Analytical design of 3-D wavelet filter banks using themultivariate Bernstein polynomial

The design of 3-D multirate filter banks where the downsampling/upsampling is on the FCO (face centred orthorhombic) lattice is addressed. With such a sampling lattice, the ideal 3-D sub-band of the low-pass filter is of the TRO (truncated octahedron) shape. The transformation of variables has been shown previously to be an effective technique for designing M-D (multidimensional) filter banks. A design technique is presented for the transformation function using the multivariate Bernstein polynomial which provides a good approximation to the TRO sub-band shape. The method is analytically based and does not require any optimisation procedure. Closed form expressions are obtained for the filters of any order. Another advantage of this technique is that it yields filters with a flat frequency response at the aliasing frequency (ω₁, ω₂ , ω₃)=(π, π, π). This flatness is important for giving regular discrete wavelet transform systems     

Flexible design of multidimensional perfect reconstruction FIR2-band filters using transformations of variables

An approach to designing multidimensional linear-phase FIR diamond subband filters having the perfect reconstruction property is presented. It is based on a transformation of variables technique and is equivalent to the generalized McClellan transformation. Methods for designing a whole class of transformation are given. The approach consists of two parts; design of the transformation and design of the 1-D filters. The use of Lagrange halfband filters to design the 1-D filters is discussed. The modification of a particular Lagrange halfband filter which gives a pair of simple 1-D filters that are almost similar to each other in their frequency characteristics but still form a perfect reconstruction pair is presented. The design technique is extended to other types of two-channel sampling lattice and subband shapes, in particular, the parallelogram and the diagonally quadrant subband cases. Several numerical design examples are presented to illustrate the flexibility of the design method.     

Design of some 2-D filters through the transformation technique

The transformation technique is a powerful tool for designing 2-D FIR filters. However, it is not useful for the design of specially shaped filters with passband/stopband regions not centred around the origin. The authors extend this technique to design two types of filters. A notch filter has a stopband centred about a small region in the 2-D frequency plane. The authors propose an extension to the transformation technique with the windowing concept to achieve the design of notch filters. A directional filter has a passband extending fully along a straight line passing through the origin. The transformation technique is further extended to yield such directional filters. Design and application examples for both these filters are also presented     

Modern network theory design of single-sideband crystal ladder filters

This paper presents modern network theory design data (rather than image parameter data) for one class of ladder network SSB crystal filters. The filter configuration involved uses crystals and capacitors only, so that the physical size of the resulting filters can be made quite small. A simple frequency transformation is first derived which enables the wealth of design data presently available for symmetrical response shape filters to be applied to the SSB response under consideration. It is shown that the transfer function being considered has n nonconjugate complex poles and n coincident zeros in the fractional bandwidth left half-plane. From the frequency transformation developed, graphs are presented for n-pole, n-coincident-zero SSB relative-attenuation shapes for n = 6, 8, 10, and 12, with a pass band peak-to-valley ratio of zero decibels; and a computation example shows how graphs may be prepared for any n, and any pass band peak-to-valley ratio. Specific design equations are presented for all the elements involved in both the upper and lower SSB filter structures, and from these an equation is derived which shows how crystal units limit the maximum fractional bandwidth which can be obtained.     

Design of Variable Two-Dimensional FIR Digital Filters by McClellan Transformation

In this paper, the technique of McClellan transformation is applied to design variable 2-D FIR digital filters. Compared with the conventional transformation, the 2-D transformation subfilter and the 1-D prototype filter are designed such that their frequency characteristics are adjustable. Moreover, they are tunable by the same variable parameter, so the variable characteristics of 1-D prototype filters are coincident with those of 2-D subfilters. Several examples, including variable fan filters, variable circularly symmetric filters, and variable elliptically symmetric filters with arbitrary orientation, are presented to demonstrate the effectiveness and the flexibility of the presented method.      

Variable digital filter design using the outer product expansion

Digital filters with adjustable frequency domain characteristics are referred to as variable digital filters. Variable filters are useful in the applications where the filter characteristics are required to be changeable during the course of signal processing. Especially in real time applications, variable filters are needed to change their coefficients instantaneously such that the real time signal processing can be performed. The present paper proposes a very efficient technique for variable 1D digital filter design. Generally speaking, the variable coefficients of variable digital filters are multidimensional functions of a set of spectral parameters which define the desired frequency domain characteristics. The authors first sample the given variable 1D magnitude specification and use the samples to construct a multidimensional array, then propose an outer product expansion method for expanding the multidimensional array as the sum of outer products of 1D arrays (vectors). Based on the outer product expansion, one can reduce the difficult problem of designing a variable 1D digital filter to the easy one that only needs constant 1D filter designs and 1D polynomial approximations. The technique can obtain variable 1D filters having arbitrary desired magnitude characteristics with a high design accuracy     

Design and performance of FSK-direct detection scheme for opticalFDM systems

The authors describe the design and performance of an optical frequency-division-multiplexing (FDM) system that uses a frequency-shift-keying (FSK) direct detection scheme. This system utilizes a Mach-Zehnder type periodic filter as a channel selective filter and optical frequency discriminator. Requirement conditions for laser diode linewidth, frequency deviation, channel frequency spacing, detuning of optical filters, and signal power variance are discussed. These characteristics are experimentally confirmed using planar waveguide type filters     

Distributed Bragg reflector active optical filters

The characteristics of active optical filters based on distributed Bragg reflector (DBR) laser diodes are described. The authors present a theoretical model for the small and large signal filter response which can be used to optimize the filter design and operating point. The large signal filter response is dominated by a variety of nonlinear effects, including a wavelength domain bistability. The authors describe a series of experiments in which the predicted filter characteristics are demonstrated. The authors present detailed measurements of the filter transfer function in both transmission and reflection modes under low and high power injection. For the small signal measurements a powerful technique which uses the swept frequency concept is presented. For the large signal regime, measurements of both frequency pulling and wavelength domain bistability are presented     

Optical half-band filters

This paper proposes two kinds of novel 2×2 circuit configuration for finite-impulse response (FIR) half-band filters. These configurations can be transformed into each other by a symmetric transformation and their power transmittance is identical. The configurations have only about half the elements of conventional FIR lattice-form filters. We derive a design algorithm for achieving desired power transmittance spectra. We also describe 2×2 circuit configurations for infinite-impulse response (IIR) half-band filters. These configurations are designed to realize arbitrary-order IIR half-band filter characteristics by extending the conventional half-band circuit configuration used in millimeter-wave devices. We discuss their filter characteristics and confirm that they have a power half-band property. We demonstrate design examples including FIR maximally flat half-band filters, an FIR Chebyshev half-band filter, and an IIR elliptic half-band filter     

A Frequency Transformation for Commensurate Transmission-Line Networks

The frequency transformation W=1/S, where S= tanh(/spl gamma/L), is investigated for commensurate transmission-line networks consisting of stubs, resistors, ideal transformers, and unit elements. This transformation takes transmission-line transformers into transmission-line lowpass filters and vice versa, Iowpass (or bandstop) filters into highpass (or band-pass) filters and vice versa, and elliptic-function bandstop filters into elliptic-function bandpass filters and vice versa. The practicality of the transformation lies in the fact that element values of the transformed network are easily related to the corresponding element values of the original network. The transformation is useful because it provides an alternative viewpoint for synthesis, and because it reduces the number of tables of designs needed for various filter types. Several examples of designs using the transformation are given. One design is an unusual narrowband 3-dB directional coupler.     

Design of zero-phase recursive 2-D variable filters with quadrantal symmetries

The digital filters with adjustable frequency-domain characteristics are called variable filters. Variable filters are useful in the applications where the filter characteristics are needed to be changeable during the course of signal processing. In such cases, if the existing traditional constant filter design techniques are applied to the design of new filters to satisfy the new desired characteristics when necessary, it will take a huge amount of design time. So it is desirable to have an efficient method which can fast obtain the new desired frequency-domain characteristics. Generally speaking, the frequency-domain characteristics of variable filters are determined by a set of spectral parameters such as cutoff frequency, transition bandwidth and passband width. Therefore, the characteristics of variable filters are the multi-dimensional (M-D) functions of such spectral parameters. This paper proposes an efficient technique which simplifies the difficult problem of designing a 2-D variable filter with quadrantally symmetric magnitude characteristics as the simple one that only needs the normal one-dimensional (1-D) constant digital filter designs and 1-D polynomial approximations. In applying such 2-D variable filters, only varying the part of 1-D polynomials can easily obtain new desired frequency-domain characteristics.     

A singular value decomposition derivation in the discrete frequencydomain of optimal noncentro-symmetric 2-D FIR filters

A new derivation is presented for the least squares solution of the design problem of two-dimensional (2-D) finite impulse response (FIR) filters by minimizing the Frobenius norm of the difference between the matrices of the ideal and actual frequency responses sampled at the points of a frequency grid. The mathematical approach is based on the singular value decomposition (SVD) of two complex transformation matrices. Interestingly, the designed filter is proved to be zero-phase if the ideal filter is so without assuming any kind of symmetry     

Digital Filter Design Techniques in the Frequency Domain

The frequency domain of wide-band linear sampleddata filters is considered. The sampled-data filter is termed ?wideband? when the frequency range of useful approximation to its continuous counterpart approaches half the sampling frequency. Sampled-data filter representations for continuous filters can be obtained using several different design procedures.[1] A particular design method utilizing the bilinear transformation is developed. The method is especially useful in designing wide-band sampleddata filters which exhibit relatively flat frequency-magnitude characteristics in successive pass- and stop-bands. Filters of this type are widely used in network simulation and data processing problems.[2] The design method possesses two chief advantages over the standard z transform.[3] The first is that the transformation used is purely algebraic in form. This means it can be applied easily to a continuous filter having a rational transfer characteristic expressed in either polynomial or factored form. The second advantage is the elimination of aliasing[4] errors inherent in the standard z transform. Thus, the sampled-data filter obtained by this design method exhibits the same frequency response characteristics as the continuous filter, except for a nonlinear warping of the frequency scale. Compenation for this warping can be made by a suitable frequency scale modification. Some of the more common filter networks to which the design method can be applied effectively are the Butterworth, Bessel, Chebyshev, and elliptic-filter structures. The design method consists first of obtaining a rational transfer characteristic for a continuous filter that satisfies the design specifications.     

A new approach to the design of critically sampled M-channeluniform-band perfect-reconstruction linear-phase FIR filter banks

A new approach is presented for the design of uniform-band M-channel perfect-reconstruction (PR) FIR filter banks employing linear-phase analysis and synthesis filters. The technique designs on the impulse responses of the analysis filters directly. The design problem is formulated as an optimization program. The filter bank's PR feature can either be implicitly enforced through a set of mathematical relationships among the analysis filters' coefficients or through a set of constraints in the optimization program. The former approach results in a filter bank whose PR feature's dependency on hardware and software is eliminated or, at least, minimized. The synthesis filters are then obtained by a set of relationships that describe each synthesis filter as a function of the analysis filters. The criterion for optimality is “least-squares,” where the square of the difference between the ideal and actual frequency responses is integrated over the appropriate frequency bands for all M analysis filters and minimized     

An Adjustable Quasi-Optical Bandpass Filter---Part I: Theory and Design Formulas

A quasi-optical bandpass filter suitable for millimeter and submilliieter wavelengths and in the far infrared region is described. It consists of three or more wire-grid polarizers with quarter-wave spacings. The filter has the advantage over conventional quasi-optical filters, e.g., Fabry-Perot filters, that its bandwidth and the shape of its frequency response are adjustable. This is achieved by changing the angular orientations of the wires of the different polarizers. The filter requires the input electric field to be linearly polarized in a direction perpendicular to the wires of the first grid. The theory of operation is presented and design formulas for the filter are given, under the assumption that ideal wire-grid polarizers are employed. The effects of using realistic grids on the performance of the filter are dealt within another paper.     

An approximation technique for the design of recursive digital and continuous filters with and without fixed zeros in the stop band

The bilinear z-transformation has been used by various authors (Golden and Kaiser 1964, Broomer 1966) to design recursive digital filters. The necessity to select an equivalent. continuous filter makes this method very laborious. Remes exchange algorithm has also been used in the design of recursive filters (Remes 1957, Deczky 1974) but not will the same success as for the design of non-recursive filters (McClennan and Parks 1973. Holt et al. 1975 ) due to its dependence on the initial conditions on optimization methods. In this paper we-present a design technique, based on Remes second exchange algorithm. This provides a versatile approximation method for the design of a wide range of minimax-type digital recursive filters. Furthermore, the method is extended to the design of continuous filters using suitable mapping methods. In the examples given, filters arc designed with various frequency characteristics for the pass band and the stop band and also with predefined zeros in the stop band.     

Optimized sub-band adaptive AEC using ear auditory feature

Acoustic echo cancellation based on sub-band filters has the characteristics of rapid con- vergence and small computational complexity. This letter analyses two different sub-band filters design methods which used in acoustic echo cancellation fields and compares them with each other. Fur- thermore, the sub-band filter construction have been optimized, which lead to the improvement of the computational efficiency. At the same time, this letter combines ear auditory feature with acoustic echo cancellation, thus improves the original algorithms by importing a new objective function creatively. At the last part, a simulation environment has been designed and a computer simulation has been carried out. The final results indicate that this method can meet the requirements of actual projects, and some improvements are demonstrated on performance and calculation quantity compared to original algo- rithms.     

Minimax design of two-dimensional parallelogram filter banks

The authors present a technique for the minimax design of two-dimensional (2-D) parallelogram filter bank (PFB) systems with linear-phase analysis/synthesis filters. To achieve perfect reconstruction, the required analysis filters must have parallelogram-shaped frequency responses. In general, the original design problem is found to be an optimisation problem with nonlinear constraints. The authors present a linearisation approach to reformulate the design problem. As a result, updating the filter coefficient vector at each iteration for the original design problem can be accomplished by searching the gradient of the linearised optimisation problem. They further present an efficient method based on a modified Karmarkar's algorithm for computing the required gradient vector and finding the required step size analytically. Therefore the filter coefficients can easily be computed by solving only linear equations at each iteration during the design process. The effectiveness of the proposed technique is shown by computer simulations     

Design of efficient M-band coders with linear-phase andperfect-reconstruction properties

This paper presents a new design technique for obtaining M-band orthogonal coders where M=2ⁱ. The structures obtained using the proposed technique have the perfect reconstruction property. Furthermore, all filters that constitute the subband coder are linear-phase FIR-type filters. In contrast with conventional design techniques that attempt to find a unitary alias-component matrix in the frequency domain, we carry out the design in the time domain, based on time-domain orthonormality constraints that the filters must satisfy. The M-band design problem is reduced to the problem of finding a suitable lowpass filter h₀(n). Once a suitable lowpass filter is found, the remaining (M-1) filters of the coder are obtained through the use of shuffling operators on the lowpass filter. This approach leads to a set of filters that use the same numerical coefficient values in different shift positions, allowing very efficient numerical implementation of the subband coder. In addition, by imposing further constraints on the lowpass branch impulse response h₀(n), we are able to construct continuous bases of M-channel wavelets with good regularity properties. Design examples are presented for four-, eight-, and 16-band coders, along with examples of continuous wavelet bases that they generate     

A Fast Windowing-Based Technique Exploiting Spline Functions for Designing Modulated Filter Banks

A very fast technique to design prototype filters for modulated filter banks without using time-consuming multivariable optimization is introduced. In the proposed method, the prototype filter is optimized by using the windowing technique, with the novelty of exploiting spline functions in the transition band of the ideal filter, instead of using the conventional brick-wall filter. A study of the optimization techniques and three different objective functions existing in the literature has been carried out, and more suitable redefinitions of these objective functions are employed to achieve as optimized prototype filters as possible. The resulting filter banks closely satisfy the perfect reconstruction property, as is illustrated by means of examples.