The multiweighted type fractional fourier transform scheme and its application over wireless communications
 Yong Li^{1, 2, 3}Email author,
 Zhiqun Song^{1, 2} and
 Xuejun Sha^{3}
https://doi.org/10.1186/s1363801810522
© The Author(s) 2018
Received: 16 October 2017
Accepted: 5 February 2018
Published: 13 February 2018
Abstract
Researching on the relationships among different multiple weighted type fractional Fourier transform (multiWFRFT) schemes, we in this paper provide the modulation order relation of different multiWFRFT in theory. Moreover, we also prove that the two matrix spaces (i.e., 4WFRFT and multiWFRFT) are equivalent. To apply multiWFRFT into the wireless communication, we design the implementation structure of multiWFRFT (in particular, for 2^{ n }WFRFT scheme). Furthermore, the generalized hybrid carrier modulation (GHCM) system has been proposed in this paper, which can be degenerated to the classical hybrid carrier modulation (HCM) system with special parameters. We also verify the modulation order relationship between multiWFRFT by numerical simulations. Leveraging the theory and structure of multiWFRFT, we finally discuss its potential applications over secure communication and multiple input multiple output (MIMO) system.
Keywords
1 Introduction
Recently, 4weighted type fractional Fourier transform (4WFRFT) has been popular in channel equalization, narrowband interference (NBI) suppression, and signal processing [1–6]. Moreover, 4WFRFTbased hybrid carrier modulation (HCM) system, due to its special structure, can achieve better performance than the traditional single carrier modulation (SCM) system and orthogonal frequency division multiplexing (OFDM) system under doubly selective (DS) channels [1, 3, 6]. The reason is that OFDM system will be plagued by intercarrier interference (ICI) due to the highly time variance under DS channels, and the effectiveness of SCM system is impaired due to intersymbol interference (ISI) cause by large timedelay spread. Meanwhile, HCM system, convergence of SCM and OFDM signal, can make the signal and residual interference evenly distribute in the timefrequency plane. In this case, it is less possible to produce the strong interference and performs better under DS channels. Although, the complexity of HCM tolerably increase by \(\mathcal {O}(K\log (K))\) comparing with OFDM system.
However, there are little reports to research on multiple WFRFT (multiWFRFT), which can be as the generalized formation of 4WFRFT on wireless communications. Furthermore, the modulation order relationship between different multiWFRFT schemes is also vague in theory. There are many different constitutions for multiWFRFT according to [5], such as the classical fractional Fourier transform (CFRFT)based multiWFRFT [7–11]. However, the standard WFRFT (i.e., 4WFRFT)based multiWFRFT is interesting due to its implement structure. Ran et al. in [5] have provided the explanation of multiWFRFT in mathematics. Unfortunately, the implementation structure of multiWFRFT and explanation in physical are still ambiguous, which will obstruct its application on wireless communications.
To apply multiWFRFT to wireless communications, we in this paper prove the relationship of modulation orders between multiWFRFT and 4WFRFT in theory, and can be described a simple formula, i.e., \(\alpha _{M}=\frac {M}{4}\alpha _{4}\) when multiWFRFT and 4WFRFT of the same signal are equal (α_{ M } and α_{4} are the modulation orders for multiWFRFT and 4WFRFT, respectively). As a corollary, we also prove the relationship of modulation orders among multiWFRFT with different M. Furthermore, some important properties of multiWFRFT have also been given in this paper. We finally prove that any N×N matrix spaces for multiWFRFT are equivalent.
We also provide the implementation structure of multiWFRFT (in special, 2^{ n }WFRFT). Furthermore, the generalized hybrid carrier modulation (GHCM) system has been proposed in this paper, which can be degenerated to the classical hybrid carrier modulation (HCM) system with special parameters. We also verify the modulation order relationship between multiWFRFT by numerical simulations. Leveraging the theory and implementation structure of multiWFRFT, we finally analyze the potential applications on wireless communications.
The rest of this paper is organized as follows. The basic knowledge of WFRFT is introduced in Section 2. Furthermore, the modulation order relationships between different multiWFRFT will be proved in Section 3. To apply multiWFRFT system into wireless communication, we provide the implementation structure for multiWFRFT system (in special, 2^{ n }WFRFT system) and its complexity requirements in Section 4. We then propose a GHCM system and provide some numerical simulations in Section 5. The potential applications of multiWFRFT is also discussed in Section 5. We conclude the whole paper in Section 6.
Notations: α_{ m } is the modulation order for mWFRFT (m≥4). And \(\mathbf {W}^{\alpha _{m}}_{m}\) denotes the mWFRFT matrix. Then, A_{ l }(α_{4})(l=0∼3) is the weight coefficient of 4WFRFT while B_{m,l}(α_{ m })(l=0∼m−1) is the weight coefficient of mWFRFT (m>4). F is the normalized discrete Fourier transform, and the elements of F, \([\mathbf {F}]_{j,k}=1/\sqrt {N}\exp (2\pi i j k /N), j,k = 0,1,...,N1\). Moreover, 〈.〉_{ N } denotes the moduloN calculation. At last, δ(.) is the Kronecker delta. In this paper, MWFRFT is used in the processing of proof; meanwhile, mWFRFT is used in other places.
2 The basic knowledge for WFRFT
where \(\mathbf {W}^{\alpha _{m}}_{m}\) denotes the N×NMWFRFT matrix with subscript of M. The definition of \(\mathbf {W}^{\alpha _{m}}_{m}\), as shown in (6), will be employed in this paper unless otherwise noted.
We have derived the relationships of A_{ l }(α_{4}) and A_{ k }(α_{4}) via (8), and the relationships of B_{ h }(α_{ M }) and B_{ j }(α_{ M }) via (9). However, the order relationships between 4WFRFT and MWFRFT are vague. We will reveal these relationships in the next section.
3 The modulation order relationships between 4WFRFT and multiWFRFT
In this section, we will reveal the modulation order relationship between 4WFRFT and multiWFRFT, which is essential for realizing the nature characteristics of WFRFT.
Theorem 1
Proof

Step 1 We firstly prove that when \(\mathbf {W}^{\alpha _{5}}_{5}=\mathbf {W}^{\alpha _{4}}_{4}\), (10) holds, i.e., \(\alpha _{5} = \frac {5}{4}\alpha _{4}\).
Inserting M=5 into (6), then$$ \mathbf{W}^{\alpha_{5}}_{5} = \sum\limits_{p=0}^{4}B_{p}(\alpha_{5})\mathbf{W}^{\frac{4p}{5}}_{4} $$(11)Note that \(\mathbf {W}^{\frac {4p}{5}}_{4} \left (p=0 \sim 4, p \in \mathbb {Z}\right)\) involves the sum of four terms and can be expressed as:
$$ \mathbf{W}^{\frac{4p}{5}}_{4} = \sum\limits_{j=0}^{3}A_{j}\left(\frac{4p}{5}\right)\mathbf{F}^{j}, p=0 \sim 4 $$(12)By substituting (12) into (11):
$$ \mathbf{W}^{\alpha_{5}}_{5} = \sum\limits_{j=0}^{3}\Theta_{j}(\mathbf{A}, \mathbf{B})\mathbf{F}^{j} $$(13)where$$ \Theta_{j}(\mathbf{A}, \mathbf{B}) = \sum\limits_{p=0}^{4}B_{p}(\alpha_{5})A_{j}\left(\frac{4p}{5}\right), j=0 \sim 3. $$(14)Comparing (2) with (13), \(\mathbf {W}^{\alpha _{5}}_{5}=\mathbf {W}^{\alpha _{4}}_{4}\) holds if and only if each coefficient of \(\mathbf {F}^{j}(j=0 \sim 3, j \in \mathbb {Z})\) is equal. For brevity, we employ the coefficient of F^{0} in (2) and (13) to obtain the modulation order relationship.
When j=0, (14) can be expressed:
$$ \Theta_{0}(\mathbf{A}, \mathbf{B}) = \sum\limits_{p=0}^{4}\Lambda_{p}, $$(15)with$$\begin{array}{@{}rcl@{}} \left \{ \begin{aligned} \Lambda_{0}=&~~~~\frac{1}{5}\ \frac{1\exp(2 \pi i \alpha_{5})}{1\exp\left(\frac{2 \pi i \alpha_{5}}{5}\right)}\\ \Lambda_{1}=&\frac{1}{20}\ \frac{1\exp(2\pi i \alpha_{5})}{\exp\left(\frac{2\pi i}{5}\right)\exp\left(\frac{2 \pi i \alpha_{5}}{5}\right)}\\ \Lambda_{2}=&~~~~\frac{1}{20}\ \frac{1\exp(2\pi i \alpha_{5})}{\exp\left(\frac{\pi i}{5}\right)+\exp\left(\frac{2 \pi i \alpha_{5}}{5}\right)}\\ \Lambda_{3}=&~~~~\frac{1}{20}\ \frac{1\exp(2\pi i \alpha_{5})}{\exp\left(\frac{\pi i}{5}\right)+\exp\left(\frac{2 \pi i \alpha_{5}}{5}\right)}\\ \Lambda_{4}=&\frac{1}{20}\ \frac{1\exp(2\pi i \alpha_{5})}{\exp\left(\frac{2\pi i}{5}\right)\exp\left(\frac{2 \pi i \alpha_{5}}{5}\right)} \end{aligned} \right. \end{array} $$(16)Through (16), we can derive that$$ \sum\limits_{p=0}^{4}\Lambda_{p} = \frac{U\sum_{j=0}^{3}(j+1)\exp\left(2j \pi i \alpha_{5}/5\right)}{20\sum_{p=0}^{4}\exp\left(2p \pi i \alpha_{5}/5\right)} $$(17)where U=1− exp(−2πiα_{5}).
Obviously, both numerator and denominator are geometric sequence or its expansion [13]. Therefore, (17) can be simplified:
$$ \sum\limits_{p=1}^{4}\Lambda_{p} = \frac{15\exp(8 \pi i \alpha_{5}/5)+4\exp(2 \pi i \alpha_{5}/5)}{20\left[1\exp(2 \pi i \alpha_{5}/5)\right]} $$(18)Finally, according to (15) and (18), Θ_{0}(A,B) can be expressed as follows:$$ \Theta_{0}(\mathbf{A}, \mathbf{B}) = \frac{1\exp(8 \pi i \alpha_{5}/5)}{4[1\exp(2 \pi i \alpha_{5}/5)]} $$(19)To explicitly compare the coefficients between 4WFRFT and MWFRFT, we provide the expression of A_{0}(α_{4}):
$$ A_{0}(\alpha_{4})=\frac{1\exp(2 \pi i \alpha_{4})}{4[1\exp(2 \pi i \alpha_{4}/4)]} $$(20)Exploiting Θ_{0}(A,B)=A_{0}(α_{4}), (19) and (20), we obtain that
$$ \alpha_{5} = \frac{5}{4}\alpha_{4} $$(21)Until now, we have completed the proof of step 1.

Step 2 Assuming that when M=k(k≥5), Theorem 1 holds [14]. That is, if \(\mathbf {W}^{\alpha _{k}}_{k}=\mathbf {W}^{\alpha _{4}}_{4}\), the following formula holds:$$ \alpha_{k}=\frac{k}{4}\alpha_{4} $$(22)
Then, we will employ (22) to prove Theorem 1 for M=k+1.

Step 3 As stated previously, we will prove when M=k+1, (10) holds. To distinguish the coefficients of kWFRFT and (k+1)−WFRFT, we in this step employ B_{k,l} and B_{k+1,l} denote the lth weight coefficient for kWFRFT and (k+1)WFRFT, respectively.
We first derive the relationship between α_{ k } and α_{k+1} when \(\mathbf {W}^{\alpha _{k}}_{k}=\mathbf {W}^{\alpha _{k+1}}_{k+1}\). \(\mathbf {W}^{\alpha _{k+1}}_{k+1}\) can be written as:
$$ \mathbf{W}^{\alpha_{k+1}}_{k+1} = \sum\limits_{q=0}^{k}B_{k,q}(\alpha_{k+1})\mathbf{W}^{\frac{4q}{k+1}}_{4} $$(23)By substituting (22) into (23), we obtain$$ \mathbf{W}^{\alpha_{k+1}}_{k+1} = \sum\limits_{q=0}^{k}B_{k,q}(\alpha_{k+1})\mathbf{W}^{\frac{kq}{k+1}}_{k} $$(24)Based upon kWFRFT, we can get the coefficients of \(\left (\mathbf {F}^{\frac {4}{k}}\right)^{0}\) for both \(\mathbf {W}^{\alpha _{k}}_{k}\) and \(\mathbf {W}^{\alpha _{k+1}}_{k+1}\).$$\begin{array}{@{}rcl@{}} {}B_{k,0}(\alpha_{k}) &=& \frac{1\exp(2 \pi i \alpha_{k})}{k[1\exp(2 \pi i \alpha_{k}/k)]}, \end{array} $$(25)$$\begin{array}{@{}rcl@{}} {}\Lambda_{0}(\mathbf{B}_{k}, \mathbf{B}_{k+1}) &=& \sum\limits_{q=0}^{k}\Gamma_{q}, \end{array} $$(26)$$\begin{array}{@{}rcl@{}} {}\Gamma_{q}&=&B_{k+1,q}(\alpha_{k+1})B_{k,0}\left(\frac{kq}{k+1}\right),\\ {}q&=&0 \sim k, q \in \mathbb{Z}. \end{array} $$(27)with$$ \Gamma_{q}=B_{k+1,q}(\alpha_{k+1})B_{k,0}\left(\frac{kq}{k+1}\right), q=0 \sim k, q \in \mathbb{Z} $$(28)With observation and complex computation, \(\sum _{q=1}^{k}\Gamma _{q}\) can be obtained:
$$\begin{array}{@{}rcl@{}} \sum\limits_{q=1}^{k}\Gamma_{q} = \frac{V \sum_{j=0}^{k1}{(j+1)\exp\!\left(\frac{2 j \pi i \alpha_{k+1}}{k+1}\right)}}{k(k+1)\sum_{q=0}^{k}\exp\!\left(\frac{2 q \pi i \alpha_{k+1}}{k+1}\right)} \end{array} $$(29)with$$ V=1\exp(2 \pi i \alpha_{k+1}) $$(30)In this case, (29) can be expressed as:
$$ {{} \begin{aligned} \sum_{q=1}^{k}\Gamma_{q} \,=\, \frac{1\,\,(k+1)\exp\left(\frac{2 \pi i (k)\alpha_{k+1}} {k+1}\right)+k\exp\left(2 \pi i \alpha_{k+1}\right)}{k(k+1)\left[1\exp\left(\frac{2 q \pi i \alpha_{k+1}}{k+1}\right)\right]} \end{aligned}} $$(31)Thus,$$ \Lambda_{0}(\mathbf{B}_{k}, \mathbf{B}_{k+1}) = \frac{1\exp\left(\frac{2 \pi i k \alpha_{k+1}}{k+1}\right)}{k\left[1\exp\left(\frac{2 \pi i \alpha_{k+1}}{k+1}\right)\right]} $$(32)Comparing (32) with B_{k,0}(α_{ k }), we can obtain:
$$ \alpha_{k+1} = \frac{k+1}{k}\alpha_{k} $$(33)
We have completed the proof of Theorem 1. □
Moreover, MWFRFT can also satisfy the boundary axiom according to the following corollary.
Corollary 1
Proof
The Corollary 1 can be straightforwardly derived via Theorem 1. Furthermore, Corollary 1 reveals a flexible method to switch between time domain and frequency domain with special modulation orders controlling.
Corollary 2
Then, these two matrix spaces are equivalent.
Proof
according to Theorem 1.
That is to say, G_{4} is the subspace of G_{ m }.
holds for any \(\mathbf {W}_{m}^{\alpha _{m}} \in \mathbf {G}_{m}\) via Theorem 1. Thus, G_{ m } is also the subspace of G_{4}.
In summary, the G_{4} space and G_{ m } space are equivalent. □
As a generalized of Corollary 2, G_{ m } and G_{ n } will be equivalent for any real m,n. This is obvious according to Corollary 2.
4 Implementation structure of 2^{ n }WFRFT and its complexity requirements
As stated previously, we have proved the modulation order relationship between 4WFRFT and MWFRFT via Theorem 1. There would be a question about the next research in our mind. What is the significance to research the MWFRFT since it can be represented by 4WFRFT? As a special case of multiWFRFT, we in this section will give the 2^{ n }WFRFT (n≥2) implementation structure and the complexity requirements.
4.1 Implementation structure of 2^{ n }WFRFT
in which P is the permutation matrix with [P]_{m,n}=δ(〈m+n〉_{ N }) (m,n=0,1,...,N−1), which can be as the inverse module. Due to the special structure of 2^{ n }WFRFT, we can design the implementation structure, which can be the generalized form of that in [15]. We can implement the 2^{ n }WFRFT using (46).
Different signals, dividing into different channels, will then be multiplied by the corresponding weight coefficients (B_{ j },j=0,...,2^{ n }−1). After taking the summation for all channels, the derived signal will be transmitted. And then, we obtain the output signal Y by parallel and serial transform.
Note that there are two time domain signals over “channel 0” and “channel 2^{n−1},” which can be as the single carrier components. Moreover, there are two frequency domain signals over “Channel 2^{n−2}” and “Channel 3×2^{n−2},” which can be as the multicarrier components [1, 3, 15]. Differing from [15], 2^{ n }WFRFT involves 2^{ n }−4 weight fractional Fourier transform domain signals over the other channels, which can be as the hybrid carrier components [1, 3]. These hybrid carrier components can be derived by different order 4WFRFT transform. Therefore, 2^{ n }WFRFT is a compatible modulation system of “single carrier,” “multicarrier,” and “hybrid carrier” modulations.
4.2 Complexity requirements
The details of complexity requirements
Operators  Complexity  Times 

4WFRFT  \(\mathcal {O}(N \log N +4N)\)  2^{n−2}−1 
DFT  \(\mathcal {O}(N \log N)\)  2^{n−2} 
Multiply  \(\mathcal {O}(N) \)  2^{ n } 
5 Simulation and potential applications analysis
5.1 Simulation and analysis
Figure 3 is the relationship between 8WFRFT and 4WFRFT of the original signal. The original signal has been shown in Fig. 3a. Figure 3b is the result of – 0.8 order 8WFRFT of original signal (here m and α are 8 and 0.8, respectively). The recover signal is derived by a 0.4 order 4WFRFT. From Fig. 3c, we can clearly observe that the demodulation signal can be well coincide with the original signal. Moreover, the common communication signal to be simulated in Fig. 4a. Figure 4b is the result of – 0.6 order 16WFFRFT of original signal (here m and α are 16 and 0.6, respectively). The recover signal is derived by a 0.3 order 8WFRFT. From Fig. 4c, we can clearly observe that the demodulation signal can be well coincide with the original signal. Furthermore, according to the error between original signal and recover signal in Figs. 3d and 4d, we can also confirm the effectiveness of theorems above.
To verify the superiority of GHCM, we set the simulation under DS channels, in comparison with SCM and OFDM systems. We consider the 512 QPSK symbol for each block under DS channels. The bandwidth is 2 MHz. Moreover, the normalize Doppler frequency f_{ d }T_{ d } = 0.00384, where f_{ d } is the singleside Doppler spread in Hz and T_{ d } denotes the sampling interval of the discretetime system. The DS channel is modeled by a sixteentap WSSUS channel with an exponential multipath intensity profile [16]. In order to be fairly compared, we employ the partial FFT demodulation [1, 3] in the three systems and the division number of 16. Without loss of generality, we employ 8WFRFT in the transmitter and 4WFRFT in the receiver.
5.2 Potential applications of multiWFRFT
The communication security is a crucial problem to the wireless communication [18–21]. Two classes of methods for wireless communication security have been exploited. First, the transmitter and receiver have the different secret key and cannot be obtained by each other [21, 22]. Second, we also enhance the physical security in the wireless transmission [20]. In the multiWFRFT scheme, we can design the communication system with the secrete key of the modulation order α. According to Theorem 1, the transmitter and receiver may exploit different secrete keys to guarantee the wireless communication security. In this case, the order using at the transmitter is not necessary to be transmitted under wireless channels, it only transmits the encrypted signal and the order employing at the receiver. What is more, we can also employ the antenna array redundancy [19] to enhance the physical security due to the multiantenna transmitting characteristics of multiWFRFT. The advantage of the multiWFRFT is that, it is not necessary to transmit the order of transmitter under wireless channels. Even if the illegitimate receiver intercepts the signal encrypted by mWFRFT at the transmitter, the signal cannot be correctly recovered without the order of mWFRFT. Thus, the theorem for multiWFRFT is important for communication security.
The immediate application is to employ multiWFRFT into MultiInput MultiOutput (MIMO) system due to its multiaccess sampling characteristics for multiWFRFT, specially for 2^{ n }WFRFT scheme. At the transmitter, we employ different antennas to transmit signals in the different channels of multiWFRFT implement structure. We can receive the signals using multiantennas in the receiver. It can take advantage of the channel diverse gain [23–25] of 2^{ n }WFRFT scheme shown in Fig. 1.
4WFRFT has various applications, i.e., channel equalization [1, 3] and narrowband interference (NBI) suppression [4], which can also be generalized into multiWFRFT scheme. Besides, the multiWFRFT, as a new carrier convergence system, can also be exploited in the next generation cellular system.
6 Conclusions
In this paper, we focus on the modulation order relationship among different multiWFRFT schemes and prove that α_{ M }=(M/4)α_{4} if \(\mathbf {W}^{\alpha _{m}}_{M}=\mathbf {W}^{\alpha _{4}}_{4}\). Moreover, we prove that the equality of 4WFRFT space and multiWFRFT space through the corollary. In particular, the implementation structure for 2^{ n }WFRFT, which is the convergence of single carrier, multicarrier, and hybrid carrier components, has been well designed in this paper. According to the theorem of this paper, we propose a generalized hybrid carrier modulation (GHCM) system, which simplifies to the classical hybrid carrier modulation (HCM) system with m = n = 4. Also, the modulation order relationships between different multiWFRFTs have been further demonstrated through some numerical simulations. Furthermore, it can be demonstrated that, through numerical simulation, GHCM based on multiWFRFT performs better than SCM system and OFDM system under typical doubly selective channels. Finally, we discuss the potential applications of multiWFRFT scheme over secure communication and MIMO system.
Declarations
Acknowledgements
The authors would like to thank the editor and the anonymous referee for their valuable comments and suggestions that improved quality of this paper. This work is supported by the project funded by the China Postdoctoral Science Foundation under grant no. 2016M601286, the fund of Science and Technology on Communication Networks Laboratory under grant no. EX156410046, and the 973 Program under grant no. 2013CB329003. Moreover, this work is also supported by the fund of the National Key Laboratory of Science and Technology on Communications.
Funding
This work is supported by the project funded by the China Postdoctoral Science Foundation under grant no. 2016M601286, the fund of Science and Technology on Communication Networks Laboratory under grant no. EX156410046, and the 973 program under grant no. 2013CB329003. Moreover, this work is also supported by the fund of the National Key Laboratory of Science and Technology on Communications.
Availability of data and materials
We assert that all of the data and materials in this paper are available, and all of the experiments can be reproduced according to this paper.
Authors’ contributions
YL found the new theory and algorithm and wrote the whole paper as the main author. Prof. ZS and Prof. XS are the responsible for the revision of the whole paper and provided some useful proposal. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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