Fractional Fourier transform as a signal processing tool: An overview of recent developments

The linear canonical transform (LCT) has been shown to be a useful and powerful tool for signal processing and optics. Many reconstruction strategies for bandlimited signals in LCT domain have been proposed. However, these reconstruction strategies can work well only if there are no errors associated with the numerical implementation of samples. Unfortunately, this requirement is almost never satisfied in the real world. To the best of the author's knowledge, the statistical problem of LCTed bandlimited signal recovery in the presence of random noise still remains unresolved. In this paper, the problem of recovery of bandlimited signals in LCT domain from discrete and noisy samples is studied. First, it is shown that the generalized Shannon-type reconstruction scheme for bandlimited signals in LCT domain cannot be directly applied in the presence of noise since it leads to an infinite mean integrated square error. Then an orthogonal and complete set for the class of bandlimited signals in LCT domain is proposed; and further, an oversampled version of the generalized Shannon-type sampling theorem is derived. Based on the oversampling theorem and without adding too much complexity, a reconstruction algorithm for bandlimited signals in LCT domain from discrete and noisy observations is set up. Moreover, the convergence of the proposed reconstruction scheme is also proved. Finally, numerical results and potential applications of the proposed reconstruction algorithm are given.

Recovery of Bandlimited Signals in Linear Canonical Transform Domain from Noisy Samples

It is well known that the partial sums of a Fourier series of a non-periodic analytic function on a finite interval exhibit spurious oscillations near the interval boundaries. This phenomenon is known as the Gibbs effect. The authors show that a similar phenomenon is observed for the fractional Fourier series (FrFS) of a function with jump discontinuities. The convergence of FrFS is discussed and proved in a theorem. Specifically, the present work proves the uniform convergence of the FrFS for a non-periodic analytic function in the smooth region. The maximum amplitude of the oscillations for the FrFS remains constant (the Gibbs constant), similar to that for a classical Fourier series expansion. Finally, three numerical examples are investigated to demonstrate that the Gibbs constant for an FrFS is the same as for a Fourier series.

http://imagetech-polynomials.com/PDF/Gibbs%20phenomenon%20for%20fractional%20Fourier%20series.pdf

Gibbs phenomenon for fractional Fourier series

The discrete fractional Fourier transform (FrFT) has been suggested to enhance performance over DFT-based multicarrier systems when transmitting over doubly-dispersive channels. In this paper, we propose a novel low-complexity equaliser for inter-symbol and inter-carrier interference arising in such multicarrier transmission system. Due to a lower spreading in the FrFT-domain compared to the DFTchannel matrix as compared to the DFT domain, the equaliser can approximate the fractional-domain channel matrix by a band matrix. Further, we utilise the least squares minres (LSMR) algorithm in the calculation of the equalisation, which exhibits attractive numerical properties and low complexity. Simulation results demonstrate the superior performance of the proposed LSMR equaliser over benchmark schemes.

/pdf/low-complexity-lsmr-equalisation-of-frft-based-multicarrier-3o57ru6ee5.pdf

Low-complexity LSMR equalisation of FrFT-based multicarrier systems in doubly dispersive channels

Unified approach to extrapolation of bandlimited signals in linear canonical transform domain

Multicarrier techniques based on affine Fourier transform (AFT) have been recently proposed for transmission in the wireless channels. The AFT represents a generalization of the Fourier and fractional Fourier transform. We derive the exact and approximated interference power, upper bound and measure of applicability for the AFT based multicarrier (AFT-MC) system. It is demonstrated that the AFT-MC system effectively minimizes interference in time-varying multipath channels with line-of-sight component and narrow beamwidth of scattered components that often occurs in aeronautical and satellite communications.

Interference analysis of multicarrier systems based on affine fourier transform

The affine Fourier transform (AFT), a general formulation of chirp transforms, has been recently proposed for use in multicarrier communications. The AFT-basedmulticarrier (AFT-MC) systemcan be considered as a generalization of the orthogonal frequency division multiplexing (OFDM), frequently used in modern wireless communications. AFT-MC keeps all important properties of OFDM and, in addition, gives a new degree of freedom in suppressing interference caused by Doppler spreading in time-varying multipath channels. We present a general interference analysis of the AFT-MC system that models both time and frequency dispersion effects. Upper and lower bounds on interference power are given, followed by interference power approximation that significantly simplifies interference analysis. The optimal parameters are obtained in the closed form followed by the analysis of the effects of synchronization errors and the optimal symbol period. A detailed interference analysis and optimal parameters are given for different aeronautical and land-mobile satellite (LMS) channel scenarios. It is shown that the AFT-MC system is able to match changes in these channels and efficiently reduce interference with high-spectral efficiency.

/pdf/multicarrier-communications-based-on-the-affine-fourier-1mjioio9we.pdf

Multicarrier communications based on the affine fourier transform in doubly-dispersive channels

Due to a fast growth of broadband wireless access, a need to research new transmission technologies to keep pace with these demands becomes more significant. There are many ways to achieve high data rate and high quality of service. Since the mobility represents the main concern in modern wireless communications, in this paper we focus on the implementation of a new multicarrier (MC) technique. We propose an MC technique based on the affine Fourier transform (AFT-MC) for transmission in railway satellite (RS) channels. The key strength of the AFT-MC is the resilience to the intercarrier interference caused by the Doppler shift and Doppler spread. The AFT-MC system model is adapted to the RS channel environment and the obtained results prove good transmission performance in terms of interference cancellation, high data rate and spectral efficiency.

A new multicarrier system for satellite based communications in the railway environment

Multicarrier modulations have been recently proposed for high rate communications in underwater acoustic (UWA) channels. The UWA channels are wideband in nature, and far more challenging for multicarrier transmission than narrowband channels. In this paper, we investigate effects of Doppler scaling that wideband time varying multipath channels induce on orthogonal frequency-division multiplexing (OFDM) systems. We derive an exact interference power for multicarrier systems and an upper bound on intercarrier interference power for OFDM. Results represent wideband counterpart for narrowband cases described in literature.

Djuro Stojanovic

Papers

Interference analysis of multicarrier systems based on affine fourier transform

Multicarrier communications based on the affine fourier transform in doubly-dispersive channels

A new multicarrier system for satellite based communications in the railway environment

The effects of doppler scaling in underwater acoustic OFDM communication