3.3 Influence on the sideband level and first null beamwidth during reduction of side lobe level of the fundamental pattern

It is understood that in addition to the desired operating frequency (center frequency), TMAAs also radiate signals at the infinite number of different

AF0ð Þ¼ θ, t e

ejð Þ <sup>ω</sup>0þω<sup>m</sup> <sup>t</sup> π

ejð Þ <sup>ω</sup>0þ2ω<sup>m</sup> <sup>t</sup> 2π

tion frequency, fm, to produce resultant sideband signals.

3. Behaviors of sideband radiation (SBR)

3.1 Characteristics of harmonic radiations (HRs)

hpk ¼ 20 log <sup>10</sup> Cpk

� � �

�= Cp<sup>0</sup> � � �

where hpk is the normalized/relative harmonic radiation corresponding to the pth element. The variation of normalized harmonic power of the first three harmonics (k=1, 2, and 3) with normalized switch-on time, τp, over its complete range (0, 1) is shown in Figure 4. As can be seen, at the lower value of τp, all hpkmax are almost the same, and for τ<sup>p</sup> ! 0, all hpkmax are exactly equal to 0 (zero) dB as it is expected from the Fourier series of unit impulse function. However, at the other extremes of τp, when τ<sup>p</sup> ! 1, all hpkmax ! �∞, which is the predicted result as can be seen in Eqs. (5) to (10), with k=1, 2, and 3. Again there is no radiation at hp2 for τ<sup>p</sup> = 0.5 and at hp 3 for τ<sup>p</sup> = 0.3 and 0.66 which can also be verified from Eq. (5) with k = 1, 2, and 3. Thus, Figure 4 indicates that the contribution of the harmonic component

� <sup>¼</sup> sin <sup>k</sup>πτ<sup>p</sup>

� � kπτ<sup>p</sup>

(11)

ment is given as [15]

8

AF1ð Þ¼ θ, t

Advances in Array Optimization

AF2ð Þ¼ θ, t

jω0t X N

X N

p¼1

X N

p¼1

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 modula-

It can be observed from Eqs. (7)–(10) that, due to time modulation, the sideband signals inherently appeared around the center frequency spaced in multiples of the modulation frequency. In this section, the characteristics of harmonic signal radiated by an arbitrary time-modulated element are observed by varying the normalized switch-on time for its complete range from 0 to 1. Then by defining relative and normalized sideband power, the effects of reducing SLL on the first null beamwidth (FNBW) and maximum sideband power level are observed.

From Eq. (7), we can see that the array factor at different sidebands is the superposition of the harmonic signal radiated from the individual antenna element. Hence, sideband power pattern and total sideband power can be obtained from the harmonic characteristics of the time-modulated elements as expressed in Eq. (5). The normalized harmonic radiation of the individual time-modulated antenna ele-

p¼1

Apτpe

Ap sin πτ<sup>p</sup> � �e

Ap sin 2πτ<sup>p</sup> � �e

<sup>j</sup>ð Þ <sup>Φ</sup>pþα<sup>p</sup> (8)

�j½ � πτp�ð Þ <sup>Φ</sup>pþα<sup>p</sup> (9)

�j½ � <sup>2</sup>πτp�ð Þ <sup>Φ</sup>pþα<sup>p</sup> (10)

harmonics of the modulation frequency. When the desired power pattern is synthesized at the center frequency, the sideband power is wasted. In this section, the influences on the first null beamwidth (FNBW) and sideband radiation by reducing SLL of the center frequency pattern are observed. The SLL of the power pattern at f0 is reduced by using the conventional amplitude tapering technique, namely, Dolph-Chebyshev (DC) [1], and a heuristic search global optimization method, namely, genetic algorithm (GA) [16].
