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Digital notch filters are useful for removing sinusoidal signals with a single frequency or multi-frequencies. For some applications, the sinusoidal signals being processed may change frequencies, and thus the notch filter must also have a variable notch frequency. Such a digital filter with a variable notch-frequency (VNF) is called variable notch-frequency (VNF) digital filter. Generally speaking, a VNF filter has the ability to change its notch frequency continuously, and can remove sinusoidal signals with variable frequencies. To design a recursive VNF filter, guaranteeing its stability is the top-priority issue. In this paper, a recursive VNF filter is designed by utilizing the least Lp-norm criterion, and the stability is guaranteed through adopting the stability-guarantee methodology based on parameter transformations. To perform the parameter transformations, a function that meets a special condition is required, and such a function is called unity-bounded function. In this paper, various unity-bounded functions are developed for the parameter-transformation-based stability guarantee. The unity-bounded functions play a crucial role in the stability guarantee. This paper details how to design a stable VNF filter by incorporating the parameter transformations employing the unity-bounded functions into the Lp-norm minimization of the magnitude response. Theoretically, the stability of the recursive VNF filter resulting from the design is definitely guaranteed. Moreover, since the coefficients of the designed VNF are variable, changing the filter coefficients enables the user to get a VNF filter with updated notch frequency. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method for the design of a recursive second-order (biquadratic) all-pass phase compensator with controllable stability margin. The design idea stems from the generalized stability triangle (GST) derived by the author for the second-order biquadratic digital filter. Based on the GST, a parameter-transformation method is proposed on the transformations of the denominator coefficients of the transfer function of the biquadratic phase compensator. The transformations convert the original denominator coefficients to other new parameters, and any values of those new parameters can guarantee that the GST condition is always satisfied. Optimizing the new parameters yields a biquadratic phase compensator that definitely meets a prespecified stability margin. That is, a biquadratic all-pass phase compensator can be designed to have an arbitrarily specified stability margin. This in turn avoids the occurrence that a recursive phase compensator may become unstable in the practical applications. Thus, the resulting biquadratic phase compensator has robust stability, which is extremely important during the practical filtering operations. A design example is given to show the stability margin guarantee as well as the approximation accuracy. [ABSTRACT FROM AUTHOR]

This paper proposes a two-step strategy for designing a variable-bandwidth (VBW) digital filter through minimizing the -norm of the magnitude-response error. This -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 -norm-minimization design for . 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 -norm-minimization-based design and verifying the stability guarantee. [ABSTRACT FROM AUTHOR]

Guaranteeing the stability is one of the most critical issues in designing a variable recursive digital filter. In this paper, we first present an odd-order recursive variable model (transfer function) that is used for designing an odd-order variable-magnitude (VM) digital filter, and then we replace the original coefficients of the denominator of the odd-order transfer function with a set of new parameters. These new parameters can ensure that they can take arbitrary values without incurring instability of the designed odd-order VM filter. To make the VM filter coefficients variable, we find all the VM filter coefficients as polynomial functions of the tuning parameter, which includes two phases. The first phase designs a set of recursive digital filters with fixed coefficients (constant filters), and the second phase utilizes a curve-fitting scheme to represent each coefficient as a polynomial function. As a result, the VM filter coefficients become variable, and the proposed parameter-substitution-based denominator coefficients ensure the filter stability. This is the most important contribution of the parameter-substitution-based design scheme. This paper uses the fifth-order demonstrative example to verify the stability guarantee as well as the design accuracy of the obtained the fifth-order VM filter. [ABSTRACT FROM AUTHOR]

Stability is one of the most concerned issues in designing a recursive variable digital filter (VDF). This is because the coefficients of a recursive VDF constantly vary in the tuning process, and updating the coefficients may incur instability. Thus, an appropriate measure needs to be taken for ensuring its stability. This paper presents a new coefficient transformation (CT) method for transforming the coefficients of a recursive transfer-function denominator into a set of new coefficients. From the viewpoint of conventional constant-coefficient filter (constant filter) design, the new coefficients can take arbitrary values without incurring instability. For designing a stable VDF, we apply this CT to the variable case and approximate each transformed coefficient as a distinct polynomial in the tuning parameter. Thus, we can change the filter coefficients by changing the value of the tuning parameter, and thus tune the magnitude response. Thanks to the proposed CT, updating the filter coefficients will never incur instability. This is the core part of the CT-based design approach. In this paper, we utilise a weighting function to ignore the transition-band errors and thus enhance the design accuracy of important frequency bands (passband and stopband). Moreover, the polynomials use different degrees so as to reduce the VDF complexity. Two design examples (lowpass VDF and bandpass VDF) are provided for verifying the design accuracy and checking the stability. [ABSTRACT FROM PUBLISHER]

In the design of recursive digital filters, the stability of the recursive digital filters must be guaranteed. Furthermore, it is desirable to add a certain amount of margin to the stability so as to avoid the violation of stability due to some uncertain perturbations of the filter coefficients. This paper extends the well-known stability-triangle of the second-order digital filter into more general cases, which results in dented stability-triangles and generalized stability-triangle. The generalized stability-triangle can be viewed as a special case of the dented stability-triangles if the two upper bounds on the radii of the two poles are the same, which is a generalized version of the existing conventional stability-triangle and can guarantee the radii of the two poles of the second-order recursive digital filter below some prescribed upper bound. That is, it is able to provide a prescribed stability-margin in terms of the upper bound of the pole radii. As a result, the generalized stability-triangle increases the flexibility for guaranteeing a prescribed stability-margin. Since the generalized stability-triangle is parameterized by using the upper bound of pole radii, i.e., the stability-margin is parameterized as a function of the upper bound, the proposed generalized stability-triangle facilitates the stability-margin guarantee in the design of the second-order as well as high-order recursive digital filters. [ABSTRACT FROM AUTHOR]