Fast Design Algorithms for FIR Notch Filters

Based on symmetry of the maximally flat frequency response of a FIR notch filter the new design procedure is developed. The closed form solution provides direct computation of the frequency response, recursive computation of the impulse response coefficients, simple windowing technique, and an access to new implementation. Several examples are included. INTRODUCTION In order to remove a single frequency component from the signal spectrum the IIR notch filter is frequently used. It consists of an abridged all-pass second-order section and allows independent tuning of the notch frequency ω0 and the 3-dB attenuation bandwidth [2]. Therefore the design of a digital IIR notch filter is rather simple. Such filter also possesses infinite impulse and step responses consequently which can produce spurious signal components unwanted in various applications ( as in ECG signal processing ). A few procedures for the design of linear phase FIR notch filters are recently available [1]. The methods which lead to feasible filters are generally derived by iterative approximation techniques or by noniterartive but still numerical procedures, e.g. the window technique. In our paper we are primarily concerned with completely analytical approach to the FIR notch filter design. The solution is partially based on exact formula for the frequency response of a FIR notch filter symmetrical about ωT = π/2. Emphasizing simplicity of form for monotonic frequency response we derive the polynomials Nl,m (w) = [ m+ l 2l (1− w) l [ m+ l 2m (1 + w) m (1) in sum of Chebyshev polynomials of the first kind Tn(w) through which the transfer function H(z) is expressed. Here and in the following we often use the transformed variable w [3] w = 1 2 (z + z) → 1 2 (z + z) ∣ ∣ ∣z = e = cos ωT , (2) which transforms the z-plane onto a two-leaved w-plane. We introduce the formula for degree of a notch filter which is related to the notch frequency, the recursion formulae for polynomials Nl,m (w) and the impulse response coefficients of a moveable notch filter. The recursive formula for Nl,m (w) offers recursive evaluation of the transfer function H(z) and consequently an alternative implementation of maximally flat FIR notch filters by a structure with the multipliers coefficients of limited dynamic range. The rectangular windowing of the large extent impulse response is presented which leads to the frequency responses comparable to those designed by standard windowing technique. FREQUENCY RESPONSE, ORDER OF A NOTCH FILTER AND NOTCH FREQUENCY Let H(z) denotes the transfer function of a FIR filter of order N − 1 H(z) = N−1