Abstract
Maximally-flat (MAXFLAT) finite impulse response (FIR) filters often face a problem of the cutoff-frequency error due to approximation of the desired frequency response by some closed-form solution. So far, there have been plenty of efforts to design such a filter with an arbitrarily specified cut off-frequency, but this filter type requires extensive computation and is not MAXFLAT anymore. Thus, a computationally efficient and effective design is needed for highly accurate filters with desired frequency characteristics. This paper describes a new method for designing cutoff-frequency-fixing FIR filters through the cutoff-frequency error compensation of MAXFLAT FIR filters. The proposed method provides a closed-form Chebyshev polynomial containing a cutoff-error compensation function, which can characterize the “cutoff-error-free” filters in terms of the degree of flatness for a given order of filter and cut off-frequency. This method also allows a computationally efficient and accurate formula to directly determine the degree of flatness, so that this filter type has a flat magnitude characteristic both in the passband and the stopband. The remarkable effectiveness of the proposed method in design efficiency and accuracy is clearly demonstrated through various examples, indicating that the cutoff-fixing filters exhibit amplitude distortion error of less than 10−14 and no cut off-frequency error. This new approach is shown to provide significant advantages over the previous works in design flexibility and accuracy.
Highlights
One of the main difficulties for MAXFLAT finite impulse response (FIR) filter design is to express the objective error function in a closed form [27]. This is due to the fact that the MAXFLAT FIR filter by any closed-form polynomial does not have any design (“free”) parameters
Problems with the cut off-frequency error always arise in a MAXFLAT filter design
This is due to the fact that the maximum possible number of zeros at z = ±1 is imposed, Problems with the cut off-frequency error no always arise in aparameters
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Jeon et al have reported a closed-form least error gain method [23] for the design of FIR filters with both a flat magnitude and exact cut off-frequency This method has the disadvantage of requiring a lot of computation processing for estimating the order of flatness, so that the error frequency function is closest to a zero gain at all frequencies [23]. The objective of this paper is to present a new method for the design of cut offfrequency fixing filters using a “cut off-error-free” polynomial function For this purpose, a frequency-response error compensation function between the desired and actual frequency responses is derived through the generalization of the three closed-form polynomials [10,15,16,17] into a Chebyshev polynomial form.
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