Design of FIR/IIR Maximally Flat Fractional Delay Filters and Fractional Hilbert Transformers Using Cepstral Methods
Date Issued
2011
Date
2011
Author(s)
Lin, Huei-Shan
Abstract
In this dissertation, we introduce several designs on digital fractional delay(FD) filters and fractional Hilbert transformer(FHT), the latter of which is the generalization of the normal Hilbert transformer(HT). Both FIR and IIR designs are considered for these filters. Two major techniques employed to design these filters are maximally flat(MF) condition and cepstral analysis, which are introduced in Chapter 1 and Chapter 2, respectively.
MF filters has the advantage of possessing flat and accurate desired response around the concerned band. With MF condition only, in Chapter 3, we design six types of FIR MF FD filters, among which four types save half the multipliers, respectively appropriate for different frequency band requirements. In Chapter 4, eight types of IIR MF HTs with integer delay are designed, through Eneström-Kakeya theorem and loosening its constraint, the resultant IIR filters are stable for most cases. While compared with existing FIR and allpass Hilbert transformer, our IIR design is more flexible and capable of balancing the bandwidth requirement and the delay constraint.
Since the FD filter and the FHT are phase-oriented filters, it is very appropriate to apply cepstrum design directly on the phase response of these filters. In Chapter 5, we further apply MF condition in cepstral domain to design FD filters and FHT, where the obtained complex cepstrum(CC) is proportional to the design parameters and facilitates update of parameters. Such proportionality is further utilized in Chapter 6 to obtain several tunable structures for FIR and IIR allpass FD filters, including the well known Farrow structure. Compared with other two existing Farrow structures, our Farrow structure largely saves numbers of multipliers while remains more accurate than some non-MF-based Farrow structure.
While the involved problem is two-dimensional and above, the complexity of traditional cepstral analysis tool, specifically, the differential cepstrum(DC) technique, grows dramatically. In Chapter 7, we simplify this technique by defining a new type of cepstrum, named partially differential cepstrum(PDC). The simplicity makes PDC a more efficient tools for cepstral analysis. Besides, a PDC-based phase-unwrapping method is proposed, which is simple for implementation and robust against noisy phase.
Subjects
Maximally Flat condition
fractional delay filter
fractional Hilbert transformer
complex cepstrum
differential cepstrum
partially differential cepstrum
tunable fractional delay structure
Farrow structure
FIR filter
IIR filter
allpass filter
Type
thesis
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