Time-Modulated Receive Arrays
Randy L. Haupt
Ball Aerospace & Technologies Corp.
Westminster, CO
λ = wavelength
Abstract— Time-modulated arrays open and close switches in
order to connect and disconnect elements from the feed network
and generate an "average" low sidelobe array. This paper
presents an analysis of the instantaneous array factor, and its
effect on received signals in order to demonstrate how timemodulation causes significant fluctuations in the directivity and
does not reject sidelobe interference as well as standard static low
sidelobe approaches.
Keywords- Antenna arrays; time-modulation; antenna pattern
synthesis; adaptive arrays; arrays component
I.
INTRODUCTION
Time-modulated arrays connect and disconnect elements
from the feed network using synchronized switches in order to
create an "average" antenna pattern with low sidelobes [1].
This "average" array pattern concept is deceptive in that the
instantaneous array factor associated with the current switch
configuration actually receives the signal.
This paper analyzes the instantaneous array factor of a
time-modulated array and shows the impact on signal
reception. An instantaneous approach demonstrates the shortcomings of time modulation that make it impractical to
implement. Section II presents the time-averaged form of the
time-modulated array factors. The instantaneous array factor is
better suited for analyzing the effects of the modulation on
received signals, so Section III introduces the instantaneous
array factor. Section IV presents plots of the time-modulated
array factors as a function of time and compares them to static
uniform and Chebyshev array factors. The main beam
directivity decreases from the start to the end of the switch
period while the sidelobes dramatically vary over the same
period. Section V demonstrates the effects of time-modulation
on continuous wave signal reception. The desired signal
entering the main beam, as well as interference signals entering
the sidelobes are modulated. As a result, the signal to noise
ratio (SNR) for the time-modulated array is less than that of the
static array it is attempting to simulate.
II.
AVERAGE ARRAY FACTOR
The time harmonic form of the array factor is given by
N
AF = ¦ wn e
d = element spacing
φ = angle from positive x-axis
wn = amplitude taper
N= number of elements
Time modulated arrays open and close switches at the
elements (see Figure 1) in a prescribed sequence that causes the
average array pattern to simulate a low sidelobe array
amplitude taper. A closed switch connects an element to the
feed network ( wn = 1.0 ), while an open switch ( wn = 0.0 )
disconnects the element. The switch closed time, τ n ,
determines the average element weights. The average
amplitude weight at element n is the fraction of the period that
a switch is closed, so that
wn =
τn
Ts
= an
(2)
where an is the desired amplitude taper. The associated time
dependent array factor [1] is given by:
∞
¦e
AF =
j ª¬(ω + pωs ) t º¼
p =−∞
N
¦w
pn
e
jk ( n −1) du
,
(3)
n =1
where
w pn =
1
Ts
³
Ts
0
wn ( t )e − jpωs t dt
­1 0 ≤ t ≤ τ n
wn ( t ) = wN +1− n ( t ) = ®
¯0 τ n < t ≤ 1
2π
ωs =
Ts
ω = 2π f
jk ( n −1) d cos φ
,
(1)
f = 1 / t = frequency
n =1
Where
Equation (3) only accounts for the array factor and does not
consider time-varying signals incident on the array.
k = 2π / λ
978-1-4244-9561-0/11/$26.00 ©2011 IEEE
968
AP-S/URSI 2011
Figure 1. Switches at the elements of a time-modulated array.
III.
INSTANTANEOUS ARRAY FACTOR
An equation that takes into account the signal received by
the array is given by
N
M
d
ª
º
F = ¦ wn ( t ) ¦ sm «t − ( n − 1) cos φm »
c
¬
¼
n =1
m =1
Figure 2. Array factor as a function of time for a 30 dB Chebyshev average
taper.
(4)
where
sm = strength of signal m incident at φm
M = number of signals incident on the array
c = speed of light you.
This equation will be compared with (2) in several simulations.
IV.
TIME DEPENDENT ARRAY FACTOR
In reality, the average array pattern is just a statistical
phenomenon and not an actual array pattern. Instead, at any
instant the array factor results from only those elements that are
connected to the feed network. If the switches are
symmetrically turned off from the edge to the center, then the
resulting array factors are all uniform. If the switch positions
follow some other pattern, then the array factor corresponds to
a thinned array. This paper follows the edge to center switching
that was suggested in [1].
Consider an 8 element uniform array (d=l/2) with switches
at each element. Assume the carrier frequency is 300 MHz and
fs =15 MHz (Ts=66.7 ns). One period of the array factor of the
30 dB average Chebyshev array is shown in Figure 2. There are
4 distinct uniform array factors corresponding to 8, 6, 4, and 2
element arrays that exist for 17.5, 17.1, 19.5, and 12.5 ns,
respectively. One period of the array factor of the 30 dB
average Chebyshev array is shown in Figure 3. A 40 dB
"average" Chebyshev array has 4 distinct uniform array factors
corresponding to 8, 6, 4, and 2 element arrays that exist for 9.7,
18.1, 22.7, and 16.1 ns, respectively. The switching times are
different for the 40 dB Chebyshev array. The uniform array
factors are identical in Figrues 2 and 3, but the length of time
that they exist differs. In both cases, the instantaneous array
factor is a uniform array factor.
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Figure 3. Array factor as a function of time for a 40 dB Chebyshev average
taper.
The sidelobes, nulls, and main beam of the array factor
change as the number of active elements change. Figure 4 is a
plot of the main beam directivity as a function of time. This
plot shows that the directivity varies from 9 to 7.8 to 6 to 3 dB
corresponding to 8, 6, 4, and 2 elements. Assuming that a
desired signal enters the main beam, then the amplitude of the
desired signal is modulated by the time-varying main beam,
and, in turn, modulates the total signal received by the array.
Sidelobes and nulls in the array factor change with time as
well. Figure 5 shows the array factor at 4 angles as a function
of time. The 90o label is identical to one period of the main
beam directivity shown in Figure 4 (different scales). The
sidelobe levels at 60o, 65o, and 70o go up and down as a
function of time as the array factor changes based on the
number of element switches closed. The sidelobe level change
can modulate signals entering the sidelobes (interference) even
more than signals entering the main beam. Moving from a null
to a sidelobe peak in the 60o case demonstrates the extreme
differences in sidelobe signal reception.
lower sidelobes that keep interfering signals from
overwhelming the desired signal entering the main beam.
Figure 4. The directivity of the array factor as a function of time for a 30 dB
Chebyshev average taper.
In order to test the time modulated array, the normalized
300 MHz interference signal shown at the top of Figure 7 is
incident at f=70o. The output of the 30 dB Chebyshev array is
quite small due to the low sidelobes. The uniform array output
is high, since it is incident on a sidelobe peak. The timemodulated array output varies depending upon the sidelobe
level of the uniform array that results from the number of
elements currently turned on. Additional perturbations are
noticeable when the array changes the number of "on"
elements (see shaded regions in the first switch period). These
perturbations increase in time extent with increasing angle.
They occur, because the signal arrives at each of the 8
elements at different times.
Figure 5. Directivity at the main beam and 3 different sidelobes for a 30 dB
Chebyshev average taper.
V.
TIME DEPENDENT RECEIVED SIGNAL
Thus far only the array factor has been considered as a
function of time. In this section the effects of time-modulation
on received signals is examined. The desired signal enters the
main beam, while the interference enters the sidelobes.
A 300 MHz CW desired signal enters the main beam. A
normalized version of this signal is shown at the top of Figure
6. If the array is time-modulated to simulate a 30 dB average
Chebyshev pattern, then the signal received has a varying peak
amplitude as shown by the second plot in Figure 6. This
amplitude modulated signal has an envelope given by the
directivity plot in Figure 4. Comparing this received signal
with those of a 30 dB Chebyshev amplitude taper and a
uniform array (bottom of Figure 6), shows that the timemodulation of the array weights reduces the quality of the
received signal. The time-modulated array received signal is
always less than or equal to the signal received by the 8
element uniform array. The 8 element Chebyshev array
received signal is sometimes greater and sometimes less than
the time-modulated array received signal. A low sidelobe
array sacrifices some main beam gain and resolution to get
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Figure 6. The normalized signal is incident on an 8 element array at f=90o.
The bottom three plots show the array output when the array is time
modulated to simulate an average 30 dB Chebyshev taper, an array with a 30
dB Chebyshev amplitude taper, and a uniform array.
Figure 7. The normalized signal is incident on an 8 element array at f=70o.
The bottom three plots show the array output when the array is time
modulated to simulate an average 30 dB Chebyshev taper, an array with a 30
dB Chebyshev amplitude taper, and a uniform array.
VI.
CONCLUSIONS
Time-modulation is an intriguing idea, especially with the
continuing improvement of RF switch speeds. This paper
looked at the instantaneous array factors of a time-modulated
array. The instantaneous array factor corresponds to a uniform
array and has high sidelobes that do not attenuate the
interference signals entering the sidelobes like a low sidelobe
array factor. The analysis presented here applies to CW and
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broadband signals. The "average array pattern" produced by
time-modulated arrays does not have the sidelobe rejection
capability of either uniform arrays or low sidelobe arrays.
REFERENCES
[1]
W. Kummer, A. Villeneuve, T. Fong, and F. Terrio, “Ultra-low
sidelobes from time-modulated arrays,” IEEE AP-s Trans., Vol. 11,No.
6, 1963, pp. 633-639.