Open access peer-reviewed chapter - ONLINE FIRST

Time Modulated Linear Array (TMLA) Design

By Oussama Gassab, Arab Azrar and Sara Bouguerra

Submitted: November 14th 2019Reviewed: March 12th 2020Published: April 16th 2020

DOI: 10.5772/intechopen.92100

Downloaded: 33

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.

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]

AFθt=ej2πf0tnIngntejkzncosθE1

where f0is the center frequency; θis the elevation angle of the usual spherical coordinate; Inare the time-independent static excitation amplitude; k=2π/λis the wavenumber, in which λis the wavelength; and znis the position of the nth element of the array along the z-axis.

2.1 TMLA with single time modulation frequency (STMF)

gntare the periodic switch-on time sequence functions, and they are written for the case of STMF as [7] (see Figure 2)

Figure 2.

The periodic time sequence graph.

gnt=1,ton,n<ttoff,n0,0<tton,nortoff,n<tTpE2

It should be indicated that in the STMF, the switching period Tpis 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.

Figure 3.

Configuration of N-element TMLA with STMF using single-throw switches.

Since gntare periodic functions, they can be expanded by the Fourier series as

gnt=m=+Gnmej2πmfptE3

where fp=1/Tpis the modulation frequency, f0fp, and Gnmis the mth Fourier coefficient of the nth radio-frequency switch, and they are represented as

Gnm=1Tp0Tpgntej2πmfptdtE4
Gnm=τnsincτnmejmπ2ξon,n+τnE5

where τn=toff,nton,n/Tpand ξon,n=ton,n/Tpare the normalized switch-on duration and the normalized switch-on instant, respectively, and sincx=sinπx/πx, wherein sinc0=1. By using (3), equation (1) can be written as

AFθt=m=+nInGnmej2πf0+mfptejkzncosθE6
AFθt=m=+AFmθtE7

where

AFmθt=nInGnmej2πf0+mfptejkzncosθE8

Note that AF0θtis the array factor at the desired frequency f0and AFmθtis the sideband array factor for the case of STMF.

The array factor at the desired frequency f0and for the case In=1is given as

AF0θt=ej2πf0tnτnejkzncosθ,0τn1E9

It can be concluded that by controlling the normalized switch-on durations τn, any array pattern AF0can 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 θpand ϕp, the elevation and azimuth angles, respectively, Eq. (1) can be represented in terms of θpas

AFθpt=ej2πf0tn=1NIngntejkzncosθp=ej2πf0tAFθptE10

where zndenotes the positions of the TMLA elements along the z-axis.

Let’s consider E0and H0be 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

Et=ej2πf0tAFθptE0E11
Ht=ej2πf0tAFθptH0E12

The instantaneous Poynting vector is given as [8]

P=E×HE13

where Eand Hrepresent the instantaneous field vectors, and they are given as

Et=Reej2πf0tAFθptE0E14
Ht=Reej2πf0tAFθptH0E15

By using ReX=X+X/2and inserting (14) and (15) in (13) and proceeding with the same analysis performed in [8], the following result is obtained:

P=12ReAF2E0×H0+12ReAF2E0×H0ej4πf0tE16

Note that E0and H0are not functions of time and AFis a periodic function with period TpT0, and then the second term has zero average power; hence, the average power density is equal to

Pdensity=WavgTp0TpAF2dtE17

where Wavg=12ReE0×H0is the power density of each antenna element in the TMLA. By considering isotropic antenna elements, Wavgr2is the radiation intensity, and it is constant over all the space, and it can be taken as unity (ris the radial distance from the TMLA to the observation point at the far-field region). By using (1), (17) is written as

Pdensity=WavgTpn,k=1NInIk0TpgntgktdtejkznzkcosθpE18

The total power is given

PT=02π0πPdensityr2sinθpdθpdϕp=4πn,k=1NInIk1Tp0Tpgntgktdtsinc2λznzkE19

We should indicate that the expression (19) is a very simple formula to determine the total power radiated by the TMLA.

For the case d=λ/2and equal spacing distance between the elements, the total power can be written as

PT=4πn=1NIn21Tp0Tpgnt2dt=4πn=1NIn2τnE20

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 gntand using the Parseval’s theorem then

1Tp0Tpgntgktdt=m=+GnmGkmE21

and the total power is given as

PT=4πn,k=1NInIkm=+GnmGkmsinc2λznzkE22

It is worth noticing that the total power PTcan be written as

PT=Pf0+PSBE23

and

Pf0=4πn=1NIn2τn2+4πn,k=1knNInIksinc2λznzkτnτkE24
PSB=4πn=1NIn2m=m0+Gnm2+4πn,k=1knNInIksinc2λznzkm=m0+GnmGkmE25

where Pf0is the power radiated at the fundamental frequency f0and PSBis the power of sidebands.

The complex Fourier coefficient Gnmin (5) can be written as

Gnm=j2πmej2πmξoff,nej2πmξon,nE26

Then

GnmGkm=14π2m2ej2πmξoff,kξoff,nej2πmξon,kξoff,nej2πmξoff,kξon,n+ej2πmξon,kξon,nE27

Then, we have

m=m0+GnmGkm=2m=1+cos2πmξoff,kξoff,n2πm22m=1+cos2πmξon,kξoff,n2πm22m=1+cos2πmξoff,kξon,n2πm2+2m=1+cos2πmξon,kξon,n2πm2E28

By using the results given in [9], then

m=m0+GnmGkm=τ¯nkτnτkE29

where

τ¯nk=12ξoff,kξoff,n+ξon,kξoff,n+ξoff,kξon,nξon,kξon,nE30

It should be indicated that τ¯nkcan be interpreted as the overlapped duration between the corresponding switch-on durations τnand τk.

At the case n=k, the Parseval’s theorem (21) can be used, then

m=m0+Gnm2=1Tp0Tpgnt2dtGn02E31

and

m=m0+Gnm2=τn1τnE32

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

PT=4πn,k=1NInIkτ¯nksinc2λznzkE33

2.1.3 Directivity

The directivity at the fundamental frequency f0of TMLA with STMF is presented as [10]

Df0=4πAF0θtmax2PTE34

By considering excitations with the same amplitude, i.e., In=1as shown in Eq. (9), the directivity Df0can be written as

Df0=4πnτn2Pf0Pf0Pf0+PSBE35

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

Df0=Dconvηf0E36

where Dconv=4πnτn2/Pf0is the directivity of the conventional antenna array, i.e., without modulation switches, and ηf0is the dynamic factor. It is worth noticing that ηSB=1denotes 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 In=1is 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 fpdue 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.

Figure 4.

The array patterns of TMLA with Chebyshev weighting at the fundamental frequency and the four positive first sidebands.

Figure 5.

The periodic time sequences of each element of TMLA, where all the switch-on instants are equal to zero.

Figure 6.

Power percentage spectrum of TMLA (Chebyshev weighting with N = 30 and SLL = − 30 dB ).

It is evident that most of the power resides at the fundamental frequency f0with 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 f0is equal to 19.93dBiwhich is less than the conventional array directivity 26.24dBibecause 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 [4] 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 ξon,n.

The array factor at the mthsideband for static excitation amplitude In=1and equal spacing distance between the elements (zn=n1d) is given by (8)

AFmθt=ej2πf0+mfptn=1Nτnsincτnmej2ξon,n+τn+kn1dcosθE37

Without steering the mthsideband, the switch-on instant is zero ξon,n=0then 2ξon,n+τn=τn, which represent an original phase shift at the mth sideband. Hence, (37) can be written as

AFmθt=ej2πf0+mfptn=1Nτnsincτnmeτnej2ξon,n+kn1dcosθE38

To steer the mthsideband toward θ0, the following condition should be taken

ej2ξon,n+kn1dcosθ=1E39

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

ξon,n=kn1d2πmcosθ0+Km,KZE40

We should indicate that Kis an integer number which is chosen in order to maintain ξon,nin the region 01. It should be indicated that when the first sideband is steered toward θ0, i.e., m=1at relation (40), the mthsideband is self-steered toward a specific angle θmthat can be determined as

By substituting (40) with m=1in (38), the following relation is obtained

AFmθt=ej2πf0+mfptnτnsincτnmeτnejn1kmdcosθ0+kdcosθE41

To find θm, the following equation should be solved

ejn1kmdcosθ0+kdcosθ=1E42

Its general solution is given as

θm=arccosmcosθ0+2K,KZE43

It should be indicated that if mcosθ0>1, then Kis chosen from Z(integer numbers set) so that mcosθ0+2K1. For example, let steer the first sideband toward θ0=120°, then the 2nd, 3rd, and 4th sidebands are steered toward 180°,60°,and90°, 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 θ0=180°, then the 2nd,3rd,and4thsidebands are steered toward 90°,180°,and90°, 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., mis even, are not steered at the case θ0=180°because there exists an integer number KZso that m+2K=0in 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.

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.

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.

Figure 9.

Switch-on time sequences optimized by PSO technique in order to reduce the SBR ( N = 30 , d = 0.7 λ ) [13].

Figure 10.

Normalized array patterns (at the fundamental frequency and the two first sidebands) of the optimized TMLA by the PSO technique [13].

It can be observed that the SLLs are maintained at 20dBand the maximum SBL is 30.2dB, 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 20dB.

Figure 11.

Power percentage spectrum of the optimized TMLA by the PSO ( N = 30 , d = 0.7 λ ).

In [14], the SA method was used in order to maintain the SLL at a certain level and minimize the SBL under 30dB. 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 [14]. For more details about the optimization methods, the reader should be referred to [13, 14].

Figure 12.

Switch-on time sequences optimized by the SA technique in order to reduce the SBR ( N = 30 , d = 0.7 λ ) [14].

Figure 13.

Normalized array patterns (at the fundamental frequency and the two first sidebands) of the optimized TMLA by the SA technique [14].

Figure 14.

Power percentage spectrum of the optimized TMLA by the SA method ( N = 30 , d = 0.7 λ ).

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 Tp,n. Hence, the periodic function gntin (1) is expressed as [15]

gnt=1,ton,n<ttoff,n0,0<tton,nortoff,n<tTp,nE44

Since gntare periodic functions, they can be expanded by the Fourier series as

gnt=m=+Gnmej2πmfpntE45

where fpn=1/Tpnis the modulation frequency and Gnmis the mth Fourier coefficient of the nth radio-frequency switch, and they are represented as

Gnm=1Tpn0Tpngntej2πmfpntdtE46
Gnm=τnsincτnmejmπ2ξon,n+τnE47

where τn=toff,nton,n/Tpnand ξon,n=ton,n/Tpnare 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

AFθt=m=+nInGnmej2πf0+mfpntejkzncosθE48
AFθt=m=+AFmθtE49

where

AFmθt=nInGnmej2πf0+mfpntejkzncosθE50

It is worth noticing that AF0is the array factor at the fundamental frequency f0and AFmaccumulates 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.

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

PSB=n=1NIn2m=m0+Gnm2E51
PSB=n=1NIn2τn1τnE52

where the power radiated at the fundamental frequency is given by

Pf0=4πn=1NIn2τn2+4πn,k=1knNInIksinc2λznzkτnτkE53

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 d=λ/2the total power relations for the STMF and MTMF are identical. Another formulation for the power relation of TMLA with MTMF is given in [17], 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 [17]. 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 f0to be 2.6 GHz, the time modulation frequency fpfor the STMF case is 30 MHz, and for the MTMF, fpnare selected as [15]

fpn=30+0.5n1MHz,n=1,2,3,,NE54

The results are plotted in Figure 16, where the maximum sideband for the STMF is 12.28dB, whereas only 35.98dBis 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 d=λ/2; 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).

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 d/λ<0.5. However, the inverse occurs for d/λ>0.5.

Figure 17.

Sideband’s power percentage for TMLA with STMF and MTMF in terms of d / λ (Chebyshev weighting with −30 dB SLLs is considered).

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 20dB, where the obtained SBL was 40.70dB[16]. In order to make a comparison between the TMLA-STMF, the first sideband of STMF is also presented. Its maximum level is 24.09dB.

Figure 18.

The optimized TMLA by the DE algorithm where the desired SLL is −20 dB ( d = 0.7 λ ). (a) Normalized array patterns. (b) Optimized switch-on time of each element.

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 N=2M+1and zn=nd, then

AF0ψt=ej2πf0tn=MMτnejnψ=ej2πf0tτ0+n=1Mτnejnψ+τnejnψE55

where ψ=kdcosθand by assuming symmetrical excitations τn=τn, then

AF0ψtτ=ej2πf0tτ0+2n=1MτncosE56

Note that τ=τ0τ1τMis a row vector that wants to be determined to satisfy the desired specifications (SLL, SBL, and directivity), where τ01M. We should indicate thatcoscan be written as

cos=TncosψE57

where Tnis the first kind Chebyshev polynomial of degree n. The sidelobes’ locations of the fundamental array factor AF0can be determined by solving the equation AF0/ψ=0, where

AF0ψ=2sinψej2πf0tn=1MnUn1cosψE58

where Un1is the second kind Chebyshev polynomial and it has a relation with first kind Chebyshev polynomial

dTnsds=nUn1sE59

where

U0s=1,U1s=2sUns=2sUn1sUn2s,n=2,3,E60

The sidelobes are located at ψ0=πand ψi=arccossi, where siare the roots of the following polynomial:

Podds=n=1MnUn1sE61

By obtaining the roots si, in the region 11, of the polynomial Poddsin terms of the excitation coefficients τnand substituting ψiτ1τ2τM=arccossiinto (56), we get the following:

AF0ψitτ=ej2πf0tτ0+2n=1MτnTnsiτ1τ2τME62

where SLLi=AF0ψitτis the SLL of the sidelobe located at ψi.

For an even number of elements N=2Mand z±n=±2n1d/2, then

AF0ψtτ1=2ej2πf0tn=1Mτncosn12ψE63

where τ1=τ1τ2τM.

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

Vns=cosn+12ψ/cosψ2E64
Wns=cosn+12ψ/sinψ2E65

respectively, where s=cosψ.

The expression AF0ψtτ1and AF0/ψcan be written in terms of (64) and (65), respectively, as

AF0ψtτ1=2cosψ2ej2πf0tn=1MτnVn1cosψE66
AF0ψ=2sinψ2ej2πf0tn=1Mn12τnWn1cosψE67

As described before, the sidelobes are located at ψi=arccossi, where siare the roots, that are located in the region 11of the following polynomial:

Pevens=n=1Mn12τnWn1sE68

and the SLLs are given as

SLLi=AF0ψitτ1=2cosψi2n=1MτnVn1siτ1τ2τME69

It should be indicated that there are no sidelobes contributed by the factor sinψ/2since its roots are ±2π, 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; DSLL=22dBand DSBLSTMF=15.2. In this case, the SBL for MTMF is −26.15 and the maximum directivity that can be achieved is Dmax=16.2dB. 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 22dB. 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 ξon,n=0for all the cases.

Figure 19.

The designed array N = 9 DSLL = − 22 dB DSBL STMF = − 15.2 and Chebyshev array N = 9 SLL = − 22 dB .

Figure 20.

Designed time sequence distributions of each antenna array element for N = 9 , DSLL = − 22 dB , DSBL STMF = − 15.2 .

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:

  1. Attain high accuracy in the designed array pattern in the TMLA because the switching distributions can be controlled at very high accuracy.

  2. 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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Oussama Gassab, Arab Azrar and Sara Bouguerra (April 16th 2020). Time Modulated Linear Array (TMLA) Design [Online First], IntechOpen, DOI: 10.5772/intechopen.92100. Available from:

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