3. Behaviors of sideband radiation (SBR)

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.

### 3.1 Characteristics of harmonic radiations (HRs)

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 element is given as [15]

$$h\_{pk} = 20\log\_{10} \left| \mathbf{C}\_{pk} \right| / \left| \mathbf{C}\_{p0} \right| = \frac{\sin \left( k \pi \tau\_p \right)}{k \pi \tau\_p} \tag{11}$$

from a particular element to produce the sideband pattern depends on the on-time duration of the corresponding element. Therefore, the desired sideband power pattern can be synthesized in TMAAs by judiciously controlling the on-time

Variation of the first three harmonic powers from an antenna element with normalized switch-on time, τp.

Usually in TMAA, the radiation pattern is synthesized at center frequency by suppressing the sideband radiation level to sufficiently low value. Thus, the maximum of the power radiated at f0 is used to normalize the corresponding power pattern at center frequency. On the other hand, the sideband power is divided by the maximum power at f0 to measure the relative power level at different sidebands with respect to that of the radiation at center frequency. In this regard, the relative signal power radiated at different harmonics (k 6¼ 0) is measured as in Eq. (12):

where "SBLk" represents the relative value of sideband level at kth harmonic (k = 1, 2, … ), i.e., relative value of the array factor AFk in dB, and "max (AF0 (θ, t))" is the maximum value of the array factor at operating frequency ω0, i.e., the maximum radiation level at k=0. Thus, with k=0, Eq. (11) gives the normalized power pattern for the center frequency pattern, whereas, for the sideband radiations (with k ¼6 0), it is the relative power with respect to the maximum of the center frequency

3.3 Influence on the sideband level and first null beamwidth during reduction

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

SBLkð Þ¼ dB 20 ∗ log <sup>10</sup>ð Þ AFkð Þ θ, t = max ð Þ AF0ð Þ θ, t (12)

sequence of the time-modulated antenna elements.

Pattern Synthesis in Time-Modulated Arrays Using Heuristic Approach

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

of side lobe level of the fundamental pattern

3.2 Normalized and relative power

pattern.

9

Figure 4.

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

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

from a particular element to produce the sideband pattern depends on the on-time duration of the corresponding element. Therefore, the desired sideband power pattern can be synthesized in TMAAs by judiciously controlling the on-time sequence of the time-modulated antenna elements.
