Weighted-type Fractional Fourier Transform Research Articles

Efficient Fast-Convolution Based Hybrid Carrier System

The weighted-type fractional Fourier transform (WFRFT)-based hybrid carrier (HC) system has been proposed to bridge the gap between orthogonal frequency division multiplexing (OFDM) system and single-carrier system with frequency domain equalization (SC-FDE). In this paper, we propose a novel HC system based on efficient fast-convolution (FC) without a cyclic prefix to meet the requirements of complex scenarios towards future communications. The proposed system can achieve flexible optimization of system performance, such as minimizing out-of-band emission (OOBE), peak to average power ratio (PAPR) and bit error rate (BER). This efficient FC based hybrid carrier (FCHC) system is among the most competitive waveforms. The closed-form BER expressions are derived and analyzed over additive white Gaussian noise (AWGN) and frequency-selective fading channels. A modified equalizer over frequency-selective channels is also provided to resist the inter-block interference (IBI). Moreover, the banded minimum mean-square error (MMSE) equalization based on FC can be embedded in the proposed system over doubly-selective channels, and then the interference caused by fading channels can be further suppressed through the combination of FC and HC. The simulation results demonstrate the superiority of the proposed system over the conventional HC system in terms of OOBE, PAPR and BER. The significant flexibility and combined performance advantages make the proposed system a potential scheme for future communications.

Key Validity Using the Multiple-Parameter Fractional Fourier Transform for Image Encryption

As a symmetric encryption algorithm, multiple-parameter fractional Fourier transform (MPFRFT) is proposed and applied to image encryption. The MPFRFT with two vector parameters has better security, which becomes the main technical means to protect information security. However, our study found that many keys of the MPFRFT are invalid, which greatly reduces its security. In this paper, we propose a new reformulation of MPFRFT and analyze it using eigen-decomposition-type fractional Fourier transform (FRFT) and weighted-type FRFT as basis functions, respectively. The results show that the effective keys are extremely limited. Furthermore, we analyze the extended encryption methods based on MPFRFT, which also have the security risk of key invalidation. Theoretical analysis and numerical simulation verify our point of view. Our discovery has important reference value for a class of generalized FRFT image encryption methods.

Research on Satellite Concealment Communication Method Based on WFRFT

Satellite signal concealment technology protects the safety of signal waveform by using signals with high power and specific parameter characteristics. However, with the development of modulation recognition and Successive Interference Cancellation (SIC) technology, the cover signal using conventional modulation methods can be cracked by non-cooperative receivers. In order to meet the safety communication requirements of portable satellite communication terminals, the idea of using local signal copies for self-interference cancellation in Paired Carrier Multiple Access (PCMA) is used for reference. This paper proposes a satellite secure communication method suitable for portable terminals, and introduces Weighted-type FRactional Fourier Transform (WFRFT) as a modulation method to mask the signal, thereby increase the difficulty of non-cooperative parties in identifying and demodulating the cover signal. Simulation shows that the method proposed in this paper improves the security of signal transmission, and the loss of the system's bit error rate is within an acceptable range after signal separation, which is suitable for practical use.

BER Analysis of Integrated WFRFT-OTFS Waveform Framework Over Static Multipath Channels

Orthogonal time frequency space (OTFS) modulation has been proposed to achieve outstanding performances over delay-Doppler channels in comparison with orthogonal frequency division multiplexing (OFDM) systems. Particularly, transmitter and receiver structures of OTFS are same as those of asymmetric OFDM (A-OFDM) waveforms, and OTFS is simplified as A-OFDM over static multipath channels. In this letter, we propose an integrated waveform framework based on weighted-type fractional Fourier transform (WFRFT) with two-dimensional adjustable parameters and strong scalability, called as WFRFT-OTFS, which can be reduced to OTFS, A-OFDM, hybrid carrier (HC), OFDM and other waveforms. Basically, theoretical BER expressions of integrated WFRFT-OTFS framework over multipath channels with zero forcing (ZF) and minimum mean square error (MMSE) receivers are derived through fusing different waveform components and verified by simulation results.

A reformulation of weighted fractional Fourier transform

This paper investigates a class of weighted-type fractional Fourier transform (WFRFT), which is mainly used in signal processing and image encryption. To date, studies have primarily focused on the application of WFRFT, and few studies have examined its properties in detail. We propose a new reformulation of WFRFT, whose properties can be readily proven. Multiweighted-type fractional Fourier transform (M-WFRFT) has attracted researchers' attention as a generalized form of WFRFT. It is very difficult to prove the properties of M-WFRFT in weighted form, and researchers have assumed that M-WFRFT has the properties of WFRFT in application. The properties of M-WFRFT are proved by applying the new reformulation proposed. The results show that M-WFRFT has boundary and unitary properties, but in few cases. We analyze and discuss the conditions in which that M-WFRFT satisfies the boundary and unitary conditions, providing a theoretical foundation for application of M-WFRFT.

Novel Modulation Recognition for WFRFT-Based System Using 4th-Order Cumulants

We present a novel modulation recognition for weighted-type fractional Fourier transform (WFRFT)-based systems using the fourth-order cumulants. First, the constellation characteristics of the basic digital modulations ASK, PSK, and QAM are analyzed, and the corresponding relationships between the neighboring constellation points’ distance and the constellation size are deduced. Second, the closed-form expressions of the fourth-order cumulants ( $C_{42}$ ) for the WFRFT-based systems with ASK, PSK, and QAM are derived. Finally, through the first- and second-order derivatives of the $C_{42}$ , we prove that the optimal WFRFT order $\alpha _{r}$ can be obtained through the minimization of the $C_{42}$ . The simulation results show that the novel recognition for the WFRFT-based system is feasible and can reach an accuracy of almost 90% when energy per bit-to-noise power spectrum density ratio ( ${E_{\text {b}}}/{N_{0}}$ ) is greater than 6.

BER Analysis of WFRFT Precoded OFDM and GFDM Waveforms With an Integer Time Offset

In this paper, we investigate bit error rate (BER) performances of weighted-type fractional Fourier transform (WFRFT) precoded orthogonal frequency-division multiplexing (OFDM) and generalized frequency-division multiplexing (GFDM) waveforms with an integer time offset (TO) over additive white Gaussian noise and fading channels. First of all, theoretical BER expressions of hybrid carrier systems with TO are derived according to the linear combination characteristic of WFRFT. In addition, the fusion mechanism is also employed to calculate noise enhancement factor (NEF) of WFRFT precoded GFDM waveforms, and then analytical BER expressions are derived, including the cases with TO. Basically, analytical BER expressions of WFRFT precoded OFDM and GFDM waveforms are under the same framework, and BER of the latter could be simplified to the former when GFDM systems are reduced to OFDM systems. Through WFRFT precoding, the BER performance of GFDM waveforms could be improved by about 2 dB over fading channels with timing errors. Furthermore, using derived NEF of WFRFT precoded GFDM waveforms, theoretical BER expressions could be easily extended to space-time coded systems. Finally, when interblock interference and intercarrier interference are generated due to insufficient cyclic prefix and timing errors, BER performances of WFRFT precoded OFDM and GFDM waveforms also surpass their two special cases, i.e., nonprecoded and discrete Fourier transform precoded ones.

BER analysis of STBC hybrid carrier system based on WFRFT with frequency domain equalization

The inherent characteristic of hybrid carrier (HC) scheme based on the weighted-type fractional Fourier transform (WFRFT) is revealed.The proposed linear combination of single carrier (SC) and multi-carriers (MC) characteristics is helpful to balance the conflicting requirements and achieve the comparative analysis of SC/MC.As an example, analytical SER/BER expressions of HC system with zero forcing (ZF) and minimum mean square error (MMSE) equalization are derived.In addition, the derived BER expressions are extended to multi-input systems.A unified framework of space time block code (STBC) HC scheme is proposed with two WFRFT modules at the transmitter or receiver,where STBC-SC and STBC-MC systems are the special cases of the novel structure.

Embedding WFRFT Signals Into TDCS for Secure Communications

This paper presents a novel combinatory strategy to embed weighted-type fractional Fourier transform (WFRFT) precoding signals into transform domain communication system (TDCS). The original data can be divided into two parts. One part is modulated with the cyclic code shift keying in the traditional TDCS. The other part is first disguised as other higher order baseband modulations MPSK through WFRFT proceding process, then will further be embedded into the traditional TDCS system. A double-random scheme is designed to dynamically generate the pseudo phase for WFRFT rotation and TDCS basic function’s phase. Theoretical analyses show that the higher spectral efficiency can be obtained while causing little effect on its original communication quality. The final simulations demonstrate that the constellation precoding method can help disguise the original signals into other higher order baseband PSK modulations with a controllable Gaussian-like noise. Besides, owing to the dynamically double-random phase generation scheme, the proposed combinatory WFRFT-TDCS system can gain better anti-eavesdropping capacity while maintaining its own communications’ quality with regards to bit error rate performances.

On the Performance of Hybrid Carrier System Based on WFRFT With Power Allocation

A unified framework for bridging the power allocation (PA) strategies of single carrier (SC) and orthogonal frequency division multiplexing (OFDM) systems is proposed through the hybrid carrier (HC) scheme based on weighted-type fractional Fourier transform (WFRFT). The performance analyses of multiple indicators are easily achieved in the proposed framework, and the expected performances could be obtained according to the flexible WFRFT order. The closed BER expressions of arbitrary PA strategies in the HC scheme are derived and verified by the simulations, where the equal gain PA and minimum mean square error (mmse) PA are given as the examples. Compared with the OFDM and SC systems, the tradeoff between BER, achievable rate, and peak-to-average power ratio is achieved in the HC system. In addition, an approximate minimum bit error rate method through joining the PA and subcarrier allocation is investigated in the HC scheme as a generalization of the proposed framework. Considering the novel structure, the dynamic threshold could be established in the HC scheme, and thus the tradeoff between BER and complexity is obtained in comparison with the OFDM and SC systems. In order to illustrate the anti-interference characteristics of the proposed HC framework with PA, the BER performance with insufficient cyclic prefix is investigated. The better BER performance is captured in the HC system with the proposed mmse PA strategy.

On Narrowband Interference Suppression in OFDM-based Systems with CDMA and Weighted-type Fractional Fourier Transform Domain Preprocessing

On Narrowband Interference Suppression in OFDM-based Systems with CDMA and Weighted-type Fractional Fourier Transform Domain Preprocessing

On the performance of hybrid carrier system with spectrum precoding based on WFRFT

High out-of-band (OOB) power emission has become the main shortcoming of multicarrier scheme in regard to the requirements and challenges of 5G. In this paper, two effective OOB power reduction methods, previously applied to traditional orthogonal frequency division multiplexing (OFDM) system, are proposed in hybrid carrier (HC) scheme based on weighted-type fractional Fourier transform (WFRFT). Simulation results demonstrate that, due to the flexible selection of WFRFT order in the HC system, combined bit error rate (BER) and peak-to-average power ratio (PAPR) performance advantages are gained without sidelobe impact and significant complexity increase in comparison with the OFDM scheme with spectrum precoding. In projection precoding, a smaller error vector magnitude of precoder Gχ is obtained in the proposed hybrid carrier scheme. In SVD precoding with an orthogonal decoding at the receiver, better BER performance could also be acquired at WFRFT order over the fading channels. The proposed two WFRFT-based structures are complementary, and their potential scenarios are given. In addition, applying the feature of projection precoding to channel estimation, a novel pilot structure based on WFRFT is posed to reduce the introduced error of spectrum precoding and finally helps to improve the BER performance.

Bit error rate analysis of generalised frequency division multiplexing with weighted‐type fractional Fourier transform precoding

Generalised frequency division multiplexing (GFDM) as a non-orthogonal waveform candidate is posed to reduce the high out of band radiation of orthogonal frequency division multiplexing system via pulse shaping. In this study, an analytical model of introduced interference from neighbouring subcarriers caused by pulse shaping is firstly proposed. Signal-to-interference-plus-noise ratio (SINR) is calculated according to related GFDM models with different weighted-type fractional Fourier transform (WFRFT) precoding orders. Approximated bit error rate (BER) expressions are derived for the low complexity GFDM system over additive white Gaussian noise and fading channels based on the analysis of SINR. Particularly, the BER superiority of WFRFT precoding is highlighted over the fading channels. Furthermore, peak-to-average power ratio and out of band power suppression performances are simulated at different WFRFT orders to illustrate the advantages of WFRFT precoding.

On the probability density function of the real and imaginary parts in WFRFT signals

Recently a Hybrid Carrier (HC) scheme based on Weighted-type Fractional Fourier Transform (WFRFT) was proposed and developed, which contains Single Carrier (SC) and Multi-Carrier (MC) synergetic transmission. The wide interest is primarily due to its appealing characteristics, such as the robust performances in different types of selective fading channels and a great deal of potential for secure communications. According to the literatures, the HC signal and SC or MC signal probability distributions are different. In particular, some benefits of this HC scheme are brought by the quasi-Gaussian distribution of WFRFT signals. However, until now researchers have only presented statistic properties through computer simulations, and the accurate expressions of signals are not derived yet. In this paper, we derive the accurate and rigorously established closed-form expressions of Probability Density Function (PDF) of WFRFT signal real and imaginary parts with a large number of QPSK subcarriers, and this PDF can describe the behavior of data modulated by WFRFT, avoiding the complex computation for extensive computer simulations. Furthermore, the components of PDF expression are described and analyzed, and it is revealed that the tendency of signal quasi-Gaussian changes with the increasing of the parameter α (α in (0, 1]). To validate the analytical results, extensive simulations have been conducted, showing a very good match between the analytical results and the real situations. The contribution of this paper may be useful to deduce the closed form expressions of Bit Error Ratio (BER), the Complementary Cumulative Distribution Function (CCDF) of Peak to Average Power Ratio (PAPR), and other analytical studies which adopt the PDF.

Significance of Weighted-Type Fractional Fourier Transform in FIR Filters

The desired frequency response of a filter is periodic in frequency and can be expanded in Fourier series. One possible way of obtaining FIR filter is to truncate the infinite Fourier series. But abrupt truncation of the Fourier series results in oscillation in the pass band and stop band. These oscillations are due to slow convergence of the Fourier series by the Gibb’s phenomenon. To reduce these oscillations the Fourier coefficients of the filter are modified by multiplying the infinite impulse response with a finite weighing sequence called a window. The Fourier transform (FT) of a window consists of a central lobe and side lobes. The central lobe contains most of the energy of the window. To get an FIR filter, the desired impulse response and window function are multiplied, which results to give finite length non-causal sequence. Since Fractional Fourier Transform (FrFT) is generalization of FT. Here an attempt is to implement filters using window by using Weighted Type Fractional Fourier Transform (WFrFt), differentiator and integrator using weighted FrFt is also present.

Hybrid Carrier Modulation System with Partial FFT Demodulation Over Doubly Selective Channels in Presence of Carrier Frequency Offset

The effectiveness of communication system is impaired by the carrier frequency offset (CFO) and inter-carrier interference (ICI) due to the significant time variation under doubly selective (DS) channels. In this paper, we consider the weighted-type fractional Fourier transform (WFRFT)-based hybrid carrier modulation (HCM) system with partial fast Fourier transform (PFFT) demodulation for DS channels in presence of CFO. The influence of CFO and ICI, after PFFT demodulation and optimal channel compensation, can be further suppressed under certain WFRFT domain. Numerical results are presented to demonstrate the superiority of HCM with PFFT demodulation over the single carrier modulation and orthogonal frequency division multiplexing systems with PFFT demodulation at a very moderate complexity.

Hybrid Carrier Communication with Partial FFT Demodulation over Underwater Acoustic Channels

Recently, the partial Fast Fourier Transform (PFFT) has emerged as a promising demodulation scheme for the orthogonal frequency division multiplexing (OFDM) over underwater acoustic (UWA) channels. However, the effectiveness of OFDM in the receivers is impaired by inter-carrier interference (ICI) due to the significant time variations under UWA channels. In this letter, we consider the weighted-type fractional Fourier transform (WFRFT)-based hybrid carrier (HC) modulation system with PFFT demodulation for UWA communication. The residual ICI, after PFFT demodulation and weight compensation, can be further suppressed in the WFRFT domain. Numerical results are presented to demonstrate the superiority of HC scheme with PFFT demodulation over the single carrier (SC) and OFDM schemes with PFFT demodulation at a very moderate complexity.

WFRFT Precoding for Narrowband Interference Suppression in DFT-Based Block Transmission Systems

In this letter, we present a new approach to narrowband interference (NBI) suppression utilizing the weighted-type fractional Fourier transform (WFRFT) as a precoding scheme in the DFT-based communications systems. The proposed scheme can degenerate into a single carrier (SC) or a multi-carrier (MC) frequency division multiple access block transmission system by means of WFRFT parameter selection, which also leads to a variation of residual energy distribution of NBI following a frequency domain avoidance strategy. By virtue of parameter optimization of the WFRFT, the proposed system can outperform both SC and MC systems with regards to NBI suppression.

Hybrid Carrier Modulation with Time-Domain Windows and Iterative Equalization over Underwater Acoustic Channels

In this letter, we consider an hybrid carrier modulation (HCM) scheme based on weighted-type fractional Fourier transform (WFRFT) for communications over underwater acoustic channels (UAC). Time domain windowing and a frequency domain iterative MMSE equalization (FD-IMSE) are employed in the HCM-underwater-acoustic system. Numerical results manifest that the HCM outperforms both the multi-carrier modulation and the single carrier modulation in terms of bit-error-ratio over UAC, though with an acceptable increase in computation burden.

Digital computation of the weighted-type fractional Fourier transform

The paper reveals the time-frequency symmetric property of the weighted-type fractional Fourier transform (WFRFT) by investigating the original definition of the WFRFT, and proposes a discrete algorithm of the WFRFT based on the weighted discrete Fourier transform (WDFT) algorithm with constraint conditions of the definition of the WFRFT and time-domain sampling. When the WDFT is considered in digital computation of the WFRFT, the Fourier transform in the definition of the WFRFT should be defined in frequency (Hz) but not angular frequency (rad/s). The sampling period Δt and sampling duration T should satisfy \(\Delta t = {T \mathord{\left/ {\vphantom {T N}} \right. \kern-\nulldelimiterspace} N} = {1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}\) when N-point DFT is utilized. Since Hermite-Gaussian functions are the best known eigenfunctions of the fractional Fourier transform (FRFT), digital computation based on eigendecomposition is also carried out as the additional verification and validation for the WFRFT calculation.