4.5 Quantized aperture size (QAS)

In Section 3, the different patterns of desired values of SLL at f0 are obtained by making the on-time sequence equal to the Dolph-Chebyshev coefficient of the corresponding patterns. Thus the appropriate set of on-time sequence is required to generate the desired pattern even in uniformly excited TMAAs.

termed as uniform period modulation (UPM). Time-modulated antenna array (TMAA) based on UPM is commonly known as uniform TMAA (UTMAA). On the other hand, if the antenna elements of the array are time-modulated with different modulation frequencies as shown in Figure 12, it is defined as time modulation with nonuniform period modulation (NPM), and the corresponding array is defined as nonuniform TMAA (NTMAA) [23–24]. Let us consider that the antenna elements are modulated with different modulation periods Tp : ∀p∈½ � 1, N having modulation

Time-modulated array architecture with NPM switching strategy where f1 6¼ f2 6¼ … 6¼ fN.

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

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

period and frequency of the pth antenna element of the array. The periodic

0; 0 <t≤ t

And finally, Fourier coefficient at the kth harmonics for the pth element is

2 <sup>p</sup> � t 1 p

<sup>k</sup><sup>π</sup> <sup>e</sup>

n o � �

1; t 1 <sup>p</sup> <t≤ t 2 p

sin kπf <sup>p</sup> t

switching pulse of the pth antenna element Upð Þ<sup>t</sup> is written as

(

UpðÞ¼ t

Cpk ¼

Apτpe

X N

p¼1 Ap on

then the corresponding array factor expression is written as

<sup>j</sup>ð Þ <sup>Φ</sup>pþα<sup>p</sup>

sin kπf <sup>p</sup>t

where ω<sup>0</sup> ¼ 2πf <sup>0</sup> denotes the center frequency of the array.

on p n o

�jkπf <sup>p</sup> 2t 1 <sup>p</sup>þt on ð Þ <sup>p</sup> e

<sup>k</sup><sup>π</sup> <sup>e</sup>

time modulate the pth antenna element.

: ∀p∈ ½ � 1, N , where Tp and fp, respectively, denote the modulation

<sup>p</sup> denote the on-time and off-time instant of the RF switch used to

�jkπf <sup>p</sup> t 2 <sup>p</sup>þt

<sup>p</sup> denote the normalized switch-on time duration, and

<sup>j</sup>2<sup>π</sup> <sup>f</sup> <sup>0</sup>þkf ð Þ<sup>p</sup> <sup>t</sup>

e <sup>j</sup>ð Þ <sup>Φ</sup>pþα<sup>p</sup>

<sup>1</sup> ð Þ<sup>p</sup> (24)

(23)

(25)

1 <sup>p</sup> or t 2 <sup>p</sup> < t≤Tp

frequency <sup>f</sup> <sup>p</sup> <sup>¼</sup> <sup>1</sup>

Figure 12.

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

obtained as [24].

Let τ<sup>p</sup> ¼ f <sup>p</sup> t

AFð Þ¼ θ, t e

17

2 <sup>p</sup> � t 1 p � � <sup>¼</sup> <sup>f</sup> <sup>p</sup><sup>t</sup>

jω0t X N

<sup>þ</sup> <sup>X</sup><sup>∞</sup> k¼�∞, k6¼0

p¼1

Tp

In this section, to generate different patterns in time-modulated antenna arrays (TMAAs) instead of considering continuous value of on-time duration [22], the modulation period is divided into a number of equal steps as in BOTS. However, in BOTS, multiple switching of on–off over the modulation period is considered. Such multiple changes of switching states over the modulation period need fast and complex switching circuit. Unlike BOTS, in this modulation scheme, the on–off states of the switches are assumed to change once over the complete modulation period like VAS. However, the on–off states of the switches are rounded off to the nearest quantization step to obtain quantized on-times (QOTs) of the corresponding elements as shown in Figure 11. In this time-modulation scheme, the time-modulation period,Tm, is quantized into "Q" number of discrete levels. At qth quantization level, the value of tq is given by q\*(Tm/Q), where q = 1, 2 … Q. The allowable on-time t on <sup>p</sup> of the <sup>p</sup>th array elements is taken as tq with q = 1, 2 … Q during each modulation period. Similar to the previously reported VAS time-modulation technique in which the continuous values of on-time durations are optimized to synthesize the desired pattern, this approach is defined as VAS with quantized on-time (VAS-QOT) or simply "quantized aperture size" (QAS) time modulation as the aperture size changes with quantized values of on-time durations of the elements.

#### 4.6 Nonuniform period modulation (NPM)

In all of the abovementioned switching strategies, all antenna elements are modulated with the same modulation frequency, ωm, and such time modulation is

Figure 11.

The proposed time-modulation approach for the quantized on-time of the switches.

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

Figure 12. Time-modulated array architecture with NPM switching strategy where f1 6¼ f2 6¼ … 6¼ fN.

termed as uniform period modulation (UPM). Time-modulated antenna array (TMAA) based on UPM is commonly known as uniform TMAA (UTMAA). On the other hand, if the antenna elements of the array are time-modulated with different modulation frequencies as shown in Figure 12, it is defined as time modulation with nonuniform period modulation (NPM), and the corresponding array is defined as nonuniform TMAA (NTMAA) [23–24]. Let us consider that the antenna elements are modulated with different modulation periods Tp : ∀p∈½ � 1, N having modulation frequency <sup>f</sup> <sup>p</sup> <sup>¼</sup> <sup>1</sup> Tp : ∀p∈ ½ � 1, N , where Tp and fp, respectively, denote the modulation period and frequency of the pth antenna element of the array. The periodic switching pulse of the pth antenna element Upð Þ<sup>t</sup> is written as

$$U\_p(t) = \begin{cases} 1; & t\_p^1 < t \le t\_p^2 \\ 0; & 0 < t \le t\_p^1 \text{ or } \ t\_p^2 < t \le T\_p \end{cases} \tag{23}$$

where t 1 <sup>p</sup> and t 2 <sup>p</sup> denote the on-time and off-time instant of the RF switch used to time modulate the pth antenna element.

And finally, Fourier coefficient at the kth harmonics for the pth element is obtained as [24].

$$C\_{pk} = \frac{\sin\left\{k\pi\mathfrak{f}\_p\left(t\_p^2 - t\_p^1\right)\right\}}{k\pi} e^{-jk\pi\mathfrak{f}\_p\left(t\_p^2 + t\_p^1\right)}\tag{24}$$

Let τ<sup>p</sup> ¼ f <sup>p</sup> t 2 <sup>p</sup> � t 1 p � � <sup>¼</sup> <sup>f</sup> <sup>p</sup><sup>t</sup> on <sup>p</sup> denote the normalized switch-on time duration, and then the corresponding array factor expression is written as

$$\begin{split} AF(\theta, t) &= \sigma^{j\text{out}} \sum\_{p=1}^{N} A\_p \tau\_p \mathfrak{e}^j(\Phi\_{\mathbb{P}} + a\_{\mathbb{P}}) \\ &+ \sum\_{k=-\infty,\ \ \text{ $p=1$ }}^{\infty} \sum\_{p=1}^{N} A\_p \frac{\sin\left\{k\pi f\_p t\_p^{\text{out}}\right\}}{k\pi} e^{-jk\pi f\_p \left(2t\_p^1 + t\_p^{\text{on}}\right)} \mathfrak{e}^{j2\pi \left(f\_0 + 4f\_p\right)t} \mathfrak{e}^j(\Phi\_{\mathbb{P}} + a\_{\mathbb{P}}) \\ &\neq 0 \end{split} \tag{25}$$

where ω<sup>0</sup> ¼ 2πf <sup>0</sup> denotes the center frequency of the array.

increased design complexity, because realization of the number of unequal

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

to generate the desired pattern even in uniformly excited TMAAs.

nearest quantization step to obtain quantized on-times (QOTs) of the

operation.

Figure 10.

allowable on-time t

elements.

Figure 11.

16

on

4.6 Nonuniform period modulation (NPM)

4.5 Quantized aperture size (QAS)

Advances in Array Optimization

subsections with smaller section over the modulation period needs faster switching

In Section 3, the different patterns of desired values of SLL at f0 are obtained by making the on-time sequence equal to the Dolph-Chebyshev coefficient of the corresponding patterns. Thus the appropriate set of on-time sequence is required

In this section, to generate different patterns in time-modulated antenna arrays (TMAAs) instead of considering continuous value of on-time duration [22], the modulation period is divided into a number of equal steps as in BOTS. However, in BOTS, multiple switching of on–off over the modulation period is considered. Such multiple changes of switching states over the modulation period need fast and complex switching circuit. Unlike BOTS, in this modulation scheme, the on–off states of the switches are assumed to change once over the complete modulation period like VAS. However, the on–off states of the switches are rounded off to the

corresponding elements as shown in Figure 11. In this time-modulation scheme, the time-modulation period,Tm, is quantized into "Q" number of discrete levels. At qth quantization level, the value of tq is given by q\*(Tm/Q), where q = 1, 2 … Q. The

each modulation period. Similar to the previously reported VAS time-modulation technique in which the continuous values of on-time durations are optimized to synthesize the desired pattern, this approach is defined as VAS with quantized on-time (VAS-QOT) or simply "quantized aperture size" (QAS) time modulation as

In all of the abovementioned switching strategies, all antenna elements are modulated with the same modulation frequency, ωm, and such time modulation is

the aperture size changes with quantized values of on-time durations of the

The proposed time-modulation approach for the quantized on-time of the switches.

<sup>p</sup> of the <sup>p</sup>th array elements is taken as tq with q = 1, 2 … Q during

The first summation indicates that the signals radiated at the center frequency ω<sup>0</sup> are accumulated in the space, whereas the second summation is due to the signals radiated at different harmonics. Now, if the modulation frequencies of the antenna elements are selected in such a way that f1 = f<sup>2</sup> = … = fN = fm, then the scenario becomes UTMAA, and the term kfp in the second summation becomes kfm that means that the kth-order harmonics of all the elements appeared at the same frequency. The scenario is the same for all other order of harmonics. As a result, radiated signals at the same frequency are accumulated in space, which in turn increases the resultant SBL.

of the main beam at operating frequency and low value of maximum sideband level (SBLmax) for synthesizing pencil beam pattern while one more objective is low

TMAA synthesis problem is non-convex and nonlinear in nature. A number of numerical techniques as already mentioned—Dolph-Chebyshev and Taylor series [1]—are available to synthesize pencil beam power pattern in conventional antenna arrays (CAAs). Also, some analytical methods are reported to generate shaped beam patterns and phase-only controlled multiple power patterns in CAAs [27, 28, 29]. Durr et al. described a modified Woodward-Lawson technique to design phasedifferentiated multiple pattern antenna arrays with prefixed amplitude distributions [27]. The analytical technique reported in [28] is used to determine the nonlinear phase distribution of linear arrays. A method based on projection approach [29] is proposed to synthesize reconfigurable array antennas of a cosecant<sup>2</sup> beam and a flattop beam (FTB) by using a common amplitude with

phase-only control of analog phase shifters. Though these numerical and analytical techniques can also be applied to determine the nonlinear distributions of dynamic excitation coefficient and phase to synthesize power pattern at operating frequency of TMAAs, such methods have no control on sideband power level. Therefore, the powerful global stochastic optimization tools such as genetic algorithm (GA) [30], differential evolution (DE) [4–5, 31, 32], particle swarm optimization (PSO) [7], simulated annealing (SA) [6, 33], and artificial bee colony (ABC) [22, 34] are essentially required to solve such multi-objective TMAA synthesis problems.

Most of the TMAA synthesis problems are solved by applying single-objective optimization method where all the objectives are added with different weighting factors to form a single cost function and the cost function is minimized by

employing heuristic evolutionary algorithms. The different stochastic optimization techniques are used with the objective to synthesize desired patterns at the operating frequency by reducing SLL and SBL. One of the commonly used techniques to define the cost function of such conflicting multi-objective TMAA synthesis prob-

where χis the set of unknown parameters, termed as optimization parameter vector which is to be determined by the used evolutionary algorithm; δ<sup>h</sup> with h = 0, 1, 2, … .V are the different parameters of the desired patterns; and δhd are the desired values of the specific parameters. For example, δ<sup>0</sup> is the maximum SLL (SLLmax) of the pattern at f0, δ<sup>1</sup> is the maximum of sideband radiations (SBRmax) among the first five sidebands, and δ<sup>2</sup> represents FNBW. "Wh" is the weighting factors for the corresponding terms. Hð Þ: is the Heaviside step function. "ρ" is any natural number. It can be seen from Eqs. (13) and (26) that when the obtained values of δ<sup>h</sup> are close to their desired values, the cost function value is moving toward zero. Thus, reaching zero value of the cost function confirms that the synthesized pattern satisfies the requirements in terms of the desired values of the intended synthesizing parameters. To illustrate the effectiveness of the cost

Wh:Hð Þ <sup>δ</sup>hd � <sup>δ</sup><sup>h</sup> :ð Þ <sup>δ</sup>hd � <sup>δ</sup><sup>h</sup> <sup>ρ</sup> (26)

ripple level for synthesizing shaped beam patterns.

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

5.3 The need of evolutionary algorithm

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

5.4 Cost function with multiple objectives

ψ χð Þ¼

h X¼V h¼0

lem is as expressed in Eq. (26):

19

But in the case of NTMAA, the modulation frequencies are selected in such a way that f1 6¼ f<sup>2</sup> 6¼ … 6¼ fN. So, due to different modulation frequencies of different antenna elements, the signals radiated from different harmonics appeared at different frequencies, and the term kfp in the second summation of [25] becomes different for different elements. That means the kth-order harmonics of different elements appear at different frequencies and the scenario is the same for all the other order harmonics. So, unlike UTMAA, the harmonic signals appeared at different frequencies and are distributed in space, which in turn decreases the resultant SBL [23]. Recently, some research works have reported the calculation of the sideband power of NTMAA [24–25], and also the reduction of the sideband power losses using NTMAA is investigated [26].
