H∞-Optimal Fractional Delay Filters with Application to Pitch Shifting*

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.