## Abstract

In this paper, we propose an adaptive optical self-interference cancellation using regular triangle algorithm for in-band full-duplex systems. By using this algorithm, the manual adjustment of the tunable optical time delay line and attenuator is replaced with the adaptive program to change the delay and attenuation for achieving optimal cancellation point. The adjustment process is simplified as a convex function problem. We choose to attain the optimal cancellation point by directly and continuously sampling the power of the signal after cancellation and in turn adjust the time delay and attenuation according to the algorithm. In this way, the two paths in the self-interference cancellation system are precisely and automatically matched. By using our proposed algorithm, the interference signal over 300-MHz wideband is diminished to the noise floor, attaining 20-25 dB cancellation depth adaptively. Compared with other existing algorithms in both the experiment and simulation, our proposed regular triangle algorithm reaches the optimal point faster with 10-30% less number of samples from the near start region, and lowers 40-60% less number of samples from the farther start region.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In-band full-duplex (IBFD) communication has emerged as an attractive solution for increasing the throughput of wireless communication systems and networks [1–3]. Compared to frequency-division duplex (FDD) and time-division duplex (TDD) mode, IBFD operation utilizes one channel for both transmission and reception in the same frequency band, which allows doubling spectral efficiency and improving flexibility [4–6]. However, IBFD has not been widely used due to the challenges from the strong in-band self-interference (IBSI). A self-interference cancellation (SIC) system is needed to enable the deployment of IBFD system. The self-interference (SI) signal is removed in the SIC system by subtracting a copy of the undesired nearby SI and recovering the signal of interest (SOI) [7,8]. Compared with electronic SIC schemes, the optical SIC method attains higher level of cancellation, including the broader cancellation bandwidth and lower cost, due to the accurate channel tracking of the two paths and more precise adjustment accuracy with optical devices [9].

Typical optical self-interference cancellation (OSIC) schemes based on the Mach-Zehnder modulator (MZM) [10–12], the electro-absorption modulator (EAM) [7,13,14], the directly modulated laser (DML) [15] and the electro-absorption modulated distributed feedback laser (EML) [16] have been proposed and demonstrated thoroughly in the previous work. However, as for the adjustment process of time delay and attenuation in those schemes, they are all accomplished by manual manipulation and waveform or spectrum observation. This process takes a lot of time and limits the practical use of the SIC system. This limitation is solved by replacing the manual adjustment process with the automatic program adjustment process using the adaptive algorithm. In this way, the optimal cancellation point is achieved by SIC system itself. This scheme has been proposed by Chang et al. and the Nelder–Mead Simplex (NMS) algorithm is used to seek the optimal precise point of the time delay line and attenuator [17]. The center frequency and bandwidth of the signal are 1.963-GHz and 20-MHz, respectively. However, this NMS algorithm takes 60-70 number of samples to converge, which is of high cost and time-consuming in the real application, and it is also carried out within a limited, narrow bandwidth.

In this paper, we firstly propose the adaptive regular triangle algorithm to reach the optimal time delay and attenuation point so that the two paths in the OSIC system are precisely and automatically matched. Both the simulation and experiment are conducted to verify the accuracy of the algorithm. By using our proposed algorithm, the SI is successfully diminished to the noise floor, attaining 20-25 dB cancellation depth over 300-MHz wideband at the center frequency of 800-MHz. Besides, compared with other existing algorithms, including Hooke-Jeeves, Rosenbrock and Nelder–Mead method [18–20], which take too much sample costs and times to converge, our proposed regular triangle algorithm reaches the optimal point faster with less number of samples, lowering about 10-30% from the near start region, 40-60% from the farther start region. Therefore, it is more appropriate for the application of adaptive OSIC system.

## 2. Principle of adaptive OSIC

#### 2.1 EML based adaptive OSIC

Figure. 1 depicts the architecture of our proposed adaptive OSIC system. The SI signal, which is modulated by the EML from the electrical to optical domain, is sent through the upper path and enters the balanced photo-detector (BPD) directly. The reference signal, which is the same as the SI, is sent through the variable optical attenuator (VOA) and tunable optical time delay line (TODL) deployed on the lower path. These two devices are set to precisely adjust the amplitude and phase of the reference and SI signal before subtraction and they have the tunable range of 0-25 dB and 0-660 ps, respectively. The corresponding principle of this EML-based OSIC system has been studied exhaustively in our previous work [16].

When the OSIC system is at its optimal cancellation point, the reference signal should be matched precisely with the SI and the output signal is ought to be close to the noise floor. At this point, the value of the optical time delay and attenuation is referred to as optimal cancellation value. The principle of the adaptive OSIC is to continuously and adaptively adjust the value of the optical time delay and attenuation. After gathering and calculating the power of the current output signal, the control unit judges whether the power is close enough to the power of noise floor, normally within 1 dB difference. When this specified threshold is attained, the program stops running, ceases the adjustment process and outputs the optimal cancellation value of the time delay and attenuation that corresponds to the minimum power. Otherwise, the algorithm keeps running itself and continues the adjustment process until the optimal cancellation point of the system is obtained.

Based on the theory proposed in our previous work [15], the relationship between the power of the signal after OSIC at the left side of the equation, the time difference and gain difference between the upper and lower path at the right side, is concluded as the Eq. (1).

In the Eq. (1), *P _{r}* denotes the power of the subtracted signal after cancellation,

*P*denotes the power of the signal before cancellation.

*α*and

*t*represent the attenuation and time delay of the lower path, respectively. The minimum value of

*P*corresponds to the optimal self-interference cancellation effect. The power ratio of

_{r}*P*to

_{r}*P*is the cancellation depth. According to the Eq. (1) and the analysis in the [15], the adaptive control scheme of the OSIC is simplified as a convex programming problem and is expressed as the Eq. (2).

As shown in the Eq. (2), *α _{0}, α_{1},* denote the lower and upper boundary of the attenuation, and

*τ*,

_{0}*τ*denote the lower and upper boundary of time delay parameters. The optimal cancellation point is obtained by solving the convex problem to reach the optimal attenuation and time delay within the boundaries. These boundaries are decided by the tunable ranges of the optical devices in the experimental system. Therefore, this convex problem should be regarded as an unconstrained optimization problem within the maximum variation ranges of the two parameters. It should be noted that the value of the objective function in the Eq. (2), the power, cannot be accurately expressed by the specific formula in this convex problem. So the power of the signal has to be sampled real time. Under this consideration, the commonly used gradient descent methods that calculate the value of the gradient are not appropriate to be used in this system. Based on this situation, the unconstrained optimization direct method is more suitable in the power-based adaptive OSIC system because it doesn’t need to calculate the gradient of the objective function and it is based on the sample value of the objective function [21].

_{1}#### 2.2 Regular triangle algorithm

There are three main algorithms of unconstrained optimization direct method, including Hooke-Jeeves, Rosenbrock and Nelder-Mead algorithm. Although these three algorithms are able to successfully converge at the optimal point in the unconstrained optimization problem, they take too much sample costs and times, which are verified in the following experiment and simulation. Thus, they are not efficient and applicable for the practical use of the adaptive OSIC system. Besides, it should be noted that the adjustment process of the adaptive OSIC needs to be carried out by sampling the power of the signal at the specific time delay and attenuation point. But the adjustment of the optical devices takes some time and it is much longer than the execution time of the program. So in order to reduce the time that spends on the adjustment of the devices and improve the efficiency of the system, the sample cost of the algorithm should be given the top priority. Under these considerations, a two-dimensional unconstrained optimization algorithm called regular triangle algorithm, is proposed for the adaptive OSIC system with less number of samples.

Here, the principle of the regular triangle algorithm is introduced. In order to make it easier to understand, an example in the Fig. 2 is used to illustrate the exploration and convergence process of the algorithm. In this example, it is assumed that the origin of the coordinate system is the optimal value point. The power of the point turns higher as the point becomes farther from the origin point in the coordinate system. The process of the algorithm is divided into two steps: regular triangle test and plus or minus 30 degrees advancement.

For the first step, regular triangle test, the system tests the power of the three vertices of a regular triangle. This regular triangle is built on the center of the current iteration point, which is the point 1 in the Fig. 2. The distance from three vertices to the center of the triangle is the current step length. In a two-dimension coordinate system, three points are the minimum number of points that cover the 360 degrees plane. So the scheme of testing the power of three points has a near-minimum sample cost. According to the design of the algorithm, an initial regular triangle is built on the center of the start point within the random 12 directions.

The second step, plus or minus 30 degrees advancement, is that after testing the power of all the three vertices, a certain vertex that has the minimum power among the three vertices is regarded as the next iteration point, which is the point 2 in the Fig. 2. Then the two points that are located at the direction plus or minus 30 degrees of the line connecting the two iteration points are tested. The distance from the current iteration point to the new two tested points remains unchanged, just as same as the current step length.

Since the regular triangle test and the plus or minus 30 degrees advancement are used in the first and second step, all the directions involved in the algorithm are the integer multiple of 30 degrees based on the X-axis. Thus, 12 directions are included in the 360 degrees plane. As long as the current direction is known, the angle of the next direction can be obtained easily. Besides, when the current step length is known, the coordinate value of the next tested point is calculated based on the coordinate of the current iteration point.

After the power of the two new points on the direction of plus or minus 30 degrees are tested, the point with the minimum power among three points, including the current iteration point, is regarded as the next new iteration point, which is the point 3 in the Fig. 2. Then the plus or minus 30 degrees advancement is continued. When the two tested points’ power values are both higher than the current iteration point, just at the 5th point, the step length is reduced to 0.386 times the current one. A new regular triangle is built to restart the first step of the algorithm with the new step length.

The red dots in the Fig. 2 indicate the points at which the tests are successful, that is, the points of each iteration. The brown dots indicate the points at which the tests fail, which have higher power value. There are two stopping conditions for the algorithm. One is that the power of the current tested point is close enough to the noise floor, the other is that the step length has been reduced to less than or equal to the specified minimum threshold. As shown in the Fig. 2, when the stop condition is met at the 7th point, the program stops, and the optimal value is reached, which is within the nearby area of the origin. Algorithm 1 gives the detailed description of the regular triangle algorithm.

## 3. Experimental setup

To verify the availability and superiority of the regular triangle algorithm, we carry out both the experiment and simulation to prove the accuracy and advantage of regular triangle algorithm compared with other three methods. Table 1 gives the parameters for the experiment setup. For sake of fair comparison, four algorithms have the same stopping criteria, and are also equipped with the same start settings. As shown in Table 1, the values of start step length, attenuation factor, acceleration factor and reflection factor, are finally obtained after so many times of MATLAB simulation tests and searches. Compared with other different values, these values are able to make the OSIC system have the least number of samples to converge. Here, the “L” is referred to as the normalized distance from the start point to the optimal point and it is calculated through the Euclid distance between the normalized time delay and attenuation points in the coordinate system.

The experiment is conducted corresponding to the architecture in Fig. 1. Figure. 3 depicts the actual platform of our adaptive OSIC system. An arbitrary waveform generator (AWG, Tektronix AWG7122C) plays the role as signal generator to provide the SI and reference signal. The oscilloscope (LeCroy SDA845Zi-A) receives and samples the subtracted signal after the BPD. The Ethernet switch is employed to connect all the related instruments in Fig. 3 to make the transmitting and receiving under the control of MATLAB console. It also ensures the adaptive process and stops the adjustment in time. The VOA and TODL are both controlled by MATLAB through serial ports. The power value of the subtracted output Orthogonal Frequency Division Multiplexing (OFDM) signal is gathered by the oscilloscope, calculated and judged by MATLAB. The reason why choosing the OFDM signal is that it is a popular modulation format with high bandwidth efficiency and has been regarded as air interface waveform technology in the future mobile communication system [22]. The adjustment process is started by MATLAB, and the algorithm is run to adjust the value of the VOA and TODL to pursue the optimal cancellation point.

The objective function in the experiment is the real-time power sampled by oscilloscope. One stopping condition is to judge whether the current power of the signal is close enough to the power of noise floor, specifically within 1 dB difference. The other is the step length threshold. When there is no cancellation part in the system, the power of the received signal is approximately **-**7 dBm. The power of the noise floor is **-**28 or −29 dBm, thus the optimal cancellation effect of the system is about 20-25 dB cancellation depth.

## 4. Results and discussion

In the experiment, the optimal time delay and attenuation point of the actual system is regarded as the origin of the coordinate system. Five start areas are set to start the adaptive process. The normalized Euclid distances from the known optimal point to the start areas are 10, 15, 20, 25, and 30 in the coordinate system. These distances, literally radius, are calculated based on the coordinate system. It should be noted that the power of the signals that start at each radius are basically the same value. The power of the signals increases as the radius becomes larger. Besides, the number of samples, which is important for improving the efficiency of the adaptive OSIC system, is referred to the numbers that the oscilloscope is controlled to sample the power of the signal at each delay and attenuation point through the whole adjustment process.

All the four algorithms are run in the experiment, and the number of samples of each method is calculated. As depicted in the Fig. 4, regular triangle algorithm in the black line has the least number of samples compared with other three algorithms, lowering about at least 30%. As the start distance becomes larger, the number of samples of the proposed regular triangle algorithm remains relatively stable and low, while the other three algorithms’ grow gradually, especially the Nelder-Mead method. At the largest start distance with relatively higher power, the regular triangle algorithm has reduced the number of samples 60% compared with the Nelder-Mead method, showing great advantages.

Besides, we select two specific start points that are at the start distance of 10 and 30, respectively, to observe the descending trend of four algorithms. These two start points have different time delay and attenuation value, and therefore the different power value. As shown in the Fig. 5(a), when the power of the start point is closer to the noise floor, which is – 28 or −29 dBm in the system, the proposed regular triangle method has the least number of samples with the cancellation depth of almost 10 dB. In the Fig. 5(b), when the start point has a higher power, the regular triangle method shows great advantage at the number of samples with the cancellation depth of 20 dB. Besides, as shown in both Fig. 5(a)-5(b), the spectrum of the OFDM signal is successfully diminished to the noise floor through the process of convergence, showing optimal cancellation effect.

Figure. 6 demonstrates the convergence path in the normalized coordinate system of the experiment’s time delay and attenuation. It has the same start point as in the Fig. 5(a). Here, the normalized attenuation and time delay range are 500 and 66, respectively, which correspond to the x-axis and y-axis in the Fig. 6. This conclusion is concluded through the comparison of the power sampled from the oscilloscope. The power of the sampled signal changes the same value when either the attenuation or the time delay is changed by the same value in the normalized coordinate system. In the coordinate system, the origin is the optimal point of the system and all the four algorithms start at the same point. It is obviously shown that the regular triangle method in the bold black line spends the least number of samples to reach the optimal region successfully.

To further analyze the probability of each algorithm’s number of samples at different start distances and avoid the lack of the testing times in the experiment, the probability distribution for 10,000 results are calculated under different start distances through the simulation. The simulation is carried out by MATLAB and all the parameters set in the simulation are the same as the values in the Table 1.

The simulation of the four algorithms is based on the following assumptions. The origin of the coordinate system is the theoretical optimal and the system starts from the points that have different distances from the optimal point. The start distances used in the simulation are as same as the experiment. According to the Eq. (1), which is the objective function in the simulation, the objective function value of the current point turns larger as the Euclid distance becomes larger.

The stopping condition in the simulation is that the objective function value is reduced more than 25 dB from the start point as well as the step length threshold. Four start areas are set in the simulation and the distances are 10, 15, 20, and 30, respectively. The number of samples’ probability distribution is shown in the Fig. 7(a)-7(d).

From the four graphs, it is found that the probability distribution of the Nelder-Mead method varies greatly with the start distance. When the start distance is lower, the Nelder-Mead method converges around 10-20 times with a large probability. As the start distance becomes farther, the number of samples of the Nelder-Mead method with a larger probability is also gradually increasing. The probability distributions of the other three algorithms are more stable with the change of the start distance. Compared with the other three algorithms, the sample cost of the regular triangle algorithm is concentrated in the vicinity of 15-17 times with a larger probability. This number is quite lower than the other three methods and shows great performance improvement. Besides, the advantage of the regular triangle algorithm becomes more evident when the start distance is larger. The result in the simulation is as same as the conclusion in the experiment.

## 5. Conclusion

In this paper, we propose, simulate and experimentally demonstrate an adaptive OSIC system using regular triangle algorithm. By using the regular triangle algorithm to replace the manual adjustment process of the optical time delay line and attenuator, the adaptive OSIC system converges automatically to its optimal cancellation point. The SI and reference signal are subtracted automatically and completely and the power of the output signal is close to the noise floor. The proposed adaptive OSIC system successfully cancels the SI with at least 20 −25 dB cancellation depth over 300-MHz wideband. Compared with other three existing algorithms, regular triangle algorithm reaches the optimal point faster with 10-30% less number of samples from the near start region, and lowers 40-60% less number of samples from the farther start region. It successfully converges at the nearby region of the optimal point. In this way, the practical application of the OSIC system is conceivable since the precise match of the two paths doesn’t need the manual adjustment and it is finished by the system itself.

## Funding

National Natural Science Foundation of China (61775137, 61431009, 61271216, 61221001); the National “863” Hi-tech Project of China.

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