7.3.1 MIMO-Based Array Antenna

An eight-element MIMO-based antenna including four transmit and four receive antennas is shown in Figure 7.2. Each of transmit and receive antennas can be viewed as a sparse array. The normalized separation distance between Rx and Tx antennas are 6 and 9, respectively, as shown in Figure 7.2. Dimension is not important, but centimeter can be assumed as example. Therefore, the sparse array antennas have constant separation of 3 cm. This means that the Rx array is equivalent to a sparse array of seven elements with a separation distance of 3 cm and the active and passive elements distribution of

The same assumption is valid for the Tx array with the following topology for the active and passive elements and again with a separation distance of 3 cm:

It is envisioned to show the transition between the MIMO theory and its equivalent virtual array antenna without being deeply involved with the mathematical calculations [11]. In Figure 7.2, each Tx antenna can independently transmit its signal and all the Rx receive antennas collect the backscattered signal. Based on the array theory, it can be shown that the array factor of each array antenna is equivalent to a polynomial of N − 1 order on which N is the number of array elements [12, 13]. Therefore, the related polynomials of receive and transmit arrays are as follows, respectively:

7.1

It can be mathematically proven that the two sets of Tx and Rx antennas on MIMO-based configuration is equivalent to a virtual array antenna. This means that instead of analyzing the MIMO-based antenna, one may consider the equivalent virtual array and expect almost the same radiation characteristics as the MIMO-based antenna. The details of expressions for relation between the MIMO-based system and the virtual array are not discussed here. It has been shown that the convolution of the Tx and Rx arrays defines their equivalent virtual array antennas and subsequently specifies the position of the active elements in the virtual array configuration [6, 11]. Therefore, two sets of the Tx and Rx antennas that have been considered as sparse arrays are mathematically linked to their virtual array antenna. Based on this assumption, the convolution of the Tx and Rx antennas in the time domain is equivalent to multiplication in the frequency domain. Hence, one may simply find the equivalent virtual array's polynomial through the following relation [6]:

7.2

where P_v is the polynomial of the virtual array. This polynomial that is in the order of 15 relates to a 16-element linear array with the following active/passive topology:

In the above configuration, “0” means that position has no antenna and “1” represents an active antenna. The bold “1” show duplicate element position. With this procedure, the MIMO-based structure of 4 + 4 physical elements is related to a virtual array of 4 × 4 elements. The actual MIMO based system and its equivalent virtual array antenna are presented simultaneously in Figure 7.3, in which the physical MIMO-based system and its equivalent virtual array antenna are presented simultaneously.

To have a more physical sense of evolution from MIMO-based system to the array antenna concept, one can track the signal route when it is transmitted by a specific transmit antenna (T_i) and received by all receive antennas. Let consider d_r and d_t as the separation distance between the Rx and Tx antennas, respectively. The relative position of each Tx antenna to the center point of the transmit array antenna is denoted by d_tn and the relative position of each Rx antenna to the center point of receive array is shown by d_rm, where m, n = [1, 4].

When one of the transmit antennas is active, then the received signals to all receive antennas can be weighted by

7.3

This clearly shows that the relative positions of each set of transmit and receive antennas and the relative phase shift in its equivalent virtual array. To establish a unique phase center for all communication links between the transmit and receive antennas, one can shift the position of transmitters to a fixed point, the center of transmit array, for instance, and then find the phase shift occurring in all receive array elements. This is shown for the two cases of T₁ and T₂ in Figure 7.4. While T₁ antenna is d_t1 shifted to the right side, all its related Rx antennas are equivalently shifted to the left side for the same distance. The same happens for case T₂. However, the distance of movement is now d_t2. If the shifted Rx antennas for these two cases are mapped to each other, they fill the gaps among physical elements. This is shown in Figure 7.4 by the dashed arrows and circles among active antennas. If the same procedure continues for T₃ and T₄, then the virtual array element is completed and would be similar to the declared virtual array polynomial and the position of the antenna in the virtual array as follows.

7.3.2 MIMO-SAR Array for Chipless RFID Imaging

To design the MIMO-based phased array antenna for the proposed chipless RFID system, the process described in Ref. [14] is followed. In the proposed configuration, every two Rx antennas are linked with five Tx antennas. Then the structure is cascaded to provide the larger required equivalent array antenna. Figure 7.5 shows the structure of this MIMO-based antenna configuration. It is important to find the relation between the separation distance of Rx and Tx antennas and their equivalent virtual array. The angles θ₁ and θ₂ can be linked to other parameters as follows:

7.4

where the parameters of (7.4) are defined in Figure 7.5. To find the position of the points a and b, one can write the following expressions:

7.5

Therefore, the separation distance of Rx antennas can be easily linked to the separation among the virtual array elements, ΔX_v, through

7.6

If the same approach is followed for θ₃ and the distance between points b and c is found, one can easily reach the following relation:

7.7

We have already discussed that the reader antenna has to illuminate the tag in every 2 mm. Therefore, the inner space of virtual array element shall be 2 mm and equivalently, the physical Rx antennas have to be 4 mm and Tx antennas 8 mm apart from each other.

The total number of sampling points also governs the total number of required physical antennas. We have already discussed that “m + n” elements in Rx and Tx configuration may provide a virtual array of “m × n” elements. It is important to mention that this is a theoretical criterion and shows the minimum number of required actual Tx and Rx antennas. For example, 8 and 13 physical antennas are utilized in Ref. [14] for a virtual array with only 44 elements while based on the theoretical relation, less number of antennas are expected for the virtual array with 44 elements. For the case of RFID application, it was shown earlier that the reader shall stop in 100–150 individual points and illuminates the tag and capture the backscattered signal. Thus, the average number of virtual array elements can be assumed as 125 points; however, it is possible to have slightly higher or even less number of array elements. Therefore, 125 points can be seen as a rough estimate. The proposed MIMO-based system topology for the Rx antenna in Ref. [14] includes two active antennas and then four passive elements in the sparse array configuration, 11000011 array polynomial. The Tx array has one active and one passive element scheme, 101010, for instance. Based on this assumption, the antenna distribution scheme is as follows:

1. For Tx array antenna: 11000011000011000011
2. For Rx array antenna: 1010101010101010101010101

Based on this array distribution, the following polynomials can be related to the Rx and Tx antennas, respectively, for the 44 elements requirements:

7.8

Obviously, the convolution of the above polynomials results in a virtual array of degree 43. If the same topology is followed for the case of 125 array elements, one may suggest the following polynomial for the Rx array antennas including 24 active antennas:

7.9

Based on the above assumption for the Rx configuration, the related Tx antennas can be defined through the following polynomials with 31 physical Tx antennas:

7.10

The convolution of (7.9) and (7.10) results in a polynomial of degree 127, which includes all the required elements.

7.11

In Equation (7.11), coefficients a_i may be different from “1.” In this case, it is not required to consider all relevant combinations between an individual Tx and Rx antennas. Therefore, through (31 + 24) physical antenna, it is possible to implement a virtual array with 127 elements. Through this configuration, the total number of array elements drops from 127 × 2 = 254 to only 31 + 24 = 55 elements. Moreover, the reading time in the proposed technique would significantly enhance while no complex system structure is utilized. The conventional SAR technique through physical antennas and its equivalent MIMO-SAR are shown in Figure 7.6. In the conventional SAR, a pair of Tx and Rx shall move around the tag and stop in 127 positions and perform the reading process. Equivalently, 127 pair of antennas can be used for fast reading process. However, in the MIMO-SAR technique, only 55 physical elements are utilized with no requirement for physical movement of antennas.

The summary of the MIMO-based system can be compared with a physical array and the conventional SAR technique as mentioned in Table 7.1. The conventional SAR technique has the minimum system complexity due to the usage of only one antenna pair. However, it provides a very slow imaging process in conventional approach or fairly slow imaging speed if a stepper motor is utilized. It is possible to develop an array antenna for fast imaging process while it requires a large number of elements, which results in significant system cost. Instead, one may use the MIMO-SAR system with much lower number of elements and the requirement for a switching network.

7.3.3 Simulation Result

Figure 7.7 shows the proposed structure of the MIMO-based system. 24 Rx antennas with 4 mm separation distance are linked with 31 Tx antennas with a separation distance of 8 mm. Hence, the total aperture size is less than 25 cm (31 × 0.8 = 24.8 cm) that is completely matched with the aperture sizes introduced before. The virtual array antenna with 127 elements is located between the Rx and Tx antennas. No phase shifter or amplitude tapering network is required. Only a switching network is utilized to connect one Tx antenna with its associated Rx antennas at a time.

To test the validity of the proposed technique of a MIMO-based reader, a 6-bit tag is selected as example. The tag's image is simulated through a MIMO-SAR system and then compared with the image through normal SAR technique. The result of signal processing is shown in Figure 7.8. In the SAR technique that includes 125 individual send and receive process, the EM image adequately shows the encoded data. If the same tag is processed through a MIMO-SAR system that utilizes 55 physical antennas, then the image is depicted in Figure 7.8(c). The EM image seems to be somehow noisier than that for the conventional SAR. Moreover, the imaged column of meander lines shows a minor inclined nature so slightly degrades the azimuth resolution. One reason may be suggested for the degraded image of the tag through MIMO-SAR technique. The equivalence between the virtual antenna and its associated MIMO system is an approximated relation. Each set of Tx and Rx antennas is associated with one single element in the virtual array. However, based on the physical distance of Tx and Rx antennas, a certain amount of error due to the phase center is expected. This error is reflected in the nonperfect image of the tag through MIMO-SAR technique. However, much faster imaging time of MIMO-SAR system compensates the noisier image of the tag. Moreover, it is possible to use higher number of physical antennas on the MIMO-based system and improve the image quality. Alternatively, by optimization of the antennas distribution on the MIMO-basis system, one may expect a higher quality of the imaged tag. The optimization process of the antennas distribution in the MIMO-SAR system is introduced in the following section.

7.4.1 Analytical Approach

First, it is suggested to represent the problem analytically and explore if there is any mathematical approach to find the minimum number of physical antennas in the MIMO-SAR configuration. The aforementioned situation for the MIMO-SAR can be analytically shown as

7.12

In the above set of expressions, the product of P_r × P_t shall produce a complete polynomial of order “n.” The number of active elements in the Rx and Tx series is shown by “k” and “t,” respectively. There are some other limitations that shall be considered in the optimization process. First, as it is mentioned before, the relation between any Tx and Rx antennas with their equivalent virtual element is an approximate relation. This relation is more accurate if the distance between the Tx and Rx antennas is negligible with respect to the distance of the target in front of the antennas. This implies an additional limitation in the above sets of formula in Equation (7.12). Every Rx antenna shall be connected to its neighboring Tx elements and not to the Tx antennas that are physically located far from the intended Rx antenna. Otherwise, the results of analytical optimization may be mathematically accurate but results in a system with inaccurate performance. Second, the goal is to minimize k + t, while it is also desirable to have the minimum physical space occupied by the MIMO-based system. The other parameters of the system shall also be considered in the above sets of formula. Considering the limitations set out for the optimization problem, any analytical approach would be very complex if any existed. The authors are not aware of any analytical approach for solving the above-described problem.

7.4.2 Numerical Optimization Process

The second approach is the numerical optimization process. In the analytical approach, the suggestion is to consider the distribution of the Tx and Rx antennas as a uniform sparse array antenna for simplicity. In the numerical approaches, however, such a suggestion is not mandatory. This means that nonuniform sparse array structures can also be considered, which may result in a lower number of antennas or smaller array size.

Before selection of any specific optimization technique, it is useful to explore the dimension of the problem. This helps selecting the best technique with least complexity. The maximum length of the demanded virtual array for the proposed application, as discussed before, is 30 cm. It is necessary to find out what is the length of Tx and Rx arrays in the MIMO system. Referring to Figure 7.3 and by following the approach of shifting the Rx array for the relative position of each Tx antenna, as described in Section 7.3.1, one may easily prove

7.13

where R_L, T_L, and V_L are the physical length of Tx, Rx, and virtual array, respectively. To find out R_L and T_L, another equation is required. This means that the relative length of Tx and Rx arrays should be known. Although no specific relation can be defined for the length of Tx and Rx arrays, the ratio of R_L to T_L that has been suggested by Lincoln Laboratory [11] may provide a practical value:

7.14

Considering Equations (7.13) and (7.14), one may easily find the total length of Rx and Tx as

The maximum length of the virtual array is 30 cm and it was already shown that the reading process shall be occurred in each 2 mm. Therefore, one may see the virtual array as a uniform array of 150 elements with a uniform distribution and an element spacing of 2 mm. Referring to Figure 7.5, the minimum element spacing of the Rx and Tx arrays shall be 2 mm or lower. For simplicity of optimization, the 2 mm is selected. Consequently, the Rx and Tx arrays on MIMO basis can be viewed as nonuniform sparse array of 60 (12 × 5) and 90 (18 × 5) elements. The total number of physical antennas on Tx and Rx is restricted by

7.15

on which k and t are the total number of real Rx and Tx antennas as already defined in Equation (7.12). A good suggestion for k and t would be 11 and 14, respectively. Therefore, the optimization problem has been simplified as follows.

The total number of 11 and 14 antennas will be distributed appropriately on sparse arrays with 60 and 90 elements, respectively, on which two arrays are connected on MIMO basis. The distribution of antennas on each array will be in such a way to result in a complete virtual array of 150 elements. Therefore, the total possible option for each case is as follows:

7.16

Therefore, the total number of possible cases would be more than 10⁴⁵. This shows very wide dimension of the answer region for the suggested problem. Obviously, it would not be possible to verify all the possible cases even with the most advanced computer systems. Instead, an efficient optimization process is required to find the solution in a reasonable time frame.

Luckily, this is a common issue nowadays in many research fields, problems with widely extended feasible region without any known classical approaches. Applying the numerical techniques seems to be the only possible method. Moreover, classical numerical methods fail to provide solution in many cases as the target function does not have specific characteristics, being a continuous function or having derivatives, for example. In such cases, global optimization approaches are usually the best option. Another benefit of global optimization approach is their ability on finding the global optimum of a function. One of the main downsides of the classical optimization techniques is their failure on finding the best possible solution for a certain problem as they can easily be entrapped in local minima. Moreover, these techniques cannot generate or even use the global information needed to find the global minimum for a function with multiple local minima. The results of classical optimization approaches normally depend on the initial suggestion or starting point.

Mentioning the drawback of the global optimization techniques, one may refer to their computationally costly process due to the slow convergence. This means that irrespective of their significant ability of directing the optimization process toward the global optimum solution, they are normally inefficient to find the final answer. One of the main reasons for their slow convergence is that they may fail to detect promising search directions especially in the vicinity of local minima due to their random constructions. Combining global optimizations with local/classical search methods is a practical remedy to overcome the drawbacks of slow convergence and random constructions of global optimization techniques, hybrid methods. The advantages of global approaches are used to direct the process to the vicinity of global optimum and a classical technique is normally added at the final stage of the search.

In the global optimization techniques, evolutionary algorithms (EAs) are an important subset that generate solutions to optimization problems using techniques inspired by natural evolution. There are many techniques in EA category in which the genetic algorithm (GA) is one of the most significant approaches with the most effective results for a wide range of problems. GA is specifically very effective if no real-time solution is required [2, 15]. GA is proposed for the current problem of MIMO-SAR technique. The rest of this section briefly introduces GA and some of its main functions. Readers are suggested to refer to special references on GA for more detailed information.

7.4.3 Genetic Algorithm

Genetic algorithm that is been inspired by natural evolution encodes a potential solution to a specific problem on a simple chromosome-like data structure and applies recombination operators to these structures so as to preserve critical information. There are some parameters and key components in GA that will be defined and introduced. The most important ones are as follows:

1. Genes: Genes are the essential blocks of the GA that carry the genetic information of an individual. Normally, genes represent the coded version of the physical parameters.
2. Chromosome (genotype): A chromosome is a certain set of genes with a known length that define a proposed solution to the problem that the GA is aimed to optimize.
3. Population: A group of chromosomes is called population.
4. Generation: GA works on a specific population at each specific time that is called generation.
5. Parents: Two or more chromosomes from the current population that have been selected to transfer their genetic characteristic to the next generation.
6. Children: Referring to a chromosome belonging to the next generation and is the result of crossover among parents in the current population.
7. Fitness: It is a number that is assigned to a chromosome and shows how well the chromosome satisfies the requirement of the problem.

There are four main operators in GA that control the flow of optimization process. First is the transformation and mapping of the real problem to the chromosome environment. The chromosome space is normally a binary-type data that shows the parameters and variables of the real problem. Then a certain number of chromosomes are selected to create the current population. After the construction of a population, parents have to be nominated for getting the chance of preserving their critical information. Normally, chromosomes with the highest fitness will have more chance of being selected as parent. However, this does not mean that the weaker individuals should have no chance for passing their genes to the next generation. Parents then do crossover and create children that belong to the next generation. There is the possibility of mutation in the children genes as well. This process will continue until one chromosome satisfies the requirement of the problem or a certain number of generation is passed. The main block diagram of the GA is shown in Figure 7.9. Readers may refer to the special references [15, 16] for better and more detailed explanations about each operator of the GA.

7.4.3.1 Chromosome and Population Structure for MIMO-SAR

As discussed before, Rx and Tx arrays are assumed as sparse arrays with minimum interelement of 2 mm and maximum length of 12 and 18 cm, respectively. Each of Rx and Tx is sparse array with 60 and 90 elements, respectively, which includes certain number of physical antennas. Therefore, each chromosome shows one possible distribution of the physical antennas on the Rx and Tx arrays. The dimension of each chromosome is selected to be a matrix of [1,150]. The first 60 elements show how the physical antennas are distributed on Rx array and the rest, elements 61–150, show the scattering of physical antennas on Tx sparse array. A sample of chromosome is shown in Figure 7.10.

0	0	1	0	0	1	0	0	1	0	0	0	0	0	0	1	0	0	0	0
1	0	0	0	0	1	0	0	0	0	0	0	0	0	0	1	0	0	0	0
1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	1	0	1	0
1	0	0	0	0	0	0	1	0	0	0	1	0	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
1	0	0	1	0	0	0	0	0	0	0	0	0	1	0	0	1	0	0	0
0	0	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0	1	1	0
0	0	0	0	0	0	0	1	0	0

In this figure, the first 60 numbers relate to Rx array and show how the physical antennas are distributed. “1” means that position in the sparse array contains an active/physical antenna and “0” shows a blank point on the sparse configuration. As one may notice, there are only 11 physical antennas on the Rx section and the rest are zeros. The same is true for the Tx array that is shown for elements 61–150 with 14 “1” showing the physical antennas.

The next step is to create the first population. If the size of population is selected to be 50, then 50 random chromosomes similar to the structure shown in Figure 7.10 will be created. Therefore, the population has the dimension of 50 × 150 as shown in Figure 7.11.

7.4.3.2 Fitness Calculation

Each individual/chromosome in the population suggests a certain arrangement of the physical antennas for the Rx and Tx array structures. It is required to evaluate and quantify the suitability of each individual by assigning a value as the fitness to each chromosome.

For calculation of the chromosome's fitness, the related polynomial for each of Rx and Tx arrays will be calculated. As an example, the related Rx and Tx polynomials for the chromosome of Figure 7.10 are as follows:

7.17

As mentioned before, the virtual array is the result of Rx and Tx convolution in the time domain or multiplication of their polynomials in the frequency domain. Therefore, two polynomials of Equation (7.17) are multiplied to result in:

7.18

In an ideal scenario, Equation (7.18) is a complete polynomial of order 149 without any zero coefficient. However, as the sample chromosome that resulted in Equation (7.18) is not the best individual, hence Equation (7.18) includes some zero coefficients.

Based on this, one may define the fitness in such a way to show the number of zero coefficients of the resulting polynomial for each chromosome. Therefore, the lower is the fitness means less number of zero coefficients and hence the better the chromosome.

After calculating the fitness of all individuals in the initial population, the chromosomes in the population are rearranged based on their fitness. The best chromosome, which has the lowest fitness value, goes to the top and the worst chromosome with highest fitness value will be located at the end of the population. This is required for selection operator, which is discussed in the following section.

7.4.3.3 Selection Operator

After arranging the population based on the chromosomes' fitness, some individuals are to be selected as parents. Obviously, chromosomes selected as parents will have the chance of transferring their genes to the next generation. The nonselected chromosomes will die and their genetic information no longer existed in the optimization process. Therefore, it is important to make sure that the algorithm provides a rational process for preserving the critical genetic information of all chromosomes by giving them enough chance of being selected as parents based on their fitness values. There are many suggested approaches for the selection operator, of which the most common approaches are introduced briefly in the following text.

1. Population Decimation. This is the simplest strategy for selection of parents. After arranging the population based on the fitness, a random value for fitness is selected. All the individuals who have the fitness worse than the threshold will be deleted. The parents are selected randomly from the remaining individuals. Irrespective of the easy implementation process, this selection strategy deletes all the weak individuals without giving them any chance of participating in the formation of the next generation.
2. Proportionate Selection. This is one of the most commonly used techniques for the selection operator. Sometimes, it is also referred to as roulette wheel approach. In this approach, the chance of selection of each individual relates to its fitness by
   7.19

   which f_parents shows the fitness of ith parents. The implementation of this approach is similar to a roulette wheel on which each chromosome occupies a certain area on the wheel surface based on its fitness as shown in Figure 7.12. The proportionate technique is efficient for the case when the number of population is high enough. Otherwise, some errors are expected for low population sizes. Moreover, the weak chromosomes have a small chance of transferring their genetic characteristics to the next generation.

3. Tournament Selection. The tournament approach is normally believed to be the most common technique for selection operators. First, a group of chromosomes are randomly selected from the current population. Then the best individual in the selected bunch is the winner and hence will be selected as parent. The selected group will be returned to the population and the process repeats again. Tournament technique provides a satisfactory GA performance and has simple implementation process.

   Variable Window Width. The variable window is one of the most effective techniques for the selection operator [15]. After rearranging the population based on their fitness, a window with variable length is applied to the population. Hence, the best chromosome is located at the beginning of the window and the end of the window includes the weakest chromosome in the selected window. Then, one individual is randomly selected from the chromosomes in the window as parent. The function that controls the window width is very important in this approach, and the total performance of the algorithm highly depends on this function [2]. The formula that has been suggested in Ref. [15] is as follows:

4. 7.20

   where W(L) is the window width, Int shows the integer value of the parameter, Pop is the total population number, L is a random value that controls the window width, and t is a parameter that has to be selected in such a way to create W_max = Pop. Implementing the abovementioned procedure for the selection process, however, did not result in satisfactory operation of the algorithm. Investigating the proposed formula for the selection operator reveals the reason for this unsuccessful performance. If Equation (7.20) is plotted as shown in Figure 7.13, one may notice the reason. For most of the cases (L > 0.2), the first 25 chromosomes are having the same chance of being selected as parent. The first chromosome and the 25th individual normally have noticeable fitness value differences. This is almost inconsistent with the overall expectation from selection operator.

   

   

Therefore, a new formula is suggested that shows better performance for the GA. The new function is Equation (7.21) and has two parts. When L, which is a random variable, is less than a certain value, 0.3, for example, the window width linearly depends on L. For L > 0.3, an exponential function has been suggested.

7.21

The above-suggested function for the window width is depicted in Figure 7.14. As one may easily see, while the better chromosomes, the first 30% of population, are having a higher chance in the selection process, the weaker chromosomes also have the opportunity to transfer their genetic information to the next generation. It is, however, important to mention that the suggested function in Equation (7.21) is only based on different functions and evaluation of their performances. This simply means that it is possible to find another function that works better for another optimization problem.

7.4.3.4 Crossover

When two chromosomes are selected as parents, they will do crossover, exchanging their genetic information and creating children. Again there are many approaches for doing crossover. The technique that is being used in the proposed application is called “multipoint crossover” technique. In this approach, four random numbers are selected between 1 and 60 and four numbers in the range 61–150. Then each of the parents changes their chromosomes between these breaking points. The crossover operator based on multipoint is figuratively shown in Figure 7.15. For the specific case of the MIMO-SAR, one major issue normally happens during the crossover that should be considered carefully; otherwise, the final result of optimization is completely wrong. As one may notice, the number of “1” in the Rx and Tx arrays is very important as it shows the physical antennas in each array. The important issue is therefore to make sure that when crossover is happening between two parents, the number of “1” in each section, 1–60 and 61–150, should remain unchanged. As an example, if two “1s” are in the genes (R₁, R₂) of Parent 1 and Parent 2 has only one “1” in its genes between R₁ and R₂, then each of the offspring, Child 1 and Child 2, will have wrong number of 1s in their Rx array section after multipoint crossover. Therefore, instead of simply changing the genes of R₁ and R₂ between Parent 1 and Parent 2, the adjustment of 1s should be maintained. For the suggested example, Child 1 will have lower number of “1s” in its Rx array. It is possible to add the required number of 1s in its genes (R₁, R₂). On the other hand, Child 2 acquires more 1s in its Rx array section. Hence, one of the 1s will be randomly deleted. The accommodation of this process in the algorithm will be carefully considered.

7.4.4 Results

Based on the described GA algorithm in the previous section, the MIMO-SAR system is optimized aiming for minimum number of physical antennas in the Rx and Tx arrays, while the resulting virtual array is a complete array of 150 elements. It was already shown that 25 (11 + 14) elements is the minimum theoretical number of physical antennas for the virtual array of 150 elements based on the discussion in Section 7.3.

The written MATLAB codes are able to calculate any number of Rx and Tx arrays. The codes are run for 11 Rx and 14 Tx arrays as already considered in the discussion. The final result of the MIMO system is shown as follows. It is important to mention that few changes and extra steps are also added in the coding process to shift any zero coefficients in the beginning (or ending) of the virtual array. This means that analytical errors happened during the optimization will have less practical effects for the physical arrays.

7.4.4.1 Example 1

In this example, the result of optimization suggests a structure for the Rx and Tx arrays that results in a virtual array of 150 elements with 10 zero coefficients. The suggested Rx and Tx polynomials are as follows:

7.22

The virtual array is shown in Figure 7.16. One may notice, 10 zeros are found in the resulted virtual array coefficient on which 9 of them located at one end of the vector. This zeros have no effect on the array performance except reducing the length of the virtual array from 30 to 28.2 cm. There is only one missing coefficient, A₁₄₂, which affects the performance of the MIMO system for the SAR purpose. The coefficients that are “2” can be simply converted into “1” by not considering one of the products in the switching network that created that coefficient.

7.4.4.2 Example 2

The second example covers the same optimization problem; 11 Rx and 14 Tx arrays for a virtual array with 150 elements. The revealed result is as follows:

7.23

Based on the suggested configuration for each of the real arrays, the resulting virtual array is shown in Figure 7.17. The resulting virtual array has seven zero coefficients out of which five are forced at each end of the array. This simply means that the virtual array length is 28.6 cm. Therefore, it is almost long enough to decode the maximum length of the tags with 17 bits of data (see Figure 6.11). However, there are two positions in the virtual array with zero coefficients that may affect the MIMO system performance. Again, the coefficients “2” will be deleted in the system implementation.