Linear-phase Filters Research Articles

Design of linear-phase quadrature mirror filters with powers-of-two coefficients

This paper considers the design of linear-phase quadrature mirror filters (QMF's) with powers-of-two coefficients. A design method based on a recently developed weighted least-squares (WLS) algorithm is presented. First, we design QMF's with continuous coefficients using the WLS algorithm. This design process provides two favorable design results that the prototype analysis filter has quasi-equiripple stopband response and the resulting QMF bank shows quasi-equiripple reconstruction error behavior. To avoid the effect of nonuniformly distributed coefficient grid due to discretization of the filter coefficients, a procedure is then performed for obtaining an appropriate filter gain factor. Finally, utilizing the resulting error weighting function obtained by the WLS algorithm, we propose an efficient discrete optimization process to obtain a discrete solution in the minimax optimal sense. Design examples demonstrating the effectiveness of the proposed method are included.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of FIR linear phase filters with discretecoefficients using Hopfield neural networks

A novel method is presented for designing FIR linear phase filters with discrete coefficients using Hopfield neural networks. The proposed procedure is based on the minimisation of the energy function of the Hopfield neural network, and can produce a good solution to the design of FIR linear phase filters with discrete coefficients.

A unified and efficient least-squares design of linear-phase nonrecursive filters

Abstract A method is described which can be used to design a wide class of nonrecursive linear-phase filters including those with arbitrary magnitude specifications and with time-domain constraints. The approach is based on formulating the weighted mean-square error between the amplitude responses of the practical and ideal filters as a quadratic function. The filter coefficients are obtained by solving a set of linear equations. This method leads to a lower weighted mean-square error and is computationally more efficient than both the eigenfilter method and the method based on the Remez exchange algorithm. However, the main advantage of our method lies in its computational efficiency. Design examples of many different types of filters are provided.

Investigation of a pulse compression technique for medical ultrasound: a simulation study.

Pulse compression techniques that are capable of producing a large signal-to-noise (SNR) enhancement, have been used successfully in many different fields. For medical applications, frequency-dependent attenuation in soft tissue can limit the usefulness of this method. In the paper, this issue is examined through model-simulation studies. Frequency-modulation (FM) chirp, considered in the study, is just one form of pulse coding technique. Pulse propagation effects in soft tissue are modelled as a linear zero phase filter. A method to perform simulations and estimate the effective time-bandwidth product K is outlined. K describes the SNR enhancement attainable under limitations imposed by the soft-tissue medium. An effective time-bandwidth product is evaluated as a function of soft-tissue linear attenuation coefficient alpha o, scatterer depth z and the bandwidth of the interrogating FM pulse, under realistic conditions. Results indicate that, under certain conditions, K can be significantly lower than its expected value in a non-attenuating medium. It is argued that although limitations exist, pulse compression techniques can still be used to improve resolution or increase penetrational depth. The real advantage over conventional short-pulse imaging comes from the possibility that these improvements can be accomplished without increasing the peak intensity of the interrogating pulse above any threshold levels set by possible bio-effect considerations.

Adaptive McClellan transformations for quincunx filter banks

A technique is developed for the design of 2-D nonseparable two-channel filter banks for a quincunx sampling lattice, where the isopotentials of the frequency response can be optimized and adapted to the input signal's statistics. By employing known odd-length symmetric linear phase filter banks as the l-D prototype filters for 2-D filters parameterized by the McClellan transformation, conditions are derived such that the resulting 2-D two-channel filter bank retains the perfect-reconstruction or aliasing-free properties of the 1-D prototype two-channel filter bank. A particular two-parameter transformation function is developed that has sufficient flexibility to adapt its orientation in any direction and whose optimization involves a simple constrained least-squares problem in which the feasible set lies within a circle. The results have practical applications in many areas of image and video processing where multirate filter banks are used. >

Efficient 2-D and multi-D symmetric FIR filter structures

In literature, the symmetries are usually used to reduce the number of multiplications without affecting the number of additions. The author shows that the symmetries can be exploited not only to reduce the number of multiplications as much as usual but also to reduce significantly the number of additions. Moreover, the required number of delay units remains minimum. A number of 2-D FIR filter structures are proposed for linear-phase filters having: (1) centrosymmetry, (2) quadrisymmetry, (3) octosymmetry. The same methodology is also applicable to multi-D symmetric filters. >

Generalised lapped orthogonal transforms

A class of linear-phase paraunitary filter banks is developed, possessing fast implementation algorithms based on the discrete cosine transform. In this formulation, the lapped orthogonal transform is a particular case which was extended to accommodate the overlap of any number of blocks. Optimised design examples are presented.

McClellan transform based design techniques for two-dimensional linear-phase FIR filters

This paper considers the design problem of two-dimensional (2-D) linear-phase FIR digital filters through the use of the McClellan transform method. We present two methods for determining the coefficients of the McClellan transform when designing 2-D linear-phase FIR digital filters with continuous and powers-of-two coefficients, respectively. Based on the proposed methods, the contour approximation for 1-D (one-dimensional) to 2-D digital filter transformation and finding the band-edge frequencies of the corresponding 1-D FIR digital filter can be achieved simultaneously. This results in that the required complexity of the designed 2-D digital filter satisfying the design specifications can be minimized. Moreover, the proposed methods allow us to employ the McClellan transform with order more than one for enhancing the capability of the original McClellan transform. Considering the determination of the McClellan transform coefficients and the band-edge frequencies for the design, we formulate the design problem as a linear programming optimization problem. Then, an efficient design procedure is presented to avoid the use of time consuming simplex algorithms. For the powers-of-two coefficient design, a discrete design procedure is presented to avoid the use of time-consuming mixed integer linear programming algorithms. Computer simulations for showing the effectiveness of the proposed design techniques are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Frequency dependent transmission line loss in quasitransversal microwave filters

A previously proposed synthesis procedure allows filters with almost arbitrary phase and amplitude responses to be designed as delay-line filters. Multiple reflections within the delay line are efficiently allowed for using a time-domain algorithm. The procedure has now been improved to allow for inherent frequency-dependent loss and dispersion in the transmission line. This overcomes a significant potential limitation of the synthesis method, particularly for devices with large TB product. Simulated results, in which synthesised filters are analysed using standard transmission matrices, are presented for chirp filters, a linear phase filter and a binary pseudo-noise generator.

Simple and robust method for the design of allpass filters using least-squares phase error criterion

We consider a simple scheme for the design of allpass filters for approximation (or equalization) of a given phase function using a least-squares error criterion. Assuming that the desired phase response is prescribed at a discrete set of frequency points, we formulate a general least-squares equation-error solution with a possible weight function. Based on the general formulation and detailed analysis of the introduced error, we construct a new algorithm for phase approximation. In addition to iterative weighting of the equation error, the nominal value of the desired group delay is also adjusted iteratively to minimize the total phase error measure in equalizer applications. This new feature essentially eliminates the difficult choice of the nominal group delay which is known to have a profound effect on the stability of the designed allpass filter. The proposed method can be used for highpass and bandpass equalization as well, where the total phase error can be further reduced by introducing an adjustable-phase offset in the optimization. The performance of the algorithm is analyzed in detail with examples. First we examine the approximation of a given phase function. Then we study the equalization of the nonlinear phase of various lowpass filters. Also, a bandpass example is included. Finally we demonstrate the use of the algorithm for the design of approximately linear-phase recursive filters as a parallel connection of a delay line and an allpass filter.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A stereo audio sigma-delta A/D-converter

A stereo sigma delta A/D-converter for audio applications is presented. In this converter, two identical cascaded fourth-order sigma-delta modulators and a sophisticated multistage linear-phase FIR decimation filter with oversampling ratio of 64 are implemented on the same die. The analog part is designed to operate at a low voltage with a low power consumption. Techniques to achieve simultaneously a high performance and a low power consumption are discussed in details. The minimum stopband attenuation of the decimator is more than 120 dB and the passband ripple of the overall converter is less than 0.0003 dB. The first decimation stage is a special tapped comb filter, whereas the remaining stages are realized without general multipliers by simultaneously implementing all the filter coefficients by using special bit-serial networks. For the integrated overall stereo converter, the power consumption and the signal-to-noise ratio are 180 mW and 97 dB (85 mW and 95 dB) for a 5 V (3 V) power supply. The circuit die area is only 4.7 mm/spl times/5.5 mm using a 1.2 /spl mu/m double-poly BiCMOS process.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A digital filter chip for ECG signal processing

A VLSI implementation of a linear-phase digital filter for ECG signal processing has been designed. With a sampling rate of 100 Hz, the passband is from 0.5 Hz to 49.5 Hz with 0.5-dB ripple. The filter architecture is based on the use of recursive running-sum blocks, resulting in a very low computational complexity. Module generators have been used in the layout design for high integration density. The circuit has been designed for a 2.0-/spl mu/m double-metal CMOS technology, having about 34000 transistors and a 15.43-mm/sup 2/ chip area.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A unified approach to the design of quadrantally symmetric linear-phase two-dimensional FIR digital filters by eigenfilter approach

Instead of the design procedures restricted to same special filter types, a unified approach is proposed to design sixteen possible cases for quadrantally symmetric linear-phase 2-D FIR filters. The original forms of the sixteen possible cases are formulated properly such that the eigenfilter method can design them with the same approach. The conversions between the filter coefficients are tabulated in a complete table. Also two numerical examples are shown to demonstrate the usefulness and flexibility of the present approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Unitary FIR filter banks and symmetry

Recently, unitary FIR filter banks with linear-phase have been completely parameterized by exploiting the eigenstructure of the exchange matrix. This correspondence gives new characterizations of polyphase matrices of filter banks with various types of symmetry on the filters. Using matrix extensions of the well-known hyperbolic and orthogonal lattices, we give a alternative proof for the parameterization of linear-phase unitary filter banks. A complete parameterization of FIR unitary filter banks with each of the different types of symmetries considered (not just linear-phase) is also given. These results can also be used to generate non-unitary filter banks with symmetries, though no completeness results can be obtained. In some cases implicit, and in others explicit parameterization of wavelet tight frames associated with these filter banks are also given. This paper only considers filter banks with an even number of channels. A similar theory can be developed if the number of channels is an odd integer. >

An Approach to Realizing Linear Phase Filters Based on the Lagrangean B-Filters

An Approach to Realizing Linear Phase Filters Based on the Lagrangean B-Filters

Optimization approach to the design of frequency sampling filters

Many digital signal processing applications require linear phase filtering. For applications that require narrow-band linear phase filtering, frequency sampling filters can implement linear phase filters more efficiently than the commonly used direct convolution filter. In this paper, a technique is developed for designing linear phase frequency sampling filters. A frequency sampling filter approximates a desired frequency response by interpolating a frequency response through a set of frequency samples taken from the desired frequency response. Although the frequency response of a frequency sampling filter passes through the frequency samples, the frequency response may not be well behaved between the specific samples. Linear programming is commonly used to control the interpolation errors between frequency samples. The design method developed in this paper controls the interpolation errors between frequency samples by minimizing the mean square error between the desired and actual frequency responses in the stopband and passband. This design method describes the frequency sampling filter design problem as a constrained optimization problem which is solved using the Lagrange multiplier optimization method. This results in a set of linear equations which when solved determine the filter's coefficients.

Optimal and suboptimal design of SAW bandpass filters using the Remez exchange algorithm

Optimal and suboptimal design techniques for surface acoustic wave (SAW) linear phase filters based on both the Remez exchange algorithm and the McClellan computer program (J.H. McClellan et al., 1973) are considered. The optimal synthesis provides uniquely the best fit to a design target, but its major drawback is an excessive amount of computation. The suboptimal synthesis technique proposed allows a considerable reduction of the amount of computations without significantly sacrificing the approximation accuracy. Thus computer runtime and storage are greatly reduced compared to the optimal synthesis. The detailed suboptimal theory and some practical design aspects are discussed. Design examples are presented which confirm the efficiency and the flexibility of the synthesis techniques proposed.

An optimization approach to the design of frequency sampling filters implemented with finite word length architectures

Many digital signal processing applications require linear phase filtering. Under certain conditions, a frequency sampling filter can implement a linear phase filter more efficiently than an equivalent filter implemented by a direct convolution structure. However, frequency sampling filter structures require pole-zero cancellations on the unit circle. For practical implementations of frequency sampling filters, finite word length effects usually prevent exact pole-zero cancellation. An uncancelled pole on the unit circle causes the filter to be unstable. Therefore, designers move the filter's poles and zeros off the unit circle and onto a circle of radius r where r < 1, by replacing z −1 with rz −1 in the filter's system function. As a result of this modification, the frequency response of the new filter using rz −1 where r < 1 is different from the frequency response of the original filterwhere r = 1. As r decreases, the frequency response of the modified frequency sampling filter increasingly differs from the frequency response of the original filter, and thus values of r close to 1 are usually chosen. However, it has been shown that the filter's output roundoff noise decreases as r decreases. Thus, there is a need for a design technique which chooses a value of r which renders an acceptable roundoff noise level while satisfying various frequency response design constraints. This paper develops an optimization method for designing a frequency sampling filter with r < 1 such that a linear combination of the mean square error between the desired and actual frequency responses in the passband and stopband and the sum of the square error of the impulse response symmetry is minimized while the stopband frequency samples are constrained to zero.

Linear phase paraunitary filter banks: theory, factorizations and designs

M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. The authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. They begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, they develop a minimal factorization for a large class of such systems. This factorization will be proved to be complete for even M. Further, they structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. They then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory. >

Optimization of linear multiresolution transforms for scene adaptive coding

Scene adaptive coders are constituted by the cascade of a linear transform, scalar quantization, entropy coding and a buffer controlled by a feedback loop for bit rate regulation. The main contribution of this paper is to derive an analytical criterion evaluating the performances of any perfect reconstruction linear transform in the frame of scene adaptive coding. This criterion is, thereafter, used in order to optimize linear multiresolution transforms. The optimization adapts the filters parameters to the codec features and to the statistics of the 2-D sources; so, the authors call these transforms adapted multiresolution transforms (AMTs). The transforms under study are implemented by a cascade of separable perfect-reconstruction (PR) FIR two-band filter banks that can change at each resolution level. Two types of filter banks are envisaged: the PR orthogonal quadrature mirror filter (QMF) bank, which allows to implement the orthogonal AMT and the PR linear-phase filter (LPF) bank, which implements the biorthogonal AMT. They perform the optimization of the filters in their factorized lattice form, taking the finite length of the multipliers into account. Their criterion also allows them to show the performances achieved by these two linear multiresolution transforms compared to other linear (multiresolution) transforms. >