## Abstract

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.

### Keywords

- antenna array
- time modulated linear array (TMLA)
- time modulation
- power radiation
- directivity
- sidelobe level (SLL)
- sideband level (SBL)
- electronic beam steering
- single time modulation frequency
- multiple time modulation frequency

## 1. Introduction

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 [4]. 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 [5] and applied to antenna array in order to achieve ultralow sidelobe level by Kummer et al in [6]. 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 [7]

where *n*th 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

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

where

where

where

Note that

The array factor at the desired frequency

It can be concluded that by controlling the normalized switch-on durations

#### 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

where

Let’s consider

The instantaneous Poynting vector is given as [8]

where

By using

Note that

where

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

#### 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 the total power is given as

It is worth noticing that the total power

and

where

The complex Fourier coefficient

Then

Then, we have

By using the results given in [9], then

where

It should be indicated that

At the case

and

It should be indicated that (29) and (32) can be used in (25) in order to obtain the closed-form expression for the sideband power.

It is worth noticing that the total power expression (22) can be written as

#### 2.1.3 Directivity

The directivity at the fundamental frequency

By considering excitations with the same amplitude, i.e.,

It can be written in the following form [10]:

where

#### 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

It is evident that most of the power resides at the fundamental frequency

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

Without steering the *m*th sideband. Hence, (37) can be written as

To steer the

The general solution of Eq. (39) is given as

We should indicate that

By substituting (40) with

To find

Its general solution is given as

It should be indicated that if

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 [14], 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

In [14], the SA method was used in order to maintain the SLL at a certain level and minimize the SBL under

The multiple time modulation frequency (MTMF) was proposed to reduce SBL of TMLA because of avoiding the accumulation of the sidebands in the space [15]; however, the SBR power was not decreased by using MTMF. In [16], 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)

In TMLA with MTMF, each antenna element has its time modulating switching period

Since

where

where

In the case of MTMF, the array factor can be written as

where

It is worth noticing that

#### 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

#### 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

The results are plotted in Figure 16, where the maximum sideband for the STMF is

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

The optimization techniques were used in order to reduce the SBLs and the SBRs, e.g., the DE method was applied in [16], and very good results were obtained. In [16], 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

## 3. Reducing SLLs and SBLs in TMLA

In this section, an analytical method is used to minimize the SLLs and SBLs in TMLA [18]. The array pattern of the TMLA can be written in the following forms:

For an odd number of elements

where

Note that

where

where

where

The sidelobes are located at

By obtaining the roots

where

For an even number of elements

where

The Chebyshev of 3rd and 4th kinds are given as

respectively, where

The expression

As described before, the sidelobes are located at

and the SLLs are given as

It should be indicated that there are no sidelobes contributed by the factor

Now let’s design TMLA with nine elements to satisfy the specifications;

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.

## 4. Conclusion

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.