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2090 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
Fig. 1. System model of a wireless communications system with an adaptive
array.
II. THE INSTANTANEOUS OUTPUT SINR OF
AN ADAPTIVE ARRAY
A. System Model
A wireless communications system with an adaptive array in
the receiving mode is shown in Fig. 1. The source information
bits are ﬁrst passed through the channel encoder and modulator.
If the channel encoder and modulator are combined together, the
two form a TCM. Then the training sequence is added in front
of all the symbols before they are transmitted via the antenna.
The signal are corrupted by the interferences at the same fre-
quency and additive white Gaussian noise (AWGN). In urban
areas, signals are also affected by different types of fading. In
this paper, an AWGN channel is used to investigate the system
performance. The received signal is ﬁrst processed by the adap-
tive beamformer to combat interference, then the demodulator
and decoder are used to extract the original information bits.
B. The Instantaneous Output SINR and BER
The received signal of an antenna in a AWGN channel is
where is the signal and is the noise. The BER for
coherent detection of a BPSK system is given by [3]
(1)
where is the complementary error function, is the signal
energy per binary symbol and is the variance of the
noise. In an AWGN channel, . The one antenna
system does not work well with co-channel interference because
the interference can be viewed as the noise and thus leads to very
low SNR.
Similarly, the received signal of the th element in an -ele-
ment antenna array with interferers is given by
(2)
where is the coefﬁcient of the channel toward ,
is the direction of the desired signal , is the
th interferer from the direction , and is the re-
ceiver noise of the th element. In an AWGN environment,
the channel coefﬁcients are deterministic and assumed to
have the same amplitude but only differ in phase, which de-
pends on the geometry of the array and direction. Therefore,
is the same as the
array vector in an AWGN channel, where indicates matrix
transpose. Thus, the output of the adaptive array is
(3)
where is the weight vector given by the
adaptive algorithm, and denotes the matrix hermitian.
The interference component in (3) can be viewed as noise.
Therefore, if BPSK is used in the system, the BER performance
is still given by (1) except that the SNR needs to be modiﬁed
accordingly. Assuming that the desired signal and interferers
are independent with zero means, the instantaneous SNR of the
adaptive array system is then given by
(4)
where the noise in each antenna element is assumed to be i.i.d
AWGN. It is obvious that the instantaneous SNR will be max-
imized when there is no interference component in the output
signal. In addition, based on the Cauchy-Schwartz inequality, it
can be expressed as
(5)
Hence the upper bound of the instantaneous SNR is given by
(6)
It can be shown that . Therefore, in
an AWGN channel, the maximum SNR gain of an -element
adaptive array over a one-antenna system is .
The upper bound is achieved with two conditions:
• there is no interference or the interference is totally
suppressed.
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HUANG AND BALANIS: BER OF ADAPTIVE ARRAYS IN AWGN CHANNEL 2091
TABLE I
THE STATE TRANSITION A CODED 8-PSK
• the weight vector is the hermitian of the channel coefﬁcient
vector; i.e., the steering vector.
The minimum BER of an -element adaptive array with BPSK
is then given by
(7)
III. SYSTEM CONFIGURATION
A. Adaptive Algorithm
The MMSE algorithm is one of the popular adaptive beam-
forming algorithms. By minimizing the MSE cost function, it
attains the optimum weights which are given by
(8)
where is the covariance matrix of the received signal, and
is the cross-correlation vector between the received and
training signals. It is usually difﬁcult to estimate (8) directly.
Thus, adaptive MMSE algorithms were developed, including
the LMS and RLS. Ideally, both algorithms converge to the
optimum weights after sufﬁcient iteration steps. The optimum
BER of the MMSE beamformer in AWGN channel with BPSK
is given by substituting (8) and (4) into (7).
B. Trellis Coded Modulation
The TCM is a scheme that combines a multilevel/phase
signal constellation with a convolutional encoder. Due to
code redundancy, the error performance is determined by the
minimum Euclidean distance, , between all possible pairs
of code sequences. Following the channel signal assignment
rules developed by Ungerboeck [21], is larger than the
minimum distance between signal points for an uncoded system
with the same data rate and average power. For example, an
8-state 8-PSK TCM system with maximum likelihood decoder
(MLD) has 3.6 dB gain over the uncoded QPSK system [3].
It has been shown by experiments that the 8-PSK code de-
picted in Table I is optimum for an 8-state TCM [21]. The ﬁrst
column contains the current state, while the ﬁrst row displays the
next state. represents the input data with output symbol
. The symbol indicates that the noted state transition is not
allowed.
When TCM is used, the Viterbi decoder (VD) [22], a max-
imum log-likelihood decoder, can be utilized to perform the de-
modulation. Comparing with the MLD, the VD greatly reduces
the computational load by eliminating the unlikely paths earlier.
Fig. 2. Geometries of the 9-element URA, UCA and UCA-CE.
TABLE II
CONFIGURATIONS OF URA, UCA AND UCA-CE
C. Geometries of the URA, UCA and UCA-CE
In order to perform fair comparisons, the number of elements
in each geometry should be the same. In addition, it should be
noted that geometries, with small number of elements, do not
differ appreciably. For example, a 4-element URA is very sim-
ilar to a 4-element UCA. A URA can only have el-
ements, where is an integer; i.e., . Hence,
9-element arrays are chosen for the comparisons.
The geometries of 9-element URA, UCA and UCA-CE are
shown in Fig. 2. The conﬁguration parameters of these arrays are
shown in Table II. Since the inter-element spacing is more crit-
ical than the array size, it is kept the same for all arrays. They are
implemented in wireless systems with an 8-state 8-PSK TCM
shown in Table I and soft-decision Viterbi algorithm to investi-
gate the inﬂuence of the array geometries on the BER.
Arrays with the same area were also investigated. Since sim-
ilar performance trends among the URA, UCA, and UCA-CE
were observed, they are not presented in this paper due to space
limitations. As the area decreases and the interelement spacing
is smaller than , the performance degrades. Increasing the
areas beyond those shown in Table II, without adding more el-
ements, will also degrade the performance because they intro-
duce aliasing. Mutual coupling was not considered in this paper
because it has been shown that mutual coupling slightly inﬂu-
ences the performance of MMSE algorithm as long as it can be
modeled as a multiplication of an invertible matrix [23].
IV. TCM GAIN IN A UCA WIRELESS SYSTEM
A UCA is shown to have great performance in adaptive beam-
forming. Thus, the BER performance of a UCA wireless system
is of interest. It is examined with an uncoded BPSK system and
a TCM system. It should be pointed out that the uncoded BPSK
and uncoded QPSK have the same BER performance [3].
The 9-element UCA with inter-element spacing is
placed on the - plane, as shown in Fig. 2. In order to evaluate
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2092 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
TABLE III
WELL SEPARATED SIGNALS
Fig. 3. BER of a 9-element UCA with uncoded BPSK.
the best BER performance in an AWGN channel, the LMS al-
gorithm is utilized with and a 1,000 training symbol
sequence to assure optimum weights. Equal power signals, 1
signal-of-interest (SOI) and 4 signals-not-of-interest (SNOI),
are impinging on a UCA from directions shown in Table III.
First, the uncoded BPSK modulation is implemented for the
UCA system. Each of the interferences has equal power as the
desired signal. The AWGN is modeled with input SNR ranging
from to . The BER performance of the UCA with
uncoded BPSK is shown in Fig. 3. The BER curve of an uncoded
BPSK one-antenna system, without interference, is also plotted
and then shifted to the left by ‘9.54’ dB. This is done because,
according to (7), the shifted curve is the optimum BER curve of
a 9-element array. This curve is very close to, if not the same,
as that of a 9-element UCA without interference. This is due
to the LMS algorithm which leads to an optimum solution and
achieves maximum array gain, , in a
9-element array.
The adaptive UCA is then implemented with an 8-state 8-PSK
TCM and soft-decision Viterbi decoder in a wireless system.
The simulations results are shown in Fig. 4. Comparing the re-
sults achieved by the uncoded BPSK and TCM, it is evident that,
in the presence of interferences, the BER of the UCA with TCM
is much better than that with BPSK at high SNRs. It is observed
that the TCM system has more than 3 dB gain over the BPSK at
BER of when there is no interferer or one interferer. Since
the BPSK and QPSK have the same BER performance, a 3-dB
gain is slightly smaller than the 3.6 dB gain that an 8-state 8-PSK
TCM achieves over an uncoded QPSK in the SISO. When there
are more interferers, the TCM still improves the BER, but not
as well as without interferers. Since TCM has greater gain over
Fig. 4. BER of a 9-element UCA with the TCM.
the uncoded BPSK/QPSK in a wireless system with an adap-
tive array, it will be used as the default system to compare other
cases.
With either modulation scheme, the UCA has greater perfor-
mance over traditional one receiver wireless system. It is known
that a system without an adaptive array cannot recover the de-
sired signal when there are strong interferers in the same fre-
quency. However, it is obvious that the adaptive UCA works
well with intensive co-channel interference. Simulations with
different number of elements and different direction signals, not
necessarily in the same plane, conﬁrmed the same observations.
V. BERS OF THE URA, UCA, AND UCA-CE
Since the URA, UCA, and UCA-CE are practical 2D geome-
tries and they have similar beamforming performances, it is of
interest to compare system performance of these arrays using
the BER as a metric.
A. BERs of Arrays With the LMS
Because the LMS is one of the popular adaptive algorithms
with low computational complexity, the LMS with
is ﬁrst used in the comparisons. The same signals with AWGN
are imposed on all geometries; the UCA, URA and UCA-CE.
1) Well Separated Signals in an Azimuthal Plane: The BER
performance of adaptive arrays in a well-separated signals envi-
ronment is examined. The signals are assumed to be of equal
power and in the azimuthal plane, , with directions
shown in Table III.
First, the effect of the training sequence length is investigated.
A temporal adaptive algorithm uses the reference symbols to
train the receive symbols. Meanwhile, since there are encoders
and modulators in a wireless system, the number of bit-data is
different from the number of transmission-symbols. Therefore,
the number of training symbols (TS) is more directly related
to the convergence, and thus it is used in this paper. Since an
8-state 8-PSK TCM is used, every transmission symbol repre-
sents 2 bits of input data. Thus, if 100 training symbols are used,
it takes 200 bits of data.
5.
HUANG AND BALANIS: BER OF ADAPTIVE ARRAYS IN AWGN CHANNEL 2093
Fig. 5. BER of 9-element arrays with the LMS at 50 TS and 200 TS.
Fig. 6. BER of 9-element arrays with the LMS at 500 TS and 1000 TS.
In the simulations, different number of training symbols are
examined. Some of the results are shown in Figs. 5 and 6. It
can be seen that the URA converges faster than the UCA and
UCA-CE. However, all of them reach stability after around
1,000 TS. When there is only 50 TS, all adaptive arrays with
the LMS algorithm do not work very well, as expected. The
LMS adaptive array system needs more training symbols to
achieve better performance. It is observed that more training
symbols do improve the BER, especially from 50 to 500 TS.
However, the LMS adaptive array improves very little from 500
to 1,000 TS. More simulations indicate that the system exhibits
no improvement beyond 1,000 TS. Simulations with different
signal sets lead to the same observation. Therefore, when 1
SOI and 4 SNOIs are well-separated in the azimuthal plane, the
LMS achieves sufﬁciently good result with 500 TS and the best
BER with 1,000 TS.
Second, the BERs of the UCA, URA and UCA-CE with
different number of interferers are examined. The signals uti-
lized in the simulations are still those in Table III. However, the
Fig. 7. BER of multiple interferers with a 9-element UCA.
Fig. 8. BER of multiple interferers with a 9-element URA.
number of interferers ranges from 1 to 4. A 1,000 TS training
sequence is used to assure convergence so that the best BER
performances are attained. The BERs of the adaptive arrays are
shown in Figs. 7–10.
It is obvious that all of these arrays work well with multiple
interferers. It is observed that the BER performance of the UCA
degrades as the number of interferers increases. There is around
a 2-dB difference between the BERs of a 1-SNOI scenario and
that of 4-SNOIs scenario at a . However, it is in-
dicated that the BER of the URA is only slightly inﬂuenced by
the number of interferers. Its performance is excellent even with
4 SNOIs. There is only about a 0.5-dB difference between the
BERs with 1 SNOI and that of 4 SNOIs at a . The
UCA-CE performs similarly as the URA. The intuitive explana-
tion is that the geometry of the 9-element UCA-CE is very close
to that of a 9-element URA, as shown in Fig. 2. This compar-
ison among UCA, URA and UCA-CE shows that the URA has
the best performance, UCA-CE is second and the UCA is the
worst. For example, as shown in Fig. 10, the BER curve of the
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2094 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
Fig. 9. BER of multiple interferers with a 9-element UCA-CE.
Fig. 10. BER comparison in a 4-SNOIs scenario.
URA has around a 0.2-dB gain over a UCA-CE and a 1.5-dB
gain over a UCA at a .
2) Mean BER in an Azimuthal Plane: Based on the speed
of convergence and the ability to combat multiple interferers,
it seems that the URA is the best for well-separated signals.
However, this conclusion is only based on the signal set shown
in Table III. Different signal sets should be investigated to justify
this result. Therefore, the signal sets shown in Table IV are used
to test the system performance. There are a total of 360 signal
sets where the signals are of equal power in every signal set. The
ﬁrst (original) signal set is the same as that described in Table III.
The th signal set is attained via rotating the original signal set
by . Thus the angular separations among the signals are the
same for all signal sets. The view of the rotation of the signals
with the UCA, URA, and UCA-CE, at two snapshots (original
set and one after 60 rotation), is shown in Fig. 11. It is obvious
that the ﬁeld-of-view changes as the impinging directions of the
signals rotate. The rotation has the most effect on the ﬁeld-of-
view of the URA, less on that of the UCA-CE, and least on that
Fig. 11. Rotation of well-separated signals in an azimuthal plane.
TABLE IV
AZIMUTHAL ANGLES () OF WELL SEPARATED SIGNAL SETS AT = 90
Fig. 12. Mean BER of well-separated signals in an azimuthal plane.
of the UCA. Without a loss of generality, the 360 signal-sets are
assumed to have equal probabilities, 1/360. The average BER is
referred to as the mean BER of the “original” signal set in an
azimuthal plane.
Fig. 12 shows the mean BERs of different arrays with the
signal sets described in Table IV. It can be seen that the mean
BER of the UCA is much better than those of the URA and the
UCA-CE. In addition, the BER of the URA is slightly better than
that of the UCA-CE with low input SNR, smaller than .
Beyond that, the UCA-CE performs better and improves faster
as the input SNR increases.
Comparing the results of Fig. 12 with those in Fig. 10, the
performances of the URA and the UCA-CE are inﬂuenced by
the rotation of the signal sets. However, the mean BER perfor-
mance of UCA is even better than that in Fig. 10 because the
UCA has the best symmetry in the azimuthal plane, which treats
7.
HUANG AND BALANIS: BER OF ADAPTIVE ARRAYS IN AWGN CHANNEL 2095
Fig. 13. BER of well-separated signals in an azimuthal plane.
each azimuthal angle basically equally. Therefore, the perfor-
mance of the UCA is not severely affected by rotating the signal
sets. However, the BER of the URA is inﬂuenced by the rotation
because the URA does not possess symmetry in the azimuthal
plane.
Details of the BERs at an input , as a function
of rotation angle, are shown Fig. 13. As expected, the BER per-
formances of the arrays are oscillating with different amplitude.
The log scale BER of the UCA changes similarly as a sinusoid
function of the rotation angle with a period of around 20 . The
maximum BER is around while the minimum is around
. The rotation has a small inﬂuence on the UCA because it
has symmetry in the azimuthal plane. However, the rotation has
strong impact on the BER of the URA and the UCA-CE. It is
obvious that the log scale BERs of the URA and the UCA-CE
ﬂuctuate with much larger amplitude than those of the UCA.
The worst BER of the URA is around 0.5 and the best one is even
lower than . The UCA-CE achieves slightly better perfor-
mance with an upper bound of BER around 0.5 and a
lower bound of around . The BER of the URA has a period
of 90 and the BER of UCA-CE has a period of 45 .
More simulations with different well-separated signal sets
led to the similar trends. It should also be noted that the UCA
outperforms the URA when both have the same area. There-
fore, for a 9-element array, it is clear that the UCA, in general,
outperforms the URA. Since their geometries are similar, the
9-element UCA-CE has similar performance as the 9-element
URA.
B. Average BERs of Arrays in Random Scenarios
In Section V-A, the mean BER is used to evaluate the average
performance of a signal set rotating in an azimuthal plane, and it
is more meaningful than the BER of a speciﬁc signal set. In this
section, random in direction signals with uniform distribution
in 2D space are generated to investigate the statistical system
Fig. 14. BERs of the arrays with random signals in 2D (LMS with 50 and
500 TS).
performance of the URA, UCA and UCA-CE. The BER is the
average value of scenarios, which is given by
(9)
where is the BER of each scenario, and is the probability
of each scenario, which is for uniform distribution. 5,000
scenarios are used in this simulation. Five equal power signals,
1 SOI with 4 SNOIs, are randomly generated in the azimuthal
plane, , for each scenario. The LMS algorithm with
is used.
Fig. 14 displays the UCA, URA, and UCA-CE system per-
formance with 50 and 500 TS. It is obvious that the average
performance of the system in a random signal scenario is not as
good. For the best case, the UCA with the LMS algorithm and
500 TS, the average BER is around . All the other systems
have BERs ranging from 3 to . In all simulations,
the UCA has the best performance with different numbers of
TS. The UCA-CE is second and the URA is the worst in perfor-
mance, except that the URA is slightly better than the UCA-CE
with 50 TS. In addition, the average BER improves very slowly
as the SNR increases, and it seems that all average BERs have
ﬂoors. According to (9), the average BERs mainly depends on
the number of bad scenarios, where the directions of the inter-
ferers are close the the desired signal direction. For example, if
the system has the worst in half of the scenarios,
the ﬂoor of the average BER is 0.25.
C. BERs of Arrays With the RLS
Although the LMS algorithm performs well in adaptive array
systems, it is slow in convergence. In order to accelerate the con-
vergence, a larger step size, , can be used in the LMS. How-
ever, a larger degrades the beamforming performance, and the
system may become unstable. Typical values used in practical
systems range between 0.001 and 0.01. Since the convergence
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2096 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
Fig. 15. BERs of arrays with the
= 0:999 RLS at 50 TS and 200 TS.
is critical for some systems, such as those with real-time appli-
cation, the RLS algorithm is an alternative because it has much
faster convergence.
Similar as the in the LMS algorithm, there is a parameter,
1— , which controls the convergence of the RLS algorithm
[24]. is referred to as the forgetting factor, and it is bounded
by . As increases, more emphasis is placed on
the values of the previous symbols. With , the RLS algo-
rithm requires inﬁnite memory. Therefore, different number of
training symbols and different number of interferers are applied
in many simulations to determine the most suitable value. It
has been shown that achieves sufﬁciently fast con-
vergence and relatively light computational complexity.
The convergence of the RLS algorithm with well-separated
signals is examined ﬁrst. Some of the results are shown in
Figs. 15 and 16. Comparing them with the results of the LMS
shown in Fig. 5, it is obvious that the RLS algorithm performs
much better with short training sequences. When there are only
50 TS, the UCA with the RLS has around a 3-dB gain over the
UCA with the LMS at a . In addition, the BER
curve of the UCA with the LMS at 200 TS is slightly better
than that with the RLS at 50 TS. Similar results are observed in
the URA and UCA-CE.
It also indicated that the RLS algorithm almost approaches
stability with around 200 TS. The BERs of all arrays improve
very little when the length of training sequence increases from
200 to 500 TS. There is no improvement beyond around 500 TS;
the system performance with 1,000 TS is identical to that with
500 TS. At last, it should be pointed out that although the RLS
achieves better performance than the LMS in cases with short
training sequence, their best performance, after both of them
converge, are the same. This is indicated by comparing the re-
sults of Fig. 6 with those of Fig. 16.
Since the best BER performance of an array with the RLS
algorithm is the same as that of the LMS, the simulation re-
sults of the array with different number of interferers are not dis-
played here. In addition, the mean BERs of the URA, UCA and
Fig. 16. BERs of arrays with the
= 0:999 RLS at 500 TS and 1000 TS.
Fig. 17. BERs of the arrays with random signals in 2D (RLS with 50 and
500 TS).
UCA-CE are examined with the RLS and the signal sets shown
in Table IV. As expected, although both the RLS and LMS al-
gorithms achieve the same performance with sufﬁcient TS, the
RLS algorithm enables the system to reach the same BERs with
much shorter training sequence. More speciﬁcally, it takes the
RLS algorithm around 200 TS to achieve the same results when
the LMS algorithm needs around 500 TS. Therefore, when a
system can afford the computational complexity, the RLS algo-
rithm is more suitable for real-time applications. It can provide
sufﬁcient good BER performance with short training sequences,
such as with only around 50 TS.
The scenarios with random signals in 2D space are also exam-
ined with the RLS algorithm, and some of the results are shown
in Fig. 17. Again, fast convergence is indicated; the BER of the
RLS with 50 TS is the same as that of the LMS with 500 TS.
In all simulations with the training sequence ranging from 50
to 500 TS, the UCA is the best, the UCA-CE is second and the
URA is the worst.
9.
HUANG AND BALANIS: BER OF ADAPTIVE ARRAYS IN AWGN CHANNEL 2097
VI. CONCLUSION
In this paper, the BER of a wireless system with an adaptive
array is investigated. The adaptive array has excellent BER per-
formance in an AWGN channel. The gain of the TCM in a wire-
less system with an adaptive array is veriﬁed. The BERs of 9-el-
ement 2D geometries, including the UCA, URA and UCA-CE,
are also examined and compared. With sufﬁcient training sym-
bols, the URA has the best performances, the smallest BER and
the fastest convergence, in some speciﬁc well-separated signal
sets. However, its performance degrades slightly as the number
of interferers increases. Since the 9-element UCA-CE has sim-
ilar geometry as the 9-element URA, they have similar perfor-
mances. The UCA performs the worst in these speciﬁc signal
sets. However, in general, the UCA has much better average per-
formance. It has much smaller “mean BER” than the URA and
UCA-CE in well-separated signal scenarios. The same results
are observed in random-direction signal scenarios.
Both the LMS and RLS algorithms were implemented in
the system. The RLS algorithm, as expected, converges much
faster than the LMS, and thus achieves better BER with shorter
training sequences. The optimum BERs, achieved after they
reach stability, are the same.
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Zhiyong Huang (S’04) received the B.Sci. degree in
electronic and information engineering from the Bei-
jing University of Posts and Telecommunications,
Beijing, China, in 2001, and the M.S.E.E. degree
from Ohio University, Athens, in 2003. He is cur-
rently working toward the Ph.D. degree in electrical
engineering at the Arizona State University, Tempe.
His current research interests include the design
and development of curvilinear adaptive antenna
array, channel models, OFDM, and MIMO.
Constantine A. Balanis (S’62–M’68–SM’74–F’86–
LF’04) received the B.S.E.E. degree from the
Virginia Institute of Technology (Virginia Tech),
Blacksburg, in 1964, the M.E.E. degree from the
University of Virginia, Charlottesville, in 1966, and
the Ph.D. degree in electrical engineering from Ohio
State University, Columbus, in 1969.
From 1964 to 1970, he was with the NASA Lan-
gley Research Center, Hampton, VA, and from 1970
to 1983, he was with the Department of Electrical
Engineering, West Virginia University, Morgantown.
Since 1983, he has been with the Department of Electrical Engineering, Arizona
State University, Tempe, where he is now Regents’ Professor. His research inter-
ests are in computational electromagnetics, smart antennas, and multipath prop-
agation. He is the author of Antenna Theory: Analysis and Design (Wiley, 1982,
1997, 2005), Advanced Engineering Electromagnetics (Wiley, 1989), Introduc-
tion to Smart Antennas (Morgan and Claypool, 2007), and Editor of Modern
Antenna Handbook (Wiley, 2008) and the Morgan Claypool Publishers Series
on Antennas and Propagation, and Series on Computational Electromagnetics.
Dr. Balanis is a Life Fellow of the IEEE. In 2004, he received an Honorary
Doctorate from Aristotle University of Thessaloniki, Greece, the 2005 IEEE
Antennas and Propagation Society Chen-To Tai Distinguished Educator Award,
the 2000 IEEE Millennium Award, the 1996 Graduate Mentor Award, Arizona
State University, the 1992 Special Professionalism Award from the IEEE
Phoenix Section, the 1989 IEEE Region 6 Individual Achievement Award, and
the 1987–1988 Graduate Teaching Excellence Award, School of Engineering,
Arizona State University. He has served as Associate Editor of the IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION (1974–1977) and the IEEE
TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING (1981 to 1984), and
was Editor of the Newsletter for the IEEE Geoscience and Remote Sensing
Society (1982 to 1983). He was Second Vice-President (1984) and member
of the Administrative Committee (1984 to 1985) of the IEEE Geoscience
and Remote Sensing Society, and as Distinguished Lecturer (2003 to 2005),
Chairman of the Distinguished Lecturer Program (1988 to 1991) and member
of the IEEE Antennas and Propagation Society AdCom (1992 to 1995 and
1997 to 1999).
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