Reconstruction of a continuous time signal from its periodic nonuniform samples and multi-channel samples is fundamental for multi-channel parallel A/D and MIMO systems. In this paper,with a filterbank interpretation of sampling schemes,the efficient interpolation and reconstruction methods for periodic nonuniform sampling and multi-channel sampling in the fractional Fourier domain are presented. Firstly,the interpolation and sampling identities in the fractional Fourier domain are derived by the properties of the fractional Fourier transform. Then,the particularly efficient filterbank implementations for the periodic nonuniform sampling and the multi-channel sampling in the fractional Fourier domain are introduced. At last,the relationship between the multi-channel sampling and the filterbank in the fractional Fourier domain is investigated,which shows that any perfect reconstruction filterbank can lead to new sampling and reconstruction strategies.
Multi-channel sampling for band-limited signals is fundamental in the theory of multi-channel parallel A/D environment and multiplexing wireless communication environment. As the fractional Fourier transform has been found wide applications in signal processing fields, it is necessary to consider the multi-channel sampling theorem based on the fractional Fourier transform. In this paper, the multi-channel sampling theorem for the fractional band-limited signal is firstly proposed, which is the generalization of the well-known sampling theorem for the fractional Fourier transform. Since the periodic nonuniformly sampled signal in the fractional Fourier domain has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived multi-channel sampling theorem and the specific space-shifting and phase-shifting properties of the fractional Fourier transform. Moreover, by designing different fractional Fourier filters, we can obtain reconstruction methods for other sampling strategies.
The cyclic filter banks, which are used widely in the image subband coding, refer to signal processing on the finite field. This study investigates the fractional Fourier domain (FRFD) analysis of cyclic multirate systems based on the fractional circular convolution and chirp period. The proposed theorems include the fractional Fourier domain analysis of cyclic decimation and cyclic interpolation, the noble identities of cyclic decimation and cyclic interpolation in the FRFD, the polyphase represen-tation of cyclic signal in the FRFD, and the perfect reconstruction condition for the cyclic filter banks in the FRFD. Furthermore, this paper proposes the design methods for perfect reconstruction cyclic filter bank and cyclic filter bank with chirp modulation in the FRFD. The proposed theorems extend the multirate signal processing in the FRFD, which also advance the applications of the theorems of filter bank in the FRFD on the finite signal field, such as digital image processing. At last, the proposed design methods for the cyclic filter banks in the FRFD are validated by simulations.
The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its potential appears to have remained largely unknown to the signal processing community until 1990s. The fractional Fourier transform can be viewed as the chirp-basis expansion directly from its definition, but essentially it can be interpreted as a rotation in the time-frequency plane, i.e. the unified time-frequency transform. With the order from 0 increasing to 1, the fractional Fourier transform can show the characteristics of the signal changing from the time domain to the frequency domain. In this research paper, the fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view. Our aim is to provide a course from the definition to the applications of the fractional Fourier transform, especially as a reference and an introduction for researchers and interested readers.
The sampling rate conversion is always used in order to decrease computational amount and storage load in a system. The fractional Fourier transform （FRFT） is a powerful tool for the analysis of nonstationary signals, especially, chirp-like signal. Thus, it has become an active area in the signal processing community, with many applications of radar, communication, electronic warfare, and information security. Therefore, it is necessary for us to generalize the theorem for Fourier domain analysis of decimation and interpolation. Firstly, this paper defines the digital frequency in the fractional Fourier domain （FRFD） through the sampling theorems with FRFT. Secondly, FRFD analysis of decimation and interpolation is proposed in this paper with digital frequency in FRFD followed by the studies of interpolation filter and decimation filter in FRFD. Using these results, FRFD analysis of the sampling rate conversion by a rational factor is illustrated. The noble identities of decimation and interpolation in FRFD are then deduced using previous results and the fractional convolution theorem. The proposed theorems in this study are the bases for the generalizations of the multirate signal processing in FRFD, which can advance the filter banks theorems in FRFD. Finally, the theorems introduced in this paper are validated by simulations.
Oversampling is widely used in practical applications of digital signal processing. As the fractional Fourier transform has been developed and applied in signal processing fields, it is necessary to consider the oversampling theorem in the fractional Fourier domain. In this paper, the oversampling theorem in the fractional Fourier domain is analyzed. The fractional Fourier spectral relation between the original oversampled sequence and its subsequences is derived first, and then the expression for exact reconstruction of the missing samples in terms of the subsequences is obtained. Moreover, by taking a chirp signal as an example, it is shown that, reconstruction of the missing samples in the oversampled signal is suitable in the fractional Fourier domain for the signal whose time-frequency distribution has the minimum support in the fractional Fourier domain.
