Fir notch filter design: a review

Abstract

Notch filters are invariably used in communication, control, instrumentation and bio-medical engineering, besides a host of other fields, to eliminate noise and power line interferences. Digital notch filters can be designed as infinite impulse response (IIR) as well as finite impulse response (FIR) structures. As compared to the latter, IIR filters have the advantage that they require lower orders for efficient approximation of a given set of specifications. However, IIR filters are potentially unstable and do not provide linear phase characteristics, in general. FIR filters, on the other hand, are unconditionally stable and can be designed to give exact linear phase characteristics. We, in this review paper, focus our attention to the recent design techniques proposed by us for FIR notch filters. Standard FIR filter design methods, such as windowing, frequency sampling and computer-aided/optimization may be used for designing FIR notch filters. However, most of these methods result in ripples in the passbands. In many situations, maximally at (MF) filters are preferred since they have maximum attenuation in the stopband and hence can yield the best signal-to-noise ratio. A number of methods are available in the literature for designing MF digital filters. We, in this paper, review the design techniques for computing the weights of MF FIR notch filters. A number of design methodologies have been highlighted that lead to either recursive or explicit formulas for the computation of weights of FIR notch filters. Procedures for the design of FIR notch filters with maximal flatness of the amplitude response (in the Butterworth sense) at ω = 0 and ω = p have been given. Empirical formulas for finding the filter length N have also been proposed. By relaxing the linear phase property, it is possible to reduce the filter order required for a given magnitude response specifications. An FIR filter (with non-linear phase) can be derived from a second order IIR notch filter prototype. Explicit mathematical formulas for computing the weights for such FIR notch filters have been given. Design approaches based on the use of (i) Bernstein polynomials, and (ii) lowpass filter design have also been exploited to obtain maximally at FIR notch filters.