Performance Analysis on Spatial MFSK Modulation with Energy Detection

of its detection performance needs to be solved. This paper analyzes the theoretical symbol error rate (SER) performance of the Spatial MFSK modulation with energy detection. The noise of the MIMO system by energy detection conform to the generalized gamma distribution. Based on this distribution, the optimal decision rule of the system and the theoretical SER formula are derived. Numerical results show that the theoretical SER formula fits well with the simulation results of the system under the condition of high signal-to-noise ratio (SNR).

coherent systems. On the contrary, noncoherent systems have the advantages of low complexity, low power consumption, insensitivity to channel changes and high mobility. Nevertheless, there is a performance loss of the MIMO based on energy detection at low SNR comparedwith coherent detection, so the theoretical performance analysis of its capacity and SER becomes a problem to be solved objectively.
Noncoherent MIMO is insensitive to time-varying channels, and the receiving equipment is simple, which has become one of the main research fields of MIMO in recent years. The main research directions of noncoherent MIMO focus on noncoherent Grassmannian MIMO [1][2][3][4][5], differential detection MIMO [6][7][8][9][10][11], and energy detection MIMO [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Among them, Grassmannian MIMO demands no or only partial CSI for demodulation, which applies to high SNR and high-speed mobile scenarios [1]. However, the capacity of Grassmannian MIMO is limited by channel coherence time and complex encoding [2]. In [3], a systematic unitary space-time code was proposed by continuously rotating the data signal in a high-dimensional space, while its design and decoding complexity are relatively high. In [4], an optimal design for the unitary space-time code was presented by iteratively searching for unitary space-time code contained in the Grassmann manifold set, while the traversal is difficult and the decoding complexity increases exponentially as the bit transfer rate accelerates. For 2x2 MIMO system, a structured unitary space-time code was proposed to obtain the full diversity gain, lower encoding and decoding complexity. However, the encoding fails to take energy efficiency into account, resulting in low encoding gain [5]. Noncoherent MIMO with differential detection could also reduce the dependence on the channel information. Differential spacetime modulation (DSTM) is a noncoherent MIMO scheme that does not require accurate CSI [6]. This technique is an extended form of differential phase shift keying (DPSK) modulation in a multi-antenna system, which uses the phase difference between two adjacent transmission code blocks to carry the transmitted information and completes the transmission of information without the CSI [7]. The research on differential space-time code mainly bases on orthogonalized differential space-time block code (DSTBC), which has low computational complexity because of orthogonality. Differential decoding at the receiver is considered as detecting a single symbol and does not require estimating CSI [8]. In [9], a multi-symbol differential detection technique was presented, the corresponding decision matrix was designed and its BER performance was analyzed. Unfortunately, differential detection requires a quasi-static fading channel with consistent channel states between adjacent symbols, which is not necessarily true in high-speed mobile environments. The Doppler shift brought about by high-speed mobility will cause the phase of the far-field MIMO channel to change rapidly. Noncoherent MIMO with energy detection is proposed to reduce the sensitivity of Doppler. It is insensitive to phase rotation and has high robustness, low complexity, and low synchronization requirements in non-ideal CSI demodulation. Existing research on noncoherent MIMO based on energy detection mainly focuses on how to obtain the maximum diversity gain [10]. On the other hand, the MIMO with energy detection of spatial multiplexing has gradually begun to attract attention since its performance improvement [11], which can improve the communication spectrum efficiency under low complexity conditions. Noncoherent MIMO based on energy detection has better robustness to synchronization and Doppler shift in high-speed mobility, which can avoid frequent channel estimation problems under time-varying channels. The modulation methods of noncoherent MIMO based on energy detection mainly include multi-ary amplitude shift keying (ASK), frequency shift keying (FSK), and pulse position modulation (PPM) [12] [13], where amplitude modulation is mostly adopted. In [14]- [17], closed-form BER expressions were given for a single-input multiple-output (SIMO) communication system modulated by ASK in additive white Gaussian noise (AWGN) channels, lognormal channels, and Rayleigh channels, respectively. In [18], the optimal system performance was obtained by optimal constellation design in the MIMO Rayleigh channels when the amplitude of the multilevel ASK is approximate as a geometric sequence at high SNR. In [19], a noncoherent system based on energy detection was presented, which uses nonnegative pulse amplitude modulation (PAM) to decode the sending signal by averaging the reception signal energy of all antennas. The SER of a SIMO system with energy detection was deduced, on this basis, a minimum distance constellation was proposed [20]. [21] presented an energy-based or equivalent amplitude-based encoding and decoding process to design the transmit constellation points of the SIMO system. This scheme can significantly improve the performance of the system compared to conventional ASK modulation or equidistant power level modulation in [22]. In [23], two MIMO systems with energy detection were proposed. One is the instantaneous channel energy based on the probability density function, which uses noise hardening and the decision threshold to demonstrate the performance of instantaneous channel energy is close to coherent detection at high SNR. Whereas, the other is the average channel energy based on the chisquare cumulative distribution function, which has better performance when the spatial degrees of freedom are large enough. In a massive MIMO system based on energy detection, Gaussian approximation is performed on the gamma distribution subject to the received signal. The distribution curve by Gaussian approximation has a good match with the curve of the original received signal. Furthermore, the system capacity boundary of Gaussian approximation [24] and the theoretical BER formula in the closed form [25] are deduced.
The paper is organized as follows: Section 2 summarize the method and contributions of this paper. Section 3 describes the channel environments in which the system is in a highspeed mobility scenario and the theoretical basis for the system to have better Doppler robustness. Section 4 describes the system model and derives the equivalent system model through energy detection. Furthermore, the signal and noise by equivalence are analyzed separately. Section 5 analyzes the distribution of noise by energy detection. Under this condition, the theoretical SER formula of the system is derived. Numerical results are presented in Section 6. Section 7 gives some concluding remarks.

Method
The MIMO based on energy detection by ASK modulation has limited spatial performance since the limited range of amplitude modulation. In this paper, we adopt the spatial MFSK modulation with better performance and analyze it theoretically. We equivalent the system model through nonlinear processing. The receiving end performs noncoherent detection through real channel equivalent channel parameters. The distribution of noise by energy detection and the theoretical expression of the average symbol error probability are derived. The optimal decision rule of the system is deduced, however, for the high complexity of ML detection with optimal judgment, ML is simplified through multi-antennas joint minimum Euclidean distance detection. At the same time, the distribution problem of signal-dependent noise that accompanies minimum Euclidean distance detection is analyzed.

Channel environment
when a=1 , the receive signal is the multipath scattering part of the transmitted signal, and the channel model is the Rayleigh channel. When a=0 or = a 1 , the receive signal is regarded as the direct part of the transmitted signal, and the channel characteristics tend to be stable [27]. Considering a far-field environment where the antenna array size is much smaller than the propagation distance, it can be deemed that all concurrent transmission channels of the system experience the approximate same Doppler shift [28], that is where , D ln f represents the Doppler shift from the nth transmitting antenna to the lth receiving antenna.
At a=0 or = a 1 , where l h is the lth row of H( ) t for L ,, r = lN 1 . The fading coefficient ln jθ ln he is independent and identically distributed ( iid ), there is The Doppler shift is supposed to be approximately constant within one symbol period of the transmit signal, it is shown that The result of energy detection of the at a frequency matching its frequency point is one. On the contrary, the result is zero. Through the square-law processing, the random phase interference from the carrier is eliminated, in the meantime, the signal orthogonal characteristic minimizes the inter-signal interference. So that it has the characteristic of anti-Doppler frequency offset and phase offset in high-speed mobile environments. Of course, the Doppler shift will also cause the signal frequency offset, which can be ignored under the condition that the Doppler frequency deviation is far less than the symbol rate.

System model
Consider an energy detection system based on spatial MFSK modulation with t N transmit and r N receive antennas. The transmitted data stream is modulated by MFSK after V-BLAST encoding, upconverted and sent in parallel through different transmitting antennas. The transmitted signal is received in a noncoherent manner at the receiving end. Within a symbol interval, the  r N 1 received signal vector is represented by where H is the  where nw b f denotes the frequency of the wth symbol on the nth transmitting antenna, indicates the transmitted symbol mapped by V-BLAST encoding. Moreover, the modulated information-bearing symbol matrix is where s n represents the lth column of the sending symbol matrix.
The equivalent signal part by energy detection can be obtained as In (15), the lth row vector of the equivalent channel matrix can be expressed as and the equivalent transmitting data vector is where  ( , ) , i nw f X 01 .According to (13)-(17), the transceiver system by energy detection at i f can be equivalent to a real model, which is given by furthermore, there is The equivalent noise component by energy detection is given by (14), there is Therefore, the diagonal and off-diagonal elements of the covariance matrix of the equivalent noise vector u are derived as where  () var denotes variance operator and  () cov denotes covariance operator. In summary, the energy detection system based on spatial MFSK modulation is equivalent to a real system model, which is written as

Analysis of detection performance
The energy detection system based on spatial MFSK modulation is equivalent to a real system model. Next, we perform a theoretical analysis of the detection performance of the system. The transmitted data stream is mapped to a symbol vector m c by V-BLAST encoding, and its signal set is represented as where where the m c is modulated by MFSK into a frequency modulation signal vector m s , as in (10). According to (6) and (7), the m s is correlated at i f to obtain a vector with elements of 0 or 1, that is where ()  I 0 denotes the modified Bessel function of the zeroth order and the first kind. Owing to the frequencies of MFSK are orthogonal to each other, the likelihood functions at each frequency are independent of each other. Thus, the joint likelihood function is the product of the likelihood functions at each frequency. The m c that maximizes the sum of the logarithmic joint likelihood function is the signal that is correctly decided, that is, (33) Therefore, the (33) is the optimal decision rule for the energy detection system based on spatial MFSK modulation. Since the Bessel function is nonlinear, the implementation complexity of the receiver is high. Approximate processing of the Bessel function at high SNR can simplify the optimal decision rule. ()  I 0 is expanded as (44) Applying the Gaussian distribution normalized result of (44) to (43), the probability of correct decision can be represented as

Results and discussion
In this section, the theoretical average SER compares with the numerical results. The simulation demonstrates the accuracy of the received signal distribution by energy detection. Furthermore, we analyze the signal-dependent noise problem of minimum Euclidean distance detection. The (33) is the optimal decision rule. It is optimal to use the obeyed distribution to make a decision, but the complexity is high. Thus, it can be done by using the minimum of the joint Euclidean distance of multiple antennas at M frequency points, which can be defined as When using the minimum Euclidean distance to make a decision, it is necessary to consider the distribution of signal-dependent noise. Figure 1 and Figure 2 are the distributions of the received signal when the transmitted signal is 1 s , 2 s and 3 s . According to the optimal decision rule, when the received signal falls in the left region of the decision boundary b1, the transmitted signal is judged to be 1 s . When the received signal falls in the region between the decision boundary b1 and b2, the judgment transmitted signal is 2 s . When the received signal falls in the area to the right of the judgment boundary b2, it is decided to send the signal as 3 s . At low SNR, the variance of each distribution is relatively large, so that there are obvious overlap regions between adjacent distributions. At the same time, large noise will cause deviations in the center position of each distribution, causing the distance between the centers of each distribution to change based on the original distance. The above changes affect the accuracy of the minimum Euclidean distance detection. When the received signal falls at position A, according to the optimal decision rule, the transmitted signal should be determined as 3 s . While according to the minimum Euclidean distance, the influence of large noise makes d1<d2. Therefore, it will be misjudged as 2 s . As shown in Figure 2, the reduction of variance at high SNR makes distributions more concentrated, and the overlap regions between neighboring distributions become smaller, which can improve the overall performance of the minimum Euclidean distance detection. Of course, the isometric constellation design of the transmitting signal can reduce the overlap regions between neighboring distributions and improve the accuracy of the minimum Euclidean distance detection.
In the energy detection system based on spatial MFSK modulation with 3 transmitting and receiving antennas, all transmitting signals are correlated at i f to obtain 8 signal vectors with elements of 0 and 1. Figure 3 and Numerical results show that the simulation and theoretical formula match well. It can be illustrated from the figure that as the SNR increases, the overlap regions between adjacent distributions decrease, which can improve the overall performance of the minimum Euclidean distance detection. Figure 5 shows the comparison between the numerical results using the minimum Euclidean distance detection (48) and the theoretical formula of average symbol error probability (47) in the Spatial 4FSK modulation with energy detection. Numerical results show a good match between the simulation and the theoretical formulation at high SNR. Under the condition that the number of MFSK modulation is constant, the SER performance gradually improves as the number of antennas increases. The reason is that as the number of receiving antennas increases, the diversity gain of the receiving end will increase. The decision criterion is to adopt multi-antenna joint detection, as the number of antennas increases, the reliability of detection will be enhanced. Therefore, the SER of the system will be improved. Figure 6 shows the performance comparison of the MIMO system using 4FSK and 4ASK modulation with energy detection. The spatial performance of the system is limited since the limited distance space of amplitude modulation. Therefore, the performance of MIMO-4FSK based on energy detection is better than MIMO-4ASK.
Under the high-speed railway channel, Figure 7 shows the Doppler robustness analysis of the spatial 4FSK modulation with energy detection and spatial QPSK modulation with coherent detection with the number of transmitting and receiving antennas are 4. As the Doppler shift increases, the performance of the MIMO-QPSK deteriorates sharply. In a high-speed mobile environment, the channel changes rapidly, and the channel coherence time becomes shorter, which makes real-time channel estimation difficult. The error performance of coherent detection will rapidly decline because of the lack of real-time tracking of phase changes. In comparison, the MIMO-4FSK based on energy detection has better Doppler robustness. The square-law energy detection can eliminate the phase interference, so that those have little effect on the energy detection system. But a large Doppler shift will cause the signal frequency offset, resulting in frequency mismatch, the error performance of the system will also decrease.

Conclusion
This paper analyzes the robustness of Spatial MFSK modulation with energy detection to Doppler in high-speed mobile scenarios. Energy detection can avoid the phase synchronization problem, reduce the complexity of the training sequence, and improve the robustness of the system to Doppler. Through the nonlinear processing of energy detection, the system is equivalent to a real model, and the equivalent signal and noise are analyzed. The theoretical formula of the distribution of the received signal by energy detection is simulated numerically, which demonstrates that the numerical results and the theoretical formula match well. The theoretical formula of average symbol error probability is derived, which fit well with the numerical results of minimum Euclidean distance detection at high SNR. Please see the Manuscript PDF le for the complete gure caption Figure 2 Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Simulated and theoretical results for SER of spatial 4FSK modulation with energy detection. The gure compares the SER performances by the theoretical analysis and the simulation.  Performance comparison between noncoherent MIMO-4FSK and coherent MIMO-QPSK in high-speed railway channels. The gure compares the Doppler robustness of the spatial 4FSK modulation and spatial QPSK modulation under high-speed railway channels.