©2017 Published in 5th International Symposium on Innovative Technologies in Engineering and Science 29-30 September 2017 (ISITES2017 Baku - Azerbaijan)

*Corresponding author: Address: Faculty of Engineering, Department of Electrical and Electronics Engineering

Gazi University, 06570, Ankara TURKEY. E-mail address: ertugrulaksoy@gazi.edu.tr, Phone: +903125823356

Sideband Suppression Through the Frequency of Sinusoidal

Variable Pulse Amplitude Time Scheme in TMAs

Yasin Yavuz, *Ertugrul Aksoy and Mert Karahan

Faculty of Engineering, Department of Electrical and Electronics Engineering

Gazi University, Ankara, Turkey

Abstract In time modulated arrays, excitation strategies affect the characteristics of the array. Instead of using

rectangular pulses, to have more stable response in frequency domain, sinusoidal signals may be used

as excitation scheme. In this paper, variable pulse amplitude excitation strategy is combined with

sinusoidal signals and the frequency of sinusoidal signal is appeared as a new degree of freedom.

Radiation pattern may easily be shaped via any optimizer and sideband suppression may be achieved

only controlling the frequency parameter. At the end of this study, an explanatory example using

Differential Evolution Algorithm is given to discuss the results.

Key words: Differential Evolution, optimization, TMLA, sideband suppression

1. Introduction

The time modulation concept in antenna arrays is achieved by switching each element with high

speed RF switches and, as an inevitable result, switching causes infinite number of harmonic

radiations which are called as sidebands. Indeed, some of the power are being transferred to these

sidebands [1]. Since the received or transmitted power is limited with regulatory restrictions, in

literature, there are many studies to suppress the unwanted sideband radiations so as to transfer

maximum power to the main radiation [2-11].

On the other hand, instead of suppressing these unwanted radiations, it is asserted that sideband

radiations may be eligible to use in some applications such as direction finding, secure

communication and steering the harmonic beams and use these radiations as second, or even third

if there is enough power, receiver which enables communication over sidebands [12-16].

Since excitation strategy is directly affects the antenna array factor, also there are some studies in

order to find the best scheme which can easily be produced in time domain and have more stable

frequency response allowing some windowing functions [17,18]. In this paper, a new excitation

scheme, which can be obtained easily in time domain and have more stable frequency response

with respect to rectangular pulse, using sinusoidal signal with DC offset is proposed and

exemplified. The rest of the paper organized as follows: next section gives a brief theory about

time modulation and proposed excitation strategy. Third and fourth sections describe optimization

process and array design with results, respectively. At the end, the study is concluded.

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1150

2. Brief Theory

2.1. Time Modulation

Once the basic structure of time modulated antenna array is considered, every element of array is

individually switched and, naturally, array factor is being affected by this switching operation. If

𝑁 isotropic elements are considered to be located along 𝑧- axis, the array factor may be defined as:

𝐴𝐹(𝜃, 𝑡) = ∑ 𝐼𝑛𝑒𝑗𝛽𝑛

𝑁

𝑛=1

𝑈𝑛(𝑡)𝑒𝑗𝑘𝑧𝑛 cos𝜃 . (1)

Here, 𝐼𝑛, 𝛽𝑛, 𝑧𝑛 and 𝜃 represents the excitation amplitude, the excitation phase, the Euclidian

distance to the system origin of nth element and the elevation angle according to z- axis,

respectively. 𝑘 = 2𝜋 𝜆⁄ denotes the wavenumber and 𝑈𝑛(𝑡) is the periodic switching function.

The most basic switching scheme which is named as variable aperture size (VAS) may be

expressed as:

𝑈𝑛(𝑡) = {1, 𝑡 ≤ 𝑡𝑛

1 ≤ 𝑇𝑝0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

, (2)

where 𝑡𝑛1 and 𝑇𝑝 denotes the switched–on duration of related element and the switching period,

respectively. Since (2) is periodic in time and as every periodic function may be decomposed into

complex Fourier series. After the decomposition and basic calculations, the most general form of

the time-modulated antenna array factor may be written using (1) as:

𝐴𝐹(𝜃, 𝑡) = ∑ ∑𝐼𝑛𝑒𝑗𝛽𝑛𝐶𝑚,𝑛𝑒

𝑗𝑘𝑧𝑛 cos𝜃

𝑁

𝑛=1

∞

𝑚=−∞

𝑒𝑗(𝑤0+𝑚𝑤𝑝)𝑡, (3)

under far field and 𝑓0 ≫ 𝑓𝑝 approximations. Here, 𝐶𝑚,𝑛, 𝑤0 = 2𝜋𝑓0 and 𝑤𝑝 = 2𝜋𝑓𝑝 denotes the

complex Fourier coefficient of nth element in mth harmonic level, the angular operating frequency

and the angular switching frequency, respectively.

2.2. Sinusoidal Variable Pulse Amplitude (Sine-VPA)

Every periodic function may be used as switching function and in literature most common

strategies use on-off switching scheme. However, in order to prevent some disadvantages of on-

off switching such as array silencing which means all elements are switched to zero and no

communication at that instant, switching between amplifier or attenuator is proposed by Aksoy in

[14]. Switching function of variable pulse amplitude (VPA) for two elements can be defined as:

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1151

𝑈𝑛(𝑡) = {𝐾𝑛1 , 0 < 𝑡 ≤ 𝑡𝑛

1

𝐾𝑛2 , 𝑡𝑛

1 < 𝑡 ≤ 𝑇𝑝, (4)

where 𝐾𝑛1 and 𝐾𝑛

2 represent nth element relative pulse amplitudes of adjacent pulses and 𝑡𝑛1

represents the amplitude reversal time instant.

Here, we introduce a new excitation strategy applying sinusoidal signal with DC offset to adjacent

pulses and in Fig. 1, VPA and Sine-VPA excitation schemes for nth element are shown. With this

approach, frequency response of the switching function becomes more stable [18] and sideband

radiation (SR) becomes dependent to frequency of the sine function which may be used as a new

degree of freedom.

Periodic switching function of Sine-VPA may be written as:

𝑈𝑛(𝑡) = {𝐾𝑛1 + C cos(2𝜋𝑓𝑛𝑡) , 0 < 𝑡 ≤ 𝑡𝑛

1

𝐾𝑛2 + C cos(2𝜋𝑓𝑛𝑡) , 𝑡𝑛

1 < 𝑡 ≤ 𝑇𝑝, (5)

where 𝑓𝑛 denote sine function frequency of nth element and |C| < 1, C ∈ ℝ. In order to compute

array factor, complex Fourier coefficients for Sine-VPA should be calculated for all harmonic

levels. After required calculations which can be seen in Appendix A, the coefficients may be

defined as (A.4) and (A.9):

Figure 1. Excitation Scheme of nth element.

Left – Standard VPA

Right – Sinusoidal VPA (𝐶 = 0.1)

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1152

𝐶𝑚,𝑛 =

{

∆𝑛𝜏𝑛 + 𝐾𝑛

2 +C

2𝜋𝑓𝑟𝑎𝑡𝑛 [sin(2𝜋𝑓𝑟𝑎𝑡

𝑛 )], 𝑚 = 0

∆𝑛𝑚𝜋

sin(𝑚𝜋𝜏𝑛) 𝑒−𝑗𝑚𝜋𝜏𝑛 +

C

2𝜋[(𝑓𝑟𝑎𝑡𝑛 )2 −𝑚2]

{𝑗𝑚 + 𝑓𝑟𝑎𝑡𝑛 sin(2𝜋𝑓𝑟𝑎𝑡

𝑛 ) − 𝑗𝑚𝑐𝑜𝑠(2𝜋𝑓𝑟𝑎𝑡𝑛 )}, |𝑚| > 0

. (6)

Here, ∆𝑛= 𝐾𝑛1 − 𝐾𝑛

2, 𝜏𝑛 = 𝑡𝑛1/𝑇𝑝 and 𝜔𝑛 = 2𝜋𝑓𝑛 represent DC offset difference of adjacent pulses,

normalized pulse reversal time and angular sine function frequency of nth element. Additionally,

𝑓𝑟𝑎𝑡𝑛 = 𝑓𝑛/𝑓𝑝 stands for sine function frequency to switching frequency ratio.

3. Optimization

As it can be clearly understood from (6), only 𝑓𝑟𝑎𝑡𝑛 may be used by any optimizer in order to

suppress SR. In this study, Differential Evolution (DE) using “DE/best/1/bin” scheme is applied

with mutation factor, crossover factor and population size of 0.6, 0.95 and 50, respectively. The

algorithm may be summarized as:

Step 1 – Randomly create population matrix with respect to parameter definitions.

Step 2 – Calculate “cost” functions for population.

Step 3 – Create a mutant vector using parameter vectors.

Step 4 – Create a trial vector with crossover.

Step 5 – Calculate “cost” function for trial vector.

Step 6 – Compare “cost” functions for jth element of the population and choose the best

one for new population.

Step 7 – Go to Step 3 until j reaches population index.

Step 8 – Create new population and go to Step 3 until iteration limit and/or stop criterion is

reached.

The cost function is calculated according to steer main radiation to broadside and to reduce side

lobe level (SLL) and SR below -20dB. An iteration limit (75 iterations) is set as the stop condition.

The hardware of PC which is used for optimization process consist of Intel i7 M640 2.80GHz CPU,

8GB RAM and 128GB SSD with Windows 10 and MATLAB R2016b softwares.

4. Array Design and Results

As an explanatory example, an array whose 20 identical elements located along z- axis with 𝑑 =0.5𝜆 𝑠pacing may be considered. Without loss of generality, other parameters are selected as

follows: 𝐼𝑛 = 1, 𝛽𝑛 = 0, ∆𝑛= 0.5, C = 0.65 and 𝐾𝑛2 = 0.2. In addition, normalized pulse reversal

time of each element is selected as corresponding Chebyshev polynomial coefficients. After 75

iterations, according to optimized frequency ratio parameters of each element which are visualized

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1153

in Fig. 2 and in Fig. 3, main radiation and fundamental harmonic radiation patterns are being

shown. Fig. 4 also describes maximum sideband radiation levels for first ten harmonic level.

Figure 2. Optimized 𝑓𝑟𝑎𝑡𝑛 Values (Normalized)

Figure 3. Normalized radiation patterns for main and fundamental harmonic radiation related to Fig. 2

(Max. SLL=-20dB and Max. SR=-22.55dB)

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1154

Once the results are examined, it is clearly seen that main radiation is directed to broadside with

low SLL (< −20𝑑𝐵) and unwanted sideband radiation is reduced lower than -20𝑑𝐵 only

optimizing the frequency ratio. After 50 independent experiments, it is also seen that iteration count

may be decreased till 40 iterations to have nearly same results with decreased operation time. 75

iterations take 56.11 seconds and for 40 iterations elapsed time is 30.87 seconds which is nearly

half of former.

Instead of having fixed DC offset difference, ∆𝑛, the study is being repeated with random ∆𝑛 values

using same iteration counts and the results are nearly identical with acceptable SLL and SR level.

Using variable ∆𝑛 values allow compensating real time error which may be occurred on switching

attenuator or amplifier.

Figure 4. Maximum levels of first ten harmonic radiations

Conclusions

In this study, a modified variable pulse amplitude excitation strategy which may be named as

Sinusoidal VPA is introduced and it is shown that this new approach enables an easily optimizable

parameter. In less than fifty iterations and in very short time range, radiation patterns may be

optimized with DE and results are quite satisfactory. In addition, errors of amplifiers and

attenuators used in switching devices may be tolerated using different offset difference scheme.

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1155

Appendix A

In this appendix, in order to maintain paper’s readability, steps for calculating complex Fourier

coefficients of Sine-VPA is being shown. Complex Fourier coefficients are calculated with:

𝐶𝑚,𝑛 =1

𝑇𝑝∫ 𝑈𝑛(𝑡)𝑒

−𝑗𝑚𝜔𝑝𝑡𝑑𝑡.

∞

−∞

(𝐴. 1)

Using (5), for 𝑚 = 0:

𝐶𝑚,𝑛 =1

𝑇𝑝[∫ (𝐾𝑛

1 + Ccos(𝜔𝑛𝑡))𝑑𝑡 +

𝑡𝑛1

0

∫(𝐾𝑛2 + Ccos(𝜔𝑛𝑡))𝑑𝑡

𝑇𝑝

𝑡𝑛1

] , 𝜔𝑛 = 2𝜋𝑓𝑛. (𝐴. 2)

𝐶𝑚,𝑛 =1

𝑇𝑝[(𝐾𝑛

1𝑡 +C

𝜔𝑛𝑠𝑖𝑛(𝜔𝑛𝑡)) |

𝑡𝑛1

0+(𝐾𝑛

2𝑡 +C

𝜔𝑛𝑠𝑖𝑛(𝜔𝑛𝑡)) |

𝑇𝑝𝑡𝑛1] . (𝐴. 3)

𝐶𝑚,𝑛 = 𝐾𝑛1𝜏𝑛 +𝐾𝑛

2 − 𝐾𝑛2𝜏𝑛 +

C

𝑇𝑝𝜔𝑛𝑠𝑖𝑛(𝜔𝑛𝑇𝑝) = ∆𝑛𝜏𝑛 + 𝐾𝑛

2 +C

𝑇𝑝𝜔𝑛𝑠𝑖𝑛(𝜔𝑛𝑇𝑝) . (𝐴. 4)

For |𝑚| > 0:

𝐶𝑚,𝑛 =1

𝑇𝑝[∫(𝐾𝑛

1 + C 𝑐𝑜𝑠(𝜔𝑛𝑡))𝑒−𝑗𝑚𝜔𝑝𝑡𝑑𝑡 +

𝑡𝑛1

0

∫(𝐾𝑛2 + C 𝑐𝑜𝑠(𝜔𝑛𝑡))𝑒

−𝑗𝑚𝜔𝑝𝑡𝑑𝑡

𝑇𝑝

𝑡𝑛1

] . (𝐴. 5)

𝐶𝑚,𝑛 =1

𝑇𝑝

[

∫𝐾𝑛1𝑒−𝑗𝑚𝜔𝑝𝑡𝑑𝑡 +

𝑡𝑛1

0

∫𝐾𝑛2𝑒−𝑗𝑚𝜔𝑝𝑡𝑑𝑡 + C∫cos(𝜔𝑛𝑡) 𝑒

−𝑗𝑚𝜔𝑝𝑡𝑑𝑡

𝑡𝑛1

0⏟ 𝐼1

+ C ∫cos(𝜔𝑛𝑡) 𝑒−𝑗𝑚𝜔𝑝𝑡𝑑𝑡

𝑇𝑝

𝑡𝑛1⏟

𝐼2

𝑇𝑝

𝑡𝑛1

]

. (𝐴. 6)

𝐶𝑚,𝑛 =1

𝑇𝑝[−

𝐾𝑛1

𝑗𝑚𝜔𝑝𝑒−𝑗𝑚𝜔𝑝𝑡 |

𝑡𝑛1

0±

𝐾𝑛2

𝑗𝑚𝜔𝑝𝑒−𝑗𝑚𝜔𝑝𝑡 |

𝑇𝑝𝑡𝑛1+ C𝐼1 + C𝐼2] . (𝐴. 7)

𝐶𝑚,𝑛 =𝐾𝑛1

𝑗𝑚2𝜋(1 − 𝑒−𝑗𝑚𝜔𝑝𝑡𝑛

1) −

𝐾𝑛2

𝑗𝑚2𝜋(1 − 𝑒−𝑗𝑚𝜔𝑝𝑡𝑛

1)+C

𝐼1𝑇𝑝+ C

𝐼2𝑇𝑝. (𝐴. 8)

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1156

After 𝐼1 and 𝐼2 calculated using Euclidean equivalent of cosine function, with some basic

mathematical operations coefficients may be expressed as follows:

𝐶𝑚,𝑛 =∆𝑛𝑚𝜋

𝑠𝑖𝑛(𝑚𝜋𝜏𝑛) 𝑒−𝑗𝑚𝜋𝜏𝑛 +

C

2𝜋[(𝑓𝑟𝑎𝑡𝑛 )2 −𝑚2]

{𝑗𝑚 + 𝑓𝑟𝑎𝑡𝑛 𝑠𝑖𝑛(2𝜋𝑓𝑟𝑎𝑡

𝑛 ) − 𝑗𝑚𝑐𝑜𝑠(2𝜋𝑓𝑟𝑎𝑡𝑛 )}. (𝐴. 9)

References

[1] Bickmore, RW, Hansen, RC. Time Versus Space in Antenna Theory. New York:Academic

Press; 1966

[2] Yang S, Gan YB, Qing A. Sideband suppression in time modulated linear arrays by the

differential evolution algorithm. IEEE Antennas Wirel Propag Lett 2002;1;173-175.

[3] Fondevila J, Bregains JC, Ares F, Moreno E. Optimizing uniformly excited linear arrays

through time modulation, IEEE Antennas Wirel Propag Lett 2004; 3;298–301.

[4] Aksoy E, Afacan E. Thinned nonuniform amplitude time modulated linear arrays. IEEE

Antennas Wirel Propag Lett 2010;9;514-517.

[5] Poli L, Rocca P, Manica L, Massa A. Handling sideband radiations in time-modulated arrays

through particle swarm optimization. IEEE Trans Antennas Propag 2010;58(4);1408-1411.

[6] Aksoy E, Afacan E. Generalized representation of sideband radiation power calculation in

arbitrarily distributed time-modulated planar and linear arrays. PIERS Proceedings 2011;368–371.

[7] Bregains JC, Fondevila J, Franceschetti G, Ares F. Signal radiation and power losses of time-

modulated arrays. IEEE Trans Antennas Propag 2008;56(6);1799–1804.

[8] Aksoy E, Afacan E. Calculation of sideband power radiation in time-modulated arrays with

asymmetrically positioned pulses. IEEE Antennas Wirel Propag Lett 2012;11;133–136.

[9] Bekele ET, Poli L, Rocca P, D’Urso M, Massa A. Pulse-Shaping Strategy for Time Modulated

Arrays–Analysis and Design. IEEE Trans Antennas Propag 2013;61(7);3525–3537.

[10] Aksoy E, Afacan E. An inequality for accelerating the calculation of relative maximum

sideband level in time-modulated linear and planar arrays. IEEE Trans Antennas Propag 2014;

62(6);3392–3397.

[11] Aksoy E. Calculation of Sideband Radiations in Time-Modulated Volumetric Arrays and

Generalization of the Power Equation. IEEE Trans. Antennas Propag 2014;62(9);4856–4860.

[12] Tennant A, Chambers B. A two-element time-modulated array with direction-finding

properties. IEEE Antennas Wirel Propag Lett 2007;6;64-65.

[13] Zhu Q, Yang S, Yao R, Nie Z. Directional modulation based on 4-d antenna arrays. IEEE

Trans Antennas Propag 2014;62(2);621–628.

[14] Aksoy E. Time modulation through variable pulse amplitude in 4D arrays. TELE–INFO

2014;12–20.

[15] Aksoy E. Harmonic beam steering in 4D linear arrays through pulse difference. TELFOR

2014;22;792–794.

E. AKSOY et al./ ISITES2017 Baku - Azerbaijan 1157

[16] Yavuz Y, Aksoy E, Karahan M. A dual channel AM receiver structure in 4D arrays. AP/S

Proceedings 2015; 804–805.

[17] Maneiro-Catoira R, Bregains J, Garcia-Naya J, Castedo L. Time modulated arrays with sum

of weighted cosine pulses. IEEE Int. Symp. Antennas Propag USNC/URSI Nat Radio Sci Meet

2016;697–698.

[18] Maneiro-Catoira R, Bregains J, Garcia-Naya J, Castedo L. Enhanced Time-Modulated Arrays

for Harmonic Beamforming. IEEE J of Sel Topics In Sig Proc 2017;11(2);259-270