Weighted Least-squares Design Research Articles

Matrix-Based Algorithms for the Optimal Design of Variable Fractional Delay FIR Filters

This paper investigates the weighted least squares (WLS) and minimax design problems for variable fractional delay (VFD) FIR filters. For the WLS design, the general case of incorporating arbitrary nonnegative weighting functions is considered. The optimal solution is characterized by two matrix equations. An efficient algorithm using conjugate gradient (CG) techniques is proposed to solve the WLS solution. The proposed algorithm is guaranteed to converge to the optimal solution in a finite number of iterations. Moreover, an iterative reweighted least squares (IRLS) algorithm that uses the proposed CG algorithm as its iteration core is developed for the minimax design problem. In both of the algorithms, the filter coefficients are arranged as matrices, achieving a great saving in computations and memory space. The associated computational complexity is analyzed. Some design examples are provided and comparisons with existing methods show that the proposed ones are either computationally more efficient or can obtain the better filter performance, or both.

Minimax and WLS Designs of Digital FIR Filters Using SOCP for Aliasing Errors Reduction in BI-DAC

This paper presented the optimal minimax and weighted least squares (WLS) methods for designing digital finite impulse response (FIR) filters to reduce the aliasing errors generated by the non-ideality of analog filters and mixers in bandwidth interleaving digital-to-analog converter (BI-DAC). To satisfy the given expected spurious free dynamic range (SFDR), we formulated these optimal designs of digital FIR filters in BI-DAC as a convex optimization problem-second-order cone programming (SOCP) that allowed the linear equality and convex quadratic inequality constraints including the magnitude flatness and the peak aliasing errors constraints to be merged. Furthermore, we derived the computational complexity of our presented optimal design. Several design examples were given to evaluate the performance of our presented unconstrained and constrained minimax and WLS designs using SOCP including their effectiveness and computational complexity. The simulation results showed that, in our presented unconstrained minimax and WLS designs using SOCP, the maximum distortion errors were all around 0.02 dB. The maximum aliasing errors were-73.9 and -80.5 dB, which satisfied the expected SFDR of a 12-bit BI-DAC system. In addition, we analyzed the influence of different values of the nonnegative weighting function on our presented unconstrained minimax and WLS designs using SOCP, and we found that there was a tradeoff among the nonnegative weighting function's value, and the distortion and aliasing errors. Moreover, when the constraints were imposed in our presented constrained minimax and WLS designs using SOCP in the selected frequency bands, the distortion errors were equal to zero and the aliasing errors were reduced below -110 dB, but the expense was that the larger distortion and aliasing errors achieved out of these selected frequency bands. Finally, we gave the computational complexity comparisons among our presented unconstrained and constrained minimax and WLS design using SOCP, we also compared the influence of the digital FIR filters' length on our presented designs' worst-case passband ripple and stopband roll-off, and we found that there was a tradeoff among the digital FIR filters' length, the passband ripple, the stopband roll-off, the computational complexity, and the actual hardware cost.

The Lp-Norm-Minimization Design of Stable Variable-Bandwidth Digital Filters

This paper proposes a two-step strategy for designing a variable-bandwidth (VBW) digital filter through minimizing the [Formula: see text]-norm of the magnitude-response error. This [Formula: see text]-norm design can be regarded as a generalized version of the existing weighted-least-squares (WLS) design. Equivalently, the WLS design is a special case of the [Formula: see text]-norm-minimization design for [Formula: see text]. This paper discusses the design of the recursive VBW filter with the transfer function whose denominator is expressed as the product of the second-order sections. As long as all the second-order sections are stable, the recursive VBW filter is also stable. To ensure that the designed recursive VBW filter is stable, we adopt the coefficient-conversion strategy that constrains all the denominator-parameter pairs of the second-order sections within the stability triangle. This paper also proposes a novel conversion function for performing the coefficient conversion. As a consequence, the designed VBW filter is definitely stable. A bandpass VBW filter is designed for showing the feasibility of the [Formula: see text]-norm-minimization-based design and verifying the stability guarantee.

Simple derivation and closed-form frequency-domain design of wideband length-8 interpolation kernel

This paper proposes a frequency-domain method for designing a wide-band length-8 interpolator in the weighted-least-squares (WLS) sense. The length-8 interpolator is even-symmetric and it consists of the fourth-order piecewise polynomials. In this paper, we first present a simple method for deriving its closed-form frequency response, and then formulate a frequency-domain WLS design with the C0-continuity constraint. Since the length-8 interpolator contains a total of 20 parameters, and the C0-continuity imposes 8 constraints on the design, the remaining 12 parameters can be taken as free parameters. After expressing the frequency response as the function of the 12 free parameters, the optimal free parameters can be found in the frequency-domain by minimizing the weighted squared error between the ideal and actual frequency responses, where a non-negative weighting function is adopted for emphasizing important frequency bands and ignoring unimportant bands. Consequently, the WLS design can yield an extremely accurate wide-band length-8 interpolator. Three interpolation examples are given to verify that the designed length-8 wide-band interpolator can achieve the highest accuracy in both the frequency-domain and time-domain as compared to other existing interpolators.

Neural Network-Based IIR All-Pass Filter Design

This paper presents a neural network-based Lyapunov energy function for the weighted least-squares design of IIR all-pass filters. In the proposed method, the error reflecting the difference between the desired phase response and the phase of the designed IIR all-pass filter is formulated as a Lyapunov error criterion. Based on the neural network architecture and suitable Hopfield parameters, the optimal filter coefficients can be obtained when convergence is achieved. Furthermore, a weight updating function is proposed to achieve accurate approximation of the equiripple response. The simulation results indicate that the proposed technique can achieve high performance in a parallel manner.

Efficient 2-D based algorithms for WLS designs of 2-D FIR filters with arbitrary weighting functions

The impulse response coefficients of a two-dimensional (2-D) finite impulse response (FIR) filter naturally constitute a matrix. It has been shown by several researchers that, two-dimension (2-D) based algorithms that retain the natural matrix form of the 2-D filter's coefficients are computationally much more efficient than the conventional one-dimension (1-D) based algorithms that rearrange the coefficient matrix into a vector. In this paper, two 2-D based algorithms are presented for the weighted least squares (WLS) design of quadrantally symmetric 2-D FIR filters with arbitrary weighting functions. Both algorithms are based on matrix iterative techniques with guaranteed convergence, and they solve the WLS design problems accurately and efficiently. The convergence rate, solution accuracy and design time of these proposed algorithms are demonstrated and compared with existing algorithms through two design examples.

WLS design of FIR Nyquist filter based on neural networks

In this paper, we devised an analog circuit for the weighted least-squares (WLS) design of FIR Nyquist filters by using a Hopfield neural network (HNN). The approach is based on formulating the error function in the optimization of the FIR Nyquist filter as a Lyapunov energy function to find the Hopfield related parameters. By using these parameters and input to the network, the optimal filter coefficients of the FIR Nyquist filter can be derived when the network achieves its convergence. The proposed technique is regular and simple to implement the problems of filter optimization without having the convergence problem as compared to the previous neural-based method. Additionally, the structure proposed can be implemented by using analog VLSI technology in real-time. Simulation results are offered as a suggestion for illustrating the usability of the new method.

Weighted least squares near-equiripple approximation of variable fractional delay FIR filters

The weighted least squares (WLS) method is a well-known method for designing a finite impulse response (FIR) filter. And some authors have reported that if a suitable frequency response weighting function is used to design the filter, the WLS method can produce an equiripple result. However, the weighting function for minimax optimality of WLS design is hard to derive analytically. By an iterative method with an adjustable elaborately constructed weighting function, this idea is extended to design a near-equiripple variable fractional delay FIR filter. The proposed method is superior to the fixed-weighting WLS design in the peak absolute error by about 6.6874 dB. The algorithm converges very rapidly. From the simulation, it typically produces a design which is only about 1 dB away from the truly equiripple solution in two iterations and converges to within 0.0056 dB in eight iterations.

Weighted-Least-Squares Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetry

Our previous work has shown that the coefficient symmetry can be efficiently exploited in designing variable finite-impulse-response (FIR) filters with simultaneously tunable magnitude and fractional-delay responses. This paper presents the optimal solutions for the weighted-least-squares (WLS) design of variable fractional-delay (VFD) FIR filters with same-order and different-order subfilters through utilizing the coefficient symmetry along with an imposed coefficient constraint. In deriving the closed-form error functions, since the Taylor series expansions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$sin(omega p)$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$cos(omega p)$</tex> are used, the numerical integrals using conventional quadrature rules can be completely removed, which speeds up the WLS design and guarantees the optimality of the final solution. Two design examples are given to illustrate that the proposed WLS methods can achieve better design with significantly reduced VFD filter complexity and computational cost than the existing ones including the WLS-SVD approach. Consequently, the proposed WLS design is the best among all the existing WLS methods so far.

Design and Experimental Evaluation of Improved Least Squares and Weighted Least Squares Quadrature Mirror Filters

The least squares (LS) and the weighted least squares (WLS) algorithms are well known procedures that are used in the design of quadrature mirror filters (QMFs). It is known that these design techniques suffer from pass-band anomaly under certain conditions. A recent method, that overcomes pass-band anomaly for LS QMFs using a frequency sampling design for the initial filter, is extended to WLS design in this letter. A comparison between the modified LS and WLS designs based on experimental results is presented. Although WLS designs have been reported to have superior near-equiripple stop-band performance as compared to LS designs, we find that this is not always true. Specifically, LS designs, with inherent computational and robustness advantages, also have better peak stop-band ripple and transition bandwidth at higher cut-off frequencies than WLS.

SVD-Based Design and New Structures for Variable Fractional-Delay Digital Filters

This paper shows that the problem of designing one-dimensional (1-D) variable fractional-delay (VFD) digital filter can be elegantly reduced to the easier subproblems that involve one-dimensional (1-D) constant filter (subfilter) designs and 1-D polynomial approximations. By utilizing the singular value decomposition (SVD) of the variable design specification, we prove that both 1-D constant filters and 1-D polynomials possess either symmetry or anti-symmetry simultaneously. Therefore, a VFD filter can be efficiently obtained by designing 1-D constant filters with symmetrical or antisymmetrical coefficients and performing 1-D symmetrical or antisymmetrical approximations. To perform the weighted-least-squares (WLS) VFD filter design, a new WLS-SVD method is also developed. Moreover, an objective criterion is proposed for selecting appropriate subfilter orders and polynomial degrees. Our computer simulations have shown that the SVD-based design and WLS-SVD design can achieve much higher design accuracy with significantly reduced filter, complexity than the existing WLS design method. Another important part of the paper proposes two new structures for efficiently implementing the resulting VFD filter, which require less computational complexity than the so-called Farrow structure.

Improved weighted least squares algorithm for the design of quadrature mirror filters

This paper presents an improved weighted least squares (WLS) algorithm for the design of quadrature mirror filters (QMFs), First, a new term is incorporated into the objective function that effectively prevents an optimization algorithm from producing suboptimal QMFs. These suboptimal QMFs exhibit a transition band anomaly; the frequency responses of the filters have large oscillatory components in the transition band. The new term can be applied to the WLS design of any FIR filter to prevent a similar transition band anomaly. Next, we present an algorithm to obtain the QMF coefficients that minimize the objective function incorporating the new term. The computational requirement of this algorithm is also briefly discussed. Last, we include a set of practical design rules for use with our algorithm. These rules simplify the design process by providing good estimation of the design parameters, such as the minimum filter length, to meet a given set of QMF specifications.

Computationally efficient weighted least-squares design of FIR filters satisfying prescribed magnitude and phase specifications

An efficient method for the design of FIR filters satisfying prescribed magnitude and phase specifications is described. The method involves formulating an objective function in a quadratic form. The filter coefficients are obtained by solving a system of linear equations which involves either a Toeplitz matrix or a Toeplitz-plus-Hankel matrix. The computational complexity associated with such systems is only O( N 2).

An efficient weighted least-squares design of linear-phase nonrecursive filters

An accelerated procedure for the design of linear-phase nonrecursive filters using a weighted least-squares technique is described. This procedure is based on formulating the error reflecting the difference between the desired amplitude response and the amplitude response of the practical filter in a quadratic form. The coefficients of the filter are obtained by solving a system of linear equations involving a Toeplitz-plus-Hankel matrix. Such a system of linear equations can be solved by computationally efficient algorithms having only O(N/sup 2/) complexity. By choosing the appropriate frequency-dependent weighting function, a filter with either a least-squares or an equiripple error variation can be designed. >

Three different criteria for the design of two-dimensional zero phase FIR digital filters

An error criterion for the design of FIR filters is proposed. Filters with relatively many free filter coefficients are designed using the Chebyshev, the weighted-least-squares (WLS), and a new partitioned minimax error criterion, and the performance of the filters is compared. A general and fast technique for the WLS design is also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>