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Abstract
In fiber-terahertz integrated communication systems, nonlinear distortion and inter-symbol interference (ISI) will degrade transmission performance. Pre-compensation is an efficient method to handle the channel distortion as it can avoid noise boosting during channel compensation and reduce receiver side signal processing algorithmic complexity at user-end (UE) considering the asymmetric access scenario. In this paper, we propose and experimentally demonstrate a neural-network (NN)-based carrier-less amplitude phase (CAP) modulated signal generation and end-to-end optimization method for a fiber-terahertz integrated communication system. The CAP signal is generated directly from quadrature amplitude modulation symbols and pre-compensated through a transmitter NN, which allows the receiver to demodulate the signal with simple linear digital signal process (DSP). In generating the CAP signal, the NN based transmitter learns a group of filters, which can generate, up-convert, and pre-compensate the signals. Based on the proposed method, a fiber-terahertz integration access system at 220 GHz is demonstrated and a sensitivity gain of 1.2 dB is achieved at a transmission speed of 50 Gbps and the forward error correction (FEC) bit error rate (BER) threshold of 1 × 10−2 compared with the baseline after 10-km fiber transmission and 1-m wireless delivering.
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1. Introduction
Over the past few decades, wireless data traffic has increased dramatically. With the development of data-intensive applications such as Internet of Everything (IoE) and virtual reality (VR), wireless capacity requirements will continue to increase in six generation (6 G) [1]. According to [2], 6 G needs to deliver 1000x more capacity in network compared with 5 G. In physical layer, Teraherz (THz) band communication has been emerged as a promising solution to support radio access networks (RANs) in 6 G [3]. Fiber-THz integrated communication system is based on optical heterodyne detection and has many advantages, such as richer spectrum resources, broad service coverage, low latency and the ability to seamlessly connect to fiber networks [4]. This makes the antenna structure of remote radio units (RRUs) simple and cheap to deploy. To realize this communication requirement, we need higher modulation order and more suitable modulation format. Carrier-less amplitude phase (CAP) modulation has attracted extensive attention and research [5][6]. It supports high modulation order and has low peak-average power ratio (PAPR). However, there are many nonlinear devices such as photodiode (PD), fiber, and power amplifiers (PA) and multi-path fading problem in fiber-THz integrated communication system. They will introduce nonlinear distortion and inter-symbol interference (ISI) in signal. Effective digital signal processing (DSP) schemes are needed to deal with them.
In previous studies, post-compensation shows good performance on nonlinear distortion and ISI compensation, such as Volterra-based equalizer [7] and neural network (NN)-based equalizer [8]. However, post-compensation will raise the system noise [9] and increase algorithmic complexity at the UE. Pre-compensation not only compensates for nonlinear distortion and ISI, but also avoids raising the system noise [9–12]. Article [9] proposed a time-domain pre-compensation scheme based on the adaptive equalizer at the receiver side, and carried out theoretical derivation and experimental verification. Article [10] studied the performance and complexity of different nonlinear and linear joint pre-equalization. The effectiveness of joint pre-equalization is proved and the advantages and disadvantages of various methods are analyzed. The end-to-end (E2E) communication system optimization method has also aroused the wide interest of researchers [13–20]. Different from the traditional methods optimize each module independently, the E2E optimization methods use artificial neural network (ANN) to replace the original signal processing steps for global optimization [14]. E2E communication system was first studied in wireless systems [14]. In [15], the E2E communication in the optical fiber system was realized, and the modulation and demodulation are realized using the E2E method. Article [16] improved the performance by using E2E geometric shaping method in optical wireless system. E2E optimization methods are based on the appropriate channel model to carry out closed-loop feedback optimization. In [17], a two-tributaries heterogeneous neural network (TTHnet) based channel model, which is very similar to the residual neural network, was proposed according to the difference between nonlinear distortion and linear distortion of the channel. This method improves the accuracy of channel modeling and establishes a better foundation for the E2E optimization method. In [18], according to the characteristics of visible light channel, the in-band neural network channel model was proposed and the E2E visible light communication system was experimentally demonstrated.
In this paper, we propose and experimentally demonstrate a neural network-based CAP (NN-CAP) modulated signal generation and E2E optimization method for fiber-terahertz integrated communication system. NN-based CAP transmitter that can directly convert quadrature amplitude modulation (QAM) symbols into pre-compensated CAP signal is trained. When training the NN channel model, random numbers are fed into real system channel as transmitting signal. Then transmitting and receiving signals are used as inputs and outputs of the NN channel model. Then NN-based CAP transmitter can be E2E trained through the NN channel model. It can maintain the same algorithmic complexity as traditional CAP while compensating for nonlinear distortion and partial ISI. Users at the receiver side only need to carry out simple linear equalization and demodulation. The overall performance of the approach is studied in a 220-GHz fiber-terahertz system and compared with traditional CAP signal and CAP signal with linear pre-equalization. Finally, a sensitivity gain of 1.2 dB was achieved at a transmission speed of 50 Gbps and a forward error correction (FEC) bit error rate (BER) threshold of 1 × 10−2 compared with the traditional method after 10-km fiber transmission and 1-m wireless delivering.
2. Principles
As shown in the Fig. 1, NN-based transmitter is employed to generate and pre-compensate the CAP signal within the fiber-terahertz integrated communication system. This NN-based transmitter undergoes E2E training based on the channel model, enabling it to effectively pre-compensate for nonlinear distortion and partial ISI. This method can decrease the algorithmic complexity of the receiver side. It only needs to go through some linear DSP such as match filtering (MF) and symbol level least mean square (LMS).
2.1 Principle of NN-based pre-compensated CAP signal generation method
The principle of the traditional CAP signal generation is shown in Fig. 2(a). The QAM symbols are up-sampled. The up-sampled signals are separated into the in-phase (I) parts and the quadrature (Q) parts. The I/Q signals are then individually filtered by pulse shaping filters (PSFs) and summed to obtain a digital signal for transmission. The PSFs are expressed as:
in which $A(t)$ is square root raised cosine (SRRC) filter, v is the center frequency of the signal. And v is represented as: where $\beta$ is the roll-off factor of the SRRC filter, ${R_s}$ is the baud rate of signal. The CAP signal is expressed as: where $I(t)$ and $Q(t)$ are I parts and Q parts of I/Q signal.When the signal is convolved with the digital filter, the convolution kernel will slide the window. multiplication is performed at corresponding positions, and the results are then added together. This process resembles the operation of neural network neurons. As shown in Fig. 2(b), the process of weighting the output results from the previous layer of neurons is equivalent to the weighted operation with a single window slide. And the weight vector W of the neural network is equivalent to the convolution kernel in the convolution operation.
This method can directly map QAM symbol to waveform. The filter factors of PSFs and signal pre-compensation filters can be learned into the neural network parameters together. Nonlinear activation functions are used to compensate for nonlinear distortion during signal generation. It also allows massive parallel processing of the single blocks at deployment time to generate pre-compensated CAP signal in parallel.
The structure of E2E NN-based transmitter training framework is illustrated in Fig. 3. The NN channel model has been trained by using real data from communication system. Sliding window processed QAM sequences are used as inputs and CAP signals are used as label. The structure of QAM sequence can be expressed as:
As shown in Fig. 2, the CAP label should be CAP signal with partial ISI. Because when signal compensation is performed, post-compensation usually will raise the high-frequency component noise, especially in a bandwidth-limit system. Pre-compensation methods are proposed to avoid the increase in system noise [9]. However, a trade-off exists between the depth of pre-compensation and the reduction in signal-to-noise ratio (SNR) [21,22]. The pre-compensation approach will amplify the high-frequency components of the transmitted signal. ISI will induce significant attenuation in high-frequencies, which will lead to a substantial in the received power of signals with a fixed transmission power compared to without pre-compensation. Therefore, compensation factor should be considered in practical implementation.
Therefore, a modified label for our proposed NN is utilized to generate CAP signal with a certain pre-compensation factor α (0<α<1). The label can be expressed as:
where $X(t)$ represents the original CAP signal, ${Y_l}(t)$ is the CAP signal with only linear distortion, and $H(t)$ is the filter factor to simulate the linear response of the channel. The filter factor is obtained by a LMS algorithm in this experiment.Based on this modified label, the proposed NN has the capability to generate a CAP signal pre-compensated for all non-linear distortions and α linear distortions.
2.2 Principle of channel model
To generate a pre-compensated CAP signal, we need to know the channel response. The real channel blocks the gradient back propagation, which prevents E2E optimization of NN-based transmitter. Therefore, NN channel model is used to simulate the channel response, which enables the channel to achieve gradient backward propagation for E2E optimization. In [17], TTHnet based channel model is proposed. The TTHnet principle is similar to residual neural network, one tributary is used to simulate nonlinear distortion in the channel and the other tributary is used to simulate linear distortion in the channel as shown in Fig. 4. The double tributary design can prevent the neural network from being unable to map back to the linear region after nonlinear mapping [23]. The equation for TTHnet can be written as:
In real system channel, bandwidth limitation will cause high-frequency component polluted. Therefore, high-frequency component is useless for channel learning. To decrease the influence of this problem, we resample the training signal to reduce the energy of the high-frequency component of the signal. After this, the signal energy is concentrated in the low-frequency component where the channel condition is good.
NN channel model needs to reflect the response of the channel accurately. LMS algorithm is used to measure the ISI influence range of the channel. If the best performance was achieved with N taps at the symbol level, and the signal is up-sampled by a factor of U, the ISI range at the waveform level was $N \times U$ . Based on this ISI range, we set the input neuron of the channel model to R and the output neuron to U. For channel nonlinear distortion, the memory range is often smaller than the ISI. Therefore, the number of input neurons does not need to adjust.
2.3 Complexity analysis
The previous section described the principle and advantages of NN-CAP. Next, we will compare and analyze its algorithmic complexity with traditional CAP. Since multiplication is the main cause of algorithmic complexity, only the number of multiplication operations is counted here.
The complexity of generating a traditional CAP signal mainly lies in the convolution process between the PSFs and the baseband signal. The one-dimensional convolution complexity is calculated by the following formula:
in which M and N represent the lengths of the two convolution sequences. For generating a traditional CAP signal, the length of the shaping filter is determined by the up-sampling factor U and the span length L. And each signal segment has the same length as the up-sampling factor. Since the QAM signal is divided into I and Q components, two convolution processes are required. Therefore, the complexity of traditional CAP can be calculated using the following formula:Then, we carried out the algorithmic complexity analysis on NN-CAP. The formula for calculating the complexity of neural network is as follows:
From the above equations, we can see that NN-CAP does not require signal processing after up-sampled when generating CAP signals, which makes the algorithmic complexity lower than traditional CAP. The presented complexity formulas provide a theoretical framework for evaluating the algorithmic complexity of the proposed methods. The specific numerical values will be determined and calculated based on the relevant parameters during the simulation and experiment.
2.4 Numerical simulation
In this section, we will validate the capability of using NN to generate traditional CAP signal, linear pre-compensated CAP signal and nonlinear pre-compensated CAP signal through simulation. We compare and analyze the time-domain waveforms, constellation diagrams, and frequency spectrums of them. And the algorithmic complexity under the simulation parameters is analyzed by using the formula in section 2.3. In section 3. Experimental setup, the sampling rate of AWG we used in the experiment is 60 GSa/s. The signal bandwidth is only 10 GHz. To meet the system sampling-rate requirements, the signal needs an up-sampling factor of 6.
A simulation comparison of the CAP signals is conducted firstly. The roll-off factor is set to 0.2, the span of CAP is 10, and the up-sampling factor is 6. The NN-based method is also utilized to generate NN-CAP signals that are identical to standard CAP signals. In this case, the number of neurons in the input layer was set to (10 + 1) × 2 = 22, which corresponds to the span of the CAP signal. The number of neurons in the output layer was set to 6, which matches the up-sampling factor.
To verify the feasibility of NN-CAP, we plotted the partial time-domain waveforms of CAP, NN-CAP, and their MSE curve. As shown in Fig. 5, the time-domain waveforms of the two signals overlap almost perfectly, and the MSE curve approaches zero. Frequency spectrums and constellations diagram of them are on the sides. Two signals remain consistent in both frequency domain and constellation diagram. According to the complexity calculation formulas provided in the previous section, the complexity of generating traditional CAP is determined to be 2 × (6 × 10 + 1) × 6 = 732, while the complexity of NN-CAP is 22 × 6 = 132. NN-CAP demonstrates lower algorithmic complexity while maintaining same performance.
Next, we conducted simulations to analyze the ability of the NN-CAP to generate CAP signals and perform linear pre-equalization. We employ random numbers with uniform distribution as training signal to train the channel model. Subsequently, we construct an NN transmitter with input layer consisted of 30 neurons, output layer consisted of 6 neurons and no hidden layer. The QAM symbols are employed as inputs to the neural network, while the CAP signal served as the network's outputs. Keeping the channel model fixed, we train the parameters of the NN-based transmitter. After the training is completed, transmit the obtained NN-CAP and traditional CAP through a Gaussian white noise channel with ISI. Traditional CAP is linear pre-compensated by LMS. And the signal-to-noise ratio of channel is 30 dB in Fig. 6.
As shown in Fig. 6, we plot the received waveforms of NN-CAP and standard CAP. The NN-CAP can closely matches the baseline with MSE under 1 × 10−3. On the other hand, the traditional CAP exhibits a larger MSE compared to the baseline because of the effects of noise and ISI. Comparing the constellation diagrams of them, NN-CAP with linear pre-equalization exhibits a more concentrated distribution of constellation points, with minimal impact from noise. In contrast, the constellation points of the traditional CAP show significant spreading due to ISI and noise. Their Amplitude/Amplitude (AM-AM) plots show that the NN-CAP is ISI-resistant. From the frequency spectrum, we can see that the NN-CAP effectively compensates for the high-frequency attenuation caused by ISI according to the channel model. With pre-compensation, the received frequency spectrum of NN-CAP is essentially the same as the frequency spectrum of traditional CAP. And the complexity of NN-CAP is 30 × 6 = 180, which remains significantly lower than 732 algorithmic complexity of standard CAP.
Finally, we conducted simulation analysis to evaluate the capability of NN-CAP in generating CAP and pre-compensating for nonlinear distortion. The method used to construct the channel model is the same as before. The NN-based transmitter is constructed by using a three-layer neural network architecture. The input layer is consisted of 22 neurons, the hidden layer is consisted of 25 neurons, and the output layer is consisted of 6 neurons. During training, the channel model is kept fixed, QAM symbols and standard CAP signals are used as the input and output, respectively. Then, we employ the trained NN-based transmitter to generate NN-CAP signals. NN-CAP and traditional CAP are sent to the Gaussian white noise channel with nonlinear distortion.
As shown in Fig. 7, the NN-CAP waveform is more consistent with the standard CAP after passing through the channel. The traditional CAP significantly affected by nonlinear distortion exhibits significant discrepancies, especially in regions with large waveform amplitudes. The MSE curves between received waveforms and baseline are also plotted, which further shows the effectiveness of NN-CAP compared to the standard CAP. The AM/AM figures of the received waveforms and the baseline also demonstrate the excellent nonlinear resilience of NN-CAP. NN-CAP exhibits nearly a linear response, while the traditional CAP suffers severe nonlinear distortion. In the constellation diagram, the outer QAM symbols of traditional CAP are noticeably compressed. This is because that the QAM symbols with larger signal amplitude after modulation suffer severe nonlinear distortion However, NN-CAP has a favorable distribution of constellation points, because it has pre-compensated for nonlinear distortion based on the NN channel model when generating signals. Their frequency spectrums show that the frequency spectrum of NN-CAP after passing through the nonlinear channel is similar to the frequency spectrum of standard CAP, which demonstrates good resistance to nonlinear distortion. At this point, the algorithmic complexity of generating NN-CAP is 22 × 25 + 25 × 6 = 700, which remains lower than that of traditional CAP.
3. Experimental setup
Figure 8, shows the experimental setup of the fiber-THz integrated communication system at 220 GHz with the NN-CAP modulated signal generation and E2E optimization method. On the transmitter side, the pre-compensated CAP signal is generated offline by the NN-based transmitter with python. The driving signal is digital-to-analog converted by the AWG with a 60-GSa/s sampling rate. After amplified by an electrical amplifier (EA), the signal is used to drive Mach-Zehnder Modulator (MZM) with a wavelength at 1550.00 nm and average power of 8 dBm continuous wave (CW) light. After 10-km standard mode fiber (SMF) transmission, the optical signal is sent to a photo-detector for detection. Then, the IF signal is up-converted to 220 GHz by using a two-order harmonic mixer with 105-GHz local oscillator (LO) and sent to antenna. After 1-m wireless transmission, the signal is received by a 220-GHz antenna on the receiver side. The 220-GHz signal is amplified by a low noise amplifier (LNA) and recovered to an IF signal by the two-order harmonic mixer with 105-GHz LO. The LOs on the both transmitter and receiver side are generated by 6x frequency multiplier from 17.5-GHz RF. The IF signal is captured by a digital storage oscilloscope (OSC) with 80-GSa/s sampling rate. Finally, the captured signal is offline processed by traditional CAP signal DSP steps such as matched filtering, down-sampling, linear equalization, constellation mapping. The NN channel model has 174 neurons in the input layer and 6 neurons in the output layer. The nonlinear tributary consists of two hidden layers with 64 and 128 neurons, respectively. The linear tributary has one hidden layer with 128 neurons. The NN transmitter has 58 neurons in the input layer, 12 neurons in the hidden layer, and 6 neurons in the output layer.
We compare the performance of our NN-CAP in different scenarios with traditional CAP and linear pre-equalized CAP. When generating them, bit data is firstly mapped into 32-QAM symbols. Then, QAM symbol-level LMS pre-equalization is performed to compensate ISI if needed. The pre-equalized QAM symbols are then up-sampled. The up-sampled QAM symbols are separated into the I part and Q part, filtered by PSFs, whose span is 10, roll off factor is 0.2. Finally, separated I and Q signals are added up as the transmitting digital signal. On the receiver side, we use the same DSP steps as for NN-CAP.
4. Experimental results and performance analysis
4.1 Performance of channel model
2891904 data sets are used for training the channel model. 90% of the data is used as training set and 10% as validation set. The data used for evaluation is different from the data used for training the channel model. We change the random seed and regenerated the data for evaluation. The new data is transmitted through the channel model and the real channel, respectively. Then, the two sets of data are received and compared. In Fig. 9, we plot partial time-domain received waveforms of the real channel and the TTHnet channel model. The received signals overlap almost completely, indicating that the TTHnet channel model response is consistent with the actual channel. We have also plotted the frequency-domain comparison and error between them. We calculated the spectrum error according to:
where $\mathop Y\limits^\sim $ represents the output of the channel model and Y represents the output of the actual channel. From the error figure, we can see that most of the errors are caused by random thermal noise and frequency leaking. Only a few other errors exist in the low frequency range.4.2 Performance of NN-CAP and E2E optimization method
In this section, we will compare the performance of NN-CAP, traditional CAP and linear pre-equalized CAP (LP-CAP). Since pre-compensation on higher-frequencies will cause high PAPR and reduces the SNR on lower-frequencies, there is a trade-off on the pre-compensation strength [9,10]. Therefore, we first studied the linear pre-equalization factor α for LP-CAP and NN-CAP, where α represents how much ISI can be compensated, with a maximum value of 1. And the ISI of the communication system is calculated by the LMS algorithm.
As shown in Fig. 10, the best ISI pre-equalization factors for traditional CAP are achieved at α=0.4, while NN-CAP achieves the best performance when α=0.1. This is because LP-CAP only compensates for ISI, NN-CAP signal needs to use more signal power to compensate for nonlinear distortion. So, the best ISI pre-equalization factor for NN-CAP is smaller than that of traditional CAP. At this point, the NN-CAP transmitter compensates for system nonlinear distortions and also ISI. The device bandwidth limitations in the system from both transmitter and receiver, and also fiber dispersion will cause ISI in the system. Frequency spectrums of NN-CAP and pre-equalized CAP are also plotted. The traditional linear pre-equalized CAP signal has more high-frequency region compensation. While the NN-CAP signal high-frequency region has less compensation and is compensated based on the non-flat frequency response by the NN channel model. The pre-compensation factors used in subsequent experiments were all based on the determined best factors.
Figure 11 shows the BER results of CAP signal, LP-CAP, LP-CAP with nonlinear equalization (NE) and NN-CAP under different peak-to-peak voltage (Vpp). The received optical power (ROP) are set to 0.5 dBm in the back-to-back and wireless integrated (B2B-wireless) case. With the increase of Vpp, NN-CAP signal shows good anti-nonlinear distortion performance, because NN-CAP has pre-compensated nonlinear distortion through NN channel model. And NN-CAP signal reaches the best performance at 350 mV Vpp. This method also has better performance than the traditional method at low Vpp, as besides the Vpp-sensitive power amplifier, other components such as MZM and PD in the communication system will also introduce nonlinear distortion. LP-CAP with NE also exhibits some resistance to nonlinearity, but its performance is not as good as NN-CAP. Because compensating for nonlinear distortion by post-equalization will be affected by noise, which in turn affects the bit error rate performance. Moreover, compensating for nonlinear distortion at the receiver side will significantly increase the DSP burden for UE.
1997424 data sets are used for training the NN-CAP transmitter. 40% of the data is used as training set and 60% of the data is used as validation set. The data used for evaluation is different from the data used for training the NN-CAP transmitter. After the completion of training NN-CAP transmitter, we change the random seed and regenerated the QAM symbols for transmission, which are then compared with other methods.
A three order Volterra Nonlinear Equalizer is used for LP-CAP w/NE, its mathematical model can be expressed as:
In this moment, the number of neurons in the input layer is 58, the hidden layer has 12 neurons, and the output layer has 6 neurons. According to complex calculation formula, the algorithmic complexity is determined to be 58 × 12 + 12 × 6 = 768, which is not significantly different from traditional CAP with a complexity of 732. This algorithmic complexity can be reduced by decreasing the number of neurons in the input layer, and the ISI lost in this way can be compensated in post-equalization.
Figure 12 illustrates the BER performance of three types of CAP signals as a function of ROP. For NN-CAP, we investigated two scenarios: one where we train the NN channel model only once, and another where we retrain the NN channel model every time the ROP is changed. The signal Vpp for the NN-CAP and other CAPs is set to 0.35 V and 0.3 V, respectively. In B2B-wireless case, NN-CAP can achieve 1.2-dBm optical power sensitivity gain compared to the traditional CAP method. With 10-km fiber in the system, NN-CAP can achieve 1.1-dBm sensitivity gain. For the scenario where we don’t train channel model at every ROP point, we conducted one channel training at an optical power of 0.5dBm and then used the result for other ROP points. As shown in the Fig. 12, good performance can still be maintained when the optical power is relatively stable. When channel is significant changes, the NN channel model is no longer accurate, causing the transmitter can’t accurately compensate for all nonlinear distortion. Therefore, it is necessary to retrain the channel model when there is a large change in ROP. In this scenario, methods like transfer learning can be employed to quickly adapt to the response of new channel.
We also test the BER performance of NN-CAP at different bandwidths. ROP is set to 0.5 dBm and Vpp is set to their respective optimal values. It can be seen from Fig. 13, the NN-CAP signal still maintains great performance compared with CAP and LP-CAP. When the bandwidth is greater than 10 GHz, the BER of the three methods increase rapidly and tend to be consistent, because this fiber-THz integrated system bandwidth limitation.
5. Conclusion
In this paper, we demonstrated a fiber-THz integrated communication system at 220 GHz with the NN based CAP signal generation method. To reduce the receiver side DSP algorithmic complexity, NN based transmitter that can directly convert QAM symbols into pre-compensated CAP signal was trained, which doesn’t need up-sampling, matching filtering, ISI and nonlinear pre-compensation or other processes. This NN-based transmitter learns a group of filters that can generate CAP signal, up-convert CAP signal and pre-compensate CAP signal. This approach can maintain the same even less algorithmic complexity than traditional CAP. It shows significant performance improvements in both nonlinear and linear regions, and a 50-Gbps net data rate is achieved. Compared with traditional CAP signals and CAP signals with linear pre-equalization, over 1.2 dB sensitivity gain was achieved at a FEC BER threshold of 1 × 10−2 compared with the traditional method. Currently in our system, the end-to-end optimization process is demonstrated with a static channel response. For dynamic linear and nonlinear impairments, a re-train process or transfer learning [24] can be used, which will be studied in our future work.
Funding
National Key Research and Development Program of China (2022YFB2903600); National Natural Science Foundation of China (62235005, 62071444, 61925104, 62171137); Natural Science Foundation of Shanghai (21ZR1408700); Major Key Project PCL.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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