3-dB couplers, filters, and multiplexers made with silicawaveguides on silicon

The performance of adiabatic 3-dB couplers and V-branches is reported. These devices are broadband and divide power equally. They have no observable polarization dependence. Typical excess losses relative to a straight waveguide is 0.1-0.2 dB for the 3-dB couplers and about 0.4 dB for the V-branches. Fiber to fiber insertion loss of 0.31 dB was measured for a 2.5-cm straight waveguide. The devices were used to fabricate transmission filters peaked at 1.55 μm and power combiners having two channels at 1.48 and 1.55 μm. The device fabrication was improved by use of a flowable top cladding layer containing boron and phosphorous which easily filled-in between closely spaced waveguides     

On fast FIR filters implemented as tail-canceling IIR filters

We have developed an algorithm based on synthetic division for deriving the transfer function that cancels the tail of a given arbitrary rational (IIR) transfer function after a desired number of time steps. Our method applies to transfer functions with repeated poles, whereas previous methods of tail-subtraction cannot. We use a parallel state-variable technique with periodic refreshing to induce finite memory in order to prevent accumulation of quantization error in cases where the given transfer function has unstable modes. We present two methods for designing linear-phase truncated IIR (TIIR) filters based on antiphase filters. We explore finite-register effects for unstable modes and provide bounds on the maximum TIIR filter length. In particular, we show that for unstable systems, the available dynamic range of the registers must be three times that of the data. Considerable computational savings over conventional FIR filters are attainable for a given specification of linear-phase filter. We provide examples of filter design. We show how to generate finite-length polynomial impulse responses using TIIR filters. We list some applications of TIIR filters, including uses in digital audio and an algorithm for efficiently implementing Kay's optimal high-resolution frequency estimator     

Designing lexicographic codes with a given trellis complexity

We generalize constructions of lexicographic codes to produce locally optimal codes with a desired trellis decoding complexity. These constructions are efficient for high-rate codes and provide a means for automated code design. As a byproduct, we improve known bounds on the parameters of lexicodes     

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.     

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     

Equiripple phase for IIR all-pass filters

A method is presented for designing IIR all-pass filters, the phase of which approximates a given phase in the equiripple (Chebyshev) space. The method is exclusively based on the solution of the linear equation system.<>     

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     

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     

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 of two-dimensional IIR filters with nonseparable and separable denominator

We present algorithms for the optimization of two-dimensional (2-D) infinite impulse response (IIR) filters with separable or nonseparable denominator, for least squares or Chebyshev criteria. The algorithms are iterative, and each iteration consists of solving a semidefinite programming problem. For least squares designs, we adapt the Gauss-Newton idea, which outcomes to a convex approximation of the optimization criterion. For Chebyshev designs, we adapt the iterative reweighted least squares (IRLS) algorithm; in each iteration, a least squares Gauss-Newton step is performed, while the weights are changed as in the basic IRLS algorithm. The stability of the 2-D IIR filters is ensured by keeping the denominator inside convex stability domains, which are defined by linear matrix inequalities. For the 2-D (nonseparable) case, this is a new contribution, based on the parameterization of 2-D polynomials that are positive on the unit bicircle. In the experimental section, 2-D IIR filters with separable and nonseparable denominators are designed and compared. We show that each type may be better than the other, depending on the design specification. We also give an example of filter that is clearly better than a recent very good design.     

On an adaptive algorithm for IIR filters

The purpose of this communication is to discuss an IIR adaptive filter algorithm developed by Stearns [1], in terms of an example that appeared in a recent article [2]. The example concerns the approximation of a fixed second-order filter by a first-order adaptive filter, when subjected to a white noise input.     

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.     

A WISE method for designing IIR filters

The problem of designing optimal digital IIR filters with frequency responses approximating arbitrarily chosen complex functions is considered. The real-valued coefficients of the filter's transfer function are obtained by numerical minimization of carefully formulated cost, which is referred here to as the weighted integral of the squared error (WISE) criterion. The WISE criterion linearly combines the WLS criterion that is used in the weighted least squares approach toward filter design and some time-domain components. The WLS part of WISE enforces the quality of the frequency response of the designed filter, while the time-domain part of the WISE criterion restricts the positions of the filter's poles to the interior of an origin-centred circle with arbitrary radius. This allows one not only to achieve stability of the filter but also to maintain some safety margins. A great advantage of the proposed approach is that it does not impose any constraints on the optimization problem and the optimal filter can be sought using off-the-shelf optimization procedures. The power of the proposed approach is illustrated with filter design examples that compare favorably with results published in research literature     

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.     

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.     

Bias removal in equation-error adaptive IIR filters

In the equation-error formulation of adaptive IIR filters, the estimated parameters contain bias when there is noise in the desired response. A method that can eliminate this bias is investigated. The idea is to maintain a quadratic constraint on the feedback coefficients so that the noise contributes only a constant term to the mean-square error. This term does not affect minimization and thus the bias is eliminated. A quadratically constrained stochastic gradient search method is applied for optimization and convergence behavior, when the noise is white, is analyzed. Adaptation of the feedback FIR filter in second-order cascade form, useful for stability monitoring, is also considered. When the noise is nonwhite, the technique requires an adaptive whitening filter. Simulation results are included to demonstrate the bias removal capability of the method, corroborate the theoretical developments, and compare with existing techniques     

Designing nonstandard filters with differential evolution

An alternative method for nonstandard filter design has been described. This method recasts the filter design problem as a minimization problem and solves the minimization via the DE minimizer, for which public domain software has been made available previously. The advantages of this method are its simplicity as well as the capability to design unconventional filter types. A great asset of this approach is that it can be applied with minimal knowledge of digital filter design theory.