Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Communications <s> Traditional multicarrier techniques perform a frequency-domain decomposition of a channel characterized by frequency-selective distortion in a plurality of subchannels that are affected by frequency flat distortion. The distortion in each independent subchannel can then be easily compensated by simple gain and phase adjustments. Typically, digital Fourier transform schemes make the implementation of the multicarrier system feasible and attractive with respect to single-carrier systems. However, when the channel is time-frequency-selective, as it usually happens in the rapidly fading wireless channel, this traditional methodology fails. Since the channel frequency response is rapidly time-varying, the optimal transmission/reception methodology should be able to process nonstationary signals. In other words, the subchannel carrier frequencies should be time-varying and ideally decompose the frequency distortion of the channel perfectly at any instant in time. However, this ideally optimal approach presents significant challenges both in terms of conceptual and computational complexity. The idea disclosed in this work is that a nonstationary approach can be approximated using signal bases that are especially suited for the analysis/synthesis of nonstationary signals. We propose in fact the use of a multicarrier system that employs orthogonal signal bases of the chirp type that in practice correspond to the fractional Fourier transform signal basis. The significance of the methodology relies on the important practical consideration that analysis/synthesis methods of the fractional Fourier type can be implemented with a complexity that is equivalent to the traditional fast Fourier transform. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Communications <s> Recently, innovative multicarrier schemes have been proposed that exploit the transmission of chirp-shaped waves e/sup -j2/spl pi/ct2/ by optimally choosing the chirp parameter c on the basis of the channel characteristics and are more robust to time-varying channels than ordinary OFDM schemes. This concept was applied to continuous-time and to discrete-time systems. In the present paper, we aim at developing those ideas using the affine Fourier transform (AFT), which is a very general formulation of chirp transforms. We present a multicarrier modulation based upon the discrete form of the AFT that is therefore inherently discrete and strictly invertible. Moreover, it allows to define a circular prefix concept that is coherent with the chirp nature of the transmission. The system can be efficiently implemented by adding a simple phase-correction block to standard OFDM modulators/demodulators and can effectively combat interchannel interference when the propagation channel is made of few multipath components affected by independent frequency offsets. Our discrete-time multicarrier scheme is an improved version of Martone's approach (as we also show by simulation results), and exhibits analogous characteristics to Barbarossa's continuous-time system. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Communications <s> 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. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Communications <s> In this work the performance of a Fractional Fourier transform (FrFT) based Minimum Mean Squared Error receiver for MIMO systems with space time processing over Rayleigh faded channels is presented. The proposed receiver called Optimum FrFT based MIMO receiver (OFMR) shows improved performance outperforming the simple MMSE receiver in Rayleigh faded channel. <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Communications <s> The fractional Fourier transform (FRFT) is one among the most efficient techniques for digital signal processing of linear frequency modulated (LFM) signal applications. Chirp (or LFM) modulation spread spectrum (CMSS) with FRFT technique is employed for demodulation of the multiple access System. The concept of multiple access is based on the use of multiple chirp signals, with different chirp rate, as carrier signal for transmission. Corresponding receiver is designed with optimal ordered FRFT block with multiplicative filter to reduce the interference. Additive white Gaussian noise (AWGN) is considered as a channel noise for evaluation of performance. Bit error rate (BER) is taken as a measure of performance with different signal to noise ratio and the results are compared with those reported in literatures. The details of the simulation techniques employed and the results of the novel receiver are presented and discussed. <s> BIB005
Traditional multicarrier (MC) systems are designed with the DFT based schemes. The main representative of these techniques is the orthogonal frequency-division multiplexing (OFDM). However, for doubly selective channels (channels with selectivity in both time and frequency) this technique produces non-satisfactory results. Martone published a corner-stone paper where the discrete FRFT is used instead of the DFT in multicarrier systems BIB001 . For channels with fast variations with available the line-ofsight component (usually associated with mobile stations mounted on high speed carriers and/or non-urban environments) this scheme significantly outperforms DFT-based counterparts. Performance of this technique is significantly improved since the time-frequency plane can be adjusted (rotated) in a way to compensate undesired modulation of the signals introduced by high velocity of participants and/ or by multipath component shifted from the line-of-sight components. This technique is generalized for general linear canonical form of the transform (affine Fourier transform-AFT) in BIB002 . This form has shown an improved flexibility with respect to the FRFT-based scheme. These chirp based modulations and associated AFT schemes have been identified as a suitable basis for multicarrier communications such as aeronautical and satellite BIB003 . Table 4 provides a comparison between the AFT-MC technique and the standard OFDM for four typical scenaria that can be observed in the case of the aeronautical channels. The best results for the AFT-MC techniques are observed for en-route scenario while in the worst case for the parking scenario, the AFT-based technique behaves the same as the OFDM. Then, in our opinion the AFT technique are important candidates for novel digital standards for the aeronautical communications with solved problem of the fast realizations of the AFT (FRFT). Similarly, other contributions discussed the application of the FRFT in communication systems. In a very recent publication (e.g., BIB004 ), a minimum mean squared error receiver based on the FRFT for MIMO systems with space time processing over Rayleigh faded channels was presented. The numerical analysis of the proposed receiver showed improved performance; outperforming the simple minimum mean squared error receiver in Rayleigh faded channel. Furthermore, chirp modulation spread spectrum based on the FRFT was recently proposed for demodulation of multiple access systems BIB005 . The numerical analysis showed that the FRFT based receiver is more flexible and efficient system for multiple access by reducing the designing complexity of the system. The authors also argued that receivers based on the FRFT are more sensitive than earlier coherent receivers for chirp signals.