In order to verify the relationship between m-WFRFT and n-WFRFT, we first provide a generalized hybrid carrier modulation system (GHCM) as shown in Fig. 2. The original signal *f*, assumed to be in the *α* order m-WFRFT domain, can be converted back to time domain through a −*α* order m-WFRFT at the transmitter. Generally, the received signal *f*^{′}, at the receiver, can be converted back to the original m-WFRFT domain by a *α* order m-WFRFT. However, we can also obtain *f*^{′}, according to Corollary 1, by a *β* order n-WFRFT with *β*=*n*/*m**α*. This is a simple model without any equalization modular. Besides, the GHCM system model, with *m*=*n*=4, can be degenerated to the classical hybrid carrier modulation system (HCM) [1, 3, 6]. The primary contribution of GHCM system is that the modulation orders (i.e., *α* and *β*) at the transmitter and receiver can be flexibly selected, which can expand the applications over communications (i.e., the security communication and MIMO systems). Finally, we also discuss the potential applications of multi-WFRFT over wireless communication.

### Simulation and analysis

First, we will provide some simulations to verify the theorem on the relationship of modulation orders. Assuming the original signal is *x*=sinc(*t*), where *t*∈[−5,5]. Here, we mainly want to verify the relationship between m-WFRFT and n-WFRFT. Therefore, the channel is assumed to be ideal without any interference and noise. The simulation results have been demonstrated from Figs. 3 and 4.

Figure 3 is the relationship between 8-WFRFT and 4-WFRFT of the original signal. The original signal has been shown in Fig. 3a. Figure 3b is the result of – 0.8 order 8-WFRFT of original signal (here *m* and *α* are 8 and 0.8, respectively). The recover signal is derived by a 0.4 order 4-WFRFT. 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 16-WFFRFT of original signal (here *m* and *α* are 16 and 0.6, respectively). The recover signal is derived by a 0.3 order 8-WFRFT. 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 single-side Doppler spread in Hz and *T*_{
d
} denotes the sampling interval of the discrete-time system. The DS channel is modeled by a sixteen-tap 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 8-WFRFT in the transmitter and 4-WFRFT in the receiver.

The simulation results have been shown in Fig. 5. The modulation order *α* of 1, here *m* = 8 and *n* = 4 via Fig. 2, is selected in the transmitter. According to Theorem 1, the demodulation order *β*, in the receiver, should be chose as 0.5. The selection of modulation order can be found in [17]. It is demonstrated that the superiority of the GHCM is obvious in comparison to OFDM systems under this DS channel. Moreover, GHCM performs better than SCM system when *E*_{
b
}/*N*_{0}≥ 15 dB. The GHCM system can be degenerated to HCM system [2] when *m* = *n* = 4 via Fig. 2. However, the communication security performance of GHCM can be enhanced as the different order at the transmitter and receiver, in comparison to the HCM system. The reason will be provided in the next section.

### Potential applications of multi-WFRFT

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 multi-WFRFT 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 multi-antenna transmitting characteristics of multi-WFRFT. The advantage of the multi-WFRFT 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 m-WFRFT at the transmitter, the signal cannot be correctly recovered without the order of m-WFRFT. Thus, the theorem for multi-WFRFT is important for communication security.

The immediate application is to employ multi-WFRFT into Multi-Input Multi-Output (MIMO) system due to its multi-access sampling characteristics for multi-WFRFT, specially for 2^{n}-WFRFT scheme. At the transmitter, we employ different antennas to transmit signals in the different channels of multi-WFRFT implement structure. We can receive the signals using multi-antennas in the receiver. It can take advantage of the channel diverse gain [23–25] of 2^{n}-WFRFT scheme shown in Fig. 1.

4-WFRFT has various applications, i.e., channel equalization [1, 3] and narrow-band interference (NBI) suppression [4], which can also be generalized into multi-WFRFT scheme. Besides, the multi-WFRFT, as a new carrier convergence system, can also be exploited in the next generation cellular system.