Adaptive Antenna Arrays Using a Genetic
Algorithm
Randy L. Haupt
The Pennsylvania State University
Applied Research Laboratory
P. O. Box 30
State College, PA 16804-0030
E-mail: haupt@ieee.org
Phone: 814-865-7299 x210, Fax: 814-863-6239
Abstract—This paper presents the application of a genetic
algorithm (GA) to adapting an antenna array's response in
order to reject interference. Since the GA reduces the total
power output of the array, constraints are used to prevent
nulling of the desired signal received by the main beam.
Constraints take the form of using only the least significant
bits or a subset of the elements to perform the nulling.
Examples demonstrate adaptive nulling using amplitude and
phase, phase only, and amplitude only weights.
I. INTRODUCTION
Adaptive antennas enhance desired signal reception by
placing nulls in the antenna pattern in the directions of the
interference while minimally perturbing the main beam.
The original adaptive antenna consisted of a highly
directional reflector antenna with a small omni-directional
antenna. This sidelobe canceller [1] adjusted the output of
the omni-directional antenna until it cancelled the
interference entering a sidelobe. Since the omni-directional
antenna gain is small compared to the main beam of the
high gain antenna, there was not much impact on the main
beam.
Adaptive arrays improved on the sidelobe canceller by
adjusting the weights of an array to both receive the desired
signal as well as cancel the sidelobe interference [2]. The
problem with most adaptive algorithms is that they need a
receiver at each element in order to get the amplitude and
phase of the signals to form the covariance matrix. The
receivers are expensive and require regular calibration.
Another approach to adaptive nulling is to minimize the
total output power of the array [3]. Since the total output
power consists of both the desired signal and interference
signals, some constraints are needed to insure that only the
sidelobes are nulled and not the main beam. This paper
shows how the constraints are implemented through using
the least significant bits of the amplitude and phase weights
or only a fraction of the elements in the array. As with a
sidelobe canceller, the signal strength associated with the
1-4244-0166-6/06/$20.00 ©2006 IEEE.
least significant bits is too small to create a null in the main
beam. A genetic algorithm (GA) performs the adaptation
by manipulating the weights until the total output power is
minimized. Results have been previously reported for a
computer model of phase-only adaptive nulling with linear
arrays [4] and experimental results for amplitude and phase
adaptive nulling with an eight element conformal array [5].
In both cases, GAs with small population sizes were used to
create deep nulls in the antenna patterns.
II. COST FUNCTION
A GA massages the variables of a cost function until the
output or cost is minimized. In this case, the cost function is
a linear array with variable amplitude and/or phase weights,
and the cost is the total output power. The linear array
model of isotropic point sources is shown in Fig. 1.
The elements in the array are often approximated by
isotropic point sources which have a corresponding array
factor given by
N
AF = ∑ wn e
jk ( n −1) d cos φ
(1)
n =1
where
wn = an e jδn
k = 2π / wavelength
d = spacing between elements
φ = angle relative to x-axis
N = number of array elements
Controlling the weights modifies the main beam peak and
nulls. Usually, the array weights are digital and have a finite
number of settings.
The problem with this cost function formulation is that
the desired signal and the interfering signals are mixed
together. Minimizing the output power will decrease the
desired signal in addition to the interfering signals unless
the desired signal is assumed to enter the main beam and the
adaptive weights are constrained to small values that cannot
place a null in the main beam.
Fig. 2. Array factor at the main beam (φ=90ο) and a sidelobe (φ=42ο) when
all combinations of 1 through 4 least significant bits out of a total of 6 bits
in the phase shifters are tried.
III. THE ADAPTIVE ALGORITHM
Fig. 1. Diagram of an adaptive array controlled by a GA.
If a six element array has uniform amplitude weights and
six bit phase shifters, then the six bit correspond to the
following six phase settings:
bit
1
2
3
4
5
phase
π
π
π
π
π
32 16
8
4
2
6
π
(2)
The least significant bits have small phase values that have
correspondingly small effects on the main beam. Assume
that the array is symmetric about its center and the center
two elements have zero phase. The two phase shifters on the
right side of the array have the negative phase of the two
phase shifters on the left side of the array. Fig. 2 shows the
array output in dB at the main beam ( φ = 90o ) and at a
sidelobe ( φ = 45o ) as a function of phase shifter settings
using up to four out of six bits. Note that at least four bits
are needed to place a null in the sidelobe. If this were a low
sidelobe array, then fewer bits may also place a null in the
sidelobe. Also note that using up to 4 least significant bits
will not significantly perturb the main beam. If five or six
bits are used, then a null can be placed in the main beam.
The cost function must be selected so that nulls can be
placed in the sidelobes while nulls cannot be placed in the
main beam. In this way, the desire signal will be minimally
perturbed while the interfering signals will be nulled. As a
result, this adaptive nulling configuration is simply
minimizing the total array output power.
Each chromosome in the GA population represents a few
of the least significant bits of the phase and/or amplitude
settings at each element in the array. Adjusting these
settings has a small effect on the main beam but can place
nulls in the sidelobes. The goal of the GA is to minimize the
total output power of the antenna by adjusting these array
settings. Since the algorithm must be fast and a global
minimum is not necessary, the GA uses a small population
size. Fig. 3 is a flowchart of the GA used for adaptive
nulling.
Each chromosome represents the least significant bits of
the phase shifters in the antenna array. The least significant
bits are sent to the antenna array and the output power is
measured. In this way, every population member has an
associated cost. Members of the population with high costs
are discarded. The surviving members form a mating pool.
The parents are combined in some manner (such as single
point crossover as shown in Fig. 4) to produce offspring.
The offspring replace the discarded chromosomes. Fig. 4
demonstrates the process of parent selection and the use of
single point crossover to create two new offspring.
The next step randomly mutates a certain percentage of
the population by changing bits from one to zero or from
zero to one (Fig. 5). Normally, the best chromosome is not
mutated. After mutation, the process repeats by measuring
the output power associated with the new population.
they require many function calls to find an acceptable
solution. These slow algorithms will not work for real time
applications like adaptive nulling. There has been strong
evidence that GAs with small population size and high
mutation rates find good solutions fast [6], [7]..
IV. AMPLITUDE AND PHASE ADAPTIVE NULLING
Fig. 3. Flow chart of the GA.
Fig. 4. The process of natural selection and mating.
Fig. 5. After random mutations occur in the population, the output power is
measured for each chromosome and the process repeats.
Most GAs have a large population size and low mutation
rate. Although these implementations have been successful,
The GA functions as an adaptive antenna algorithm by
minimizing the total output power of the array. This
approach only works if the desired signal is not present or if
the adaptive algorithm is constrained to making small
amplitude and phase perturbations at each element.
Although adapting while the desired signal is absent may
work for a stationary antenna with relatively stationary
interference sources, it would fail to provide reasonable
protection for most communications and radar systems.
Limiting the amplitude and phase perturbations at each
element is actually quite easy to do, especially with a GA,
since the GA inherently constrains its variables. Most phase
shifters and attenuators are digital, so a binary GA naturally
works with binary control signals.
The first example has a 20 element array of isotropic
point sources spaced 0.5λ apart. Each element has six bit
amplitude and phase weights. There is a 20 dB, n = 3 low
sidelobe Taylor amplitude taper with two least significant
bits of the amplitude weights and three of the phase weights
dedicated to adaptive nulling. The desired signal has an
amplitude of 0 dB and is incident on the peak of the main
o
beam at φ=90 . Two 30 dB interference sources are incident
at 111o and 117 o . GA parameters include a population size
of 8, a 50% selection rate, roulette wheel selection, uniform
crossover, and a 10% mutation rate. The quiescent and
resulting adapted patterns appear in Fig. 6. Deep nulls are
created in the pattern with little perturbation to the main
beam. The nulls come at a cost of increased average
sidelobe level. Convergence of the algorithm is shown in
Fig. 7. The GA uses elitism, so the maximum number of
output power measurements is 8 + 17 × 7 = 127 . The power
output decreases monotonically, but the individual
interference source received power varies. At times, the
power received by the array from one interference source
will go up or down relative to the power received from the
second interference source. The output power of the desired
signal shows little variation, because the main beam
remains virtually unperturbed. Fig. 8 is a graph of the signal
to interference ratio for the best chromosome in each
generation.
A second example has two 30 dB interference signals at
o
o
50 and 130 . The desired signal is at 0 dB and is incident at
the peak of the main beam. More bits were needed to
perform the nulling in this case. Consequently, four
amplitude and three phase least significant bits were used.
Fig. 9 shows the resulting adapted pattern. Pattern distortion
is more noticeable than in the previous non-symmetric
example.
Fig. 6. The solid line is the adapted array factor and the dashed line is the
quiescent array factor.
Fig. 9. Adapted pattern when there are two symmetric interference sources.
V. PHASE ONLY ADAPTIVE NULLING
Fig. 7. GA convergence for amplitude and phase adaptive nulling.
Fig. 8. Signal to interference ratio as a function of generation for amplitude
and phase adaptive nulling.
Phased arrays have variable phase weights to steer the
main beam, but rarely have variable amplitude weights.
Thus, it is very desirable to do the adaptive nulling using
only digital phase shifters. The same idea applies where
only a few of the least significant bits of the phase shifters
are used for nulling.
The non-symmetric example is repeated for the phase
only case. The three least significant bits from the six bit
phase shifters were used by the same GA to minimize the
output power from the array. Fig. 10 shows the adapted
array factor. The resulting pattern has little main beam
perturbation. Sidelobes are generally higher, though.
Convergence of the phase only GA is shown in Fig. 11.
Both interference signals are minimized. Fig. 12 is a plot of
the signal to interference ratio. The GA improves the ratio
by about 20 dB.
Nulling interference sources that are at symmetric angles
about the main beam is more difficult for phase only nulling
than for amplitude and phase nulling. Using three least
significant phase bits only place one nulling in the pattern.
Adding a fourth bit produced the adapted pattern shown in
Fig. 13. Although the desired nulls appear in the pattern, the
main beam gain dropped slightly, and the sidelobes
increased significantly.
Fig. 10. The solid line is the adapted array factor and the dashed line is the
quiescent array factor.
Fig. 13. Adapted pattern when there are two symmetric interference
sources.
VI. AMPLITUDE ONLY ADAPTIVE NULLING
Fig. 11. GA convergence for phase only adaptive nulling.
Amplitude only adaptive nulling is not normally
considered a viable option, because arrays have adjustable
phase shifters and not adjustable amplitude weights. A new
concept where smart materials are used to adjust the
amplitude of the array elements provides adequate
justification for trying this approach.
The approach to the adaptive nulling is slightly different
here. Instead of using only the least significant bits of the
variable amplitude weights, only a few of the edge element
are given variable amplitude weights. The GA can use the
full range of the amplitude weights without limiting to small
amplitude values. Because only some of the elements are
adaptive, the main beam receives limited perturbations.
As an example, consider an array of 16 elements that are
spaced λ/2 apart. Three edge elements on both ends of the
array have continuous variable amplitude settings. A
continuous variable GA is used in place of the binary GA to
do the adaptation. The array is assumed to start with a
uniform amplitude distribution.
There are two interference signals: one 20 dB signal
incident at 72o and one 24 dB signal incident at 133o. The
resulting adapted amplitude weights are given by
w = [.064 .68 .69 1 1 1 1 1 1 1 1 1 1 .69 .68 .064]
Fig. 12. Signal to interference ratio as a function of generation for phase
only adaptive nulling.
Fig. 14 is the adapted pattern corresponding to these
weights. On the one hand, sidelobes go down due to the
effective amplitude taper that places the nulls. On the other
hand, the main beam gain goes down and the beam width
expands. Limiting the number of elements with variable
amplitude weights is an alternative to using only the least
significant bits of the amplitude weights. Both approaches
place nulls without undue main beam variations.
constraining the size of the weight perturbations allows the
GA to minimize the output power to reject the interference
while at the same time not placing a null in the direction of
the desired signal.
References
[1]
[2]
[3]
[4]
Fig. 14. Array factor due to amplitude only nulling with six out of sixteen
elements.
VII. CONCLUSIONS
The GA has proven quite successful as an adaptive
antenna algorithm for arrays. Using a small population size
and high mutation rate helps the GA to quickly place nulls.
In this paper several examples were presented to show how
[5]
[6]
[7]
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