Phase-Only Adaptive Nulling with a Genetic
Algorithm
Randy L. Haupt
HQ USAFADFBE
2354 Fairchild Dr, Suite 2F6
USAF Academy, CO 80840-6236
719-333-3191
email: hauptrl.dfee@ usafa.af.mil
Abstract-This
paper describes a new
approach to adaptive phase-only nulling with
phased arrays. A genetic algorithm adjusts
some of the least significant bits of the beam
steering phase shifters in order to minimize the
total output power. Using a few bits for nulling
speeds convergence of the algorithm and limits
pattern distortions. Various results are
presented to show the advantages and
limitations of this approach.
TABLE OF CONTENTS
1. INTRODUCTION
2. PROBLEM FORMULATION
3. THE ADAPTIVEALGORITHM
4. RESULTS
5. CONCLUSIONS
1. INTRODUCTION
Low sidelobes don't guarantee adequate
reception of a desired signal in the presence of
interfering
sources.
Adaptive
nulling
complements the low sidelobe antenna by
placing nulls in a few low sidelobes to reject
the strongest interfering sources. An ideal
adaptive algorithm for a phased array antenna
has the following desirable characteristics:
Sue Ellen Haupt
HQ USAFA/DFP
2354 Fairchild Dr
USAF Academy, CO 80840
719-333-3055
email: hauptse.dfp@usafa.af.mil
Places multiple deep nulls in the directions
of interference,
Rejects interference over the bandwidth of
the antenna,
Places the nulls very quickly,
Complements existing phased array
technology, and
Minimizes pattern perturbations.
An adaptive algorithm possesses some of these
characteristics, but no adaptive algorithm
meets all the characteristics. Selection of the
adaptive algorithm, hence the desirable
characteristics, depends upon the antenna, the
cost, the performance requirements, and the
interference environment.
An off-the-shelf adaptive algorithm is not
usually suitable for use with an off-the-shelf
phased array antenna. Many adaptive antenna
array algorithms originate from the generic
signal processing literature [ 13 and require
modification of the algorithm and the antenna
in order to have a working adaptive array. An
adaptive algorithm developed for data
transmission requires hardware and software
modification in order for it to successfully
place nulls in the far field pattern of an array.
151
0-7803-3741-7/97/$5.00 0 1997 IEEE
152
Most adaptive antenna algorithms multiply the
quiescent weights by the inverse of the
sampled covariance matrix to get the adapted
weights. The resulting complex weights place
nulls in the far field pattern in the directions of
interference. A sampled covariance matrix is
formed from the complex signals received at
each element in the array. Although
mathematically elegant and fast these methods
have two impractical hardware requirements
on the antenna array. First, the array must have
an expensive receiver or correlation at each
element. Most arrays have a single receiver at
the output of the summer, so the antenna must
be designed especially for the algorithm. Not
only are multiple receivers expensive, but the
receivers require a sophisticated method for
calibration [2]. Second, the array must have
variable analog amplitude and phase weights at
each element. Usually, a phased array has only
digital beam steering phase shifters at the
elements. The feed network determines
amplitude weights. There are two problems
fkom an algorithmic standpoint as well. First,
digital phase shifters only approximate the
phase calculated by the adaptive algorithms.
The weight quantization error limits null
placement. Second, these algorithms get stuck
in local minima [3]. As a result, they do not
find the optimum weights to reject the
interference at hand. Some common adaptive
algorithms include Least Mean Square
Algorithm and Howells-Applebaum Adaptive
Processor, and examples can be found in
references [3] and [4]. These methods are very
fast but the difficulties mentioned prohibit their
wide-spread use, particularly for arrays with
more than a handful of elements.
amplitude weights or correlators. Their
drawbacks include slow convergence and
possibly high pattern distortions.
Another class of algorithms adjusts the phase
shifter settings in order to reduce the total
output power from the array [5], [61, [71.
These algorithms are cheap to implement
because they use the existing array architecture
without expensive additions, such as adjustable
1. They require an expensive receiver at each
element - makes array impractical to build;
2. They get trapped in local minima - don't
use full potential of the antenna to reject
interference;
This class has four approaches, the last of
which is the topic of this paper. The first
approach is the random search algorithm [3].
Random search algorithms randomly sample a
small fraction of all possible phase settings in
search of the minimum output power. The
search space for the current algorithm iteration
can be narrowed around the regions of the best
weights of the previous iteration. This
approach is usually too slow for beam steering
and radar applications. It is less likely to get
stuck in a local minimum and does not require
an expensive receiver at each element. A
second approach forms an approximate
numerical gradient and uses a steepest descent
algorithm to find the minimum output power
[8]. This approach has been implemented
experimentally but is slow and gets trapped in
local minima. As a result, the best phase
settings to achieve appropriate nulls are usually
not found. The third approach is a beam space
algorithm that assumes the location of the
interference is known. This algorithm forms a
cancellation beam in the direction of the
interference. The height of the cancellation
beam is adjusted to cancel the sidelobes and
place a null in the interference direction. This
approach is fast but requires knowledge of the
interference locations and a reasonably
accurate estimate of the amplitude and phase
weights at each element.
Serious drawbacks
algorithms include:
to
current
adaptive
3. They slowly converge - often not useful for
radar or scanning applications;
4. They can't be implemented on existing
antennas--they require adjustable amplitude
weights and receivers at every element in
addition to beam steering phase shifters;
5. They cause the main beam to move from
its desired pointing direction; and
6. They significantly raise the sidelobe levels
of a low sidelobe array.
This paper describes a simple technique
suitable for implementation on existing phased
arrays. The approach combines a genetic
algorithm with the hardware limitations of the
array to place nulls in the directions of
interference with small perturbations to the far
field pattern. Excellent nulling results are
possible for most interference scenarios.
incident
field
elements
\/
Y
Figure 1. Diagram of a phase-only adaptive
linear array
2. PROBLEM FORMUL,ATION
A linear array antenna is a group of equally
spaced antennas arranged along a line and
whose outputs are added together to provide a
single output. Figure 1 shows a diagram of
such an array. Mathematically, the array far
field pattern is given by [11
The amplitude weights are fixed. Lowering the
sidelobe levels requires an even phase shift
about the center of the array [9], while nulling
requires an odd phase shift [lo]. Since nulling
is of importance here, (1) simplifies
to
N
M ( u ) = 2sin @ x u , cos[(n
,
- 1)Y + A,, + 6
1
n=l
where
wn = ant?" complex weight at element n
2N = number of elements in the array
Y =kdu+ A
k =2 d A
A = wavelength
d = spacing between elements
U = cos$
$ = angle of incidence of electromagnetic
plane wave
A = beam steering phase
(2)
Note that Y = kdu and no longer includes the
beam steering phase. Since the beam steering
phase is quantized, it differs from (n - l)A and
must be represented by A,,. The steering
phase, A,,, is calculated first and the beam
steered to the proper angle, before the nulling
phase, 6 , , is found. Figure 1 is a diagram of a
phase-only adaptive array with A,, 4.
The digital phase shifters have B bits. B needs
to be as small as possible to reduce the cost of
the phase shifter but should be large enough to
maintain low sidelobes over the scan angles of
153
the array. The quantized steering phase at
element n is given by
solution by simulating evolution in nature. In
this application the phase shifter settings
evolve until the antenna pattern has nulls in the
directions of jammers. A genetic algorithm was
chosen for this application because it is an
whereas the nulling phase is represented by
efficient method to perform a search of a very
P
large, discrete space of phase settings to
6 , = 2 a x b, 2-’
(4) achieve the minimum output power of the
p=l
array. An adaptive phase-only array has 2NB
where
possible phase settings, many corresponding to
local “ain the total power output. Such a
B = total number of bits in phase shifters
large number of phase settings and local
P = number of phase bits used for nulling
minima make random search and gradient
based algorithms impractical to use.
[ b,b, ...b p ]= vector containing the nulling bits
representing 6 ,
round{* } = round * to the nearest integer
rem{* } = takes only the digits to the right of
the decimal point of *
In order to minimally perturb the array pattern,
the adaptive algorithm assumes P<B.
Fill phase settings
matrix with random
1’s and 0‘s
3. THE ADAPTIVEALGORITHM
A phase-only adaptive algorithm modifies the
quantized phase weights based on the total
output power of the array. If no interference is
present, then the algorithm tries to minimize
the desired signal. To prevent desired signal
degradation, the algorithm should only be
turned on when the desired signal becomes
swamped by the interference or the nulling
phase shifts should be small. This potential
problem is discussed in more detail in the next
section. The disadvantages of the phase-only
algorithms make them unlikely candidates for
use with antenna arrays. This section presents
a method that is as fast as the beam space
algorithm, doesn’t easily get stuck in local
minima, and limits pattern distortion.
The adaptive algorithm is based on a genetic
algorithm and uses a limited number of bits of
the digital phase shifters. A genetic algorithm
is a computer program that finds an optimum
154
Figure 2. Flow chart of the genetic
algorithm used for adaptive nulling.
r, nulling- bits
phase shifter
Y
Y
Y
Y
output
power
Y
5 9 9 E E
E
3
0
E E E
2
232
6 ) b )
b)
E
2
6)
Figure 3. A population of phase shifter
settings with corresponding output powers.
Figure 2 shows a flowchart of the adaptive
genetic algorithm. It begins with an initial
population consisting of a matrix fiUed with
random ones and zeros. Each row of the
matrix (chromosome) consists of the nulling
bits for each element placed side-by-side.
There are NP columns and M rows. The
output power corresponding to each
chromosome in the matrix is measured and
placed in a vector (Figure 3). M must be large
enough to adequately search the solution space
and help the genetic algorithm arrive at an
excellent solution. On the other hand, M needs
to be small, so the algorithm is fast. The speed
of the algorithm is also a function of N and P.
As N and P increase in size, M needs to be
larger to keep the algorithm out of local
minima, and the number of iterations required
for convergence increases. The output power
vector and associated chromosomes are ranked
with the lowest output power on top and the
highest output power on the bottom. The next
step discards the bottom X% of the
chromosomes, because they have the greatest
output power. The algorithm generates new
nulling chromosomes from the chromosomes
that were kept to replace those discarded
(Figure 4). Two chromosomes are selected at
random. Chromosomes with lower output
power receive a correspondingly higher
probability of selection. Next, a random point
is selected and bits to the right of the random
point are swapped to form two new
chromosomes. These new chromosomes are
placed in the matrix to replace two settings
that were discarded, and their output powers
are
measured.
When
enough
new
chromosomes are created to replace those
discarded, the chromosomes are ranked and
the process repeated. A small number (less
than 1%) of the nulling bits in the matrix can
be randomly switched from a one to zero or
visa versa. These randomly induced errors
(mutations) allow the algorithm to try new
areas of the search space, while it converges
on a solution. Usually, the best phase setting is
not randomly altered. More general
descriptions of genetic algorithms can be found
in [ll] and [12]. The next section shows
results for determining the best values for P
and M.
-
phase shifter
settings
-
. output
power
1
000 001 000 001 000
001 000 000 001 000
000000000001001
001 000 001 000 001
r
000 001 001 000 001
m
cd
c
e,
9
d
Figure 4. Two partners are selected from the
mating pool to produce two offspring that are
put back into the population to replace those
chromosomes that were discarded.
155
4.RESULTS
The genetic algorithm used here begins with
20 chromosomes. Only 10 are kept during the
25 iterations of the simulation. Each iteration,
the bottom 6 chromosomes are discarded and
replaced by chromosomes generated from the
top 4 chromosomes. These numbers are small
enough to keep the algorithm fast but large
enough to place the nulls.
The first example array modeled in this paper
has 40 elements and a 30 dB Chebychev
amplitude taper. Elements are spaced 0.5A0
apart at the center frequency fo. Phase shifters
must have six bit accuracy in order to keep the
quantization lobe slightly below the general
sidelobe level over the scan angles of the array
(+-30" from broadside). The adaptive array
model was tested for five different interference
scenarios and judged based on three
performance criteria. Results appear in Table
2. The first performance characteristic is the
sidelobe reduction at the interference angle(s).
When there are two interference sources
(ui=.62 and .72), the minimum sidelobe
reduction of the two angles is listed in the
table. The second performance characteristic is
the number of power measurements required
for a 10 dB reduction in the sidelobe level.
This number directly relates to the speed of the
algorithm. The third performance characteristic
is the maximum sidelobe level, which is an
indication of the amount of pattern distortion.
A lower value for any performance
characteristic indicates better performance.
Table 1. The genetic algorithm adaptive array was tested for five scenarios. The seven columns to
the right list the number of nulling bits and the phase value of the most significant bit (MSB). The
three performance characteristics judge the algorithm for null depth, speed, and pattern distortion.
A low value for a performance characteristic is good.
156
Two important parameters are the number of
bits used for nulling and the maximum phase
shift or the phase of the most significant bit
(MSB). A genetic algorithm is often sensitive
to the number of bits in a chromosome. Thus,
comparing the performance of the algorithm
against the number of bits used for nulling is
important. The maximum phase shift impacts
the main beam and sidelobe distortion and the
null depth obtainable. Results for MSB phase
values above .n/4 produced significant pattern
distortions, so they are not listed in Table 1.
t
N
N
K
INN
11111
l5
10
N
INN
I
a,
11%
N
-lo/
N
N
M
N
NI
I
N
A gradient-based algorithm using central
difference approximations of the derivatives
would take 80 power measurements per step in
the algorithm. The genetic algorithm is a clear
winner by converging in only 32 power
measurements.
Figure 7 is a graph of the maximum and
average
sidelobe reduction
of
the
chromosomes in the population after each
iteration. Iteration 0 is the quiescent level of
the sidelobe. The average chromosome is of
importance, because it indicates how well the
antenna rejects interference during the
adaptation.
The
average
and
best
chromosomes are the same at iteration 11.
They don't remain the same in later iterations
because random mutations are introduced into
the population.
x
NNN
I
-1 5
*!UN
5
"
* I
f
10
15
20
element
25
30
35
40
Figure 5. Adapted phase weights for the 40
element (d=0.5h) low sidelobe array with 6 bit
phase shifters. Two bits were used for nulling
with an MSB of d16. Interference sources
appear at u=.62 and -72.
Scenario 1 is a simple example of two
interference sources at ui=.62 and .72.
Sidelobe reduction is best for a middle-sized
MSB and P. Convergence is fastest for a
smaller P and MSB. As expected, sidelobe
distortion increases with larger values of the
phase of the MSB. This example suggests that
P=2 and MSB phase = x/16 is the best
combination. Figure 5 shows the adapted
phase weights that produce the nulled pattern
in Figure 6. The maximum phase shift to
produce the nulling is 16.875' The maximum
sidelobe level increases to -27 dB.
-10
B
.-C
E -20
B
g
9-30
-mp!
-iJ
-40
-50
-1
-0.5
0
0.5
1
U
Figure 6. Adapted antenna pattern
corresponding to the phase shifts in Figure 5.
157
the size of the phase shift keeps the main beam
intact. As long as P and MSB are small, then
the algorithm has little effect on the main
beam. An MSB phase of d 1 6 or d 3 2 does not
require turning the algorithm off when no
interference is present, because the algorithm
can't attack the main beam.
0
5
10
15
20
25
iteration
Figure 7. Genetic algorithm convergence is
graphed over 25 iterations. The solid line
represents the output power for the best
chromosome, while the dashed line represents
the average output power for all the
chromosomes in the population.
Scenario 2 presents the algorithm with the
difficult situation where the jammers are at
angles symmetric about the main beam. In
Figure 6 notice the increase in the sidelobes
symmetrically opposed to the nulls placed by
the adaptive algorithm. This phenomena is
discussed in [14]. As noted in Table 2,
symmetric interference sources can only be
nulled with large values of phase and many
power measurements. Pattern distortions are
quite high. Performance improves as the
symmetry is broken. When the interference
sources are one sidelobe apart, performance
similar to Scenario 1 is obtained.
Scenario 3 tests the bandwidth performance of
the algorithm by placing two interference
sources close together. In this case, the smaller
P and MSB phase, the better the algorithm
performance. For the most part, null depth is
diminished for broadband jammers.
Scenario 4 checks the algorithm when no
interference is present, but the desired signal is
received by the main beam. The algorithm
should not attack the desired signal. Limiting
158
Scenario 5 has the algorithm attempt to place
a null in the quantization sidelobe. Results are
marginal for all values of MSB phase. The
quantization lobe results from the digital phase
shifter quantization.
x m
*xx
m
x
x It
15mx
10-
%
1 X!E%lK
2
50)
:
._
O - X M
mx
x m x
20
x**
x
40
x
60
element
1lx
111
Ymx
i
%!X
x y I x
80
x
* *
*sx-
100
Figure 8. Adapted phase weights for the 100
element (d=0.5h) low sidelobe array with 6 bit
phase shifters. Two bits were used for nulling
with an MSB of n/16. Interference sources
appear at u=.25 and .35.
- -lot
.-C
E -20
2
2i
.-%! -30
-me
4-
-40
-50
-1
0
-0.5
0.5
1
U
Figure 9. Adapted
antenna pattern
corresponding to the phase shifts in Figure 8.
I
I
I
Phase-only adaptive nulling works on existing
phased-array antenna designs, unlike signal
processing based adaptive arrays that require
receivers at every element. Its main
disadvantage is slow convergence time. The
genetic algorithm approach to phase-only
adaptive nulling is significantly faster than the
previous approaches of random search and
gradient methods. Thus, it advances the stateof-the-art of phase-only adaptive nulling.
[l] C. F. N. Cowan and P. M. Grant, Adaptive Filters,
Englewood Cliffs, NJ: Prentice Hall, 1985.
-60
0
and the 100 element uniform array showed fast
convergence, deep nulling capability, and small
pattern distortions. Using only a few least
significant bits and small phase values for the
MSB are keys to the algorithm's performance.
Its important advantage is that it is easy to
implement on existing phased arrays.
Disadvantages are: 1) little success at nulling
interference entering a quantization sidelobe
and 2) interference at symmetric angles about
the main beam. Increasing the bandwidth of
the interference also degrades algorithm
performance.
5
15
10
20
25
iteration
Figure 10. Genetic algorithm convergence is
graphed over 25 iterations. The solid line
represents the output power for the best
chromosome, while the dashed line represents
the average Output power for
chromosomes in the population.
all the
5. CONCLUSIONS
The genetic
performed we' for two
jammers that were: 1) separated by an angular
width of at least one sidelobe and 2) were not
symmetric in angular distance about the main
beam. Both the 40 element low sidelobe array
[2] R. Kinsey, "Array antenna self-calibration
techniques," M A Workshop: Testing Phased Arrays
and Diagnostics, San Jose, CA, Jun 89.
[3] R. A. Monzingo and T. W. Miller, Introduction to
Adaptive Antennas, New York: Wiley, 1980.
[4] R. T. Compton, Adaptive Antennas Concepts and
Performance, Englewood Cliffs, NJ: Prentice Hall,
1988.
[5] C. A. Baird and G. G. Rassweiler, "Adaptive
sidelobe nulling using digitally controlled phaseshifters," IEEE AfJ Trans., Vol 24, No. 5 , pp. 638-649,
sep76.
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[6] M. K. Leavitt, "A phase adaptation algorithm,"
IEEE AP-S Trans., Vol. 24, No. 5 , pp. 754-756, Sep
76.
[7] H. Steyskal, "Simple method for pattern nulling by
phase perturbation," IEEE AP-S Trans., Vol. 31, No. 1,
pp. 163-166,Jan 83.
[8] R. L. Haupt, "Adaptive nulling in monopulse
antennas," IEEE AP-S Trans., Vol. 36, No. 2, pp. 202208, Feb 88.
[9] J. F. Deford and 0. P. Gandhi, "Phase-only
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[ 101 R. A. Shore, "A proof of the odd-symmetry of the
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[l 11 J. H. Holland, "Genetic algorithms," Sci. Amer.,
pp 66-72, July 1992.
[12] R. L. Haupt, "An introduction to genetic
algorithms for electromagnetics," IEEE Antennas and
Propagation Magazine, Vol. 37, NO. 2, pp. 7-15, Apr
95.
[13] S . Lundgren and J. Sanford, "A new technique
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160
[141 R A. Shore, "Nulling at symmetric pattern
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Randy Haupt is a Lieutenant Colonel in the USAF
and a Professor of Electrical Engineering at the
United States Air Force Academy. He received his
BS in EE from the USAF Academy in 1978, his
MS in Engineering Management from Western
New England College in 1981, his MS in EE from
Northeastern University in 1983, and his PhD in
EE from The University of Michigan in 1987. He
worked as a project engineer on the OTH-B Radar
and as an antenna engineer for Rome Air
Development Center. Lt Col Haupt was Federal
Engineer of the Year in 1993.
Sue Ellen Haupt is a Research Scientist in the
PAOS Program at the University of Colorado,
Boulder. For the 1996-1997 academic year, she is
on a sabatical as a visiting scholar at the Physics
Department at the USAF Academy. She received
her BS in Meterology from Penn State in 1978,
her MS in Engineering Management from Western
New England College in 1981, her MS in ME from
Wochester Polytechnic Institute in 1984, and her
PhD in Atmospheric Science from The University
of Michigan in 1988.