In this chapter, time modulated linear array (TMLA) is presented and discussed in detail where all its theoretical backgrounds are derived. The difference between single and multiple time modulation frequencies of TMLA is shown, where different examples in designing them are presented. In addition, the power and directivity of TMLAs are derived in their closed form. Moreover, the relation between the steering angle of each sideband with respect to the first sideband angle is developed analytically. Also, an efficient mathematical method is presented to design TMLA with desired sidelobe (SLL) and sideband levels (SBLs) with maximum attainable directivity. It is shown that the TMLA can be designed by only controlling its time sequence distributions which is a very good advantage as compared to the conventional antenna array.
- antenna array
- time modulated linear array (TMLA)
- time modulation
- power radiation
- sidelobe level (SLL)
- sideband level (SBL)
- electronic beam steering
- single time modulation frequency
- multiple time modulation frequency
The antenna array performance can be improved by decreasing its sidelobe level (SLL) and increasing its directivity. To do that, many different methods and techniques were proposed such as genetic algorithm (GA), particle swarm optimization (PSO), and hybridization between different arrays [1, 2, 3]. However, these methods provide very satisfactory results in the designed array; the realization of the designed excitations by using conventional approaches, such as tapered amplitude distributions and amplitude attenuators, is very challenging due to the fact that any small inaccuracy in the design will cause unwanted deviations in the SLL . In order to overcome this problem, the time modulated linear array (TMLA), also called 4-D antenna array, was proposed. The main concept of this idea was used in  and applied to antenna array in order to achieve ultralow sidelobe level by Kummer et al in . The idea of TMLA is to use the time as an additional degree of freedom in the design by using radio-frequency switches that periodically modulate the elements. The concept of TMLA is to use switching modulation (on, off) in order to reduce the effects of errors because the on-off switching can be controlled at a very high accuracy level.
2. Time modulated linear array
Suppose an N-isotropic element 4-D linear array aligned along the z-axis and centered on its origin as shown in Figure 1.
The array factor of time modulated array is given by 
where is the center frequency; is the elevation angle of the usual spherical coordinate; are the time-independent static excitation amplitude; is the wavenumber, in which is the wavelength; and is the position of the nth element of the array along the z-axis.
2.1 TMLA with single time modulation frequency (STMF)
It should be indicated that in the STMF, the switching period is the same for all the antenna elements.
The topology of TMLA with STMF is shown in Figure 3, where single-throw switches are connected to each antenna so that to control the switching between the two states: on and off.
Since are periodic functions, they can be expanded by the Fourier series as
where is the modulation frequency, , and is the th Fourier coefficient of the th radio-frequency switch, and they are represented as
Note that is the array factor at the desired frequency and is the sideband array factor for the case of STMF.
The array factor at the desired frequency and for the case is given as
It can be concluded that by controlling the normalized switch-on durations , any array pattern can be generated with very high accuracy.
2.1.1 Power radiations in time domain
In this section, we outline how to obtain the generalized power expression of the TMLA. By aligning the array along the z-axis and considering spherical coordinate with and , the elevation and azimuth angles, respectively, Eq. (1) can be represented in terms of as
where denotes the positions of the TMLA elements along the z-axis.
Let’s consider and be the intensities of the electric and magnetic field (complex spatial form) radiated by a given antenna used in the TMLA, hence, the electric field and magnetic field generated by the array are given as
The instantaneous Poynting vector is given as 
where and represent the instantaneous field vectors, and they are given as
Note that and are not functions of time and is a periodic function with period , and then the second term has zero average power; hence, the average power density is equal to
where is the power density of each antenna element in the TMLA. By considering isotropic antenna elements, is the radiation intensity, and it is constant over all the space, and it can be taken as unity (is the radial distance from the TMLA to the observation point at the far-field region). By using (1), (17) is written as
The total power is given
We should indicate that the expression (19) is a very simple formula to determine the total power radiated by the TMLA.
For the case and equal spacing distance between the elements, the total power can be written as
2.1.2 Power radiations in frequency domain
In this section, the power radiation is represented in the frequency domain. By taking the Fourier series (3) of and using the Parseval’s theorem then
and the total power is given as
It is worth noticing that the total power can be written as
where is the power radiated at the fundamental frequency and is the power of sidebands.
The complex Fourier coefficient in (5) can be written as
Then, we have
By using the results given in , then
It should be indicated that can be interpreted as the overlapped duration between the corresponding switch-on durations and .
At the case , the Parseval’s theorem (21) can be used, then
It is worth noticing that the total power expression (22) can be written as
The directivity at the fundamental frequency of TMLA with STMF is presented as 
By considering excitations with the same amplitude, i.e., as shown in Eq. (9), the directivity can be written as
It can be written in the following form :
where is the directivity of the conventional antenna array, i.e., without modulation switches, and is the dynamic factor. It is worth noticing that denotes no sideband radiations (SBRs).
2.1.4 Simulation and computed results
To understand the benefits of TMLA with STMF, simulation examples should be analyzed in detail. Let’s consider 30-element Chebyshev weighting with 30 dB SLL, where is considered. The normalized array pattern at the fundamental frequency and the first four sidebands frequencies are shown in Figure 4, whereas the periodic time sequences for each element are shown in Figure 5. It is evident that the array pattern at the fundamental frequency has a Chebyshev array pattern. However, there are other array patterns at the multiple of the time modulation frequency due to the modulating switches, which cause power losses at the SBRs. The power distribution over the sidebands is shown in Figure 6, where only the positive sidebands are shown because the negative sidebands have the same power distribution as the positive ones.
It is evident that most of the power resides at the fundamental frequency with 75.94%, where the remaining sidebands have only 24.06% of the total power. It should be indicated that the directivity of the array pattern at the fundamental frequency is equal to which is less than the conventional array directivity because of the SBR losses. Many wondering facts may arise now; since the conventional array has no sideband power losses and higher directivity than the TMLA, why is the TMLA really needed? Why do we need modulating switches that generate infinite sideband at the multiple of the modulation frequency? The answer is simple; the conventional array needs tapered amplitude distributions and amplitude attenuators  in order to realize the excitation amplitudes. However, this method is not so accurate; hence, it causes deviations in the desired array pattern. Therefore, high SLLs are generated. In this case, the realistic directivity is dramatically reduced. Whereas the excitation amplitudes in TMLA can be easily generated with very high accuracy by using modulating switches, therefore, the desired array pattern is totally preserved (its SLLs are kept at their designed level).
It is worth noticing that the sideband levels (SBLs) are high at the main lobe of the fundamental array pattern as shown in Figure 4. This kind of problem can be solved by shifting the sideband arrays by controlling the normalized switch-on instants .
The array factor at the sideband for static excitation amplitude and equal spacing distance between the elements () is given by (8)
Without steering the sideband, the switch-on instant is zero then , which represent an original phase shift at the mth sideband. Hence, (37) can be written as
To steer the sideband toward , the following condition should be taken
The general solution of Eq. (39) is given as
We should indicate that is an integer number which is chosen in order to maintain in the region . It should be indicated that when the first sideband is steered toward , i.e., at relation (40), the sideband is self-steered toward a specific angle that can be determined as
To find , the following equation should be solved
Its general solution is given as
It should be indicated that if , then is chosen from (integer numbers set) so that . For example, let steer the first sideband toward , then the 2nd, 3rd, and 4th sidebands are steered toward , respectively. The results are shown in Figure 7 (a), where the corresponding switch-on time is shown in Figure 7 (b). When the first sideband is steered toward , then the sidebands are steered toward , respectively. The normalized array pattern is shown in Figure 8 (a), and the corresponding switch-on time for each antenna element is presented in Figure 8 (b). We should indicate that the even sidebands, i.e., is even, are not steered at the case because there exists an integer number so that in Eq. (43). It is worth noticing that the power distribution over the sidebands as given in Figure 6 remains unchanged for all the steering angles.
From the above results, it can be deduced that the use of periodic switches to modulate the signal generates SBRs at the multiples of the time modulation frequency, which causes power loss and low directivity. To overcome the SBR problem, the optimization techniques, such as differential evolution (DE), GA, PSO, and the simulated annealing (SA), were used to reduce the SBL as well as maintain SLL at a certain low level [11, 12, 13, 14]. In , the PSO technique was used in order to minimize the power losses and maintain the SLL and SBL at the desired level; therefore the time sequences generated by the PSO are given in Figure 9, and the corresponding array pattern is presented in Figure 10.
It can be observed that the SLLs are maintained at and the maximum SBL is , where only four elements are time modulated and the elements 1, 26, 27, and 29 are always turned off. This TMLA can be considered 26-element array with nonuniform spacing because the 4-turned off elements can be ignored. The power distribution over the sidebands is shown in Figure 11. It can be observed that most of the power resides at the fundamental frequency with 96.43% of the total power, where the remaining sidebands have only 3.57% of the total power. However, in this case, the SLL is only .
In , the SA method was used in order to maintain the SLL at a certain level and minimize the SBL under . Therefore, the obtained switch-on time sequences are shown in Figure 12, and the corresponding array patterns of the optimized TMLA are shown in Figure 13, where the power spectrum percentage is presented in Figure 14. It can be observed that only 9 elements are time modulated, where the remaining 21 elements are always turned on. In this case, the SBLs are minimized greatly, where they have only 3.96% of the total power. The PSO is more efficient than the SA one as indicated in . For more details about the optimization methods, the reader should be referred to [13, 14].
The multiple time modulation frequency (MTMF) was proposed to reduce SBL of TMLA because of avoiding the accumulation of the sidebands in the space ; however, the SBR power was not decreased by using MTMF. In , the DE was used with MTMF to suppress SLL, SBL, and SBR power, and very good results were obtained. In the following section, the MTMF is investigated in detail.
2.2 TMLA with multiple time modulation frequency (MTMF)
Since are periodic functions, they can be expanded by the Fourier series as
where is the modulation frequency and is the th Fourier coefficient of the th radio-frequency switch, and they are represented as
where and are the normalized switch-on duration and the normalized switch-on instant for each element in the TMLA, respectively.
In the case of MTMF, the array factor can be written as
It is worth noticing that is the array factor at the fundamental frequency and accumulates different sideband frequencies as it was described in [15, 16]. The idea of using MTMF is to avoid the superposing of sidebands because each array element has its corresponding switching frequency. Therefore, the sidebands of each element could not be superposed with the sidebands of another element; this concept is explained clearly in Figure 15.
2.2.1 Power radiations
The power radiation by TMLA with MTMF can be obtained by considering the following assumption:
The sidebands of each antenna element are not overlapped with the sidebands of the other elements.
In this case, the sidebands power is given as
where the power radiated at the fundamental frequency is given by
It is worth noticing that relation (51) shows that all the Fourier’s coefficients of each element are summed independently because they are located at different frequencies. Also, it should be indicated that for the total power relations for the STMF and MTMF are identical. Another formulation for the power relation of TMLA with MTMF is given in , where a prime distribution is assumed for the time modulation frequencies. For more details about time modulation frequencies with prime distribution, the reader should be referred to . It should be noted that the directivity of the TMLA with MTMF is identical to the one given in relation (35).
2.2.2 Simulation and computed results
In this section, computed results and examples are considered in order to investigate the benefits of TMLA with MTMF. The same example taken in Section 2.1.4 is considered so as to make a fair comparison between TMLA-STMF and TMLA-MTMF. Let’s consider the fundamental frequency to be 2.6 GHz, the time modulation frequency for the STMF case is 30 MHz, and for the MTMF, are selected as 
The results are plotted in Figure 16, where the maximum sideband for the STMF is , whereas only is obtained for MTMF case. It is evident that the TMLA with MTMF is more efficient than STMF in reducing the sideband levels. However, the power loss in the sideband radiations is the same for the case ; hence, the directivity has remained unchanged. It is worth noticing that the normalized switch-on time is identical to the one given in Figure 5, where, in the case of MTMF, each element has its corresponding normalizing period as shown in Eq. (47).
The sideband’s power percentages for STMF and MTMF are presented in Figure 17. It is evident that the sideband’s power of STMF is larger than the sideband’s power of MTMF for . However, the inverse occurs for .
The optimization techniques were used in order to reduce the SBLs and the SBRs, e.g., the DE method was applied in , and very good results were obtained. In , the DE method was implemented so as to maintain the SLLs at a given level, whereas the SBLs and SBRs are minimized as much as possible. Figure 18 shows the results of the DE applied to the TMLA-MTMF in order to maintain the SLLs at , where the obtained SBL was . In order to make a comparison between the TMLA-STMF, the first sideband of STMF is also presented. Its maximum level is .
3. Reducing SLLs and SBLs in TMLA
In this section, an analytical method is used to minimize the SLLs and SBLs in TMLA . The array pattern of the TMLA can be written in the following forms:
For an odd number of elements and , then
where and by assuming symmetrical excitations , then
Note that is a row vector that wants to be determined to satisfy the desired specifications (SLL, SBL, and directivity), where . We should indicate thatcan be written as
where is the first kind Chebyshev polynomial of degree . The sidelobes’ locations of the fundamental array factor can be determined by solving the equation , where
where is the second kind Chebyshev polynomial and it has a relation with first kind Chebyshev polynomial
The sidelobes are located at and , where are the roots of the following polynomial:
By obtaining the roots , in the region , of the polynomial in terms of the excitation coefficients and substituting into (56), we get the following:
where is the SLL of the sidelobe located at .
For an even number of elements and , then
The Chebyshev of 3rd and 4th kinds are given as
respectively, where .
As described before, the sidelobes are located at , where are the roots, that are located in the region of the following polynomial:
and the SLLs are given as
It should be indicated that there are no sidelobes contributed by the factor since its roots are , which are a worthy advantage in designing even number of elements in TMLA.
Now let’s design TMLA with nine elements to satisfy the specifications; and . In this case, the SBL for MTMF is −26.15 and the maximum directivity that can be achieved is . The results are plotted in Figures 19 and 20. To investigate the effectiveness of the proposed method, a comparison is performed with nine-element Chebyshev array that has SLL equal to . It has SBL equal to −12.4 and its directivity 16.42 dB which is larger than the designed array only with 0.22 dB. Note that for all the cases.
Finally, it should be indicted that the TMLA can be designed by only controlling the time sequence distributions which is a very good advantage as compared to the conventional array under the following reasons:
Attain high accuracy in the designed array pattern in the TMLA because the switching distributions can be controlled at very high accuracy.
In the conventional array, attenuators and distributors are needed for exciting the array which is not accurate method. Therefore, it causes deviation in the designed array pattern and high SLLs are generated.
In this chapter, the main backgrounds and theories of TMLA are derived where different simulation examples are presented and discussed in detail. A comparison between different results given in the previous literature is also discussed. In addition, an analytical method to reduce the SLLs and SBLs in TMLA with maximum achievable directivity has been developed. This analytical method helps us to visualize the relation between switch-on durations, SLL, and SBL, which is an advantage compared to the other designing methods. It was shown that the TMLA has better performance than the conventional array.