| Summary: | Maximally flat (MAXFLAT) half-band filters usually have wider transition-band than other filters although the frequency response is maximally flat (i.e., no ripples) in the passband and stopband. This is due to fact that the maximum possible number of zeros at z = ±1 is imposed in half-band close form solution, which leaves no degree of freedom, and thus no independent parameters for direct control of the frequency response. This paper describes a novel method for the design of FIR half-band filters with an explicit control of the transition bandwidth. The proposed method is based on a generalized Lagrange half-band polynomial (<inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-LHBP) with coefficients parameterizing a <italic>0</italic>-th coefficient <inline-formula> <tex-math notation="LaTeX">$h_{0}$ </tex-math></inline-formula> and allows the frequency response of this filter type to be controllable by adjusting <inline-formula> <tex-math notation="LaTeX">$h_{0}$ </tex-math></inline-formula>. Then, <inline-formula> <tex-math notation="LaTeX">$h_{0}$ </tex-math></inline-formula> is modeled as a steepness parameter of the transition-band and this is accomplished through theoretically analyzing a polynomial recurrence relation of the <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-LHBP. This method also provides explicit formulas for direct computation of design parameters related to choosing a desired filter characteristic (by a reasonable trade-off between the transition-band sharpness and passband & stopband flatness). The examples are shown to provide a complete and accurate solution for the design of such filters with relatively sharper transition-band steepness than other existing half-band filters.
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