The PID algorithm is a feedback control algorithm that has been widely employed in the field of industrial control owing to its simple structure, excellent stability, reliable operation, and convenient adjustability. The fundamental concept that defines this algorithm involves adjusting the system’s control input based on the disparity between the current desired output and the actual output of the system by conducting proportional, integral, and derivative control actions. This adjustment aims to minimize deviations between the system output and the desired value as much as possible by appropriately setting the proportional, integral, and derivative coefficients. Consequently, it ensures a rapid response time, offers accurate tracking capabilities, and improves overall system stability.

Furthermore, in contrast to the proportion control-based negative feedback iterative method, the PID algorithm is capable of eliminating steady-state errors [6]. Steady-state error refers to the discrepancy between the desired output value and the actual output value when the system control process approaches stability. Notably, the integral component plays a primary role in reducing steady-state error and enhancing system accuracy. The effectiveness of an integral action depends on the integration time constant—a larger integration time constant results in weaker integral action, and vice versa. Meanwhile, the derivative component reflects deviation signal trends (i.e., the rate of change). Furthermore, it introduces an early correction signal into the system before deviation signals become too large, thereby accelerating system response times and minimizing adjustment periods.

Considering this context, the relationship between the expected output from a system which denoted as ỹ(n), and the actual output is y(n), referred to as e(n), can be expressed as Eq. (1):

Therefore, according to the definition of the PID algorithm, it can be expressed as Eq. (2):

where k_p, k_i, and k_d are the coefficients for the proportional, integral, and derivative terms, respectively.

As previously mentioned, the appropriate selection of PID parameters plays a crucial role in a negative feedback system. The estimation methods for these parameters can be categorized into traditional approaches and intelligent algorithms. Common traditional methods include empirical techniques, trial-and-error procedures, and root locus analysis. These algorithms rely heavily on manual expertise, involve time-consuming trial-and-error iterations, lack global search capabilities, necessitate high accuracy of the system model, and exhibit a limited applicability range.

Intelligent algorithms, on the other hand, are capable of conducting global searches, which enables them to avoid being trapped in local optima and find a global optimal solution or a solution close to it for the PID parameters. Furthermore, they can optimize parameters automatically, thus reducing manual intervention and improving the efficiency and accuracy of parameter optimization. Additionally, they exhibit strong adaptability and a considerable degree of generalization ability, which allows them to quickly identify suitable PID parameters for different control systems, thereby enhancing control performance. Moreover, intelligent algorithms can be easily implemented in parallel computing, where they take full advantage of the multicore processors of modern computers to improve computational speed.

As shown in Fig. 2, in this study, we employed the PSO algorithm to optimize the PID parameters. PSO is a swarm intelligence-based optimization algorithm that emulates the foraging behavior exhibited by birds to identify the optimal solution for a given problem. As a result, the PSO algorithm effectively handles continuous nonlinear problems [7]. Originally proposed by Eberhart and Kennedy [8], this approach conceptualizes the entire population as a swarm, where each particle explores a predefined search space to attain its individual best solution while also collectively interacting with other particles to achieve global solutions. This algorithm comprises displacement and velocity equations, which are influenced by factors such as weight, personal best coefficient, global best coefficient, number of particles, and number of iterations.

In this heuristic search technique, each agent examines its position information within the solution space to update the personal best displacement of its optimal solution, which in turn updates the global best displacement to achieve the globally optimal solution. Meanwhile, the velocity components are calculated based on the displacement of each particle. The coefficients or acceleration constants c₁ and c₂, representing the velocity of the particles toward their individual and global best positions, respectively, determine the speed and displacement of the particles. In this context, it is crucial to maintain a balance between the constants, since a higher value would compel the particles to rapidly converge toward or surpass the target position, while a lower value may result in overshooting of the target coordinates during exploration without proper recall. Notably, the inertia value ω governs the exploration rate between an individual’s best position and the global position.

Eqs. (3) and (4) present the fundamental velocity and displacement formulas for standard swarm particles:

where ω is inertia weight; c₁, c₂ are non-negative constant acceleration; Vik is velocity of particle i in the k-th iteration; r₁, r₂ are random numbers distributed in the interval [0,1]; Pik is individual best value of particle i in the k-th iteration; and Pgk is population extremum of the k-th iteration.

To improve the balance between the global search and local search capabilities of the algorithm, Shi and Eberhart [8] proposed using a linearly decreasing inertia weight, as depicted in Eq. (5):

In this equation, ω_start represents the initial inertia weight, ω_end denotes the inertia weight at maximum iteration count, and T_max signifies the maximum number of iterations. Usually, optimal performance is achieved by the algorithm on setting ω_start = 0.9 and ω_end = 0.4.

The flowchart in Fig. 3 illustrates the procedural steps involved in PSO. Initially, the positions and velocities of the particles within a population are randomly initialized, following which the fitness value of each particle is computed to find the individual and global optima. Based on these optima, the positions and velocities of the particles are updated. This iterative process continues until a termination condition is met. Notably, each particle represents three parameters of a PID controller: k_p, k_i, and k_d. Furthermore, the fitness evaluation of each particle involves calculating the normalized mean square error (NMSE) between the actual output and the distortion-free output obtained through the PID negative feedback iteration. The NMSE is then compared against a predefined threshold to determine whether it satisfies the specified condition. If it fails to meet this condition, the particle swarm is updated. The program terminates either when the NMSE meets the set condition or when it reaches the maximum number of iterations.

The NMSE is a robust approach for assessing data fitting. It quantifies the accuracy of fitting outcomes based on root mean square deviations. As a result, the NMSE is considered the accuracy evaluation function in this paper. Eq. (6) presents the computational formula for NMSE:

Here, k denotes the k -th iteration and N represents the length of the signal. A smaller NMSE value indicates a more accurate fitting effect of the data. Furthermore, y_k(n) refers to the actual output of the model, whereas ỹ_k(n) represents the fitted value of its output.

The introduction of harmonic distortion into the output signal by DAC during the process of converting digital signals to analog ones can be attributed to the non-ideal characteristics of devices, such as nonlinearity, limited bandwidth, and noise. To mitigate the generation of harmonics, a digital predistorter can be cascaded prior to feeding the signal into the DAC for preprocessing. In this regard, traditional memory polynomials [9] can be expressed as Eq. (7):

where q represents the memory term, k denotes the nonlinear order, and c_kq signifies the model coefficients.

Notably, in Eq. (7), the k-th order higher-order term x^k(n) of the input signal encompasses not only the k-th harmonic components and their associated sum and difference frequency components but also the lower-order harmonics located in proximity and their corresponding sum and difference frequency components. As illustrated in Fig. 4, when the input signal x(n) is a two-tone signal, x⁶(n) generates the fourth harmonic, second harmonic, and related spectral components. Similarly, x⁴(n) also produces second harmonic components. However, if we intend to modify only the second harmonic component, it becomes necessary to adjust x²(n), x⁴(n), and x⁶(n). Furthermore, altering x⁴(n) and x⁶(n) would inevitably impact both the fourth harmonic and the sixth harmonic, which would compel us to deviate from our intended objective. Hence, one of the primary focuses of this paper is to modify x⁶(n) while minimizing any influence on lower-order harmonics.

First, we applied the Hilbert transform [10] to the real-valued signal x(n), as shown in Eqs. (8) and (9):

where * denotes convolution and j represents the imaginary unit. We transformed x(n) into a complex-valued signal z(n), containing pure k-th harmonics along with the surrounding parasitic components. Fig. 5 illustrates the spectra of z(n), z²(n), z³(n) and z⁴(n) for a two-tone signal. It is observed that modifying z⁴(n) does not affect the fourth or second harmonic compared to x⁴(n). However, the lower-order components are missing in z⁴(n), which requires compensation.

We exploited the spectral displacement of z^k(n) to align it with the k-th power term of the lower-order components of x(n). In digital communication systems, the center frequency f_c and bandwidth information of a digital signal sent to a DAC are often known a priori. Therefore, by applying complex conjugation to the center frequency, we obtained −f^c in the negative frequency domain, resulting in a digitally sampled −f^c(n). Following this, we calculated the spectrum shift using Eq. (10):

where m represents the number of shifts.

By multiplying z^k(n) with f_c,m(n), Eq. (11) demonstrates the completion of the frequency spectrum shift for z^k(n). Following this shift, we obtained the low-order harmonic components of x^k(n), along with the sum and difference frequency components surrounding these harmonics. Notably, to differentiate it from z^k(n), we denote the result obtained after the frequency spectrum shift as z^k_l(n). Fig. 6 presents the simulation results obtained using z⁶(n) of a two-tone signal as an example.

Using Eq. (9), we calculated z^k(n) and then multiplied it with Eq. (10) to obtain z^k_l(n), as depicted in Eq. (11). Subsequently, by optimizing x^k(n) in Eq. (7), we formulated Eq. (12). In this context, it is important to note that the optimization process for the memory term follows a similar approach as that for the current term. However, elaborating on this further is beyond the scope of this study. Notably, Eq. (12) reveals that x^k(n – q) exhibits a higher degree of parameterization:

By substituting Eq. (12) into Eq. (7) and applying the Hilbert transform to y(n), we obtained Eq. (13), which represents the HTBMP model. The flowchart for this model is illustrated in Fig. 7.

The original signal was used as the input for the predistorter, while the PSO-PID algorithm was employed to obtain the optimal input signal x_d(n) for achieving a harmonic-free DAC output. Once sufficient accuracy was achieved, the optimized signal was used as the output of the predistorter. This can be expressed as Eq. (14), where a⃗ represents the parameter vector of the HTBMP.

Where

Subsequently, the parameter vector a⃗ of the HTBMP model was extracted using the least squares method proposed in [11], formulated as Eq. (16):

We chose to conduct experimental verification in the shortwave frequency band due to its broad frequency range, spanning 1.6 MHz to 30 MHz, and the current high sampling rate of DACs that enables the efficient identification of harmonics within this range. Enhancing the SFDR of DACs in shortwave communication systems can effectively mitigate system nonlinear distortion and improve system stability, ultimately enhancing the reliability of shortwave communication. Furthermore, given that shortwave communication necessitates coverage across multiple frequency bands, increasing the SFDR of DACs will not only alleviate the pressure on analog filter groups but will also ensure higher clarity and integrity during signal transmission, in turn extending the reach of shortwave communication. Improving the SDFR of DACs may enhance interference resistance in complex electromagnetic environments while maintaining stable communication. Therefore, advancements in DAC’s SFDR have the potential to optimize spectrum resource utilization and improve the communication efficiency of shortwave communication systems.

In this study, we employed a methodology involving the measurement and averaging of multiple sine wave signals within an extended dynamic range to assess the SFDR of DACs. Experimental verification was conducted using single-tone sine wave signals. By capturing real-time data of the DAC input and output, we simulated the PSO-PID algorithm, as depicted in Fig. 8. Subsequently, a computer-generated predistorted signal based on the PSO-PID algorithm was generated and fed into an arbitrary function generator. The digital signal from this generator was then converted into an analog signal, which underwent further digitization by a digital oscilloscope for spectrum analysis. The resulting data were iteratively processed using the PSO-PID algorithm. In this context, it should be noted that during the data acquisition process, the signal coupled from the DAC output had to be adjusted to a suitable magnitude before being fed into the analog-to-digital converter (ADC) so as to let the ADC work in an ultra-linear region. Hence, the distortions caused by ADC can be ignored. The distortions included in the captured data were mainly caused by the DAC. In addition, we used a spectrum analyzer with a dynamic range of 116 dB to verify the effect of the PSO-PID HTBMP.

The actual fabricated DAC data acquisition platform is presented in Fig. 9. Initially, a sine signal was generated using MATLAB and then imported into Keysight M8190A arbitrary waveform generator. The clock of the arbitrary waveform generator was configured to internal mode, with the output level set to 600 mV. For collecting the DAC output data, Keysight M9703B digitizer and Keysight 89600VSA software were employed. To ensure synchronization between the arbitrary waveform generator and the digitizer in terms of the clock source, the digitizer’s clock was set to synchronize with the output clock of the arbitrary waveform generator. Similarly, to synchronize their sampling times, the arbitrary waveform generator was set as the trigger source for the digitizer.

After acquiring the DAC input and output data, a negative feedback iteration using PSO-PID was performed. The PID parameters were constrained within the range of −1 to 1, with the particle swarm comprising 30 particles. Both the acceleration factors, c₁ and c₂, were set to 0.8, while the maximum number of iterations for the particle swarm was limited to 60 times. Fig. 10 illustrates the simulation results of the PSO-PID parameters and NMSE variation with respect to the number of iterations. The graph shows that the PID parameters tended to stabilize and exhibit excellent convergence performance in terms of NMSE after 40 iterations. By employing the PSO-PID algorithm, we obtained three optimal parameters for the PID, as well as the harmonic distortion compensator’s output signal x_d(n). Finally, by utilizing both the original signal x(n) and signal x_d(n) as the input and output, respectively, we calculated the parameters of the predistorter model in the HTBMP model.

After extracting the harmonic digital predistortion (DPD) parameters for the DAC, the original signal was fed into the DAC equipped with a predistorter, and the resulting output signal was collected for harmonic spectrum analysis. Fig. 11 illustrates the suppression effect on harmonics in the case of a typical single-tone sinusoidal signal, demonstrating that the digital predistorter effectively mitigated the harmonic components originating from the DAC.

The effects of DPD on single-tone sine wave signals at frequencies of 2 MHz, 6 MHz, 12 MHz, 18 MHz, and 24 MHz are recorded in Table 1. The findings presented in Table 1 point to a minimum improvement of 9 dB in the second harmonic, indicating an increase in the DAC’s SFDR by at least 9 dB. Notably, improvement in the third harmonic was not obvious and was far less than in the second harmonic.

Fig. 12 traces the improvements in the DAC’s harmonics corresponding to different frequencies, as well as the improvements in its SFDR. With an increase in frequency, both harmonics and SFDR fluctuate to a certain extent. It is evident that the PSO-PID HTBMP DPD effectively suppressed the DAC’s harmonics and improved its SFDR at different frequencies.