應用倒頻譜方法設計平坦分數延遲濾波器及分數希爾伯轉換器

本文介紹數種數位分數延遲濾波器以及分數希爾伯轉換器的設計，後者亦概括普通希爾伯轉換器。在系統型態上，有限脈衝響應或無限脈衝響應的設計皆有考慮。兩大主要設計技巧為平坦條件以及倒頻譜分析，分別於第一章與第四章作介紹。平坦濾波器在頻率響應上具有於所關切的頻率附近平坦且精確的優點。在只考慮平坦條件下，第二章我們設計六型有限脈衝響應平坦分數延遲濾波器，分別適合應用於不同的頻帶需求，其中有四型在實現上可以省下一半的乘法運算；以及第三章設計了八型具有整數延遲的無限脈衝響應平坦希爾伯轉換器，透過Eneström-Kakeya定理並放寬其條件，所得到的濾波器設計在大部分狀況皆穩定，與已知的有限脈衝響應或全通希爾伯轉換器比較下，我們的無限脈衝響應型態可以做更有彈性的設計，以平衡頻寬需求與延遲限制。因為分數延遲濾波器以及希爾伯轉換器皆為相位導向的濾波器，因此很適合用倒頻譜來直接設計濾波器的相位響應。第五章我們進一步將平坦條件用在全通分數延遲濾波器與全通希爾伯轉換器的倒頻譜設計，得到的複數倒頻譜直接正比於設計參數，有利於參數更新。第六章利用此正比特性進一步設計出有限脈衝響應與全通分數延遲濾波器的可調式架構，包括熟知的Farrow架構，與已知的兩類Farrow架構比較下，我們的設計大幅節省了乘法數，而依然維持優於非平坦類型Farrow架構的精確度。當所分析問題在二維度以上時，傳統的倒頻譜分析工具，特別是微分倒頻譜技巧，複雜度將顯著提高。第七章我們對其簡化，定義一個新的倒頻譜，稱為部分微分倒頻譜，在倒頻譜分析以及相關應用上更有效率。此外亦提出一種基於部分微分倒頻譜的相位攤展演算法，不但在操作上更簡便，也能有效對抗相位雜訊。

關鍵字

平坦條件；分數延遲濾波器；分數希爾伯轉換器；複數倒頻譜；微分倒頻譜；部分微分倒頻譜；可調式分數延遲架構； Farrow架構；有限脈衝響應濾波器；無限脈衝響應濾波器；全通濾波器

並列摘要

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.

並列關鍵字

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

參考文獻

responses using the Hilbert transform,” Microwave Theory and Techniques,

[1] N. Anderson, E. B. Saff, and R. S. Varga, “On the Enestr¨om-Kakeya theorem

and its sharpness,” Linear Algebra and its Applications, vol. 28, pp. 5–16,

[2] L. B. Jackson, Digital Filters and Signal Processing, 3rd ed. Dordrecht, The

Netherlands: Kluwer Academic Publishers, 1996.

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應用倒頻譜方法設計平坦分數延遲濾波器及分數希爾伯轉換器

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