Abstract
Terahertz (THz) technology has great potential in the next generation of wireless communications due to its abundant spectrum resources. Electronic device-based systems have very limited device bandwidth when used to generate THz waves, which restricts the growth of system throughput, while photonics-aided THz systems are expensive and complex in structure. To solve this problem, a terahertz system using Delta-Sigma modulation (DSM) is proposed. In the study, a set of 0.15 THz 1024-QAM signals is generated and a 2-meter free-space wireless THz wave transmission system is successfully demonstrated. In addition, inspired by the rapid error propagation in DSM demodulation, a method combining digital signal processing (DSP) with complex-valued neural network (CVNN) is proposed to improve the bit error rate (BER) performance of 1024-QAM signals. When the BER soft-decision is 2×10-2, the sensitivity of the receiver is improved by 0.5 dB. The method proposed in this paper can effectively solve the problems caused by noise in traditional quadrature amplitude modulation (QAM) to improve the spectrum efficiency of electronic devices with insufficient bandwidth.
Introduction
Terahertz frequency bands are attractive for short-range wireless communication systems because they have abundant spectrum resources and can provide huge throughput [1]. Terahertz wireless communication technology has been put into use in many scenarios, such as 5G front-end transmission of large amounts of data. However, one of the prerequisites for high-speed transmission is a wide system bandwidth. Although many reports have claimed that photon-assisted terahertz system approaches can solve the bandwidth problem, photonic technology is still expensive and complex for commercial use. Since electrical methods not only have a simple system structure but are also easy to integrate, they have broad application prospects in terahertz systems [2-4]. Nevertheless, due to the limitation of insufficient bandwidth, generating terahertz waves based on electrical components remains a challenge.
In order to maintain the simplicity of the terahertz system structure while not being limited by the bandwidth of electrical equipment, DSM has become an attractive solution [5-7]. Generally, considering the transmission rate, single-carrier low-order QAM or orthogonal frequency-division multiplexing (OFDM) transmission will occupy a large bandwidth. With the rapid growth of transmission rate requirements, the bottleneck of device bandwidth is soon reached. Although single-carrier high-order QAM transmission can improve spectral efficiency by saving the bandwidth of electrical equipment, the linear and nonlinear noise caused by the equipment and channel will significantly affect signal recovery. DSM technology has the ability to solve these problems. By oversampling the digital-to-analog converter (DAC) and using only 1-bit quantization, the bandwidth of the transmitted signal will be compressed to a small range, so the spectral efficiency will be significantly increased [7]. In addition, after DSM processing, the quantization and nonlinear noise are significantly reduced, enhancing the system's noise resistance.
Although DSM ensures low signal bandwidth and mitigates various noises, terahertz signals still become difficult to distinguish at the receiving end during transmission. This is because even a small amount of error in demodulating DSM signals will lead to a sharp increase in error bits. Neural Networks (NN) has strong adaptability and has been widely used in wireless communication and optical fiber communication, and has achieved remarkable results [8].
Our group proposed a CVNN equalizer [9] that provides the possibility of an end-to-end machine learning (ML) implementation for directly recovering QAM modulated signals from noise. With the help of CVNN, the receiver performs better in recovering impaired signals, especially against nonlinear effects, which correspondingly reduce the receiver sensitivity.
This paper experimentally demonstrates a1024-QAM 0.15 THz signal generation and 2 m free space wireless transmission system. Based on DSM, the bandwidth limitation of electrical equipment can be solved. When the soft-decision forward error correction (SD-FEC) threshold of 0.02 is met, 15 Gbit / s 1024-QAM can be successfully transmitted in this system. In addition, with the help of CVNN, the sensitivity of the receiver is improved by 0.5 dBm when the BER soft decision is 2×10-2.
1 Experimental Setup
Fig.1DSM schematic: (a) Power spectrum; (i) Analog signal with bandwidth BW; (ii) Digital signal with quantization noise after oversampling (OSR is the oversampling ratio, and the quantization noise can be seen in the Nyquist diagram) ; (iii) After noise shaping, the quantization noise in part (ii) is pushed outside the signal bandwidth BW; (iv) The signal after passing through a low-pass filter is also the demodulated result of the DSM signal, and the quantization noise pushed out of BW is reduced; (b) Zeros and poles of the noise shaping function; (c) Frequency response power of the noise shaping function; (d) QPSK signal spectrum after DSM; (e) CVNN structure (the scatter plot shows the difference between the QPSK signal before and after CVNN) ; (f) A simple schematic diagram of the combination of DSM and CVNN in our setting.
DSM has been widely studied in mobile fronthaul applications. It can transmit analog signals through digital ports, and the demodulation process of DSM only requires a low-pass filter. Figure1 (a) shows the working principle of DSM, and the inset (i) in Figure1 (a) illustrates the required analog signal spectrum. First, as shown in the inset (ii) in Figure1 (a) , DSM oversamples the analog signal to extend the Nyquist bandwidth. Quantization noise exists everywhere in this band. Second, a noise shaping function is applied so that more quantization noise is moved out of the signal band, and the quantization noise inequality in the entire Nyquist zone is obtained, as shown in the inset (iii) in Figure1 (a) . In this way, the signal is reshaped into a1-bit On-Off Keying (OOK) signal that can be directly transmitted. Finally, after receiving the OOK signal and the decision process, the DSM signal is restored to a1-bit signal. Through appropriate low-pass filter processing, the original analog signal can be restored and the quantization noise outside the signal band can be removed.
Figure1 (c) shows the frequency response of the DSM noise shaping function, and its zero-pole diagram is shown in Figure1 (b) . From the noise frequency response, it can be seen that for a baseband digital signal whose normalized frequency starts at zero, the quantization noise close to the zero frequency is strongly suppressed, thereby improving the signal-to-noise ratio (SNR) .
Fig.2Schematic diagram and experimental setup of this article: AWG is arbitrary waveform generator; RF stands for radio frequency; LNA stands for low noise amplifier; EA is electronic amplifier; OSC is oscilloscope.
CVNN extends the traditional deep neural network (DNN) [10]. The activation function, weight initialization and batch normalization in CVNN are all forms of its complex-valued extension. When using complex-valued extension, the essential relationship between the in-phase and orthogonal components of the complex signal is intentionally preserved [9]. In this work, CVNN has two fully connected hidden layers with the same number of neurons. Each of its outputs is determined by the corresponding301 continuous inputs.
Fig.3(a) Offline DSP diagram at the Tx end; (b) Offline DSP diagram at the Rx end; (c) Representation of the signal constellation after the following processing: (i) resampling; (ii) T/2 constant modulus algorithm (CMA) ; (iii) frequency offset estimation (Frequency Offset Estimation, FOE) and Carrier Phase Recovery (CPE) ; (iv) Decision-Direct Least-Mean-Square, DDLMS) ; (v) CVNN; (vi) low-pass filter
The system setup is shown in Figure2, and the DSP is shown in Figure3. First, in the transmission of DSP, a pseudo-random binary sequence is generated; it is then used to map a1024-QAM signal with a pattern length of 16384. In order to use a root raised cosine roll-off filter, the1024-QAM data is first upsampled to two samples per symbol. After being processed by a root raised cosine roll-off filter (with a roll-off factor of 0.1) , the data is oversampled to achieve DSM modulation. The oversampled data is divided into real and imaginary parts, namely the in-phase component and the orthogonal component. These two components are independently fed into the DSM modulation with the same noise shaping function. The DSM outputs two 1-bit signals (both with a pattern length of 16384 8) . The two OOK signals are then combined into a QPSK signal, whose power spectrum is shown in Figure1 (d) . The QPSK signal is then transmitted to a Tektronix AWG for transmission. The sampling rate of the AWG can reach 12 GSa/s; when the sampling rate reaches this value, the double-side bandwidth of DSM modulation is 1.5 GHz, as shown in Figure1 (d) .
At the transmitter end there is a high frequency signal source and an integrated mixer that generates RF and multiplies RF by a fixed factor of 12. In the system setup, we set the output RF to 12.5 GHz. It will transmit 0.15 THz terahertz wave. 0.15 The THz wave is mixed with the output from the AWG and sent to free space through the antenna Tx.
In Transmission 2 m later, the terahertz wave is captured by the receiving antenna Rx. Then, the received signal is amplified by LNA. Similar to the transmitting end, another integrated high-frequency signal source outputs a 0.1392 THz wave. The amplified signal is the same as 0.1392 The THz waves mix to produce a10.8 GHz Intermediate Frequency IF) signal. After using the EA, the final signal is sent to a50 In the GSa/s OSC, the captured offline data is used by the receiving end digital signal processing (Rx DSP) .
In the receiving DSP, the received data is first digitally down-converted from IF to baseband. In order to fully recover the QPSK signal transmitted from the transmitter Tx, the down-converted signal is resampled to accommodate the double sampling rate of the AWG. With the help of a series of digital signal processing algorithms, such as T /2 CMA, FOE, CPR and DDLMS, the QPSK signal can be mostly recovered. Then we compare the quality of the QPSK signal with and without CVNN, which will be described in detail in the following content. Then, the processed QPSK signal enters the decision stage to recover the two standard OOK signals.
In the DSM demodulation stage, the two OOK signals are processed independently by resampling, i.e., inverse oversampling at the transmitter. These signals are then fed into root raised cosine roll-off filters with a roll-off factor of 0.1. DSM demodulation is performed after the low-pass filter, as previously described in Figure1 (a) (iv) , outputting the in-phase and quadrature components of the original1024-QAM signal.
2 Experimental Results and Discussion
Fig.4BER performance comparison of QPSK and 1024-QAM signals at different AWG sampling rates.
In this transmission experiment, we first evaluate the BER of QPSK signals and the bit error rate of 1024-QAM signals at different AWG sampling rates. At the same time, we observe the performance of CVNN through the bit error rate. Then, we adjust the transmit power at the Tx end to evaluate the impact of the transmit power on the receiver.
The actual transmitted QPSK baud rate is adjusted by adjusting the sampling rate in the AWG. Figure4 shows the BER performance of QPSK and 1024-QAM signals for QPSK baud rates ranging from 7 to 12. It should be noted that without the use of CVNN, the BER of the desired 1024-QAM signal is below the 0.02 SD-FEC threshold even at the most stringent 12 Gbaud case.
It is not difficult to see that the BER trend of the1024-QAM signal is closely related to the corresponding value of the transmitted QPSK signal, and the relationship between them is not linear. A slight increase in the bit error rate of the QPSK signal will result in a significant increase in the bit error rate of the1024-QAM signal. In addition, it is interesting to note that even if the bit error rate of the transmitted QPSK signal is 0, in other words, even if the QPSK signal is perfectly received without any errors, that is, at 7 Gbaud and 8 In the case of Gbaud, the bit error rate of 1024-QAM cannot be reduced to zero, but it can be reduced to a stable value (1.35×10-2 in this example) . This is due to the inherent principle of DSM. Once parameters such as the number of quantization bits are given, the noise transfer function is then determined. This noise transfer function determines the remaining quantization noise within the signal bandwidth. In this case, since the quantization noise within the signal band cannot be removed, there is no guarantee that the DSM signal can be recovered without bit errors even if the DSM QPSK signal points are received perfectly.
Fig.5BER performance of QPSK and 1024-QAM signals trained with CVNN at different batch sizes.
The batch size of CVNN is also a key parameter that can significantly affect the performance of the equalizer [10]. Since the amount of data required to transmit in the oversampling process of DSM is very large, when using CVNN, if we do not use the batch method, the amount of computation required to train the network will be very large. This is not applicable to the system scenario of this article. Therefore, we should consider the batch size parameter. A larger batch size can increase the training speed because the update of the weight value after each training can make it more accurate, thereby speeding up the training process. When the batch size is set appropriately, the gradient will become more accurate. However, an excessively large batch size will lead to a significant gap between the performance of the validation set and the training set. In this case, even increasing the batch size will not make the gradient more accurate. On the other hand, a smaller batch size will make the training process more random, that is, the gradient contains more noise during the training process.
Figure5 shows the bit error rate performance of QPSK and 1024-QAM signals with different batch sizes. As mentioned above, increasing the batch size makes the updated gradient more accurate, thereby effectively reducing the bit error rate. In Figure5, when the batch size is set to 1600, the lowest bit error rate is obtained, which is only 0.0189. Correspondingly, when the batch size is further increased, a local optimum is reached and the bit error rate increases slightly.
Fig.6BER performance of QPSK and 1024-QAM signals trained with CVNN at different hidden layer depths.
Fig.7BER performance of QPSK and 1024-QAM signals at different received powers with and without CVNN training.
Figure6 shows the effect of the number of hidden layer neurons on the bit error rate performance. If there are too few neurons in the hidden layer, it will result in underfitting. Conversely, if there are too many neurons in the hidden layer, it may cause overfitting. When the neural network has too many nodes and too strong information processing capabilities, the limited information in the training set is not enough to train all the neurons in the hidden layer, resulting in overfitting. Even if the training data contains enough information, too many neurons in the hidden layer will increase the training time, making it difficult to achieve the desired effect. Therefore, it is crucial to choose the appropriate number of hidden layer neurons. In this experiment, the optimal number of neurons in the two hidden layers was set to 100 respectively.
By reducing the transmit power at the Tx end, we obtain the BER performance of QPSK and 1024-QAM signals at different receive powers. In this case, CVNN is used to mitigate the impact of error propagation in DSM demodulation. Figure7 shows the BER performance of QPSK and 1024-QAM signals with and without CVNN equalizer. It is obvious that higher SNR improves the BER performance of 1024-QAM DSM signal. The constellation diagram shows the QPSK diagram after DSM and the1024-QAM diagram after DSM demodulation, respectively. The BER of 1024-QAM signal is significantly reduced after using CVNN equalizer. It is worth noting that with CVNN, the sensitivity of the receiver is improved by 0.5 dB when the BER is 2×10-2 .
3 Conclusion
We propose a scheme combining DSM with CVNN to achieve high spectral efficiency through 1-bit quantization. On the one hand, this scheme alleviates the impact of insufficient bandwidth of electronic devices and can further improve the data transmission rate of the system; on the other hand, DSM effectively alleviates the nonlinear problem and further handles it through complex-valued training. Based on the proposed scheme, the bit error rate performance of QPSK and 1024-QAM DSM signals at different transmission sampling rates and received powers is compared. The results show that in order to avoid the rapid increase of the bit error rate of the DSM demodulated signal, the bit error rate of the QPSK signal should be reduced as much as possible. To achieve this goal, we use a CVNN equalizer, which can significantly reduce the bit error rate of QPSK and 1024-QAM. With the help of CVNN, the sensitivity of the receiver is improved by 0.5 dB when the SD-FEC threshold is 2×10-2. In addition, we also optimize the batch size and hidden layer depth to achieve better terahertz system performance. We believe that DSM has good application prospects in future high-speed terahertz transmission with electrical bandwidth bottleneck. In the future, we will also conduct more research on improving terahertz transmission rate.