Abstract
The fractional Fourier transform (FrFT), as an extension of Fourier transform, has recently been introduced in optics, communication and signal processing areas. In the last decade, there have been several approaches on the topic of computing discrete fractional Fourier transform (DFrFT). The purpose of this work is to provide a unified approach to improve the computation of DFrFT. With introduction of block-input-block-output technique, a specific multirate filtering system can be converted to an equivalent linear time-invariant (LTI) multi-input-multi-output (MIMO) system. Other DFrFT algorithms based on matrix-vector multiplication can also be implemented by the similar structure. Therefore, a unified computation framework is developed for the multirate implementation of the DFrFT via the proposed MIMO system. Further, a corresponding fractional Fourier filter bank (FrFB) is also constructed as an extension of the conventional discrete Fourier transform (DFT) filter bank system.Multirate filter bank applications profound in many fields such as communication system, speech coding system, etc.. There has been much research work on the design of perfect reconstruction (PR) system in multirate filter bank system and applications. However, in practical applications, the noise occurs in subband is unavoidable, for example, disturbance in communication in each subband, the quantized effect can also be modeled as a uniformly distributed additive noise. A suboptimal Wiener filter in fractional domain had been proposed by Kutay et al.. Based on the equivalent polyphase transfer matrix, a set of causal stable synthesis filters are obtained. That is, an equivalent fractional Fourier bank to optimally reconstruct the noise-corrupted signal in the fractional domain has been proposed in this study. The proposed stable design also provides a feasible solution for Kutay's kernel matrix which is known as singular. And the proposed inverse has been proposed to improve the performance of the conventional noncausal fractional domain Wiener filter.Based on the fact that the signal-energy-ratio is varying with the transform parameter a. The fractional-based matched filter can be designed accordingly. Further, this approach can be applied to the problem of signal separation.