Arbitrary Filter Research Articles

Arbitrary FIR Filter Synthesis Using a Neural Network

We propose a neural network to synthesize an arbitrary FIR filter in a least square sense. The network can evolve to its steady-states or equilibrium points from any initial state in the magnitude of the circuit‘s time constant. Under the steady-state, the output of the network is just our designed FIR filter coefficient if a real, symmetric, and positive-definite matrix calculated by the design specifications is directly used as the synaptic strength matrix.

Optimal pyramidal and subband decompositions for hierarchical coding of noisy and quantized images

Optimal hierarchical coding is sought, for progressive or scalable image transmission, by minimizing the variance of the error difference between the original image and its lower resolution renditions. The optimal, according to the above criterion, pyramidal and subband image coders are determined for images subject to corruption by quantization or transmission noise. Given arbitrary analysis filters and assuming adequate knowledge of the noise statistics, optimal synthesis filters are found. The optimal analysis filters are subsequently determined, leading to formulas for globally optimal structures for pyramidal and subband image decompositions. Experimental results illustrate the implementation and performance of the optimal coders.

Nonlinear Extensions of Classical Controllers Using Symbolic Computation Techniques: A Dynamical Systems Unifying Approach

AbstractIn previous works we reported the development of symbolic computation tools to automate the design of nonlinear state feedback controllers [1], nonlinear PID controllers [2], nonlinear lag‐lead compensators [3], and nonlinear obsevers [4] using the extended linearization method [5]. In this paper we show that a careful analysis of the state variables representation of these classical controllers indicates that all of them are particular cases of the nonlinear extension of a mth order linear filter, consisting of a kth order input derivative operator followed by an output mth order linear dynamical system. Using this two blocks decomposition approach, the design of nonlinear extensions of nth order controllers can be decomposed into two independent subalgorithms: a kth order PD controller algorithm, and a mth order state vector feedback algorithm. Hence, an appropriate, assembly of our symbolic computation tools NL Feedback and NLPID could, in principle, allow to use the extended linearization method to synthesize nonlinear extension of arbitrary nth order linear filter (controllers).

FIR filter designs with linear constraints using the eigenfilter approach

Conventionally, the eigenfilter approach has been successfully used to design a linear phase FIR filter without adding, any constraint, such as a low-pass filter, differentiator, and Hilbert transformer. When a constraint is incorporated in the eigenfilter design, only some special examples have been considered. The Nyquist filter is a typical example. In this paper, we first develop an eigenfilter approach to design a linear phase FIR filter in conjunction with general linear constraints. Then, we extend the proposed approach to the arbitrary FIR filter design with general linear constraints. Two design examples are presented to demonstrate the effectiveness of our approaches.

Approach to wavelet transformimplementation: wavelet-like filter banks

The authors present a new method for the design of wavelet-like filter banks using arbitrary lowpass filter response functions, thereby providing a flexible alternative to the implementation of the wavelet transform. The proposed wavelet-like filter banks are shown to offer superior reconstruction performance and possess the property of almost linear phase.

Design of filters for optimal pyramidal and subband decomposition

Properties are determined of the filters which achieve optimal construction of multiresolution sequences by minimising the variance of the error signals between successive pyramid levels. A measure of the entropy reduction achieved by the pyramid is in this way maximised. Further, filters are also determined which minimise the error committed when only one of the channels is retained from a multi-channel perfect reconstruction filter bank. The effect of this is to ensure that the lower-resolution image produced by the primary subband bears maximum resemblance to the input image. In both cases, it is assumed that additive transmission noise corrupts the downsampled signal prior to the synthesis stage. It is seen that under noiseless transmission conditions, the above two families of optimal analysis and synthesis filters coincide. Given arbitrary filters in one of the analysis or synthesis stages of the filter bank, the optimal corresponding synthesis and analysis filters are determined, respectively. Also the globally optimal pairs of analysis and synthesis filters are found to be ideal filters with passbands depending on the input power spectrum. The construction of realisable suboptimal approximations to the ideal filters is investigated with and without constraints on the linearity of the phase of the filters.

Finite impulse response utilizing the principle of superposition

A critical parameter in any finite impulse response (FIR) design is the impulse response length, which must be optimized for the given design specifications in order to reduce the size of the filter. To this end, many design algorithms have been introduced, such as Remez exchange, linear programming, and least mean squares. A new algorithm has been derived that is simple, efficient, and accurate for the design of arbitrary filter specifications and requires fewer computations than many other FIR approaches. This paper provides the definition of the basic functions used for the design process. An overview of the design process is given and the design technique used to design filters with tailored passband and stopband responses to yield a near-optimum time length is presented. This design can be very useful when compensating for the effects of a second transducer or other second order effects in surface acoustic wave (SAW) devices. The effects of monotonically increasing sidelobes on the impulse response length are discussed and illustrated. The addition of arbitrary phase response to the filter design process is discussed. The results of the current FIR approach are discussed and compared with other design techniques.

Design of complex FIR filters with arbitrary complex frequency responses by two real Chebyshev approximations

The well-known Parks-McClellan program can only design linear phase real FIR filters, and the Remez exchange algorithm is not readily applied for general complex FIR filter design. Here the problem of designing complex FIR filters is solved by dividing the original complex approximation into two real ones such that the Remez exchange algorithm can be applied by slightly modifying the Parks-McClellan-McCalling program. Arbitrary frequency response complex FIR filters and complex all-pass filters can be easily designed by this efficient technique. This approach has several practical advantages such as fast design time and easy implementation with satisfactory performance.

Evaluation and design of filters using a Taylor series expansion

We describe a new method for analyzing, classifying, and evaluating filters that can be applied to interpolation filters as well as to arbitrary derivative filters of any order. Our analysis is based on the Taylor series expansion of the convolution sum. Our analysis shows the need and derives the method for the normalization of derivative filter weights. Under certain minimal restrictions of the underlying function, we are able to compute tight absolute error bounds of the reconstruction process. We demonstrate the utilization of our methods to the analysis of the class of cubic BC-spline filters. As our technique is not restricted to interpolation filters, we are able to show that the Catmull-Rom spline filter and its derivative are the most accurate reconstruction and derivative filters, respectively, among the class of BC-spline filters. We also present a new derivative filter which features better spatial accuracy than any derivative BC-spline filter, and is optimal within our framework. We conclude by demonstrating the use of these optimal filters for accurate interpolation and gradient estimation in volume rendering.

Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator

We describe an incoherent correlator, based on the shadow-casting principle, that is able to implement any real-valued linear correlation filter. The correlation filter and the input image are displayed on commercial liquid-crystal television (LCTV) panels. Although it cannot handle high-resolution images, the incoherent correlator is lensless, compact, low cost, and uses a white-light source. A bipolar technique is devised to represent any linear filter, computed from a single reference image or composite, in the correlator. We demonstrate experimentally the efficiency of the design in the case of optimal trade-off (OT) filters and optimal trade-off synthetic discriminant function (OT-SDF) filters.

Irregular grids in seismic tomography and minimum-time ray tracing

Tomographic inversion of traveltimes is often carried out by discretizing the Earth as a grid of regular pixels. This choice simplifies the related ray-tracing algorithms, but contributes significantly to the non-uniqueness of the estimated velocity distribution. A singular value decomposition of the tomographic matrix enables one to recognize the causes of this mathematical ambiguity in the model space. This is more intuitive than introducing arbitrary damping factors and spatial filters, and allows one to control the non-uniqueness of solutions by modifying the pixel distribution or the acquisition geometry. This approach lends itself to the adoption of irregular grids and to the definition of a new ray-tracing algorithm, based on Fermat's principle of minimum time, which is able to simulate transmitted, reflected, refracted and diffracted waves. The joint tomographic inversion of these different types of waves potentially provides an additional improvement to the quality and reliability of the estimated velocities.

Noise analysis of low noise, high count rate, PIN diode X-ray detectors

A noise analysis for a planned PIN diode X-ray detector array using CMOS technology explores the affect of allowed device dimensions on attainable energy resolution using a methodology which allows low frequency, process dependent 1/f and 1/f/sup 2/ terms to be explicitly evaluated for arbitrary noise filters. We find that enhancement mode MOSFETs are too noisy for this application while, for peaking times /spl tau//sub p/<1 /spl mu/s and carefully processed buried channel depletion mode MOSFETs, both 1/f and 1/f/sup 2/ terms are negligible and the classical approaches to minimizing noise (reducing gate lengths and minimizing capacitances) remain valid. A concrete example demonstrates expected performance: namely a /spl Delta/E of 53 eV FWHM for a 180 ns /spl tau//sub p/ at 300 K using a 0.1 pF detector coupled to a 0.5 /spl mu/m gate length input transistor.

Optimal biorthogonal wavelet bases for signal decomposition

The selection of scaling functions for optimal signal representation by general multidimensional biorthogonal wavelet bases is investigated. Criterion for optimality is the minimization of the mean-square approximation error at each level of the decomposition. Conditions are given under which the approximation error of the decomposition approaches zero as the level increases. Given arbitrary synthesis filters, the optimal corresponding analysis filters are determined. Globally optimal families of filters are also found, and suboptimal linear and nonlinear-phase filters for the realization of the optimal scaling functions are explicitly determined.

Synthesis of coherent two-port optical delay-line circuit with ring waveguides

A method has already been reported by the author and others for synthesizing coherent two-port lattice-form optical delay-line circuits which have the same filter characteristics as finite impulse response (FIR) digital filters. This paper proposes a two-port circuit configuration with ring waveguides which can realize the same filter characteristics as infinite impulse response (IIR) digital filters. It also describes a synthesis method for realizing arbitrary IIR filter characteristics with the circuit configuration. This method is based on scattering matrix factorization. Some synthesis examples are demonstrated including an elliptic filter, a Butterworth filter, an optical filter with maximally flat group-delay characteristics, a group-delay dispersion equalizer, and a multichannel selector.

Performance of optically preamplified receivers with Fabry-Perot optical filters

A technique is presented for evaluating the probability of error for an optically preamplified receiver with a fiber Fabry-Perot optical filter of arbitrary bandwidth between the preamplifier and photodiode. The technique permits simultaneous consideration of the distortion due to modulator chirp and fiber dispersion, the optical filter and the baseband electrical filter. The optical filter is characterized by a Lorentzian passband response, and the quadrature noise processes due to the amplified spontaneous emission are represented by Karhunen-Loeve expansions. For an arbitrary optical filter bandwidth, the evaluation of the probability of error depends explicitly on the details of the Karhunen-Loeve expansion. The solution procedure is general and consequently can, in principle, be applied to other filter passband responses. The influence of the bandwidths of the optical and electrical filters on the performance of a 10 Gb/s system is assessed.

Design and implementation of linear‐phase two‐channel perfect reconstruction finite impulse response filter banks with equiripple stopband

AbstractThis paper presents a design and implementation method for the linear phase two‐channel perfect reconstruction finite impulse response (FIR) filter banks with arbitrary filter lengths. This method is labeled a weighted Lagrange‐Newton method; it is obtained by introducing the least squares weighting function into the Lagrange‐Newton method. Filter banks with good stopband attenuation (equiripple stopband) can be designed using this method. Furthermore, a new method is presented that makes the filter banks with a lattice structure using the property of a polyphase component matrix. the filter banks can achieve perfect reconstruction in spite of coefficient quantization.

Application of serial- and parallel-projection methods to correlation-filter design

We describe generalized projection procedures for the design of arbitrary filter functions for correlators. More specifically, serial and parallel implementations of projection-based algorithms are employed. The novelty of this procedure lies in its generality and its ability to handle wide varieties of constraints by the same procedure. The procedure is demonstrated by the design of filters for the 4-ƒ linear correlator, the phase-extraction correlator, and variants thereof. The filters are subject to a variety of constraints, including rotation-invariant pattern recognition and class discrimination. Examples are given to show the versatility, flexibility, and applicability of the design process to a variety of pattern-recognition tasks. Satisfactory results are also obtained because of the combination with the special nonlinear correlators proposed for pattern recognition.

The effect of the clutter-to-noise ratio on Doppler filter performance

The effect of the clutter-to-noise ratio on the performance of a Doppler filter is considered. Clutter is assumed to have a power level which is unknown and varies in range. The assessment of the performance of a Doppler filter is based on the gain of the filter, which is the normalized output signal-to-interference ratio improvement at a given Doppler. The gain is generally a complex function of the statistics of the clutter. New upper and lower bounds on the gain differential between the expected design point clutter-to-noise ratio and the actual clutter-to-noise ratio are found. These bounds are independent of the clutter covariance matrix and are only a function of the unknown clutter-to-noise ratio. The bounds are valid for both Gaussian and non-Gaussian noise and for arbitrary linear filters. The upper and lower bounds differ by the theoretical coherent integration gain, 10 logN dB, where N is the number of pulses. A tighter lower bound is found for the case when the filters are matched filters. A simple exact expression is found for matched filters assuming a Gaussian Markov clutter model as the clutter spectral width approaches zero. An easily implementable adaptive procedure is given which improves performance due to the unknown clutter-to-noise ratio. This work extends a previous result, valid for the Emerson filter, that shows the effect of clutter-to-noise ratio on performance in terms of an average quantity, the improvement factor. >

Displacements in wavelet decomposition of images

In the wavelet decomposition of an image, there may exist visual displacements in the decomposed image. Such displacements are dependent on the filter sequences used in the decomposition. In this paper, we first compute the supports of finite-length signals decomposed by arbitrary filter sequences. Then, based on the dominant terms of the filter sequences, we give analytic definitions of displacements, and obtain explicit expressions for them. We also propose a method to design filter sequences that do not produce any displacement. Some experimental results from our method for the decomposition of images are shown.

Generalized theory of the resolution of object tracking novelty filters

A general theory of the resolution of arbitrary novelty filters, optical as well as electronic, including the only existing theory as a special case, is presented. Resolution is calculated for the two essential processes of object tracking, i.e. object localization and separation of different objects. Algorithms for the reconstruction of the original objects from the output image are developed. Contour generation, appearing in the novelty filtering of moving objects, is used as a tool to maintain the input resolution. The possibility of image processing and novelty filtering in the Fourier plane is analytically verified. The application of the theory to photorefractive optical novelty filters is briefly discussed.