Tian-Bo Deng

Weighted least-squares method for designing arbitrarily variable 1-D FIR digital filters

Abstract Variable digital filters with variable cutoff frequencies can be designed by frequency transformation methods. Such methods are simple, but they are not applicable to the design of variable filters with arbitrarily variable frequency responses. This paper proposes an efficient method for designing variable one-dimensional (1-D) finite-impulse-response (FIR) digital filters with arbitrarily variable magnitude characteristics and specified linear or nonlinear phase responses. First, each coefficient of the variable FIR filter is assumed to be a multi-dimensional (M-D) polynomial of spectral parameters that specify different variable magnitude characteristics. Then we present a pair of least-squares algorithms for finding the optimal polynomial coefficients by minimizing the weighted squared error between the desired variable frequency response and the actual frequency response. The first one is for designing 1-D FIR filters with variable magnitude response and linear-phase, and the second one is for designing variable 1-D FIR filters with variable magnitude response and nonlinear-phase. Although the first algorithm can be regarded as a special case of the second one, using the first one in the linear case can simplify the design significantly. The proposed methods are very straightforward and efficient for designing variable digital filters with arbitrarily variable magnitude responses and specified linear or nonlinear phases.

An improved method for designing variable recursive digital filters with guaranteed stability

Digital filters with variable frequency responses are called variable digital filters. Generally speaking, variable recursive digital filters require lower orders and less computational complexity to satisfy the same desired variable magnitude responses than nonrecursive filters, but the stability of variable recursive filters is difficult to guarantee since their coefficients are also varied in some manner. This paper presents an improved version of an existing algorithm for designing variable recursive one-dimensional (1-D) digital filters with guaranteed stability, which is an improved version of the existing one. The basic idea is to find both the numerator and the denominator coefficients of the transfer function of a variable recursive filter as multi-dimensional (M-D) polynomials of the spectral parameters that define variable magnitude characteristics. To guarantee the stability, some stability constraints have to be imposed on the denominator coefficients. Therefore, we first substitute the denominator coefficients by another set of variables whose values can be arbitrary but without affecting the stability. Then both the numerator and the new denominator coefficients are determined as M-D polynomials of the spectral parameters. Compared to the existing design method, the new one is simpler, but does not degrade the final design accuracy. Two design examples are given to compare the new design method with the existing one.

Coefficient relation-based minimax design and low-complexity structure of variable fractional-delay digital filters

Although the coefficient-relation of the even-order sub-filters in the Farrow structure has been revealed and adopted in the weighted-least-squares (WLS) design of even-order finite-impulse-response (FIR) variable fractional-delay (VFD) digital filters in the literature, it has never been exploited in implementing low-complexity VFD filters. This paper aims to(i)propose a low-complexity implementation structure through exploiting the coefficient-relation such that the coefficient-relation is not only utilized in the design process, but also utilized in the low-complexity implementation; (ii)formulate the minimax design of an even-order VFD filter through exploiting the coefficient-relation; (iii)propose a new two-stage scheme called increase-then-decrease scheme for optimizing the sub-filter orders so as to minimize the VFD digital filter complexity. With the above three advances, an even-order VFD filter can not only be designed and but also be implemented efficiently by exploiting the coefficient-relation. We will use a design example to illustrate the low complexity and high design accuracy.

Improved bi-equiripple variable fractional-delay filters

This paper presents a bi-minimax method for designing an odd-order variable fractional-delay (VFD) finite-impulse-response (FIR) digital filter such that both the peak errors of its variable frequency response (VFR) and VFD response can be simultaneously suppressed. The bi-minimax design iteratively minimizes a mixed error function involving both the VFR-peak-error and VFD-peak-error subject to the second-order-cone (SOC) constraints on the VFR errors and linear-programming (LP) constraints on the VFD errors. As compared with the existing SOC-based minimax design that minimizes the VFR-peak-error only, this odd-order bi-minimax design suppresses the VFD-peak-error and flattens both the VFR errors and VFD errors simultaneously. Consequently, both the two errors are made nearly equi-ripple (bi-equiripple). An example is given for showing the simultaneous suppression of the two kinds of peak errors and verifying the effectiveness of the odd-order bi-minimax design approach.

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.

Delay-error-constrained minimax design of all-pass variable-fractional-delay digital filters

This paper first derives a simplified variable-fractional-delay (VFD) expression for the all-pass (AP) VFD digital filter, and then uses the simplified VFD expression to formulate the minimax AP-VFD filter design as a two-step linear-programming (LP) problem. To suppress the maximum error of the VFD response (VFD-peak-error), this minimax design minimizes the maximum error of the variable-frequency-response (VFR) subject to the VFD-peak-error constraint. Thus, this two-step design can minimize the VFR-peak-error with the VFD-peak-error suppressed below a prescribed upper bound. With the aid of the simplified VFD expression, the VFD-peak-error constraint can be approximately linearized as a linear one, and thus the minimax design can be solved by using the LP method. We will use an illustrative example to verify that the simplified VFD expression is almost the same as the true one, and that the proposed LP-based two-step minimax design can significantly suppress the VFD-peak-error. HighlightsWe derive a simplified group-delay expression for an all-pass fractional-delay filter.We propose a two-step method for designing all-pass fractional-delay filters.We use illustrative examples to verify the two-step minimax design scheme.The two-step minimax method can significantly suppress the maximum delay errors.

Design of 2-D variable digital filters with arbitrary magnitude characteristics

Abstract Digital filtering techniques are extensively applied in such areas as acoustic signal processing, image processing, biomedical signal processing and geophysical data processing. In many applications, the frequency-domain characteristics of digital filters are required to be changeable during the course of filtering process. The digital filters with variable frequency-domain characteristics are called variable filters. This paper proposes an efficient technique for designing 2-D variable filters with arbitrary magnitude characteristics. The technique is based on the decomposition of the given 2-D variable magnitude specifications. By this technique, we can obtain a 2-D variable digital filter by simply designing a set of 2-D constant filters and performing a set of 1-D polynomial approximations. Consequently, the original difficult 2-D variable filter design problem can be easily solved.

Stability trapezoid and stability-margin analysis for the second-order recursive digital filter

To design a variable recursive digital filter whose stability is always guaranteed, it is necessary to ensure that the stability conditions are always satisfied in the tuning process. Furthermore, it is also necessary to keep a certain margin for the stability (stability margin) in such a way that some unpredictable environmental changes and coefficient-value deviations will not cause instability. To add a stability margin to the stability of the second-order (2nd-order) recursive filter, this paper first introduces a stability trapezoid by trimming the well-known stability triangle of the 2nd-order recursive digital filter. Then, we quantitatively analyze the upper bound for the stability margin by using a stability-margin parameter. This theoretical stability-margin analysis is fundamental to the design of a variable recursive filter with an expected stability margin. Finally, we utilize a demonstrative example to verify the consistence between the theoretical upper bound of the stability margin and the computer simulation results. HighlightsWe propose a stability trapezoid for guaranteeing a specified stability-margin.We analyze the relation between stability margin and stability-margin parameter.We use a demonstrative example to verify the stability margin.We have verified the consistence between stability margin and simulation results.

New method for designing stable recursive variable digital filters

Abstract The digital filters with adjustable frequency-domain characteristics are called variable filters. Variable filters are used in many signal processing fields, but the recursive variable filters are extremely difficult to design due to the stability problem. This paper proposes a new method for designing recursive one-dimensional (1-D) variable filters whose stability is always guaranteed. To guarantee the stability, we first perform coefficient substitutions on the denominator coefficients such that for arbitrary real-valued coefficients, the stability condition is always satisfied. Then both the denominator coefficients and the numerator coefficients after substitutions are determined as multi-dimensional (M-D) polynomials of spectral parameters that specify variable magnitude characteristics. In applying the resulting variable filters, substituting different values of the spectral parameters into the M-D polynomials will obtain different filter coefficients and thus different magnitude characteristics. Two examples are given to show the effectiveness of the proposed design technique.

Design of arbitrarily variable 2-D digital filters using array-array decomposition

Abstract This paper proposes a new method for designing variable 2-D digital filters with arbitrarily variable 2-D frequency responses. The design method simplifies the difficult variable filter design problem as the relatively easier one that only requires constant 2-D filter design and multi-dimensional (M-D) polynomial fitting. It includes the following steps. 1. Construct a complex M-D array by sampling the variable 2-D frequency response specification. 2. Decompose the constructed M-D array into the sum of products of 2-D complex arrays (matrices) and M-D real arrays. 3. Approximate the 2-D complex arrays by using constant 2-D digital filters, and best fit the M-D real arrays by using M-D polynomials. 4. Interconnect the obtained constant 2-D filters and M-D polynomials to form a variable 2-D digital filter. nSince constant 2-D filters are relatively easy to design, and M-D polynomials can be obtained by least-squares fitting technique, the above steps are easy to perform except the step 2. The main content of this paper is focused on the decomposition algorithm needed in step 2. We call the decomposition the Array–Array Decomposition (AAD). Based on the AAD, the original variable 2-D filter design problem can be reduced to the easier one as mentioned above.