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Experimental demonstration of high spectral efficient 4 × 4 MIMO SCMA-OFDM/OQAM radio over multi-core fiber system

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Abstract

In this paper, 4 × 4 multiple-input multiple-output (MIMO) radio over 7-core fiber system based on sparse code multiple access (SCMA) and OFDM/OQAM techniques is proposed. No cyclic prefix (CP) is required by properly designing the prototype filters in OFDM/OQAM modulator, and non-orthogonally overlaid codewords by using SCMA is help to serve more users simultaneously under the condition of using equal number of time and frequency resources compared with OFDMA, resulting in the increase of spectral efficiency (SE) and system capacity. In our experiment, 11.04 Gb/s 4 × 4 MIMO SCMA-OFDM/OQAM signal is successfully transmitted over 20 km 7-core fiber and 0.4 m air distance in both uplink and downlink. As a comparison, 6.681 Gb/s traditional MIMO-OFDM signal with the same occupied bandwidth has been evaluated for both uplink and downlink transmission. The experimental results show that SE could be increased by 65.2% with no bit error rate (BER) performance degradation compared with the traditional MIMO-OFDM technique.

© 2017 Optical Society of America

1. Introduction

The fifth-generation (5G) mobile communication network will become commercially available around 2020 to achieve 10 times higher spectral efficiency and energy efficiency than current 4G wireless networks and support very low latency and high data rates up to 10 Gbps [1]. To achieve this goal, some physical layer technologies such as radio over fiber (RoF), massive multiple-input multiple-output (MIMO), and advanced orthogonal frequency division multiplexing (OFDM) techniques will be introduced [2]. As a kind of advanced OFDM techniques, OFDM using offset quadrature amplitude modulation (OFDM/OQAM) can get higher side lobe suppression ratio of pulse spectrum and resist inter symbol interference (ISI) and inter carrier interference (ICI) with the help of well-designed prototype filters instead of cyclic prefix (CP), leading to the increase of spectral efficiency (SE) and convenience for carrier aggregation (CA) application [3, 4]. On the other hand, massive connectivity with a large number of devices has become an important requirement in the 5G cellular system. For the traditional orthogonal frequency division multiple access (OFDMA) used in long term evolution (LTE) networks, the number of connections is limited by the time-frequency resources that can be scheduled. To solve this problem, sparse code multiple access (SCMA) as a non-orthogonal access technique has been proposed, which can support massive users by overloading data streams and improve the SE with the help of specially designed codebooks [5]. The sparseness of codebooks makes sure that we can demodulate the superimposed information correctly, and SCMA is flexible that it is easy to change the length of codebooks to support more users and get higher SE [6, 7]. Due to these excellent performances, SCMA has been already used in passive optical network (PON) to increase the SE [8]. For massive MIMO, not only electrical MIMO techniques but also optical spatial multiplexing techniques such as polarization division multiplexing (PDM) techniques [9] and wavelength division multiplexing (WDM) techniques [10] are considered as promising technologies to significantly improve the capacity and flexibility in 5G cellular system. Moreover, space-division multiplexing techniques based on multi-core fiber (MCF) and few-mode fiber (FMF) are proved to have enormous advantages such as compactness and spatial parallelism making it easier to realize massive MIMO [11, 12].

In this paper, we extend our previous work in [13] and introduce SCMA and MIMO-OFDM/OQAM techniques in MCF-based RoF system for the first time to not only further increase the SE but also support massive user’s access. In our experiment, with the help of well-designed prototype filters in OFDM/OQAM and different codebooks in SCMA, 11.04 Gb/s 4 × 4 MIMO SCMA-OFDM/OQAM signal is successfully transmitted over 20 km 7-core fiber and 0.4 m air distance in both uplink and downlink, corresponding to the SE of 22.08 bit/s/Hz. As a comparison, 6.681 Gb/s traditional MIMO-OFDM signal with the same occupied bandwidth has been evaluated for both uplink and downlink transmission, corresponding to the SE of 13.362 bit/s/Hz. The experimental results show that the SE could be increased by 65.2% by using the proposed SCMA-OFDM/OQAM technique.

2. Principle

Figure 1 shows the principle of the proposed N × N MIMO SCMA-OFDM/OQAM signal over multi-core fiber system. In the block diagram of the SCMA encoder, an example of a codebook set with 6 users is displayed. Each of the codebook has 4 multidimensional complex codewords that correspond to 4 points of constellation, and the length of each codeword is 4. For SCMA encoding, the codeword of each user is selected based on the input 2-bit pseudo random binary sequence (PRBS). At last, the codewords from different users are overlaid with each other for OFDM/OQAM modulation, and the signal constellation in each OFDM/OQAM subcarriers are shown in the insets of Fig. 1. In this way, 6 users’ data could be transmitted by only using 4 subcarriers, which means that the SE could be increased by at least 0.5 times. As we know, time and frequency resources could be allocated to different users in OFDMA, and now SCMA is help to serve more users simultaneously by allocating not only time and frequency resources but also non-orthogonally overlaid codewords compared with OFDMA, resulting in the increase of system capacity.

 figure: Fig. 1

Fig. 1 The schematic diagram of the proposed N × N MIMO SCMA-OFDM/OQAM RoF system.

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After SCMA encoding, the output signal is used for OFDM/OQAM modulation. The real and imaginary part of the generated SCMA signal are separated and the imaginary part is delayed by half symbol period. After inserting preambles for time synchronization and channel estimation, the phase of each point is shifted by ej(m+n2)π/2, where m is the index of subcarrier and n is the index of symbol. This additional phase shift makes the crosstalk between OFDM/OQAM symbols into imaginary format and maintains the orthogonality of OFDM/OQAM symbols in the real field. After inverse Fast Fourier Transform (IFFT), each symbol is filtered by a bank of FIR shaping filters, such as square root raised cosine filter. At last, the generated baseband SCMA-OFDM/OQAM signal can be expressed as

y(t)=m=0M1n=02N1amng(tnT2)ej2πmftejπ(m+n2)/2,
where amn are the generated signal after SCMA encoding, g is the designed shaping filters, T/2 represents the half symbol period delay, π(m+n2)/2 is the additional phase terms, f is the subcarrier spacing, and the condition of Tf=1needs to be always satisfied in OFDM/OQAM modulation [14].

After multi-core fiber and wireless transmission, the received N channel signals are demodulated by N-channel SCMA-OFDM/OQAM demodulators, respectively, and then the MIMO channel estimation module [15] could be used to recover N channel data streams. In each SCMA-OFDM/OQAM demodulator, time synchronization, FIR filters and FFT modules are adopted to transform the received signal into frequency domain. After N × N MIMO channel estimation and equalization, the real field orthogonality could be maintained and then the obtained offset signal could be transformed into SCMA signals just by delaying half symbol period and combing back to the complex form again.

To demodulate the obtained multi-dimensional constellations after OFDM/OQAM demodulator, a message passing algorithm (MPA) [16] is used because the SCMA codebook contains sparse code words to achieve near-optimal detection of mixed signal with acceptable complexity of processing. An example of a factor graph with 6 users and 4 subcarriers is shown in the inset of Fig. 1. There are two kinds of nodes in the factor graph: user nodes (UNs) and subcarrier nodes (SNs). MPA works by passing probabilities called messages along the edges between nodes, and updates the likelihood of the received SCMA codeword iteratively. The details of how to pass messages between UNs and SNs are described in [16, 17]. The first step of MPA algorithm is the initial calculation of the conditional probability. In the second step which is named as UNs updating block, the probabilities of SCMA codewords are passed from SNs to UNs. Relatively, in the third step which is called SNs updating block, the probabilities of SCMA codewords are passed from UNs to SNs. The second and third steps are then iteratively executed, and after N times iterations the most possible codewords could be obtained and the correct binary data are recovered.

When considering the channels noises in both optical and wireless link, and taking 4 × 4 MIMO SCMA-OFDM/OQAM signal over hybrid fiber and wireless system for example, the hybrid channel model in frequency domain could be expressed as

[y1y2y3y4]=[h11h12h13h14h21h22h23h24h31h32h33h34h41h42h43h44]([H11H12H13H14H21H22H23H24H31H32H33H34H41H42H43H44][x1x2x3x4]+[N1N2N3N4])+[n1n2n3n4],
where hmn and Hmn represent the channel response in wireless and optical link respectively. Moreover, nn and Nn are the random channel noises in wireless and optical link respectively. Due to the reason that the inter-core crosstalk of the used MCF is below −45 dB/100km, the interference terms in Hmn could be neglected [18, 19]. To estimate the hybrid 4×4 MIMO channel response, specially designed preambles of [pppp], [pppp], [pppp], [pppp] and an interference cancellation (IC) method [20] are adopted in our experiment. By the way, the first half and the second half of p are symmetrically distributed to facilitate frame synchronization. So the estimated channel response Hbmn could be expressed as
[Hb11Hb12Hb13Hb14Hb21Hb22Hb23Hb24Hb31Hb32Hb33Hb34Hb41Hb42Hb43Hb44]=[Y11Y12Y13Y14Y21Y22Y23Y24Y31Y32Y33Y34Y41Y42Y43Y44]×[p11+jX11p12+jX12p13+jX13p14+jX14p21+jX21p22+jX22p23+jX23p24+jX24p31+jX31p32+jX32p33+jX33p34+jX34p41+jX41p42+jX42p43+jX43p44+jX44]1
where Xmn are the interferences between preambles in each channel, and Ymn stand for the received training symbols [21, 22].

3. Experimental setup and results

3.1 Experimental comparison of OFDMA and SCMA-OFDM/OQAM based RoF system

To investigate the multiple access ability of OFDMA and SCMA-OFDM/OQAM techniques, a bidirectional RoF system is built up as shown in Fig. 2. For the downlink transmission, the generated RF signal by a 10 GSa/s arbitrary waveform generator (AWG, TekAWG7122B) is used to drive a directly-modulated laser (DML) with the wavelength of 1546 nm. After optical circulator (OC) and 24.5 km standard single mode fiber (SSMF), optical signal is detected by a 10 GHz bandwidth photodiodes (PD), and the generated RF signal is amplified by an electrical amplifier with 23 dB gain and then fed into an antenna. Moreover, the operation bandwidth and the gain of the transmitter and receiver antennas (AMXF-1727-3) are 1710-2700MHz and 3dBi respectively. The horizontal spacing between two Tx antennas is 0.1m, and the vertical spacing between Tx antennas and Rx antenna is 0.3m. After 0.3 m air transmission, the transmitted wireless signals are received by two antennas of two ONUs and then captured by a 20 GSa/s digital sampling oscilloscope (DSO, Tektronix DPO72504DX). Offline signal process, including resampling, time synchronization, OQAM demodulation, SCMA decoder, data mapping and BER counting, is performed. For the uplink transmission, the generated two RF signals by AWG are amplified by two low noise amplifiers (LNAs) and then fed into two antennas. After 0.3 m air transmission, these two wireless signals are detected by a receiving antenna, and then used to drive a DML with the wavelength of 1530 nm. After an OC and 24.5 km SSMF propagation, the upstream signal is captured by a 10 GHz PD and 20 GSa/s DSO. The same digital signal process used in downstream transmission is performed.

 figure: Fig. 2

Fig. 2 Experimental setup of the bidirectional SCMA-OFDM/OQAM RoF system.

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For SCMA-OFDM/OQAM testing, 128 subcarriers are divided into 32 groups, and in each group the number of codebooks is set to 6 and the size of each codebook is 4. Therefore, the maximum number of access users is 192. Therefore, 4 OFDM/OQAM subcarriers are used for each SCMA frame, and the sparse codewords are multiplexed and only three of them collide over each subcarrier. In both uplink and downlink, 6 codebooks are divided into two groups equally, and each ONU occupy 3 codebooks. It means that the generated two SCMA-OFDM/OQAM signals for these two ONUs share the same frequency and time resources, and could only be separated by different sparse codewords. Subsequently, the SCMA-encoded 128 subcarriers and 256-point IFFT are used for OFDM/OQAM modulation, noting that no CP is used here. The total net rate of two SCMA-OFDM/OQAM ONUs is 1.932 Gb/s with 350 MHz bandwidth, and the SE is 5.52 bit/s/Hz. For OFDMA testing, a data stream with a PRBS length of 2151 is mapped onto 192 subcarriers, of which each subcarrier carries 16-QAM data. These 192 subcarriers are divided equally to two ONUs and converted into the time domain by applying 256-point IFFT, and the CP length is set to 10%. The total net rate of two OFDMA ONUs is 1.527 Gb/s with the bandwidth of 525 MHz, and the SE is 2.909 bit/s/Hz. The received SCMA-OFDM/OQAM and OFDMA signal at the PD in uplink is shown in the insets of Fig. 2. Obviously, the SE of the SCMA-OFDM/OQAM signal is 0.89 times higher than that of the OFDMA signal. This could be explained by the fact that the core feature of SCMA is the number of non-orthogonally overlaid codewords, therefore SCMA can serve more users simultaneously under the condition of using equal number of time-frequency resources compared with OFDMA, resulting in the increase of transmission capacity.

Figure 3(a) shows the measured BER performance in terms of received optical power at the PD for SCMA-OFDM/OQAM signal with hybrid 24.5 km SSMF and 0.3 m air transmission in downlink. It could be clearly observed that the receiver’s sensitivity at the forward-error correction (FEC) limit (BER of 3.8x10−3) are −9 dBm and −8.8 dBm for optical back to back (OBTB) case and 25.4 km SSMF case respectively. The negligible power penalty means that the fiber dispersion induced by 24.5km SSMF has no impact on SCMA-OFDM/OQAM signal due to the effective channel estimation, and the received constellation of the downlink SCMA-OFDM/OQAM signal over 24.5 km SSMF transmission are shown in the insets of Fig. 3(a).

 figure: Fig. 3

Fig. 3 Measured BER performance of SCMA-OFDM/OQAM signal over 24.5 km SSMF and 0.3 m air transmission in downlink (a), and both SCMA-OFDM/OQAM and OFDMA with 24.5 km SSMF and 0.3 m air transmission in uplink (b).

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Figure 3(b) illustrates the measured BER performance in terms of the received optical power at the PD for both SCMA-OFDM/OQAM and OFDMA with 24.5 km SSMF and 0.3 m wireless transmission in uplink, and the BER performance for two ONUs are calculated together here. In the uplink, the spacing between two transmitter antennas is 0.1m, a little different channel response from each transmitter antenna to receiver antenna is induced by these different wireless transmission paths. For OFDMA transmission, two OFDM RF signal from two transmitter antennas are separated in frequency domain, so these two OFDM signals can be estimated by training sequences respectively. However, for SCMA transmission, the spectra of SCMA-OFDM/OQAM signals from different transmitter antennas are completely overlapped, and accuracy channel estimation cannot be achieved just by using one receiver antenna. To address this issue, the least squares (LS)-based channel estimation algorithm is combined with MPA algorithm in the SCMA-OFDM/OQAM decoder. In this way, the estimated channel responses are used to update codebooks instead of directly recovering the received signals. The updated codebooks of each ONU are then used in MPA to recover these two channel signals. However, the measured BER performance of SCMA/OQAM is still worse than that of OFDM when the optical power is higher than −7 dBm. This problem could be perfectly solved by using MIMO channel estimation algorithm in N × N MIMO scenario. When the received optical power at the PD is lower than −7dBm, the channel noises dominate and the SNR of the received signals are relatively low. The performance of the used LS-based channel estimation algorithms is significantly degraded in both SCMA and OFDMA cases. However, the combined LS-based channel estimation and MPA algorithms which are used in SCMA are help to relieve the effect of channel noises to some extent, and the BER performance of SCMA/OQAM is better than that of OFDMA. The received constellation of uplink SCMA-OFDM/OQAM signal and OFDMA signal over 24.5km SSMF transmission are also shown in the insets of Fig. 3(b).

3.2 Experimental demonstration of 4 × 4 MIMO SCMA-OFDM/OQAM based radio over MCF system

To further increase the SE and support more users, MIMO techniques in optical and wireless domain are introduced. Figure 4 shows the experimental setup of the proposed 4 × 4 MIMO SCMA-OFDM/OQAM signal over 7-core fiber system, and both uplink and downlink transmission are evaluated. Meanwhile, a traditional 4 × 4 MIMO-OFDM signal is chosen for the purpose of performance comparison. For downlink transmission, an AWG (Keysight M8195A) with 60 GSa/s sampling rate is used to generate four-channel RF signals. For the case of SCMA-OFDM/OQAM, in each channel 128 subcarriers are divided into 32 groups, and in each group, the number of codebooks is set to 6 and the size of each codebook is 4. It means that 4 OFDM/OQAM subcarriers are used for each SCMA frame, and the sparse codewords are multiplexed and only three of them collide over each subcarrier as shown in Fig. 1. After offset-QAM modulation, the SCMA-encoded 128 subcarriers are converted into time domain by applying 256-point IFFT. It should be noted that no CP is required here due to the use of raised root cosine pulse shaping filter with roll off factor of 1. For the case of traditional OFDM, a PRBS length of 2151 is mapped onto 128 subcarriers. In each subcarrier 16QAM and 256-point IFFT are used, and the CP length is set to 10%. In both SCMA-OFDM/OQAM and traditional OFDM cases, eight training sequences for every 100 payload symbols are employed, and the first training symbol is also used for time synchronization. As mentioned above, every four MIMO training symbols are full loaded in a manner of [pppp], [pppp], [pppp] and [pppp]. The generated four channel baseband signals are then up-converted to 5.5 GHz in AWG. Finally, the generated signals have a net rate of 11.04 Gb/s (4×60GSa/s60×4×64×1008100×128256) with a 500 MHz (60GSa/s60×128256) bandwidth and a SE of 22.08 bit/s/Hz for the case of 4 × 4 MIMO SCMA-OFDM/OQAM, and 6.681 Gb/s(4×60GSa/s60×4×1008100×128256×256282) with the same bandwidth and SE of 13.362 bit/s/Hz for the case of 4 × 4 MIMO-OFDM.

 figure: Fig. 4

Fig. 4 Experimental setup of the proposed 4 × 4 MIMO SCMA-OFDM/OQAM RoF system.

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Subsequently, the generated four-channel signals are used to drive four electro-absorption modulated lasers (EML) respectively, and the optical spectra of four-channel optical SCMA-OFDM/OQAM signals are shown in the inset (i) of Fig. 4. These four-channel optical SCMA-OFDM/OQAM signals are coupled into 7-core MCF by using low insertion loss and low crosstalk fan-in/fan-out devices which are developed by ourselves [23, 24]. Additionally, the inter-core crosstalk of the used MCF is lowered to −45 dB/100km and the 1st, 3rd, 4th and 6th core are chosen for four-channel optical signal transmission. After 20 km MCF propagation, the optical SCMA-OFDM/OQAM signals are detected by four PDs with bandwidth of 10 GHz, and the generated four-channel RF signals are then fed into four antennas after four electrical amplifiers with 22 dB gain. After 0.4 m air transmission, four antennas and four-channel DSO (Tektronix MSO72004C) with 25 GSa/s sampling rate are used to capture these four wireless signals. Moreover, the operation bandwidth and the gain of the transmitter and receiver antennas (DB-link) are 5150-5850MHz and 19dBi respectively. The horizontal spacing between each two Tx antennas is 0.1m, and the vertical spacing between Tx antennas and Rx antennas is 0.4m. At last, offline signal process, including resampling, time synchronization, MIMO channel estimate, OQAM demodulation, SCMA decoding and BER counting, are implemented. The electrical spectra of the received RF OFDM and SCMA-OFDM/OQAM signals are shown the inset (iii) of Fig. 4.

For uplink transmission, the generated four-channel SCMA-OFDM/OQAM RF signals with 5.5 GHz carrier frequency are fed into four transmitter antennas. After 0.4 m air transmission, the received upstream wireless signals are used to drive four EMLs. The optical upstream signals are captured by four 10 GHz PDs and 25 GSa/s DSO after 20 km MCF propagation. The same digital signal process as used in downstream transmission is performed, and the net rate of the upstream is also 11.04 Gb/s for 4 × 4 MIMO SCMA-OFDM/OQAM due to the use of the same modulation parameters.

Figure 5 shows the measured BER performance in terms of received optical power at the PD for four channel SCMA-OFDM/OQAM signals over 20 km MCF and 0.4 m wireless link in downlink. It could be clearly observed that the receiver’s sensitivity at the FEC limit are −9.35 dBm, −8.84 dBm, −8.81 dBm and −8.61 dBm for four channels of the downlink respectively. Due to the different characteristics of optical and electrical components, about 0.6 dB power penalty is achieved among four channels. Moreover, the constellations of the received four-channel SCMA-OFDM/OQAM signal are plotted in the insets of Fig. 5. For the convenience of discussion, the BER performance of four channels are calculated together in the following experiments.

 figure: Fig. 5

Fig. 5 Measured BER performances of each channel in 4 × 4 MIMO SCMA-OFDM/OQAM signals transmission system.

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Figures 6(a) and 6(b) illustrate the measured BER performance in terms of received optical power at the PD for 4 × 4 MIMO SCMA-OFDM/OQAM signal over 20 km MCF and 0.4 m air transmission in both downlink and uplink respectively, and 4 × 4 MIMO-OFDM signals are also evaluated for comparison. It could be clearly observed in Fig. 6(a) that the receiver’s sensitivity at the FEC limit is −10 dBm for 4 × 4 MIMO SCMA-OFDM/OQAM signal over OBTB and 0.4 m air transmission in downlink, and only 1 dB power penalty at the FEC limit is induced by 20 km MCF transmission. The same BER performances are achieved in both 4 × 4 MIMO SCMA-OFDM/OQAM and 4 × 4 MIMO-OFDM cases, which could be described as the fact that these two RF signals have occupied the same bandwidth. As shown in Fig. 6(b), the receiver’s sensitivity at the FEC limit is increased to −8.25 dBm for 4 × 4 MIMO SCMA-OFDM/OQAM signal over optical back to back (OBTB) and 0.4 m air transmission in uplink, and this could be attributed to the decreased signal to noise ratio (SNR) induced by the different position of wireless link. Actually, in upstream transmission, each SCMA-OFDM/OQAM signal is launched into air link and then into MCF, it should be noted that the crosstalk between four antennas is larger than inter-core crosstalk in MCF. Moreover, the different channel noise items will have different influences on the used LS MIMO equalization algorithm. 1.7 dB power penalty is observed due to the impact of inter-crosstalk in 20 km MCF. Similarly, negligible power penalty is observed between 4 × 4 MIMO SCMA-OFDM/OQAM and 4 × 4 MIMO-OFDM signal transmission, however, the SE is increased by 65.2% with the help of SCMA. The received constellation of 4 × 4 SCMA-OFDM/OQAM signal and OFDM signal are all shown in the insets of Figs. 6(a) and 6(b).

 figure: Fig. 6

Fig. 6 Measured BER performance for 4 × 4 MIMO SCMA-OFDM/OQAM and MIMO-OFDM signals over 20km MCF and 0.4m air transmission in both downlink (a) and uplink (b).

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4. Conclusions

We have experimentally demonstrated the transmission of 4 × 4 MIMO SCMA-OFDM/OQAM RF signal over MCF and air. OFDMA and SCMA-OFDM/OQAM based RoF system are contrastively investigated, and the results show that with the help of SCMA not only time and frequency resources by also non-orthogonally overlaid codewords can be scheduled to support massive connections. In our experiment, 11.04 Gb/s 4 × 4 MIMO SCMA-OFDM/OQAM signal is successfully transmitted over 20 km 7-core fiber and 0.4 m air distance in uplink and downlink alike. The experimental results show that SE could be increased by 65.2% with no BER performance degradation compared with the traditional MIMO-OFDM technique, which makes the proposed scheme more promising in the future 5G communication system.

Funding

The 863 Program of China (2015AA016904) and National Natural Science Foundation of China (NSFC) (61675083, 61505061, 61575071, 61331010).

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Figures (6)

Fig. 1
Fig. 1 The schematic diagram of the proposed N × N MIMO SCMA-OFDM/OQAM RoF system.
Fig. 2
Fig. 2 Experimental setup of the bidirectional SCMA-OFDM/OQAM RoF system.
Fig. 3
Fig. 3 Measured BER performance of SCMA-OFDM/OQAM signal over 24.5 km SSMF and 0.3 m air transmission in downlink (a), and both SCMA-OFDM/OQAM and OFDMA with 24.5 km SSMF and 0.3 m air transmission in uplink (b).
Fig. 4
Fig. 4 Experimental setup of the proposed 4 × 4 MIMO SCMA-OFDM/OQAM RoF system.
Fig. 5
Fig. 5 Measured BER performances of each channel in 4 × 4 MIMO SCMA-OFDM/OQAM signals transmission system.
Fig. 6
Fig. 6 Measured BER performance for 4 × 4 MIMO SCMA-OFDM/OQAM and MIMO-OFDM signals over 20km MCF and 0.4m air transmission in both downlink (a) and uplink (b).

Equations (3)

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y ( t ) = m = 0 M 1 n = 0 2 N 1 a m n g ( t n T 2 ) e j 2 π m f t e j π ( m + n 2 ) / 2 ,
[ y 1 y 2 y 3 y 4 ] = [ h 11 h 12 h 13 h 14 h 21 h 22 h 23 h 24 h 31 h 32 h 33 h 34 h 41 h 42 h 43 h 44 ] ( [ H 11 H 12 H 13 H 14 H 21 H 22 H 23 H 24 H 31 H 32 H 33 H 34 H 41 H 42 H 43 H 44 ] [ x 1 x 2 x 3 x 4 ] + [ N 1 N 2 N 3 N 4 ] ) + [ n 1 n 2 n 3 n 4 ] ,
[ H b 11 H b 12 H b 13 H b 14 H b 21 H b 22 H b 23 H b 24 H b 31 H b 32 H b 33 H b 34 H b 41 H b 42 H b 43 H b 44 ] = [ Y 11 Y 12 Y 13 Y 14 Y 21 Y 22 Y 23 Y 24 Y 31 Y 32 Y 33 Y 34 Y 41 Y 42 Y 43 Y 44 ] × [ p 11 + j X 11 p 12 + j X 12 p 13 + j X 13 p 14 + j X 14 p 21 + j X 21 p 22 + j X 22 p 23 + j X 23 p 24 + j X 24 p 31 + j X 31 p 32 + j X 32 p 33 + j X 33 p 34 + j X 34 p 41 + j X 41 p 42 + j X 42 p 43 + j X 43 p 44 + j X 44 ] 1
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