Enhanced Meta-Heuristic Approach for Echo Suppression in Synthetic Aperture Radar Using Non-Periodic Interrupted Sampling

Due to the increase in imaging technologies, the SAR plays an essential role in both military aspects and civil. SAR jamming includes a variety of techniques that have been developed in order to prevent the tracking of Target of Echo (TOI) by the SAR [1]. Blanket jamming and deception jamming are examples of SAR jamming techniques. Further, SAR jamming is often divided into two categories such as barrage jamming and misleading jamming [2]. The processed SAR image has been greatly impacted by barrage jamming, especially in the TOI region. A distorted image of the TOI is formed with the aid of deceptive jamming, making it difficult to figure out the TOI [3].

According to theory of echo cancellation, by finding the cancellation echo—a synthetic replica of the echo signal itself unwanted echoes may be eliminated. Suppressing the target echo at the radar receiver using echo cancellation is a good technique that protects the TOI [4]. The basic premise of echo cancellation is the generation of echo cancellation signal i.e., the copy of the target echo by an active source to the radar so that both target echo and cancellation echo reach the radar. Numerous studies regarding the use of echo cancellation have been conducted during the last few decades [5]. Over a significant number of studies, it has been taken into account as essential for synchronization in the amplitude and time is required [6]. From the previous research works on echo cancellation, The SAR has the ability to pass signals with a wide temporal bandwidth, such as LFM pulses. Thus, at the SAR receiver, a canceled signal may arrive after the target echo. As they used amplitude and time-delay modulation in the cancellation technique [7]. Since there is an amplitude mismatch, the target echo is able to be totally deleted from the SAR images and stays apparent in the SAR image [8].

In order to mitigate the issues in the SAR, a canceled echo can actually stay behind the target one [9]. This happens when SAR receives and transmits large “Time Bandwidth Product (TBP)” signals like LFM pulses, and the production phases of echo cancellation need adequate processing time. Since delayed echo signal and original echo have distinct frequencies, and this situation will lead to cancellation failure [10]. ISRJ is commonly employed to disable radar fake behavior, and it includes Inverse SAR (ISAR) and SAR [11]. Most recently, the ISRJ approach has been employed to perform delayed cancellation echo; nevertheless, this technique fails quickly since several fake targets arise while the true target echo is canceled [12].

In order to maintain good resolution and to get suppressed side lobes of the signal, one can use an improved Non-Linear Frequency Modulation (NLFM) polynomial signal. To have a low side lobe and high Peak to Average Ratio (PAR) a Polynomial NLFM signal is used [13]. This research presents a novel idea for echo cancellation in SAR using heuristic algorithms.

The most important contributions of the heuristic aided-echo cancellation model are enumerated as follows:

• To flourish the heuristic-assisted echo cancelation model in SAR to cancel the target echo at the receiver of the SAR so the chance of detection of TOI is prevented.
• To augment the performance of the heuristic assisted-echo cancelation model, an IHGSOA is offered. It is employed to optimize the rectangular envelope pulse train in NP-ISM so the PAR value is lowered.
• The results generated by the IHGSOA-based echo cancellation model are examined with the traditional echo cancellation system to determine the overall capability of the recommended echo cancellation model.
• A Polynomial Non-Linear Frequency Modulation (PNLFM) is used to reduce the peak side lobes and to increase the SNR and PAR, so that the target echo is canceled effectively.

The balance parts of the IHGSOA-based echo cancellation model are prearranged as follows. Segment II gives the prior echo cancellation models and their uses and limitations. In segment III, the basic signal model of echo cancellation based on incidence and time-delay modulation and the impacts caused by amplitude mismatch for echo cancellation is elaborately given. The NP-ISM pulse characteristics, modulated Polynomial NLFM signal characteristics and procedure of the developed model are categorized in segment IV. In segment V, conventional HGSOA, implemented IHGSOA, IHGSOA-based optimal NP-ISM is presented in a brief way. In segment VI, the results and conversation of the explored echo cancellation model in SAR are given. The final segment, VII gives the conclusion of the recommended echo cancellation model.

3.1 Basic signal model of echo cancellation based on frequency and time-delay modulation

The SAR consists of a target and a canceller. The trajectory of the SAR is identified from the dashed line. The Line of Sight (LOS) is placed in between the SAR to the target, and here canceller is placed on this LOS of the SAR geometry. The target is pointed by the receiver antenna, and the SAR is pointed by the transmitter antenna, and both of these antennas are positioned in the canceller. In SAR, the echo cancellation is completed by three steps. The goal echo is intercepted by the canceller in the initial step of this procedure. Afterward, the attributes of the signal are examined to develop the cancellation signal. To this end, the antennas are employed to determine the transmitting power, and this transmitting power is retransmitted to the SAR for the echo cancellation [20].

The LFM signal transferred to the SAR without any loss of simplification is elucidated in Eq. (1):

$s(t)=v(t) \exp \left(j 2 \pi g_0 t+j \pi l t^2\right)$     (1)

where,

$v(t)=\operatorname{rect} \frac{t}{U}$   （2）

Here, the width of the pulse is indicated as $U$, the chip rate is illustrated as $l$, the fast time is signified as $t$, and the middle of frequency is elucidated as $g_0$. The Eq. (3) shows expression for the baseband echo.

$\begin{aligned} s_e\left(t, t_n\right)= & v\left(t-\frac{2 R\left(t_n\right)}{c}\right) \exp \left(j 2 \pi g_0 \frac{2 R\left(t_n\right)}{c}\right) \\ & \cdot \exp \left\{j \pi l\left(t-\frac{2 R\left(t_n\right)}{c}\right)^2\right\}\end{aligned}$     (3)

Here, the distance in the middle of the radar and the is represented as $R\left(t_n\right)$, slow time is indicated as $t_n$, an propagation speed of the EM is indicated as $c$ [18]. symbolic form of outcome is indicated in Eq. (4):

$\begin{aligned} I_e\left(t, t_n\right)= & \sin c\left(\frac{t-\frac{2 R_{\min }}{c}}{\varpi_r}\right) \sin c\left(\frac{t_n-t_c}{\varpi_a}\right) \\ & \cdot \exp \left(-j \frac{4 \pi g_0 R_{\min }}{c}\right)\end{aligned}$     (4)

Here, when the target and the SAR are very close to each other, then the slow time is represented as $t_c$, the smallest distance in the middle of the target and the radar is signified as, $R_{\min }$ the resolutions of azimuth and the resolutions of the range is indicated as $\varpi_a$, and $\varpi_r$ respectively.

$\varpi_r=\frac{1}{C_r}$  （5）

$\varpi_a=\frac{1}{c_a}$  (6)

Here, the bandwidth of the signal is represented as $C_r$ and the bandwidth of the azimuth is signified as $C_a$.

In the receiver of the SAR, the target is canceled if the amplitude of the indication to be canceled and the amplitude of the aim echo must be in the same condition, and it is mathematically given in Eq. (7):

$s_c\left(t, t_n\right)=-s_e\left(t, t_n\right)$    (7)

Because of the unavoidable time delay $\tau_d$, the signal to be canceled is lag on the target echo, and it is elucidated in Eq. (8):

$\hat{s}_c\left(t, t_n\right)=-s_e\left(t-\tau_d, t_n\right)$ （8）

Here, the assumption is $\tau_d \neq 0$. It is not able to synchronize the signal to cancel and the target echo signal at fast time variable. If the synchronization process is not done for the cancelation signal and the target echo at the fast time variable then it causes failures in the echo cancellation process [21, 22].

At time delay $\tau_d$, the intercepted signal is accompanied by the frequency shift for canceling the baseband received signal, and it is illustrated in Eq. (9):

$\begin{gathered}s_s\left(t, t_n\right)=v\left(t-\frac{2 R\left(t_n\right)}{c}-\tau_d\right) \\ \cdot \exp \left(-j 2 \pi g_0 \frac{2 R\left(t_n\right)}{c}\right) \\ \cdot \exp \left\{j \pi l\left(t-\frac{2 R\left(t_n\right)}{c}-\tau_d\right)^2\right\} \\ \cdot \exp \left\{j \pi g_d\left(t-\frac{2 R\left(t_n\right)}{c}-\tau_d\right)\right\} \cdot \exp \left(-j 2 \pi g_0 \tau_d\right)\end{gathered}$  （9）

The results produced in pictorial form are given in Eq. (10).

$\begin{aligned} & I_c\left(t, t_n\right)=I_e\left(u-\tau_d+\frac{g_d}{l}, t_n\right) \\ & \quad \exp \left\{-j \pi\left(\frac{g_d^2}{l}+2 g_0 \tau_d\right)\right\}\end{aligned}$     (10)

The time delay $\tau_d$ and the frequency shift $g_d$ must be fulfilled if the $I\left(t, t_n\right)=0$. The time delay $\tau_d$ and the frequency shift $g_d$ are identified by Eqs. (11) and (12).

$\tau_d=-\frac{g_0}{l}+\sqrt{\left(\frac{g_0}{l}\right)^2+\frac{(2 i+1)}{l}}$   (11)

$g_d=-g_0+\sqrt{g_0^2+(2 i+1) l}$    (12)

For particular time domain, the pictorial outcome of the combined signal is synchronized. This process is done in the reverse direction; the integer is represented as i. The geometric view of the SAR is illustrated in Figure 1. 

1.png

Figure 1. Geometric view of the SAR [2]

3.2 Modulated NLFM and polynomial NLFM signal characteristics

Modulated NLFM and Polynomial NLFM signals are designed to enhance range resolution and suppress sidelobes in radar systems. Polynomial NLFM offers better control over frequency modulation, improving signal-to-noise ratio (SNR) and target detection.

3.2.1 NLFM

Utilizing the NLFM waveform may result in excellent SNR, low noise interference, and high range resolution [23]. When compared to a linear frequency modulated wave, NLFM offers a greater detection range. The NLFM signal is represented as:

$S(t)=\exp [j \emptyset(t)]$   (13)

Here, $\emptyset(t)$ is a frequency modulation function, and $f(t)$ may be obtained via differentiation phase of Eq. (1). No weighting mechanism is needed to reduce side lobes in the NLFM waveform. Improved NLFM refers to an NLFM signal that is intended for varied time sweeps. The bandwidth and overall time sweep are divided into two and three stages, respectively, via improved NLFM. The side lobe level at first peak of 17.62 dB is achieved after autocorrelation of the tri-stage NLFM signal, which is lower than the two-stage NLFM. To further lower PSL, an Improved Polynomial-I \& II NLFM is devised for different BT values. It can be seen that the lowered PSL for conventional approaches is insufficient for many applications in field of radar and sonar. In addition to Tri-stage NLFM, developed Polynomial-I NLFM and Polynomial-II NLFM signals show a drop in the first side lobe level [24].

3.2.2 Polynomial-I and polynomial-II NLFM

Two types of PNLFM waveforms Polynomial-I and Polynomial-II are considered with Uniform and Non-Uniform PRI [24, 25].

$S p(t)=\exp [j \emptyset(t)]$   (14)

where, $\emptyset(t)=2 \pi \int f(t)$.

olynomial-I NLFM:

$f p 1(t)=a_1 t^6-a_2 t^4+a_3 t^2$   (15)

Polynomial-II NLFM:

$f p 2(t)=a_1 t^2-a_2 \sqrt{1-a_3 t^2}$   (16)

where, $a_1, a_2, a_3$ are constants and $t$ are the instantaneous time.

The IHGSOA is an advanced meta-heuristic optimization technique inspired by the solubility of gases in liquids. It enhances the original Henry Gas Solubility Optimization (HGSO) algorithm by improving exploration and exploitation capabilities, allowing it to find optimal solutions more efficiently. IHGSO is particularly effective for solving complex, nonlinear problems such as signal processing in radar systems, where it optimizes parameters for tasks like echo cancellation, reducing PAR, and improving overall system performance. Its flexibility and robust search mechanisms make it well-suited for optimizing multi-dimensional and dynamic environments.

Here is a structured outline of the IHGSOA (IHGSO with key steps and mathematical formulations):

1. Initialization

• Population Initialization: Define a population of gas molecules (solutions) X_i for (i = 1, 2,…., N), where N is the population size. Each solution represents a candidate in the search space.

$X_i=\left\{x_{i 1}, x_{i 2}, x_{i 3} \ldots \ldots x_{i d}\right\}$   (22)

where, $x_{i d}$ represents the position of the $i_{t h}$ solution in the ddimensional search space.

• Objective Function: Define an objective function f(X_i ) to evaluate the quality of each solution.

2. Henry’s Law for Gas Solubility

• Henry’s Law Constant Calculation: Calculate Henry’s law constant H_c for each molecule, which determines the gas solubility and helps balance exploration and exploitation.

$H_c=H_{\min }+\left(H_{\max }-H_{\min }\right) \times \operatorname{rand}(0,1)$    (23)

where, $H_{\min }$ and $H_{\max }$ are predefined lower and upper bounds of the Henry constant, $\operatorname{rand}(0,1)$ is a random value between 0 and 1.

3. Solubility Calculation

• Solubility of Gas Molecules: Calculate the solubility S_i of each molecule based on its current position and Henry’s constant.

$S_i=H_c \times e^{-f\left(X_i\right)}$  (24)

where, $f\left(X_i\right)$ is the objective function value for solution $X_i$, and $S_i$ is the solubility representing the likelihood of exploring that region of the search space.

4. Position Update

Molecule Movement: Update the position of each molecule based on solubility. The new position is influenced by the best solution found so far, X_best and random perturbations for exploration.

$\begin{aligned} X_i^{\text {new }}=X_i^{\text {old }}+ & \operatorname{rand}(0,1) \times\left(X_{\text {best }}-X_i^{\text {old }}\right) \\ & +\alpha \times \operatorname{rand}(-1,1)\end{aligned}$   (25)

where $\left(X_i^{\text {new }}\right)$ is the updated position, $\alpha$ is a step size control parameter, and $\operatorname{rand}(-1,1)$ is a random value between -1 and 1 to introduce randomness.

5. Exploration vs. Exploitation Balance

Adaptive Parameter Tuning: Use dynamic parameter control to balance exploration (searching new regions) and exploitation (intensifying the search around the best solutions). The adjustment of Henry’s constant and solubility adapts based on the algorithm's progress.

6. Fitness Evaluation

Update Fitness: Evaluate the new solutions using the objective function $X_i^{\text {new }}$. If the new solution is better than the current best solution, update $X_{\text {best }}$.

7. Convergence Criteria

Stopping Condition: Repeat steps 3 to 6 until the stopping criterion is met (e.g., maximum number of iterations or a sufficiently small change in the best solution).

If $\left|f\left(X_{\text {best }}\right)-f\left(X_{\text {prev }}\right)\right|<\epsilon$ or Maximum iterations reached, stop.

8. Return the Optimal Solution

Final Solution: Once the stopping condition is met, return the best solution $X_{{best }}$ found by the algorithm.

This framework describes how IHGSO refines gas molecule positions in search space, progressively finding the optimal solution.

The IHGSOA is utilized in the echo cancellation model to optimize the Random rectangular envelope pulse train in the NP-ISM. So, the value of PAR is greatly lowered so the target echo can be effectively canceled. Moreover, the echo cancellation process is completed within a limited amount of time because of this optimization process. The HGSOA is a physics-inspired algorithm and it can easily defeat complex issues. Additionally, the local optimal issues are easily avoided by the HGSOA. The most important advantage of the HGSOA is it can easily handle the balance in the middle of the examination and the management phase. Yet, it gives only a single solution for the optimization problem. Additionally, it may be stuck in the local optimal condition. To deal with these issues, an IHGSOA is presented. Here, the best and the mean fitness are employed to calculate the new random number and it is specified in Eq. (26):

rand $=\frac{X_{\text {best }}}{100 * X_{\text {mean }}}$  （26）

3.png

Figure 3. Illustration of developed IHGSOA with flow chart

Here, the new random number calculated by the explored IHGSOA is characterized as and the best and the mean fitness is represented as, and respectively. Because of the explored PAR value is greatly lowered.  The process flow depiction of the IHGSOA is prescribed in Figure 3.

5.1 IHGSOA-based optimal NP-ISM

The IHGSOA-based optimal non-periodic interrupted sampling intonation process is carried out to cancel the target echo in the SAR. Here, the echo cancellation process is done in three steps. At first, the inescapable time delay that has occurred on the intercepted echo is avoided. In the second step, at the longest time filed, the cancellation signal is combined with the non-periodic interrupted sampling. Here, the echo present in the signal was canceled via the “NP-ISM” and the random rectangular envelope pulse train in the “NP-ISM” was optimized by the explored IHGSOA. At last, the loss of energy when transferring the signal to the SAR is neglected to achieve an accurate signal. The controller is attained by analyzing the intercepted target signal. Further, the attributes from the controller are used to modulate the termination signal.

The objective function of the IHGSOA-based optimal NP-ISM is used to lower the value of the PAR in the signal, and it is accustomed in Eq. (27):

$F O=\arg \min _{\{r r e p t\}}(P A R)$    (27)

Here, the objective function is delineated by the term FO, and the random rectangular envelope pulse train is signified as rrept, and it lies in the region of [1 to 512], the PAR is estimated by Eq. (28):

$P A R=\frac{P P}{A P}$    (28)

Here, the term PP gives the peak power, and the term AP gives the average power. The solution diagram of the IHGSOA-based optimal NP-ISM is indicated in Figure 4.

4.png

Figure 4. Solution diagram of the IHGSOA-based optimal NP-ISM

The statistical measures used to evaluate the performance of a heuristic-based echo cancellation model are:

i. Best (Maximum Performance)

The best performance observed among all the data points of the fitness function. This is the highest value in your dataset. The value represents the optimal performance achieved under the best conditions or settings.

$Best =\max \left(X_1, X_2, X_3 \ldots \ldots, X_n\right)$  (29)

ii. Worst (Minimum Performance)

The worst performance observed among all the data points of fitness function. This is the lowest value in your dataset. This value reflects the least favourable performance observed.

$Worst =\min \left(X_1, X_2, X_3 \ldots \ldots, X_n\right)$    （30）

iii. Mean

The sum of all performance values divided by the number of data points of the fitness function. This provides an overall average measure of performance. The mean gives a central value around which the performance data is distributed.

$Mean =\frac{1}{n} \sum_{i=1}^n X_i$   (31)

iv. Median

The middle value when the data points are arranged in ascending order. If there’s an even number of data points, it’s the average of the two middle values. The median is a measure of central tendency that is less sensitive to extreme values than the mean.

v. Standard Deviation (SD)

A measure of the amount of variation or dispersion in the performance data of the fitness function. It shows how much the performance values deviate from the mean. A low standard deviation indicates that the performance values are close to the mean, while a high standard deviation indicates greater variability.

$S D=\sqrt{\frac{1}{n} \sum_{i=1}^n\left(X_i-M e a n\right)^2}$   （32）