AbstractâWe address the design of Time-Modulated Arrays. (TMAs) excitations in the frequency domain. The design is based on the particular features of the ...
Frequency-Domain Synthesis of Time-Modulated Arrays Roberto Maneiro-Catoira, Julio Br´egains, Jos´e A. Garc´ıa-Naya, and Luis Castedo Universidade da Coru˜na (Univ. of A Coru˜na), GTEC, Fac. Inform´atica, Campus Elvi˜na, 15071 A Coru˜na, Spain. E-mail:{roberto.maneiro, julio.bregains, jagarcia, luis}@udc.es |G00|
Abstract—We address the design of Time-Modulated Arrays (TMAs) excitations in the frequency domain. The design is based on the particular features of the spectrum of the periodic pulses which modulate the excitations: a discrete spectrum with impulses at multiples of the time-modulation frequency, and whose corresponding areas are 2π times the associated exponential Fourier series coefficients. In this view, a simplified design of ad-hoc pulses for beamforming purposes is proposed by applying the Fourier series coefficients properties to preprocess conventional rectangular pulses before they are applied to the antenna elements.
|G(N-1)0|
Time-Modulated Arrays (TMAs) are based on the application of variable-width periodical pulses to the individual antenna excitations. TMA is a nonlinear beamforming method that causes the appearance of radiation patterns at the harmonic frequencies of such periodic pulses. The TMA harmonic patterns can be adapted to exploit the spatial diversity of wireless channels. As a result, TMA is an attractive beamforming solution that exhibits a good trade-off between performance and hardware complexity [1]. The solutions existing in the literature to synthesize efficient and versatile beamforming TMAs are based either on the optimization of the time parameters of rectangular pulses, usually implemented using on-off switches [1], or on the use of alternative pulse shapes with windowing features [2]. This work proposes a strategy based on the preprocessing of rectangular pulses –after a previous analysis of their frequency representation– to accomplish a twofold objective for the harmonic beamforming: (1) the suppression of those harmonics not exploited, including the fundamental pattern –due to its scanning inability, and (2) a time-domain linear control of the magnitudes of the dynamic excitations. The fact that the frequency spectrum of a periodic pulse can be represented by an infinite sequence of complex numbers, –the Fourier series coefficients– allows for a clearer detection of the limitations of rectangular pulses applied to TMA beamforming. At the same time, this analysis helps to propose solutions consisting in the modification of the Fourier series coefficients which correspond, in the time-domain, to a preprocessing of the TMA rectangular pulses. II. F REQUENCY RESPONSE OF A CONVENTIONAL TMA Let us consider a linear TMA with N isotropic elements with unitary complex static excitations In = 1, n ∈ {0, 1, . . . , N − 1}, modulated by periodic pulses gnT0 (t), being
|G10| |G11|
...
|G(N-1)1| |G(N-1)2|
|G01| |G | 02 |G12|
...
0
... |G1L|
1
q=
2 q=
L
q=
|G0L|
...
...
n=0
q
T0
G0 (ω) T0
... ...
...
...|G(N-1)L| ... n
I. I NTRODUCTION
978-1-5386-3284-0/17/$31.00 ©2017 IEEE
q=
n=1
G1 (ω)
T0
n=N-1
GN-1 (ω)
T0 Fig. 1. Frequency response magnitude of the periodic gn (t), which govern the beamforming on a TMA with rectangular pulses. |Gnq | is plotted as a discrete function (n, q, |Gnq |). The discrete character of the FT of such pulses opens new perspectives to model a more efficient and flexible TMA beamforming based on the previous preprocessing of the pulses.
T0 the fundamental period. Each gnT0 (t) corresponds to the periodic extension of a basic rectangular pulse1 gn (t) = rect(t/τn ). Accordingly, gnT0 (t) admits Fourier P∞ an exponential jq2π/T0 t series representation gnT0 (t) = where q=−∞ Gnq e Gnq = Tτn0 sinc(πq Tτn0 ) = ξn sinc(πqξn ) are the exponential Fourier series coefficients and ξn = Tτn0 are the normalized pulse durations. The FT of gnT0 (t) is (see Fig. 1) GTn0 (ω) = FT[gnT0 (t)] = 2π
∞ X
Gnq δ(ω − qω0 ),
(1)
q=−∞
which is expressed in terms of the angular frequency ω = 2πf , being ω0 = 2πf0 and with f0 = 1/T0 . We observe that, for a given array element n, the spectrum GTn0 (ω) is given by the discrete pairs (qω0 , Gnq ), expressed in a simpler manner as GTn0 (ω) = (q, Gnq ), q ∈ (−∞, ∞).
(2)
Dually, for a given harmonic q, the pairs (n, Gnq ) constitute the dynamic excitations which control the corresponding radiation pattern of the TMA, i.e., TMA Inq = (n, Gnq ), n ∈ {0, 1, . . . , N − 1}.
(3)
More specifically, Gnq , viewed under the perspective of (3), are also present in the spatial array factor Fq (θ) = PN −1 jkzn cos θ centered at the frequency ωc +qω0 , with n=0 Gnq e ωc being the carrier frequency [1]. 1 rect(t/τ ) = 1 for t ∈ (−τ /2, τ /2), and 0 otherwise. τ is the ON n n n n pulse duration.
1041
AP-S 2017
2) To control, through phase shifting, the phases of the excitations of such chosen harmonics. The spectrum of each individual harmonic component (a single spectral line on q) is given by 1 T0 Mnq (ω) = (q, Mnq ) = q, e−j2πqonq , q ∈ P. (5) 2
TABLE I P ROPERTIES OF THE F OURIER COEFFICIENTS USED IN THE TMA PULSES PROCESSING . Property
Periodic signal x(t), y(t) with period T0
Frequency shifting Periodic convolution Time shifting
1 T
x(t)ejM 2π/T0 t R T x(τ )y(t − τ )dτ x(t − τ )
Fourier coefficients ak , bk ak−M ak bk ak e−jk2πτ /T0
T0 By multiplying each discrete spectrum CnL (ω) and the T0 corresponding Mnq (ω), and choosing L taking all values of P, we obtain the individual spectral lines
III. B EAMFORMING D ESIGN BY T RANSFORMING THE D ISCRETE S PECTRUMS OF THE P ULSES As we focus on the preprocessing of the gnT0 (t) pulses, we fix the variable n, working with the discrete pairs (n, Gnq ) of (2). Our aim is: 1) To synthesize a beamforming with full capacity of selection of the working harmonics, i.e., one that is able to work effectively with a chosen set of harmonics of order q ∈ P while suppressing the rest. P is a set of indices exhibiting even symmetry, i.e., a positive q always coexists with its negative counterpart −q, due to the real-valued character of the pulses. 2) Apart from the effective harmonic selection, we look for a linear control of the magnitudes of the dynamic excitations through the normalized pulse durations ξn . On the other hand, we also seek independence between the phases of the dynamic excitations corresponding to harmonic patterns of different order to guarantee a suitable full independence on the scanning capacity of each harmonic beam. In this work we consider the following transformations of the discrete spectrum of a periodic rectangular pulse GTn0 (ω) = (q, Gnq ) to accomplish the previous objectives: • First of all, we see how to achieve a linear control of the dynamic excitations of a given harmonic pattern with order L. The convolution
T0 T0 T0 Rnq (ω) = (q, Rnq ) = Cnq (ω)Mnq (ω) 1 = q, ξn e−j2πqonq , q ∈ P. 2
T0 According to the second property in Table I, rnq (t) is the periodic convolution of the corresponding periodic pulses in the time-domain, i.e., Z 1 T0 rnq (t) = cT0 (τ )mTnq0 (t − τ )dτ, (7) T0 T0 nq
where mTnq0 (t) and, by considering the time shifting property in Table I, we obtain 1 δ(t − δnq ), (8) 2 the time–delays applied to each
mTnq0 (t) =
•
This is the discrete spectrum –with a number of lines equal to the number of elements in P– for the n-th pulse to be applied to the array. Such pulses provide substantial benefits with respect to conventional TMA beamforming with rectangular pulses: (1) fully independent scanning of the harmonic beams; (2) higher antenna efficiency due to the ability for suppressing unexploited harmonics; and (3) linear control of the magnitudes of the dynamic excitations of all patterns.
•
(4)
T0 leads to a new spectrum CnL (ω), whose coefficients satisfy CnL = Gn0 = ξn for n ∈ {0, 1, . . . , N − 1} by virtue of the first property in Table I, thus achieving a linear control of the magnitude of the L-th order harmonic. The corresponding pulse transformation in the time-domain, according to Table I and (4), yields 0 cTnL (t) = ejL2π/T0 t gnT0 (t). Secondly, we consider the discrete spectrum of the 0 multiplication of each periodic signal cTnL (t) by a set of individual spectral lines which correspond to pure complex-valued exponentials in the time domain. Next, all individual products are summed up. This operation has a twofold function: 1) To act as individual harmonic masks or logical AND functions, i.e., keeping each individual working harmonic while suppressing the rest.
with δnq = onq T0 impulse. Finally, we sum all the orthogonal frequency lines T0 (ω) for q ∈ P to produce the frequency–domain Rnq representation of rnT0 (t), i.e., X 1 T0 (ω) = q, ξn e−j2πqonq . (9) Rnq RnT0 (ω) = 2 q∈P
T0 CnL (ω) = GTn0 (ω) ∗ δ(ω − Lω0 )
= (q, Cnq ) = (q, Gn(q−L) )
(6)
ACKNOWLEDGMENT This work has been funded by Xunta de Galicia, MINECO of Spain, and FEDER funds of the EU under grants ED431C 2016-045, ED341D R2016/012, and TEC2016-75067-C4-1-R.
R EFERENCES [1] R. Maneiro-Catoira, J. Br´egains, J. A. Garc´ıa-Naya, L. Castedo, P. Rocca, and L. Poli, “Performance analysis of time-modulated arrays for the angle diversity reception of digital linear modulated signals,” IEEE Journal of Selected Topics in Signal Processing, vol. PP, no. 99, pp. 1–1, 2016. [2] R. Maneiro-Catoira, J. Br´egains, J. Garc´ıa-Naya, and L. Castedo, “Enhanced time-modulated arrays for harmonic beamforming,” IEEE Journal of Selected Topics in Signal Processing, vol. PP, no. 99, pp. 1–1, 2016.
1042
