Anti-Error Robust Pattern Synthesis Algorithm in Time-Modulated Array Antenna

Compared with the traditional antenna array, the time-modulated array can increase the flexibility of the array by adding time switches. However, due to the introduction of time switches, the time-modulated array antenna is more sensitive to random errors than the traditional array. Besides random errors caused by unavoidable factors such as array deformation and defects of processing and assembly in the traditional array, time-modulation parameter errors will be introduced by time switches. As errors in the traditional array can degrade the performance of the array, time-modulation parameter errors can lower the performance of the array pattern as well, which will lift the side-lobe and increase the dynamic range of the side-lobe. Aiming at the time-modulation parameter errors in the time-modulated array, an anti-error robust pattern synthesis algorithm (AERPS) based on the convex optimization (CVX) is proposed in this paper. In the algorithm, the optimization model for the pattern synthesis of the time-modulated array is established and analyzed. Then the model is divided into two sub-models of the center frequency and the first sideband, and the convex optimization solution is first performed on the center frequency model. After the center frequency model is solved, the convex optimization solution is used to solve the first-order sideband model to obtain the final results. The simulation results of the time-modulated array in this paper show that the algorithm can reduce the average value and the dynamic range of the normalized side-lobe level, thus verifying the effectiveness and the robustness of the algorithm.


I. INTRODUCTION
Array antenna can achieve higher performance than a single antenna by combining signals with each other. However, due to the manufacturing mistakes and other unavoidable factors, array antenna will be affected by various kinds of errors, such as the position error of elements, the amplitude and phase excitation error, the interval error of elements, etc. These errors will raise the side-lobe of the pattern of arrays, resulting in degradation of the radiation pattern performance. In order to reduce the negative impact of errors, it is necessary to make analysis of errors.
Many researchers have studied the amplitude and phase excitation error in array antenna. Appasani et al. proposed a pseudo measurement technique for detection and correction of excitation error of the array antenna [1]. Zahedi and Arand proposed a phase compensation method based on the The associate editor coordinating the review of this manuscript and approving it for publication was Hisao Ishibuchi . genetic algorithm to realize the phase error correction for large phased array antennas [2]. Huang et al. proposed a non-iterative technique for the self-calibration of the amplitude and phase error of the sensors of the airborne array antenna with unknown source location [3]. Aiming at the amplitude and phase error of the microwave feed network of the phased array radar, Xie et al. proposed an array monitoring method based on the compensation and correction technology [4]. Zhang et al. proposed an improved calibration method using coefficient matrix for the amplitude and phase error in the phased array radar [5]. Dai et al. proposed a new method that does not require a priori calibration information for the problem of amplitude and phase error calibration for large-scale planar arrays [6]. Wang et al. proposed a calibration method for circular array antennas with mutual coupling and amplitude and phase error, and obtained a more accurate DOA (Direction of Arrival) estimate [7]. Rocca et al. proposed a statistical method based on the interval analysis to analyze the influence of the amplitude and phase excitation with random errors on the radiation pattern of the phased array [8]. He et al. introduced how to use the probabilitybased method and method based on the interval arithmetic to evaluate the effects of the phased array antenna with excitation errors [9].
There are also many literatures that have analyzed and studied the position error of array elements. For the array antenna with element position errors, Lou et al. considered the mutual coupling effect in the traditional compensation method to reduce the influence of the array deformation [10]. Yoo et al. studied the phased array radar where the antenna feed position is not in the center of the antenna, and proposed a compensation technique that does not collect information at the center of the antenna [11]. Ye et al. proposed an equivalent method for estimating signal parameters based on the single-signal source rotational invariance technique, which can effectively balance the position error caused by the vibration deformation of the array [12]. Sippel et al. analyzed the antenna array with mutual coupling, unknown time delay and position errors, and proposed a calibration method using an incoherently transmitting beacon [13]. Wang et al. proposed the coupled structure-electromagnetic statistic model for the randomness of the position of elements in the antenna array, which can quickly and effectively obtain the antenna performance [14]. Sa et al. proposed a hybrid tolerance design method for antenna array with geometric errors, which improved the performance of the antenna [15]. Wang et al. proposed a prior knowledge-based algorithm for large array antennas with interval errors, phase excitation errors and position errors, which improved the robustness of the antenna array [16]. Zhang et al. analyzed the position error existing in the conformal array and proposed a position error calibration algorithm to reduce the influence of the error [17].
In addition to the excitation error and position error mentioned above, other kinds of errors (including interval error, beam pointing error, etc.) have also been studied in related literature. Zhou et al. analyzed the joint array antenna with the array pointing error, and found the error will increase the side-lobe of the array pattern and reduce the gain [18]. Frid et al. analyzed installation errors such as platform reflection, diffraction from metal edges, and reflection and refraction in the radome, and proposed a post-processing method to improve DOA estimation accuracy [19]. Guo et al. studied large-scale array antennas containing assembly errors. They established an accurate gain prediction model based on the improved extreme gradient boosting algorithm and transfer learning method, and proposed a compensation method to control the error [20]. Aiming at the beam pointing error of phased arrays, Zhang et al. proposed a sequential phase feeding algorithm with the minimum beam pointing error for centrally symmetric planar array antennas, and deduced the theoretical expression of the beam pointing error caused by phase quantization errors [21]. Lee and Song conducted a detailed mathematical analysis of the array antennas with the quantization error and interval error, and theoretically  obtained the statistical characteristics of the radiation pattern, which is in good agreement with the simulation results [22].
Compared with the traditional antenna array, the timemodulated antenna array deploys the time-modulation switches, so it will introduce time-modulation parameter errors. These errors in time-modulated arrays have also been studied by some literature. Yurtsev et al. analyzed the time interval random errors in the time-modulated array and pointed out that the side-lobe level of the pattern increases faster than that of the array without time-modulation [23]. Amjadian and Fakharzadeh studied the effect of non-ideal time switches on the performance of time-modulated arrays, and pointed out that time error has a little effect on beam pointing and beam width, but more on main-lobe level [24]. However, there is still a lot of room for research on various kinds of errors in time-modulated arrays.
Aiming at the time-modulation parameter errors in the time-modulated array, this paper proposes an anti-error robust pattern synthesis algorithm (AERPS) to optimize timemodulation parameters and the excitation coefficients of array elements, so as to minimize the influence of the errors on the array pattern and further improve the low side-lobe and robust performance. The structure of this paper is as follows: Chapter II introduces the pattern synthesis model of the time-modulated linear array under error conditions. The proposed AERPS algorithm is discussed in detail in Chapter III. Numerical simulation results are presented in Chapter IV to demonstrate the effectiveness and robustness of the proposed algorithm. Finally, conclusions are given in Chapter V.

II. MATHEMATICAL MODEL OF TIME-MODULATED LINEAR ARRAY PATTERN SYNTHESIS UNDER ERROR CONDITIONS
Time-modulated linear array composed of N isotropic array elements is shown in Fig 1 and the far-field radiation intensity can be expressed as where f 0 , x n and w n represent the center frequency of the array, the position and complex excitation coefficient of the n−th element in order. k = 2πf 0 c is the carrier propagation coefficient in free space, and c is the speed of light. As shown in Fig 2, in the time-modulation repetition period T r , u n (t) is the rectangular function, and its mathematical expression is where t n and τ n are the opening time and duration time, all satisfying 0 ≤ t n , τ n ≤ T r . Expanding the Fourier series of the u n (t), we have where f r = 1/T r , and a nm is the expansion coefficient, whose expression is therefore formula (1) can be further expressed as where α n = τ n f r and β n = t n f r are the normalized duration time and opening time, respectively, satisfying 0 ≤ α n , β n ≤ 1. The radiation intensity expressions of the array at the center frequency f 0 and the m − th sideband f 0 + mf r are Ignoring the common factor terms e j2πf 0 t and e j2π(f 0 +mf r )t , the above expressions are rewritten as Next we will discuss the effect of errors on the radiation intensity. Array errors usually contains the following types: excitation errors of the array elements, position errors of the array elements, and time-modulation parameter errors. Only the time-modulation parameter errors are studied in this paper, that is, the opening time error and duration time error are studied.
Assuming that the time switch of the n − th element has normalized opening time error β n and normalized duration time error α n , the modified expression of the array antenna's m − th radiation intensity is × e jkx n sin θ −jmπ(2β n +α n +2 β n + α n ) = n w * n α n sinc (mπ α n ) e −jmπ(2β n +α n ) The part in {} is the sum of the error terms caused by the time-modulation parameter errors α n , β n .
For the pattern synthesis of the time-modulated linear array affected by the errors, we have studied the following problems: 1)Beamforming in the given direction at the center frequency f 0 . 2)Beamforming with given parameters on the first sideband f 0 + f r . 3)Making the peak side-lobe level (PSLL) as low as possible at the center frequency f 0 and first sideband f 0 + f r .
In summary, the comprehensive optimization model of the pattern synthesis is given below In the model, α = α 1 · · · α n T , β = β 1 · · · β n T are the time modulation parameter vectors that need to be optimized, and they forms the optimal solution vectors of the optimization problem together with the complex excitation coefficient vector w = w 1 · · · w n T . F 0 (θ 0 ) represents the radiation intensity in the beam direction θ 0 at the center frequency f 0 . F 1 (θ 1 ) represents the radiation intensity in the beam direction θ 1 at the first sideband f 0 + f r , and µ 1 is its maximum value. F 0 (θ s ) and F 1 (θ s ) represent the radiation intensity of the sampling direction θ s in the side-lobe area s at the center frequency and first sideband, respectively, and ε 1 , ε 2 are the side-lobe constraint parameters.

III. ANTI-ERROR ROBUST PATTERN SYNTHESIS OF TIME-MODULATED LINEAR ARRAY
The model (11) will be divided into two sub-models of the center frequency f 0 and first sideband f 0 + f r . At the center frequency m = 0, according to (10), we have Defining a new parametric variable χ n = w * n α n and its vector form χ = χ 1 · · · χ n T = w * · α which need to be optimized. Denoting b θ = e jkx 1 sin θ · · · e jkx N sin θ T as the steering vector at the sampling direction θ. At the same time 57194 VOLUME 10, 2022 Assuming X is a random vector, and E (X ), D (X ) are the expectation and variance of X , respectively. Under the assumption of α n i.i.d.
∼ N 0, σ 2 1 , b is a complex Gaussian random vector and the expectation and variance are According to (13),the error term in F 0 (θ) is χ T b ,whose statistical average power is therefore when the parametric vector χ remains unchanged, increasing α n can reduce the error power. At the same time, for further analysis, according to the triangle inequality in vector form |X| − |Y| ≤ |X + Y| ≤ |X| + |Y| and the Cauchy inequality in vector form X T Y ≤ X 2 Y 2 , there is We further introduce the positive real parameter η which is the upper bound of the error term, and for the random vector b , P { b 2 ≤ η} = 1 statistically holds. We intend to enhance the robustness of the pattern under the error condition by setting the upper bound η. With (15), we have The term χ T b θ in the above formula is exactly the radiation intensity without error, that is E 0 (θ), therefore the upper/lower bound of the radiation intensity under the error condition is obtained can be selected as a typical value. In summary, the anti-error robust pattern synthesis problem at the center frequency can be solved by two steps: 1)Enhance the robustness of the array pattern with the parameter η introduced, whose model is where b 0 = e jkx 1 sin θ 0 · · · e jkx N sin θ 0 T is the steering vector in the main radiation direction θ 0 , and b s = e jkx 1 sin θ s · · · e jkx N sin θ s T is the steering vector in the sampling direction θ s in the side-lobe area.
2)In the case of keeping χ solved at the previous step unchanged, increase α n to reduce the error power P 0 (θ).
From above we solve the pattern synthesis problem of the center frequency, and next we will focus on the pattern synthesis problem of the first sideband. At the first sideband m = 1, according to (10), we have × e jkx n sin θ−jπ(2β n +α n +2 β n + α n ) In the following of this part, with χ calculated, only the influence of β n is considered, and similarly we assume β n i.i.d.
Denoting h n = w * n α n sinc (π α n ) e −jπ(2β n +α n ) , (20) can be transformed into Further writing (21) in vector form, we have For the convenience of analysis, we define h = h 1 · · · h n T and b = e −j2π β 1 −1 e jkx 1 sin θ · · · e −j2π β N −1 e jkx N sin θ T which is the term related to the error. The steering vector is also represented by b θ = e jkx 1 sin θ · · · e jkx N sin θ T . Analyzing the error term b , we have where I N ×N represents identity matrix. It can be seen from (22) that the error term in F 1 (θ) is h T b , whose statistical average power is So when the parametric vector h is unchanged, the greater the variance σ 2 2 is, the greater the error power P 1 (θ) is. Furthermore So we get P 1 (θ) = 1 − e −4π 2 σ 2 2 n χ 2 n sinc 2 (π α n ) (26) VOLUME 10, 2022 With σ 2 2 and χ n unchanged, increasing the normalized duration time of array elements can reduce the error power.
Defining a new parametric vector g = g 1 · · · g n T of which the n−th entry is g n = sinc (πα n ) e −jπ(2β n +α n ) , so h = χ · g holds. The robust pattern synthesis problem at the first sideband can be solved through the following optimization model.
where χ is a known parametric vector solved by (18), b 1 = e jkx 1 sin θ 1 · · · e jkx N sin θ 1 T is the steering vector of the main radiation direction of the first sideband, and g is the parametric vector we need to obtain. µ 1 is the maximum radiation intensity at the radiation direction of the first sideband, and ε 2 is the side-lobe constraint parameter. µ 2 is the control parameter, and the smaller µ 2 is, the smaller sinc (πα n ) and error power are obtained, so that the side-lobe performance of the first sideband can be better.
Denoting the amplitude and phase of g n are a gn and ϕ gn , respectively, after obtaining g by the convex optimization method, we can calculate α n and β n through the following relationship α n = 1 π sinc −1 a gn β n = − 1 2π ϕ gn + sinc −1 a gn + 2πp (28) In the above formula, it's necessary to take an appropriate integer p to make 0 ≤ α n , β n ≤ 1 , and the complex excitation coefficient of each element can be obtained by w * n = χ n α n .

IV. SIMULATION
In order to verify the performance of the proposed algorithm,

A. COMPREHENSIVE ANALYSIS OF DUAL-BAND PATTERN UNDER TIME-MODULATION PARAMETER ERRORS
This section describes the effect of time-modulation parameter errors on the dual-band pattern. It can be seen from (12) that the radiation intensity of the center frequency is only related to the normalized duration time error α n and has nothing to do with the normalized opening time error β n . Fig 3 shows the influence of different normalized duration time errors on the center frequency pattern. It can be seen that due to the normalized duration time error α n , the side-lobe performance deteriorates, and as the variance of the error σ 2 1 increases, the rise of the side-lobe level is more obvious. Table 1 shows the statistical data of the 500 repeated simulation results. It can be seen that the average value of the PSLL has risen from the error-free −28.64dB to −13.80dB, −19.00dB, −23.95dB, −26.06dB, and −27.28dB, respectively, which is consistent with the above conclusion.
As for the first-order sideband, it can be seen from (20) that when χ is determined, the radiation intensity is affected by the normalized opening time error β n . Fig 4 shows the influence of different normalized opening time errors on the first sideband pattern. As the variance of the error σ 2 2 increases, the side-lobe level rises more obviously.   Table 2 shows the statistical results of 500 experiments. It can be seen from Table 2 that with the increase of σ 2 , the maximum radiation intensity of the main-lobe is decreasing, while the mean of PSLL is increasing.

B. ANALYSIS OF ANTI-ERROR ROBUST PATTERN SYNTHESIS OF CENTER FREQUENCY
In this section, we will study the influence of the parameter η on the performance of the center frequency pattern according to (18). Fig 5 shows the influence of different η on the center frequency pattern with σ 1 = 0.05. It can be seen that as η increases, the maximum of the radiation intensity increases, and the PSLL also increases. In order to further obtain the statistical performance of the algorithm we proposed, we performed 500 repeated simulations on multiple sets of parameters, and the results are shown in Table 3 and Table 4. Table 3 shows the statistical average of the normalized side-lobe level in 500 random experiments under different σ 1 conditions when η takes a specific value, from which we can see: 1)When keeping η unchanged, such as the row of η = 3, the normalized side-lobe level increases as the parameter σ 1 increases. 2)When keeping σ 1 unchanged, for example, in the column of σ 1 = 0.05, from a statistical point of view, the VOLUME 10, 2022  normalized side-lobe level increases with the increase of η, which shows that setting the upper bound parameter η cannot improve the deterioration of side-lobe level. Table 4 shows the statistical root of variance of the normalized side-lobe level, that is, the deviation degree of the normalized side-lobe level in 500 random experiments under the different σ 1 conditions when η takes the specific values, respectively. From this table we can see clearly: 1)When keeping η unchanged, such as the row of η = 3, the degree of deviation increases with the increase of σ 1 . 2)When keeping σ 1 unchanged, such as the column of σ 1 = 0.05, the degree of deviation decreases as η increases, which shows that setting the upper bound parameter η can reduce the side-lobe deterioration range.
Combining the conclusion of the Table 3 and Table 4, it can be seen that although the upper bound parameter η cannot lighten the uplift of the side-lobe caused by the error, it can reduce the dynamic range of its fluctuation, which is exactly the meaning of the so-called 'robustness' in the algorithm.
Furthermore, in order to make the side-lobe performance better, we can reduce the statistical average and the statistical root of variance of the normalized side-lobe level by increasing max (α n ). Table 5 shows the statistical results of 500 repeated simulations of the center frequency pattern when α n is increased proportionally under the setting of η = 3, σ 1 = 0.02.
However, increasing α n proportionally will affect the radiation characteristics of the first sideband. As shown in Fig 6,   the first sideband need to be further optimized for robust pattern synthesis of dual frequency bands.

C. ANALYSIS OF ANTI-ERROR ROBUST PATTERN SYNTHESIS OF FIRST SIDEBAND
It can be seen from the previous part that increasing α n will worsen the radiation characteristics of the first sideband. So it is necessary to adjust the parameter µ 2 , according to (27), to further control the value of α n to improve the radiation characteristics of the first sideband f 0 + f r . This part shows the influence of µ 2 on the first sideband pattern. Fig 7 shows that as µ 2 takes the value 0.5, 0.6, 0.7, 0.8, 0.9 and 0.99 in turn with σ 2 = 0.05, the statistical average value of the maximum radiation intensity of the main-lobe and PSLL are increasing. Furthermore, in order to obtain the statistical results of the maximum radiation intensity and PSLL of the first sideband under different parameters, we perform 500 repeated experiments on multiple sets of parameters. The results are shown in Table 6 and Table 7. Table 6 and Table 7 respectively give the statistical average values of the maximum first sideband radiation intensity and PSLL in 500 repeated experiments when the specific values of the parameter µ 2 are taken respectively under the different values of σ 2 , from which we can find : 1)When keeping the parameter µ 2 unchanged with the increase of σ 2 , the maximum radiation intensity decreases along with the increase of the PSLL.
2)When keeping σ 2 unchanged, as the parameter µ 2 increases, the maximum radiation intensity increases, and the PSLL also increases.  It can be concluded that adjusting parameter µ 2 can control the performance of the main-lobe and side-lobe under the error condition, but the performance improvement of the main-lobe and side-lobe are often contradictory. Therefore, it's necessary to select the appropriate µ 2 based on the prior knowledge of the error variance σ 2 2 to achieve a better tradeoff between the main-lobe and side-lobe performance.

D. ANALYSIS OF ANTI-ERROR ROBUST PATTERN SYNTHESIS OF DUAL-BAND
Combining the analysis results of above parts, we will take a specific setting of the parameters η = 3, µ 2 = 0.7 to verify the performance of the algorithm we proposed(another setting of the parameters can also reflect the performance of the algorithm to vary degrees), and compare with the algorithm of η = 0, µ 2 = 1 (that is, the CVX algorithm without processing). It can be seen that the proposed algorithm can make the side-lobe performance better. In order to further give the performance of the algorithm in a statistical sense, we have done 500 repeated experiments, and the results are shown in Table 8. Table 8 shows the performance comparison between the unprocessed CVX algorithm and AERPS algorithm under the specific setting of σ 1 = 0.05, σ 2 = 0.05. It can be seen that the radiation intensity of the proposed AERPS algorithm in the direction of the center frequency θ 0 = 0 • and first sideband θ 1 = 10 • all increases, and the statistical variance is significantly reduced.
The statistical variance of the maximum radiation intensity at the center frequency is reduced from 0.18dB to 0.03dB, and the statistical variance of the maximum first sideband radiation intensity is reduced from 0.17dB to 0.02dB, which embodies the significant improvement of the robustness of the pattern synthesis.

V. CONCLUSION
Aiming at the dual-band time-modulated linear array pattern synthesis model with errors, this paper proposes an AERPS algorithm, which can reduce the side-lobe lift caused by errors and improve the robustness of the pattern synthesis. Our basic idea is to divide the model into two sub-models and further solve the revised model step by step through making reasonable assumptions about the distribution of the time-modulation parameter errors. We first perform convex optimization solution for the center frequency band, and then perform convex optimization solution for the first sideband after obtaining the intermediate parametric variables, and finally obtain the complex excitation coefficient, opening time and duration time of each element. This paper shows the effectiveness and robustness of the algorithm through the simulation results of 30-element linear array. It can be seen from the results that the algorithm proposed in this paper can reduce the normalized side-lobe level in a statistical sense, and reduce the dynamic range of the normalized side-lobe level.