H ∞ -Optimal Fractional Delay Filters with Application to Pitch Shifting Masaaki Nagahara ∗ Yutaka Yamamoto ∗∗ ∗ Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, JAPAN (e-mail: nagahara@ieee.org). ∗∗ Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, JAPAN (e-mail: yy@i.kyoto-u.ac.jp). Abstract: Fractional delay filters are digital filters to delay discrete-time signals by a fraction of the sampling period. Since the delay is fractional, the intersample behavior of the original analog signal becomes crucial. In contrast to the conventional designs based on the Shannon sampling theorem with the band-limiting hypothesis, the present paper proposes a new approach based on the modern sampled-data H ∞ optimization which aims at restoring the intersample behavior beyond the Nyquist frequency. By using the lifting transform or continuous-time polyphase decomposition, the design problem is equivalently reduced to a discrete-time H ∞ optimization, which can be effectively solved by numerical computation softwares. Moreover, a closed-form solution is obtained under an assumption on the original analog signals. Using this closed-form solution, we introduce a sampling rate conversion with arbitrary conversion rate, and propose a new pitch shifting method for digital sound synthesis. Design examples are given to illustrate the advantage of the proposed method. Keywords: Fractional delay filter, sampled-data control, H ∞ optimization, digital signal processing, sampling rate conversion, pitch shifting. 1. INTRODUCTION Fractional delay filters are digital filters that are designed to delay discrete-time signals by a fractional amount of the sampling period. Such filters have wide applications in signal processing, including sampling rate conversion [Ramstad (1984); Smith and Gossett (1984)], nonuniform sampling [Johansson and L¨ owenborg (2002); Prendergast et al. (2004)], wavelet transform [Yu (2007)], digital mod- eling of musical instruments [Lehtonen and Laakso (2007); V¨ alim¨ aki et al. (2006)], to name a few. For more applica- tions, see survey papers [Laakso et al. (1996); V¨ alim¨ aki and Laakso (2000)]. Conventionally, fractional delay filters are designed based on the Shannon sampling theorem [Shannon (1949); Unser (2000)] for strictly-bandlimited analog signals. By this the- ory, the optimal filter coefficients are obtained by sampling a delayed sinc function. This ideal filter is however not realizable because of its non-causality and instability, and hence many studies have focused their attention on ap- proximating the ideal filter by, for example, windowed sinc function [Cain et al. (1995); Selva (2008)], maximally-flat FIR approximation [Hermanowicz (1992); Pei and Wang (2001); Samadi et al. (2004); Hachabiboglu et al. (2007); Shyu and Pei (2008)], all-pass approximation [Jing (1987)], weighted least-squares [Tarczynski et al. (1997); Shyu and ⋆ This research is supported in part by the JSPS Grant-in-Aid for Scientific Research (B) No. 2136020318360203, Grant-in-Aid for Exploratory Research No. 22656095, and the MEXT Grant-in-Aid for Young Scientists (B) No. 22760317. Pei (2008)], and minmax (Chebyshev) optimization [Put- nam and Smith (1997)]. Although these studies are based on the Shannon paradigm, no real analog signals are fully band-limited, and hence the assumption is not realistic. It is, therefore, necessary to design a filter that takes account of high-frequency com- ponents beyond the Nyquist frequency and the intersample behavior. In our recent study [Yamamoto et al. (2012)], we have proved that sampled-data H ∞ control theory provides an optimal platform to overcome the frequency limitation enforced by the Shannon sampling theorem. Based on this study, we formulate the design of fractional delay filters as a sampled-data H ∞ optimization problem. That is, we design a filter that minimizes the H ∞ norm of the error system between the ideal fractional delay and an ap- proximated one. In particular, the closed-form formula for the H ∞ optimal fractional delay filter is given under the assumption that the underlying frequency characteristic of the continuous-time input signal is governed by a low-pass filter of first order. By using the closed-form formula, we propose a new pitch shifting method for digital sound synthesis [Roads (1996)]. Pitch shifting is a technique for raising or lowering the orig- inal pitch of audio signals. This is often used in synthesiz- ing musical tone from a recorded signal of a musical instru- ment with a fixed fundamental frequency [Roads (1996)]. We show by simulation that the proposed method out- performs the conventional phase-vocoder method [Portoff (1976); Ellis (2002)].