Abstract
The beginning of this chapter contains general considerations about the design of a digital filter. The form in which the filter specifications must be expressed by the designer are illustrated, and the reasons why an IIR or an FIR filter might be preferred are listed. Then the discussion focuses on the design of linear phase (LP) or generalized linear phase (GLP) FIR filters, which exist in four types. Issues related to the selection of the design method and to the quantitative approximation criteria that may be established to judge the resemblance of the designed filter with the desired one are discussed. The properties of LP/GLP FIR filters are examined in detail, and a factorization of the zero-phase response, useful to unify the symmetry condition for the coefficients of the four filter types, is presented: the zero-phase response is split into a fixed factor, depending on the filter type but not on specifications, and an adjustable factor, with coefficients to be determined according to specifications. The most flexible and optimum design method for LP/GLP FIR filters is then described: this is the minimax method, which ensures the filter meets specifications with the minimum possible order. The properties of optimum FIR filters are finally studied.
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- 1.
In this discussion we use the symbol \(\omega _p\) for the upper passband limit, in place of the symbol \(\omega _c\) previously used to indicate the cutoff frequency of the ideal filter.
- 2.
In principle, we should use different symbols for the pre-fixed values of \(\delta _p\) and \(\delta _s\) and the corresponding optimized values, but this would weigh down the notation too much.
- 3.
The value \(\kappa =1\) was chosen for graphical convenience, i.e., for getting ripples in the two approximation band which are equally large and therefore equally visible.
- 4.
The process is actually iterated a maximum number of allowed times, within which hopefully convergence is achieved.
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Alessio, S.M. (2016). FIR Filter Design. In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_7
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DOI: https://doi.org/10.1007/978-3-319-25468-5_7
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