2. Theory of time-modulated antenna array (TMAA)

Let us consider a linear antenna array of N number of mutually uncoupled isotropic radiators with inter-element spacing d0. The antenna elements are placed along the x-axis with the first element at the origin of the geometrical coordinate system as shown in Figure 1. In the XZ plane (one of the vertical principle plane), the array factor expression of CAAs can be obtained as in Eq. (1) [1]:

$$AF^{\mathbf{x}} = \sum\_{p=1}^{N} A\_p e^{j\Phi\_p} e^{j[a\_0\mathbf{t} + (p-1)\beta d\_0 \cos\theta]} \tag{1}$$

where ω<sup>0</sup> = 2πf0 = 2π/T0 is the angular frequency in rad/sec for the operating signal of frequency f0 in Hz; T0 is the time period of the operating signal; β = 2π/λ is the wave number with λ being the wavelength; p = 1, … … , N represents the element number of the antenna array; Ap and Ф<sup>p</sup> ∀p ∈½ � 1, N stand for the normalized static excitation amplitudes and phases of the array elements, respectively; and θ is the angle made by the line joining the observing point and the origin with the x-axis as shown in Figure 1.

In order to control the antenna pattern by using the additional degree of freedom, namely, "time," periodically the static excitation amplitudes of the antenna element are time-modulated. The commonly used and simplest way of doing that is to insert high-speed radio-frequency (RF) switches in the feed network, just prior to radiating sources as shown in Figure 2. Each array element is assumed to be connected to the RF switches with individually controlled switching circuits. The switches are periodically "on" and "off" according to a predetermined on-time sequence t on <sup>p</sup> (0≤ t on <sup>p</sup> ≤Tm)∀p∈½ � 1, N , with time period,Tm. The switching rate, fm = <sup>1</sup>=Tm, is selected such that if the maximum frequency of the message signal is fmax (Hz), T<sup>0</sup> < <Tm ≤ <sup>1</sup> f max [14]. Thus, during each period, the on-time duration by which a switch is on, the array element connected to that switch is active for that time duration only; otherwise, it will be inactive.

Figure 1. Basic antenna array of N element with inter-element spacing of d0.

transmitter power from a single antenna element needs high-power amplification in the feed network. The high-power amplifier is not easy to design and safe to handle. Therefore, a number of antenna elements are arranged along a line, called linear antenna array (LAA), or in a plane called planer antenna array (PAA). The use of multiple antenna elements in the transmission and reception systems simplifies the power amplifier design problem by reducing the power level per transmitting antenna elements of the arrays. Some other advantages of using antenna arrays are to improve signal fading resistance or deliberately exploit the signal fading; mitigate the interfering signal coming from other directions, adaptive beam forming, and null steering at both transmitter and receiver; and increase system capacity. Due to its high gain and narrow beamwidth, the large antenna arrays also find applications in weather forecast, astronomy, image processing, and

Although the antenna array with uniform excitation amplitude and equally spaced antenna elements is the simplest one for practical implementation and also can be used to synthesize different patterns, due to the high value of peak SLL, it is impractical to use in such applications. In conventional antenna array (CAA) system, the low side lobe pattern is obtained by tapering the static excitation amplitudes. The well-known analytical techniques to taper amplitude distributions in nonuniformly excited antenna arrays are Dolph-Chebyshev (DC) and Taylor series [1]. However, the high dynamic range ratio (DRR) and complex excitation of the antenna elements are the major drawbacks of such CAA synthesis method with nonuniform excitation, because the complex excitation is practically difficult to realize and designing the practical antenna with high DRR of static amplitude tapering provides various errors such as systematic errors and random errors.

Conversely, the ultralow SLL pattern in the far-field of the antenna array can be realized even in uniform amplitude antenna arrays by exploiting "time" as a fourth dimension [2, 3]. The introduction of the additional dimension "time," into the antenna array system, results in time-modulated antenna array (TMAA). By using the fourth degree of freedom, "time" in antenna array system, various errors in realizing the low SLL pattern can be drastically reduced, and error tolerance levels become equivalent to those obtained in conventional antenna array system for the patterns of ordinary SLLs [4, 5]. Yet, the main disadvantage in TMAA is the generation of sideband signals which appeared due to the time modulation of the antenna signals by periodically commutating the antenna elements with the specified modulation frequency. Therefore, time modulation involves with the radiation or reception of electromagnetic energy at different harmonics of the modulation frequency that are termed as sidebands. In some applications where the antenna array is synthesized at center (operating) frequency, sideband signals are not useful. In such cases, sideband signals and associated power losses are suppressed to improve the radiation efficiency at the operating frequency of the antenna array [5, 6]. Presently, it is investigated that sideband signals are also effective in synthesizing multiple patterns and researchers are interested to exploit the same in some specific applications of the modern-day communication systems like harmonic beam forming [7], generation of multibeam radiation pattern [8], beam steering [9, 10], direction finding [11], wireless power transmission [12], etc. The interested readers may refer to Reference [13] for the stateof-the-art overview, applications, and present research trend on time-modulation

This chapter explains about the fundamental theory and techniques of different

time-modulation strategies and such antenna array synthesis methods using optimization algorithms. The parameters involved with the use of optimization

techniques and TMAA synthesis problem have also been presented.

biomedical imaging.

Advances in Array Optimization

theory and techniques.

4

## Advances in Array Optimization

Let us further assume that all the switches corresponding to the antenna elements in Figure 2 are on (short circuited) at the same instant of time, say at the beginning of each period "η\*Tm" with "η" being the time period number 0, 1, 2, … , by using rectangular pulses of amplitude unity. Hence, the switches which are on for the whole time period Tm as shown in Figure 3(a) can be directly connected to the signal as time modulation is not required for such cases. On the other hand, the switches remained short circuited for their specific on-time duration and open

circuited after their corresponding on-time duration (t

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

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

such that at each period

written as in Eq. (3) [2]:

technique as

where τ<sup>p</sup> ¼ t

array elements.

7

on

AFð Þ¼ θ, t

readily obtained by combining Eqs. (5) and (6) as

is obtained as in Eqs. (8), (9), and (10), respectively:

AFkð Þ¼ θ, t e

Figure 3(b) shows the on–off time sequence of the switches for the first two time periods only. The same process is repeated in the next consecutive periods. Thus the switching function of the pth element can be expressed by a periodic pulse Up (t),

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

0; elsewhere (

After the switching operation, the array factor expression of Eq. (1) can be

Upð Þt Ape

where <sup>α</sup><sup>p</sup> <sup>¼</sup> ð Þ <sup>p</sup> � <sup>1</sup> <sup>β</sup>d0f g cos <sup>θ</sup> � cos <sup>θ</sup><sup>0</sup> is the linear progressive phase shift of pth element and θ<sup>0</sup> is the direction of maximum radiation. As Up (t) in Eq. (3) is a time periodic function of periodicity Tm, it can be decomposed by applying Fourier series

N

p¼1

UpðÞ¼ t

Cpk ¼ τ<sup>p</sup>

k X ¼þ∞

jð Þ ω0þkω<sup>m</sup> t

k¼�∞

X N

p¼1

X N

Apτ<sup>p</sup>

Therefore, the array factor at the fundamental frequency, i.e., at operating frequency (for k ¼ 0) and at the first two positive harmonics (for k ¼ 1 and k ¼ 2),

p¼1

Thus, Eq. (6) expresses that the signal is not only radiated at the operating frequency, ω<sup>0</sup> for k=0, but also the signals are radiated at different harmonics of the modulating frequency, kωm, with ω<sup>0</sup> as the center frequency. The signal radiation at different harmonics is termed as sideband radiation (SBR). For such a TMLAA, the array factor expression at kth harmonic of the modulation frequency is

k X ¼þ∞

coefficient at the kth harmonics for the pth element and is obtained as [5, 14]

k¼�∞

where ω<sup>m</sup> ¼ 2π=Tm ¼ 2πf <sup>m</sup> is the modulation frequency and Cpk is the Fourier

sin kπτ<sup>p</sup> � � kπτ<sup>p</sup>

Putting Eq. (4) in Eq. (3), the array factor expression of Eq. (3) is obtained as

ApCpke

Cpke

e

<sup>p</sup> =Tm∀p ∈½ � 1, N stand for the normalized on-time durations of the

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

sin kπτ<sup>p</sup> � � kπτ<sup>p</sup>

e

AFð Þ¼ <sup>θ</sup>, <sup>t</sup> <sup>X</sup>

on

on p � �

<sup>j</sup>½ � <sup>ω</sup>0tþαpþΦ<sup>p</sup> (3)

jkωmt (4)

�jkπτ<sup>p</sup> (5)

<sup>j</sup>ð Þ <sup>ω</sup>0þkω<sup>m</sup> <sup>t</sup> (6)

�j k½ � πτp�ð Þ <sup>Φ</sup>pþα<sup>p</sup> (7)

<sup>p</sup> ) as shown in Figure 3(b).

(2)

Figure 2. Time-modulated linear antenna array (TMLAA) geometry.

#### Figure 3.

The periodic pulse sequence of the TMLAA. (a) Unit pulse of periodicity TP. (b) On–off time duration of each antenna elements for one time-modulation period TP, and it is repeated at every TP time interval.

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

circuited after their corresponding on-time duration (t on <sup>p</sup> ) as shown in Figure 3(b). Figure 3(b) shows the on–off time sequence of the switches for the first two time periods only. The same process is repeated in the next consecutive periods. Thus the switching function of the pth element can be expressed by a periodic pulse Up (t), such that at each period

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

After the switching operation, the array factor expression of Eq. (1) can be written as in Eq. (3) [2]:

$$AF(\theta, t) = \sum\_{p=1}^{N} U\_p(t) A\_p e^{j\left[a\_0 t + a\_p + \Phi\_p\right]} \tag{3}$$

where <sup>α</sup><sup>p</sup> <sup>¼</sup> ð Þ <sup>p</sup> � <sup>1</sup> <sup>β</sup>d0f g cos <sup>θ</sup> � cos <sup>θ</sup><sup>0</sup> is the linear progressive phase shift of pth element and θ<sup>0</sup> is the direction of maximum radiation. As Up (t) in Eq. (3) is a time periodic function of periodicity Tm, it can be decomposed by applying Fourier series technique as

$$U\_p(t) = \sum\_{k=-\infty}^{k=+\infty} C\_{pk} e^{jk\alpha\_m t} \tag{4}$$

where ω<sup>m</sup> ¼ 2π=Tm ¼ 2πf <sup>m</sup> is the modulation frequency and Cpk is the Fourier coefficient at the kth harmonics for the pth element and is obtained as [5, 14]

$$C\_{pk} = \tau\_p \frac{\sin\left(k\pi\tau\_p\right)}{k\pi\tau\_p} e^{-jk\pi\tau\_p} \tag{5}$$

where τ<sup>p</sup> ¼ t on <sup>p</sup> =Tm∀p ∈½ � 1, N stand for the normalized on-time durations of the array elements.

Putting Eq. (4) in Eq. (3), the array factor expression of Eq. (3) is obtained as

$$AF(\theta, t) = \sum\_{k=-\infty}^{k=+\infty} \sum\_{p=1}^{N} A\_p C\_{pk} e^{j\left(\Phi\_p + a\_p\right)} e^{j\left(a\_0 + ka\_m\right)t} \tag{6}$$

Thus, Eq. (6) expresses that the signal is not only radiated at the operating frequency, ω<sup>0</sup> for k=0, but also the signals are radiated at different harmonics of the modulating frequency, kωm, with ω<sup>0</sup> as the center frequency. The signal radiation at different harmonics is termed as sideband radiation (SBR). For such a TMLAA, the array factor expression at kth harmonic of the modulation frequency is readily obtained by combining Eqs. (5) and (6) as

$$AF\_k(\theta, t) = e^{j(\alpha\_0 + k\nu\_m)t} \sum\_{p=1}^N A\_p \tau\_p \frac{\sin\left(k\pi\tau\_p\right)}{k\pi\tau\_p} e^{-j\left[k\pi\tau\_p - \left(\Phi\_p + a\_p\right)\right]} \tag{7}$$

Therefore, the array factor at the fundamental frequency, i.e., at operating frequency (for k ¼ 0) and at the first two positive harmonics (for k ¼ 1 and k ¼ 2), is obtained as in Eqs. (8), (9), and (10), respectively:

Let us further assume that all the switches corresponding to the antenna elements in Figure 2 are on (short circuited) at the same instant of time, say at the beginning of each period "η\*Tm" with "η" being the time period number 0, 1, 2, … , by using rectangular pulses of amplitude unity. Hence, the switches which are on for the whole time period Tm as shown in Figure 3(a) can be directly connected to the signal as time modulation is not required for such cases. On the other hand, the switches remained short circuited for their specific on-time duration and open

The periodic pulse sequence of the TMLAA. (a) Unit pulse of periodicity TP. (b) On–off time duration of each

antenna elements for one time-modulation period TP, and it is repeated at every TP time interval.

Figure 2.

Figure 3.

6

Time-modulated linear antenna array (TMLAA) geometry.

Advances in Array Optimization

$$AF\_0(\theta, t) = \sigma^{javt} \sum\_{p=1}^{N} A\_p \tau\_p \sigma^{j\left(\Phi\_p + a\_p\right)}\tag{8}$$

$$AF\_1(\theta, t) = \frac{\mathcal{e}^{j(a\_0 + a\_m)t}}{\pi} \sum\_{p=1}^{N} A\_p \sin\left(\pi \tau\_p\right) e^{-j\left[\pi \tau\_p - \left(\Phi\_p + a\_p\right)\right]} \tag{9}$$

$$AF\_2(\theta, t) = \frac{\mathfrak{e}^{j(a\_0 + 2a\_m)t}}{2\pi} \sum\_{p=1}^{N} A\_p \sin\left(2\pi\mathfrak{r}\_p\right) e^{-j\left[2\pi\mathfrak{r}\_p - \left(\Phi\_p + a\_p\right)\right]} \tag{10}$$

From Eq. (8), it can be observed that τp's∀p ∈½ � 1, N provides an additional flexibility in synthesizing antenna array patterns. For example, making values of τp's∀p∈ ½ � 1, N equivalent to that of the required static excitation to synthesize Dolph-Chebyshev or Taylor series pattern, low SLL patterns can be realized even with uniformly excited array with unit static excitation Ap = 1∀p∈ ½ � 1, N . Also, Eqs. (9) and (10) indicate that the harmonics radiated from different timemodulated elements are added together at frequencies in multiples of the modulation frequency, fm, to produce resultant sideband signals.
