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.
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F1.png)
Figure 1.
Time modulated array elements positions aligned along the z-axis and centered on its origin.
The array factor of time modulated array is given by [7]
where
2.1 TMLA with single time modulation frequency (STMF)
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F2.png)
Figure 2.
The periodic time sequence graph.
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.
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F3.png)
Figure 3.
Configuration of N-element TMLA with STMF using single-throw switches.
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
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F4.png)
Figure 4.
The array patterns of TMLA with Chebyshev weighting at the fundamental frequency and the four positive first sidebands.
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F5.png)
Figure 5.
The periodic time sequences of each element of TMLA, where all the switch-on instants are equal to zero.
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F6.png)
Figure 6.
Power percentage spectrum of TMLA (Chebyshev weighting with
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
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
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F7.png)
Figure 7.
(a) The array patterns of TMLA with Chebyshev weighting at the fundamental frequency and steered four positive sidebands where the first positive side is steered toward 120°. (b) The periodic time sequences of each element of TMLA.
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F8.png)
Figure 8.
(a) The array patterns of TMLA with Chebyshev weighting at the fundamental frequency and steered four positive sidebands where the first positive side is steered toward 180°. (b) The periodic time sequences of each element of TMLA.
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.
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F9.png)
Figure 9.
Switch-on time sequences optimized by PSO technique in order to reduce the SBR (
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F10.png)
Figure 10.
Normalized array patterns (at the fundamental frequency and the two first sidebands) of the optimized TMLA by the PSO technique [
It can be observed that the SLLs are maintained at
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F11.png)
Figure 11.
Power percentage spectrum of the optimized TMLA by the PSO (
In [14], the SA method was used in order to maintain the SLL at a certain level and minimize the SBL under
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F12.png)
Figure 12.
Switch-on time sequences optimized by the SA technique in order to reduce the SBR (
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F13.png)
Figure 13.
Normalized array patterns (at the fundamental frequency and the two first sidebands) of the optimized TMLA by the SA technique [
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F14.png)
Figure 14.
Power percentage spectrum of the optimized TMLA by the SA method (
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
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F15.png)
Figure 15.
Illustration of SBL suppression in TMLA with MTMF. The SBLs in STMF are superposed because all the time modulating switchers have the same time modulating frequency, whereas the SBLs in MTMF are not superposed because each element has the corresponding time modulating frequency.
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
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F16.png)
Figure 16.
Normalized array pattern at the fundamental frequency with STMF first sideband and the MTMF maximum sideband level.
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
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F17.png)
Figure 17.
Sideband’s power percentage for TMLA with STMF and MTMF in terms of
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
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F18.png)
Figure 18.
The optimized TMLA by the DE algorithm where the desired SLL is −20 dB (
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;
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F19.png)
Figure 19.
The designed array
![](http://cdnintech.com/media/chapter/71799/1512345123/media/F20.png)
Figure 20.
Designed time sequence distributions of each antenna array element for
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.
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