Vol. 14, No. 3, December 2001, 329-336
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FIR NOTCH FILTER DESIGN - A REVIEW

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Suhash Chandra Dutta Roy, Balbir Kumar
and Shail Bala Jain

**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 flat (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
*omega=0*
and *omega=pi* 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 flat FIR notch filters.

**Key words:**
Digital filters, FIR filters, notch filters.