369 research outputs found

    On the eigenfilter design method and its applications: a tutorial

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    The eigenfilter method for digital filter design involves the computation of filter coefficients as the eigenvector of an appropriate Hermitian matrix. Because of its low complexity as compared to other methods as well as its ability to incorporate various time and frequency-domain constraints easily, the eigenfilter method has been found to be very useful. In this paper, we present a review of the eigenfilter design method for a wide variety of filters, including linear-phase finite impulse response (FIR) filters, nonlinear-phase FIR filters, all-pass infinite impulse response (IIR) filters, arbitrary response IIR filters, and multidimensional filters. Also, we focus on applications of the eigenfilter method in multistage filter design, spectral/spacial beamforming, and in the design of channel-shortening equalizers for communications applications

    Critical analysis of the eigenfilter method for the design of FIR filters and wideband beamformers

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    The least squares based eigenfilter method has been applied to the design of both finite impulse response (FIR) filters and wideband beamformers successfully. It involves calculating the resultant filter coefficients as the eigenvector of an appropriate Hermitian matrix, and offers lower complexity and less computation time with better numerical stability as compared to the standard least squares method. In this paper, we revisit the method and critically analyze the eigenfilter approach by revealing a serious performance issue in the passband of the designed FIR filter and the mainlobe of the wideband beamformer, which occurs due to a formulation problem. A solution is then proposed to mitigate this issue, and design examples for both FIR filters and wideband beamformers are provided to demonstrate the effectiveness of the proposed method

    On the design of real and complex fir filters with flatness and peak error constraints using semidefinite programming

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    This paper studies the problem of designing digital finite duration impulse response (FIR) filters with prescribed flatness and peak error constraints using semidefinite programming (SDP). SDP is a powerful convex optimization method, where linear and convex quadratic inequality constraints can readily be incorporated. This property is utilized for the optimal minimax and least squares (LS) design of linear-phase and low-delay FIR filters with prescribed magnitude flatness and peak design error, which are formulated as a set of linear equality and convex quadratic inequality constraints, respectively. A method for structurally imposing these equality constraints in the SDP formulation is also proposed. Using these results, the design approach is further extended to the design of constrained complex coefficient FIR filters and variable digital filters (VDFs). Design examples are given to demonstrate the effectiveness of the approach.published_or_final_versio

    Unified eigenfilter approach: with applications to spectral/spatial filtering

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    The eigenfilter approach is extended to solve general least-squares approximation problems with linear constraints. Such extension unifies previous work in eigenfilters and many other filter design problems, including spectral/spatial filtering, one-dimensional or multidimensional filters, data independent or statistically optimal filtering, etc. With this approach, various filter design problems are transformed into problems of finding an eigenvector of a positive definite matrix that is determined by filter design specifications. This approach has the advantage that many filter design constraints can be incorporated easily. A number of design examples are presented to show the usefulness and flexibility of the proposed approach

    Wideband data-independent beamforming for subarrays

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    The desire to operate large antenna arrays for e.g. RADAR applications over a wider frequency range is currently limited by the hardware, which due to weight, cost and size only permits complex multipliers behind each element. In contrast, wideband processing would have to rely on tap delay lines enabling digital filters for every element.As an intermediate step, in this thesis we consider a design where elements are grouped into subarrays, within which elements are still individually controlled by narrowband complex weights, but where each subarray output is given a tap delay line or finite impulse response digital filter for further wideband processing. Firstly, this thesis explores how a tap delay line attached to every subarray can be designed as a delay-and-sum beamformer. This filter is set to realised a fractional delay design based on a windowed sinc function. At the element level, we show that designing a narrowband beam w.r.t. a centre frequency of wideband operation is suboptimal,and suggest an optimisation technique that can yield sufficiently accurate gain over a frequency band of interest for an arbitrary look direction, which however comes at the cost of reduced aperture efficiency, as well as significantly increased sidelobes. We also suggest an adaptive method to enhance the frequency characteristic of a partial wideband array design, by utilising subarrays pointing in different directions in different frequency bands - resolved by means of a filter bank - to adaptively suppress undesired components in the beam patterns of the subarrays. Finally, the thesis proposes a novel array design approach obtained by rotational tiling of subarrays such that the overall array aperture is densely constructed from the same geometric subarray by rotation and translation only. Since the grating lobes of differently oriented subarrays do not necessarily align, an effective grating lobe attenuation w.r.t. the main beam is achieved. Based on a review of findings from geometry,a number of designs are highlight and transformed into numerical examples, and the theoretically expected grating lobe suppression is compared to uniformly spaced arrays.Supported by a number of models and simulations, the thesis thus suggests various numerical and hardware design techniques, mainly the addition of tap-delay-line per subarray and some added processing overhead, that can help to construct a large partial wideband array close in wideband performance to currently existing hardware.The desire to operate large antenna arrays for e.g. RADAR applications over a wider frequency range is currently limited by the hardware, which due to weight, cost and size only permits complex multipliers behind each element. In contrast, wideband processing would have to rely on tap delay lines enabling digital filters for every element.As an intermediate step, in this thesis we consider a design where elements are grouped into subarrays, within which elements are still individually controlled by narrowband complex weights, but where each subarray output is given a tap delay line or finite impulse response digital filter for further wideband processing. Firstly, this thesis explores how a tap delay line attached to every subarray can be designed as a delay-and-sum beamformer. This filter is set to realised a fractional delay design based on a windowed sinc function. At the element level, we show that designing a narrowband beam w.r.t. a centre frequency of wideband operation is suboptimal,and suggest an optimisation technique that can yield sufficiently accurate gain over a frequency band of interest for an arbitrary look direction, which however comes at the cost of reduced aperture efficiency, as well as significantly increased sidelobes. We also suggest an adaptive method to enhance the frequency characteristic of a partial wideband array design, by utilising subarrays pointing in different directions in different frequency bands - resolved by means of a filter bank - to adaptively suppress undesired components in the beam patterns of the subarrays. Finally, the thesis proposes a novel array design approach obtained by rotational tiling of subarrays such that the overall array aperture is densely constructed from the same geometric subarray by rotation and translation only. Since the grating lobes of differently oriented subarrays do not necessarily align, an effective grating lobe attenuation w.r.t. the main beam is achieved. Based on a review of findings from geometry,a number of designs are highlight and transformed into numerical examples, and the theoretically expected grating lobe suppression is compared to uniformly spaced arrays.Supported by a number of models and simulations, the thesis thus suggests various numerical and hardware design techniques, mainly the addition of tap-delay-line per subarray and some added processing overhead, that can help to construct a large partial wideband array close in wideband performance to currently existing hardware

    Design of FIR digital filters with prescribed flatness and peak error constraints using second-order cone programming

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    This paper studies the design of digital finite impulse response (FIR) filters with prescribed flatness and peak design error constraints using second-order cone programming (SOCP). SOCP is a powerful convex optimization method, where linear and convex quadratic inequality constraints can readily be incorporated. It is utilized in this study for the optimal minimax and least squares design of linear-phase and low-delay (LD) FIR filters with prescribed magnitude flatness and peak design error. The proposed approach offers more flexibility than traditional maximally-flat approach for the tradeoff between the approximation error and the degree of design freedom. Using these results, new LD specialized filters such as digital differentiators, Hilbert Transformers, Mth band filters and variable digital filters with prescribed magnitude flatness constraints can also be derived. © 2005 IEEE.published_or_final_versio

    Relaxing Tight Frame Condition in Parallel Proximal Methods for Signal Restoration

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    A fruitful approach for solving signal deconvolution problems consists of resorting to a frame-based convex variational formulation. In this context, parallel proximal algorithms and related alternating direction methods of multipliers have become popular optimization techniques to approximate iteratively the desired solution. Until now, in most of these methods, either Lipschitz differentiability properties or tight frame representations were assumed. In this paper, it is shown that it is possible to relax these assumptions by considering a class of non necessarily tight frame representations, thus offering the possibility of addressing a broader class of signal restoration problems. In particular, it is possible to use non necessarily maximally decimated filter banks with perfect reconstruction, which are common tools in digital signal processing. The proposed approach allows us to solve both frame analysis and frame synthesis problems for various noise distributions. In our simulations, it is applied to the deconvolution of data corrupted with Poisson noise or Laplacian noise by using (non-tight) discrete dual-tree wavelet representations and filter bank structures

    A new class of two-channel biorthogonal filter banks and wavelet bases

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    We propose a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters

    Optimum design of discrete-time differentiators via semi-infinite programming approach

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    In this paper, a general optimum full band high order discrete-time differentiator design problem is formulated as a peak constrained least square optimization problem. That is, the objective of the optimization problem is to minimize the total weighted square error of the magnitude response subject to the peak constraint of the weighted error function. This problem formulation provides a great flexibility for the tradeoff between the ripple energy and the ripple magnitude of the discrete-time differentiator. The optimization problem is actually a semi-infinite programming problem. Our recently developed dual parametrization algorithm is applied for solving the problem. The main advantage of employing the dual parameterization algorithm for solving the problem is the guarantee of the convergence of the algorithm and the obtained solution being the global optimal solution that satisfies the corresponding continuous constraints. Moreover, the computational cost of the algorithm is lower than that of algorithms implementing the semi-definite programming approach
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