Pulse Splitting for Harmonic Beamforming in Time-Modulated Linear Arrays

A novel strategy for harmonic beamforming in time-modulated linear arrays is proposed.The pulse splitting technique is exploited to simultaneously generate two harmonic patterns, one at the central frequency and another at a preselected harmonic of arbitrary order, while controlling the maximum level of the remaining sideband radiations. An optimization strategy based on the particle swarm optimizer is developed in order to determine the optimal parameters describing the pulse sequence used to modulate the excitation weights of the array elements. Representative numerical results are reported and discussed to point out potentialities and limitations of the proposed approach.


Introduction
The synthesis of time-modulated antenna arrays is a topic that gained much attention from the scientific community in the last years.According to the principles conceived in [1], several strategies have been proposed aimed at determining suitable modulating functions defining the timevarying excitations of the elements in order to obtain desired radiation features.In this framework, many efforts have been devoted to synthesize a desired average pattern at the antenna working frequency while minimizing the spurious harmonic radiations (i.e., the sideband radiations, SRs [2]) caused by the modulation of the excitation amplitudes in the time domain.Toward this aim, several algorithms have been employed to find the optimal parameters of the modulating waveforms by solving the synthesis problem recast to an optimization one, among which are the simulated annealing (SA) [3], the genetic algorithms (GA) [4], the particle swarm optimizer (PSO) [5][6][7][8][9], the differential evolution (DE) [10][11][12][13], and the multiobjective differential evolution (MO-DEA) [14].Although it has been shown that the SRs can be significantly reduced through a suitable shaping of the modulating functions [15], simpler periodic rectangular pulsed waveforms have been considered in all other works.Dealing with this latter kind of waveforms, conventional approaches consider a common rising instant for the pulses modulating the excitation weights [13,16].However, the shifting of the rectangular pulses has been proposed in [17] as an additional degree of freedom in the synthesis process to properly control the harmonic radiations.Further degrees of freedom can be also introduced by considering split rectangular waveforms [4,[18][19][20].The division of the pulses into multiple shifted subpulses has been preliminarily investigated in [4], where the modulation period has been quantized into uniform time steps independently handled by a GA-based optimization technique.A more general case has been proposed in [18], where durations and rise instants of the rectangular subpulses have been arbitrarily defined by means of a DE-based optimization procedure.Moreover, suitable split rectangular waveforms have been also studied in [19] with the aim of shifting the undesired power on the SRs to obtain a partial suppression of the unwanted radiation by exploiting the limited bandwidth of the real radiating elements.
A first insight into the possibility of exploiting the pulse splitting to selectively distribute the power among the sideband harmonic frequencies was given in [20], where the pulses have been partitioned in two shifted parts in order to maximize the level of the second harmonic pattern, minimizing the level of the remaining harmonics while keeping the pattern at the working/central frequency unaltered.As a matter of fact, the pulse splitting enables the possibility of effectively synthesizing a desired pattern at an arbitrary harmonic frequency in order to take advantage of these sideband radiations for enabling advanced applications.Based on this perspective, some works have been recently proposed: multiple patterns have been simultaneously synthesized at different frequencies in [21][22][23][24][25][26][27] for direction of arrival estimation [21], adaptive beamforming [23,24], and direction finding purposes [23,26,27].
This paper is aimed at taking advantage of the pulse splitting approach in order to realize an easily reconfigurable antenna system able to provide a selective beamforming on the sideband patterns for a multichannel communication.A synthesis strategy based on the PSO is proposed to generate multiple harmonic beams aimed at simultaneously receiving multiple signals from different directions.Indeed, the harmonic patterns generated due to the time-modulation technique may be used as independent "channels" in the receiving mode that may be effectively handled through a proper combination of local oscillators and low-pass filters [25].Dealing with the transmitting mode, the pulse splitting technique may also be exploited to synthesize a beam pattern at the harmonic frequency properly selected on the basis of the requirements arising from a working environment shared with other systems in order to transmit a signal on a free "channel" avoiding interferences.
The paper is organized as follows.The mathematical theory of the problem is formulated in Section 2, whereas the optimization strategy is described in Section 3. Selected numerical results are presented and discussed in Section 4. Some conclusions are finally remarked in Section 5.

Basic Mathematical Formulation
Let us consider a linear array of  elements aligned along the -axis whose uniform excitation amplitudes vary in time according to a periodic modulation function () with period   , composed of  rectangular pulses defining in which portions of the modulation period the elements are in active (on) or inactive (off ) state, as shown in Figure 1.The timevarying array factor can be calculated as where  0 is the angular working frequency of the antenna,  = 2/ is the wavenumber,  is the interelement spacing, and  ∈ [0 ∘ : 180 ∘ ] degrees.By expanding () in a Fourier series, (1) becomes where   = 2/  and the complex Fourier coefficients  ℎ (ℎ ∈ Z) can be computed as follows: where    and    are the rise instant and the fall instant normalized with respect to the modulation period   of the th subpulse related to th waveform, respectively ( = 1, . . .,  and  = 1, . . ., ).The pattern generated in far-field at the angular frequency  ℎ = ( 0 + ℎ  ) is expressed as Dealing with the central angular frequency  0 , it can be easily proved that the array factor ()|  0 turns out to be

PSO-Based Synthesis Strategy
The optimization strategy is aimed at simultaneously synthesizing a beam pattern at the working angular frequency  0 oriented along the broadside direction   0 = 90 ∘ and a beam pattern at a selected frequency  ĥ = ( 0 + ĥ  ), ĥ ∈ Z 0 , steered toward an arbitrary direction   ĥ ∈ [0 The PSO algorithm which has already provided effective results in the synthesis of time-modulated arrays [5][6][7][8][9]23] has been applied to determine the optimized split pulse sequences exciting the uniform weights of the array elements.

(i) Step 1 (parameters setup). Consider the following:
(a) definition of the desired pattern features described in terms of sidelobe level (SLL  ℎ trg , ℎ = 0, ĥ) first null beam width (Θ  ℎ trg , ℎ = 0, ĥ), desired peak level evaluated along   ℎ (Peak  ℎ trg , ℎ = 0, ĥ), and undesired sideband level (SBL (ii) Step 2 (swarm initialization).The particles of the swarm are randomly generated except for a single particle whose entries are analytically determined in order to speed up the convergence of the optimization process as follows.
(b) The split pulse sequence is then shifted in order to steer the beam pattern generated at the angular frequency  ĥ along the desired direction   ĥ : = τ  = 1, . . ., ;  = 1, . . ., where   is the interelement distance expressed in wavelength.
(iii) Step 3 (optimization process).The cost function defined in the following will be iteratively computed in order to evaluate if the shape of the multibeam pattern of the optimized configuration meets the requirements defined through the target pattern parameters initialized in Step 1(a).It can be expressed as the summation of three main terms: where Ψ ()  is the term aimed at shaping the multibeam pattern according to the parameters SLL  ℎ trg , Θ  ℎ trg , and Peak  ℎ trg (ℎ = 0, ĥ) that is defined as follows: where Υ = {SLL, Θ, Peak} and H is the Heaviside function, whose argument has inverted sign when  ← Peak; the term Ψ ()  is devoted to handling the undesired sideband radiation limiting the maximum sideband level: Finally, the term is a penalty factor introduced to discard the solutions which provide meaningless configurations (i.e., when [Σ  =1 (   +    ) > 1]).Moreover,   ( ∈ Υ),   , and   are real weighting coefficients and  is the PSO iteration index.

Numerical Results
Let us consider an arrangement of  = 16 elements displaced along the -axis and equally spaced of  = /2.In a preliminary example, the synthesis of a sum beam International Journal of Antennas and Propagation pattern at the central angular frequency  0 in the broadside direction (  0 = 90 ∘ ) and a sum beam pattern at the fundamental frequency  1 = ( 0 +   ) along   1 = 120 ∘ is proposed.The specific multibeam configuration has been already investigated in [23], but in this case and unlike [23], in order to limit the loss of power, the reduction of the sideband level associated with the undesired harmonics (|ℎ| = 2, . . ., , being in the specific case  = 10) has been also taken into account during the optimization process.The target pattern features have been selected as [23] SLL trg = 20.6 ∘ (these two latter have been chosen according to the optimized result achieved in [23], in order to take into account the unavoidable slight widening of the beam width due to the steering operation).Moreover, the undesired sideband level has been fixed to SBL The power patterns generated at  0 and  1 afforded by the optimized pulse sequence of Figure 2(a) with  = 1 are shown in Figure 2(b).As expected, the optimization of durations and time-shifts of simple rectangular pulsed waveforms allows obtaining a desired multibeam radiation pattern that fulfills the given requirements, as can be seen from the data reported in Table 1  subscript PSO is related to the technique presented in [23]) and also in terms of useful power (expressed in percentage on the total radiated power) associated with the pattern synthesized at  0 and  1 as can be seen in Figure 3 (i.e.,  The synthesis problem has been addressed also considering split pulsed waveforms, setting  = 2 or  = 3, but in the specific case no significant improvements have been obtained with respect to the result reached with the Table 1: Multibeam sum pattern synthesis at  0 and  1 ( = 16;  = /2;   0 = 90 ∘ ;   1 = 120 ∘ ) and pattern features for  = 1,  = 2, and  = 3.
Unlike the previous case, the single-pulse rectangular waveforms do not provide satisfactory results when synthesizing the second beam pattern at a harmonic angular frequency of higher order (|ℎ| > 1).Let us consider in the second example the synthesis of two sum beam patterns at the angular frequencies  0 and  3 = ( 0 + 3 ⋅   ), ( ĥ = 3).The target pattern features have been selected as in the previous case (SLL

and SBL
ℎ trg = −10 dB).The optimized solution and the associated radiation power patterns are reported in Figures 4(a) and 4(b), respectively.Although the beam patterns have been almost perfectly shaped according to the input requirements as shown in Figure 4(b), the undesired sideband level turns out to be very high, SBL  ℎ PSO,=1 = −2.77dB.Indeed, as shown in Figure 5, most of the power is lost in the unusable radiation, and the useful power associated with the patterns synthesized at  0 and  3 is limited to a small percentage of the total amount ( Dealing with the same synthesis problem, more effective results can be obtained by setting the number of subpulses to  = 3, considering split rectangular waveforms.The optimized pulse sequence turns out to be as in Figure 6(a) affording a multibeam radiation reported in Figure 6(b) that widely satisfies the initial requirements (Table 2).Figure 7(a) shows as the optimization of the split pulses allows selectively distributing the power among the harmonic frequencies, comparing the power associated with each harmonic when  = 1 (i.e., without splitting) and  = 3.Similar considerations arise from the observation of the comparison in terms of sideband level in Figure 7(b).As a matter of fact, by analyzing the undesired radiation at  1 and  2 , Table 2: Multibeam sum pattern synthesis at  0 and  3 ( = 16;  = /2;   0 = 90 ∘ ;   3 = 120 ∘ ) and pattern features for  = 1,  = 2, and  = 3.   it is possible to observe that the sideband level decreases from SBL However, it is worth noting that similar performance can be obtained avoiding the optimization of complex split rectangular waveforms.Indeed, starting from the optimized pulse sequence achieved in the preliminary example, described by the parameters τ Figure 9: Multipulse waveform,  = 2-multibeam sum pattern synthesis at  0 and  3 ( = 16;  = /2;   0 = 90 ∘ ;   3 = 120 ∘ )-plot of (a) the optimized pulse sequence and (b) corresponding power patterns generated at  0 ,  1 ,  2 , and  3 .
Therefore, the pattern previously generated at  1 will be reproduced at  ĥ, nullifying the radiation at the frequencies  ℎ ∈ {( 0 + ℎ ⋅   ); ℎ ̸ =  ⋅ ĥ; |ℎ| = 1, . . ., ∞; || = 0, . . ., ∞}, but giving rise to a spreading of the harmonic spectrum (i.e., the pattern previously generated at  ℎ , |ℎ| = 1, . . ., ∞, will be shifted to  ℎ⋅ ĥ).Dealing with the case for ĥ = 3, the analytically calculated parameters of the adapted split waveforms are reported in Figure 8 However, (12) shows that the durations of the subpulses of the split waveforms depend on ĥ, and more specifically the higher the harmonic index ĥ, the shorter the durations of the subpulses.Accordingly, pulses with very short durations (i.e., very small fractions of the modulation period   ) could arise from the necessity to shift the pattern to a harmonic frequency of high order (e.g., ĥ > 3) and could be hard to realize in practice.However, since the proposed optimization strategy does not require fixing the number of subpulses according to the harmonic index ĥ of the selected frequency, it can be profitably employed to synthesize the multibeam radiation previously investigated through simpler waveforms (i.e., with  = 2 < ĥ).In fact, the beam patterns shown in Figure 9(b) are generated by the split pulse sequence having a number of subpulses limited to  = 2 graphically described in Figure 9(a).At the expense of a higher amount of power lost in the SRs (SR PSO,=2 = 66.0%versus SR PSO,=3 = 44.7%,being the subscript PSO related to

Figure 1 :
Figure 1: Example of split pulse controlling a RF switch.
(a), whereas the radiation patterns are shown in Figure 8(b).