4.1 Variable aperture size (VAS)

This is the first type of time-modulation strategy as reported in [2] where the aperture size of the antenna array is varied with time. The time-modulation principle as discussed in Section 2 falls under this category.

#### 4.2 Time modulation through pulse shifting

In VAS time-modulation scheme, only the switch-"on" time duration is considered for deriving the array factor expression. However, when the RF switches are used to commutate the antenna elements in TMAAs, the radiation patterns at center frequency as well as at different harmonics depend not only on the switch-on time duration but also on the switch-"on" and switch-"off" time instants of the array elements [18, 19]. Thus along with the switch-on time durations as considered in VAS scheme, switch-on and switch-off time instants are also taken as another degree of freedom to control the power pattern in TMAA. For the pulse shifting strategy, periodic switching instants of the pth element over the modulation period are shown in Figure 8. In this case, both on-time instant t 1 <sup>p</sup> and off-time instant t 2 p can be controlled independently such that individually t 1 <sup>p</sup>and t on <sup>p</sup> ¼ t 2 <sup>p</sup> � t 1 <sup>p</sup> should be ≤ Tm. Thus, two situations may occur. The first case is shown in Figure 8(a) where t 1 <sup>p</sup> <t 2 <sup>p</sup> and t 1 <sup>p</sup> þ t on <sup>p</sup> ≤ Tm. Therefore, the switching function as expressed in Eq. (2) for VAS will be modified and is represented as in Eq. (14):

increases and becomes almost steady at 12.5 dB after 30 dB SLL. From Figures 6 and 7, it can be seen that for the GA-based patterns of different SLLs, SBLmax and FNBW vary randomly as in the cost function, only SLL is considered without

The plot of SBR1(max) and SBR2(max) for the different values of SLLs of the patterns at f0.

FNBW for different values of SLLs of the GA and Dolph-Chebyshev patterns at f0.

Different time-modulation strategies have been reported for synthesizing antenna arrays. These can be classified as (1) variable aperture size (VAS); (2) pulse

controlling FNBW and SBL.

Figure 6.

Advances in Array Optimization

Figure 7.

12

4. Time-modulation strategies

Advances in Array Optimization

$$U\_p(t) = \begin{cases} \mathbf{1}; & \left(\eta T\_m + t\_p^1\right) \le t \le \left(\eta T\_m + t\_p^2\right) \le (\eta + \mathbf{1})T\_m\\ \mathbf{0}; & \text{elsewhere} \end{cases} \tag{14}$$

The periodic on–off sequence of the set of time steps corresponding to the pth element is represented by the switching function Up(t). If the on–off status of qth

pattern, the optimal binary arrangement of the bit patterns, i.e., set of time steps to be under on states and off states, can be found by employing simple genetic algo-

sequence of N array elements represents the chromosome (χ) of GA and is given as

The complex Fourier coefficient of pth element at kth harmonic with the BOTS

X Q

q¼1 bq pe

Tm is the normalized time step. Thus, by incorporating more number of

kπ

the degrees of freedom as the optimization variables to the evolutionary algorithm, the radiation pattern characteristics like SLL, SBL, SBR, etc. can be controlled skillfully.

In SOTS-based switching strategy, the time-modulation period (Tm) is divided into a number of subsections with variable lengths [21]. Let us assume that Tm is divided into Q number of time steps as shown in Figure 10 for the switching strategy of pth element of the array. For the qth time step, the on and off time

qoff

qoff

The Fourier coefficient at the kth harmonics for the pth element can be written as

qoff <sup>p</sup> � t qon p � � � � <sup>e</sup>

Tm denotes the modulation frequency used for the time modulation.

<sup>Q</sup> , then SOTS takes the form of BOTS. So, SOTS-based

<sup>p</sup> ≤Tm : ∀q ∈½ � 1, Q

�jkω<sup>m</sup> t qon <sup>p</sup> þt qoff p � �

qon <sup>p</sup> and t

switching pulse Upð Þt for the different time steps over a complete modulation

qon <sup>p</sup> ≤t≤ t

sin c kω<sup>m</sup> t

It can be observed that, if the number of subsections Q is 1, then SOTS is transformed into pulse shifting-based strategy. On the other scenario, if the on-time duration at each step, i.e., the separation between the on and off time instants,

switching strategy provides more flexibility in the design of optimized time

sequences as compared to the other abovementioned switching strategies. However such improved flexibility in synthesizing the pattern is obtained at the cost of some

0; elsewhere (

q <sup>p</sup> ¼ t qoff <sup>p</sup> � t qon

<sup>p</sup>=1 and that for "off" status <sup>b</sup><sup>q</sup>

<sup>p</sup>, for which "on"

<sup>N</sup> (19)

<sup>p</sup>=0, then the set of time steps

�jkπτ0ð Þ <sup>2</sup>q�<sup>1</sup> (20)

<sup>p</sup> , respectively, and the resultant on-

<sup>p</sup> . Therefore, the periodic time

(21)

(22)

<sup>p</sup> . In order to synthesize the desired

<sup>p</sup>, as a gene, then on–off time

time step for the pth element is symbolized with a binary bit, bq

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

pb2 pb3 p⋯bQ

BOTS : Cpk <sup>¼</sup> sin ð Þ <sup>k</sup>πτ<sup>0</sup>

rithm (SGA) [16]. Considering each time step, bq

status corresponds to bq

where <sup>τ</sup><sup>0</sup> <sup>¼</sup> <sup>t</sup>

for the pth element is given as b<sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.89479

<sup>χ</sup> <sup>¼</sup> <sup>b</sup><sup>1</sup> 1b2 1b3 1⋯b<sup>Q</sup> <sup>1</sup> <sup>b</sup><sup>1</sup> 2b2 2b3 2⋯b<sup>Q</sup> <sup>2</sup> <sup>⋯</sup>b<sup>1</sup> pb2 pb3 p⋯bQ <sup>p</sup> <sup>⋯</sup>b<sup>1</sup> Nb<sup>2</sup> Nb<sup>3</sup> N⋯bQ

0

switching scheme can be obtained as [20]

4.4 Subsectional optimized time steps (SOTS)

UpðÞ¼ <sup>t</sup> 1; 0<sup>≤</sup> <sup>t</sup>

instants of the switch are denoted by t

period is represented as

Cpk <sup>¼</sup> <sup>X</sup> Q

where <sup>ω</sup><sup>m</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup>

becomes multiples of Tm

15

q¼1

t qoff <sup>p</sup> � t qon p � �

Tm

time duration at qth step is obtained as t

Hence, the normalized switch-on time duration, <sup>τ</sup>p, is given as <sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>t</sup> on p Tm <sup>¼</sup> <sup>t</sup> 2 <sup>p</sup>�t 1 p Tm . Thus, the pulse shifting strategy reduces to VAS with t 1 <sup>p</sup>=0.

Another possible situation may appear as shown in Figure 8(b) where t 1 <sup>p</sup>>t 2 <sup>p</sup> and t 1 <sup>p</sup> þ t on <sup>p</sup> ≥Tm. Under such situation, the switching operation can be expressed as in Eq. (15):

$$U\_p(t) = \begin{cases} \mathbf{1}; & \left(\eta T\_m + t\_p^1\right) \le t \le (\eta + 1)T\_m \text{ and } (\eta + 1)T\_m \le t \le \left\{(\eta + 1)T\_m + t\_p^2\right\} \\\ \mathbf{0}; & \text{elsewhere} \end{cases} \tag{15}$$

The complex Fourier coefficient for the pulse shifting strategy at kth harmonic due to the pth element under the two cases can be obtained, respectively, as [14].

$$Pulse\ shifting\ (Case\ 1)\ C\_{pk} = t\_p^{on} \frac{\sin\left(k\pi t\_p^{on}\right)}{k\pi t\_p^{on}} e^{-jk\pi\left(t\_p^{on} + 2t\_p^1\right)}\tag{16}$$

$$\text{Pulse shifting (Case 2) } C\_{pk} = \frac{1}{k\pi} \left\{ \begin{aligned} \sin\left[k\pi \left(1 - t\_p^1\right)\right] e^{-jk\pi \left(1 + t\_p^1\right)} + \\ \sin\left[k\pi \left(t\_p^1 + t\_p^{on} - T\_m\right)\right] e^{-jk\pi \left(t\_p^1 + t\_p^{on} - T\_m\right)} \end{aligned} \right\} \tag{17}$$

By taking into account the additional degree of freedom, namely, on-time instants of the antenna elements, improved array patterns can be observed. For example, more sideband reduction as compared to VAS approach is obtained when the same array pattern is synthesized at the center frequency [18–19], and electronic beam steering [9] and harmonic beam patterns of different shapes [7, 8] can be realized without phase shifters.

#### 4.3 Binary optimized time sequence (BOTS)

In binary optimized time sequence (BOTS), the switch-on time duration of an arbitrary pth element is divided into Q number of minimal time steps of equal length over a modulation time period Tm [20] as shown in Figure 9. The minimal time step, t 0 , is given by

$$t^0 = t^1 = t^2 = \dots = t^q = \dots = t^Q \tag{18}$$

Figure 9 Switching function defining binary optimized time sequence (BOTS) strategy.

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach DOI: http://dx.doi.org/10.5772/intechopen.89479

The periodic on–off sequence of the set of time steps corresponding to the pth element is represented by the switching function Up(t). If the on–off status of qth time step for the pth element is symbolized with a binary bit, bq <sup>p</sup>, for which "on" status corresponds to bq <sup>p</sup>=1 and that for "off" status <sup>b</sup><sup>q</sup> <sup>p</sup>=0, then the set of time steps for the pth element is given as b<sup>1</sup> pb2 pb3 p⋯bQ <sup>p</sup> . In order to synthesize the desired pattern, the optimal binary arrangement of the bit patterns, i.e., set of time steps to be under on states and off states, can be found by employing simple genetic algorithm (SGA) [16]. Considering each time step, bq <sup>p</sup>, as a gene, then on–off time sequence of N array elements represents the chromosome (χ) of GA and is given as

$$\chi = b\_1^1 b\_1^2 b\_1^3 \cdots b\_1^Q b\_2^1 b\_2^2 b\_2^3 \cdots b\_2^Q \cdots b\_p^1 b\_p^2 b\_p^3 \cdots b\_p^Q \cdots b\_N^1 b\_N^2 b\_N^3 \cdots b\_N^Q \tag{19}$$

The complex Fourier coefficient of pth element at kth harmonic with the BOTS switching scheme can be obtained as [20]

$$BOTS: C\_{pk} = \frac{\sin\left(k\pi\tau\_0\right)}{k\pi} \sum\_{q=1}^{Q} b\_p^q e^{-jk\pi\tau\_0(2q-1)}\tag{20}$$

where <sup>τ</sup><sup>0</sup> <sup>¼</sup> <sup>t</sup> 0 Tm is the normalized time step. Thus, by incorporating more number of the degrees of freedom as the optimization variables to the evolutionary algorithm, the radiation pattern characteristics like SLL, SBL, SBR, etc. can be controlled skillfully.

### 4.4 Subsectional optimized time steps (SOTS)

In SOTS-based switching strategy, the time-modulation period (Tm) is divided into a number of subsections with variable lengths [21]. Let us assume that Tm is divided into Q number of time steps as shown in Figure 10 for the switching strategy of pth element of the array. For the qth time step, the on and off time instants of the switch are denoted by t qon <sup>p</sup> and t qoff <sup>p</sup> , respectively, and the resultant ontime duration at qth step is obtained as t q <sup>p</sup> ¼ t qoff <sup>p</sup> � t qon <sup>p</sup> . Therefore, the periodic time switching pulse Upð Þt for the different time steps over a complete modulation period is represented as

$$U\_p(t) = \begin{cases} \mathbf{1}; & \mathbf{0} \le t\_p^{q\_m} \le t \le t\_p^{q\_{\text{eff}}} \le T\_m: \quad \forall q \in [\mathbf{1}, Q] \\\ \mathbf{0}; & \text{elsewhere} \end{cases} \tag{21}$$

The Fourier coefficient at the kth harmonics for the pth element can be written as

$$\mathbf{C}\_{pk} = \sum\_{q=1}^{Q} \frac{\left(\mathbf{t}\_p^{q\_{q\overline{f}}} - \mathbf{t}\_p^{q\_m}\right)}{T\_m} \sin c \left(k o\_m \left(\mathbf{t}\_p^{q\_{q\overline{f}}} - \mathbf{t}\_p^{q\_m}\right)\right) e^{-jk\alpha\_m \left(\mathbf{t}\_p^{q\_m} + \mathbf{t}\_p^{q\_{q\overline{f}}}\right)} \tag{22}$$

where <sup>ω</sup><sup>m</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> Tm denotes the modulation frequency used for the time modulation.

It can be observed that, if the number of subsections Q is 1, then SOTS is transformed into pulse shifting-based strategy. On the other scenario, if the on-time duration at each step, i.e., the separation between the on and off time instants, becomes multiples of Tm <sup>Q</sup> , then SOTS takes the form of BOTS. So, SOTS-based switching strategy provides more flexibility in the design of optimized time sequences as compared to the other abovementioned switching strategies. However such improved flexibility in synthesizing the pattern is obtained at the cost of some

UpðÞ¼ <sup>t</sup> 1; <sup>η</sup>Tm <sup>þ</sup> <sup>t</sup>

(

Advances in Array Optimization

t 1 <sup>p</sup> þ t on

Eq. (15):

step, t 0

Figure 9

14

UpðÞ¼ <sup>t</sup> 1; <sup>η</sup>Tm <sup>þ</sup> <sup>t</sup>

Pulse shifting Case 2 ð Þ Cpk <sup>¼</sup> <sup>1</sup>

be realized without phase shifters.

, is given by

4.3 Binary optimized time sequence (BOTS)

t <sup>0</sup> <sup>¼</sup> <sup>t</sup>

Switching function defining binary optimized time sequence (BOTS) strategy.

<sup>1</sup> <sup>¼</sup> <sup>t</sup>

(

1 p � �

Hence, the normalized switch-on time duration, <sup>τ</sup>p, is given as <sup>τ</sup><sup>p</sup> <sup>¼</sup> <sup>t</sup>

Another possible situation may appear as shown in Figure 8(b) where t

<sup>p</sup> ≥Tm. Under such situation, the switching operation can be expressed as in

The complex Fourier coefficient for the pulse shifting strategy at kth harmonic due to the pth element under the two cases can be obtained, respectively, as [14].

> on p

1 <sup>p</sup> þ t on <sup>p</sup> � Tm h i � �

sin kπ 1 � t

sin kπ t

By taking into account the additional degree of freedom, namely, on-time instants of the antenna elements, improved array patterns can be observed. For example, more sideband reduction as compared to VAS approach is obtained when the same array pattern is synthesized at the center frequency [18–19], and electronic beam steering [9] and harmonic beam patterns of different shapes [7, 8] can

In binary optimized time sequence (BOTS), the switch-on time duration of an arbitrary pth element is divided into Q number of minimal time steps of equal length over a modulation time period Tm [20] as shown in Figure 9. The minimal time

<sup>2</sup> <sup>¼</sup> … <sup>¼</sup> <sup>t</sup>

<sup>q</sup> <sup>¼</sup> … <sup>¼</sup> <sup>t</sup>

sin kπt

1 p h i � �

kπton p

> e �jkπ 1þt <sup>1</sup> ð Þ<sup>p</sup> <sup>þ</sup>

0; elsewhere

Thus, the pulse shifting strategy reduces to VAS with t

1 p � �

Pulse shifting Case 1 ð Þ Cpk ¼ t

kπ

8 ><

>:

0; elsewhere

≤t ≤ ηTm þ t

2 p � �

> 1 <sup>p</sup>=0.

≤t≤ð Þ η þ 1 Tm and ð Þ η þ 1 Tm ≤t≤ ð Þ η þ 1 Tm þ t

on p � �

e �jkπ t on <sup>p</sup> þ2t

> e �jkπ t 1 <sup>p</sup>þt on ð Þ <sup>p</sup> �Tm

≤ ð Þ η þ 1 Tm

(14)

2 p

(15)

(17)

on p Tm <sup>¼</sup> <sup>t</sup> 2 <sup>p</sup>�t 1 p Tm .

n o

<sup>1</sup> ð Þ<sup>p</sup> (16)

<sup>Q</sup> (18)

9 >=

>;

1 <sup>p</sup>>t 2 <sup>p</sup> and

Figure 10.

The schematic of the periodic pulse sequence for SOTS switching strategy of the pth element of TMLAA.

increased design complexity, because realization of the number of unequal subsections with smaller section over the modulation period needs faster switching operation.
