IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008 2089
BER of Adaptive Arrays in AWGN Channel
Zhiyo...
2090 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
Fig. 1. System model of a wireless communica...
HUANG AND BALANIS: BER OF ADAPTIVE ARRAYS IN AWGN CHANNEL 2091
TABLE I
THE STATE TRANSITION A CODED 8-PSK
• the weight vec...
2092 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
TABLE III
WELL SEPARATED SIGNALS
Fig. 3. BER...
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 2...
2094 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
Fig. 9. BER of multiple interferers with a 9...
HUANG AND BALANIS: BER OF ADAPTIVE ARRAYS IN AWGN CHANNEL 2095
Fig. 13. BER of well-separated signals in an azimuthal plan...
2096 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008
Fig. 15. BERs of arrays with the 
 = 0:999 R...
HUANG AND BALANIS: BER OF ADAPTIVE ARRAYS IN AWGN CHANNEL 2097
VI. CONCLUSION
In this paper, the BER of a wireless system ...
Upcoming SlideShare
Loading in...5
×

BER of Adaptive Arrays in AWGN Channel

107

Published on

For more projects please contact us @ www.nsrcnano.com

Published in: Technology, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
107
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Transcript of "BER of Adaptive Arrays in AWGN Channel"

  1. 1. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 56, NO. 7, JULY 2008 2089 BER of Adaptive Arrays in AWGN Channel Zhiyong Huang, Student Member, IEEE, and Constantine A. Balanis, Life Fellow, IEEE Abstract—A system model is presented of adaptive arrays with trellis coded modulation (TCM) to investigate the bit-error-rate (BER) of a wireless system in a co-channel interference environ- ment. Adaptive arrays with different geometries, uniform circular array (UCA), uniform rectangular array (URA) and UCA with center element (UCA-CE), are implemented with the least mean square (LMS) algorithm to combat interference at the same frequency. The wireless systems are investigated in well-separated signals and random direction signals scenarios with additive white Gaussian noise. The URA and UCA-CE have faster convergence and their BERs, in some cases, are slightly influenced by the number of interferers. However, although the UCA converges slowly and its performance degrades as the number of interferers increases, it generally has the best BER. The recursive least square (RLS) algorithm is also investigated. It achieves much better per- formance than the LMS algorithm when the training sequences are short. However, the optimum BERs attained by these two algorithms are the same. Index Terms—Adaptive array, additive white Gaussian noise (AWGN), bit-error-rate (BER), least mean square (LMS), recur- sive least square (RLS). I. INTRODUCTION IN a traditional wireless system one transmitter and one re- ceiver are used, which is usually referred to as single-input- single-output (SISO) system [1], [2]. In a SISO system, different terminals utilize different frequency channels to communicate with others. The same frequency channels are reused when the systems are sufficiently separated to limit co-channel interfer- ence. Thus the SISO system has to lower the signal power or to increase the separation distance of channels with the same frequency. The SISO system has been commercially used for a long time, and its performance has been examined and docu- mented [1]–[3]. A major limitation of the SISO system is that its performance degrades as the co-channel interference increases. Nowadays, wireless systems with multiple-input-single- output (MISO), single-input-multiple-output (SIMO), and multiple-input-multiple-output (MIMO) have become more popular because they greatly increase system capacity [1]. The adaptive antenna array system is one of such systems, and it has received much attention [4]–[6]. Different from traditional systems, the adaptive array can work efficiently when there are Manuscript received December 19, 2007; revised March 14, 2008. Published July 7, 2008 (projected). This work was supported by the National Science Foundation under Grant 0355255. The authors are with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-5706 USA (e-mail: zhiyong.huang@asu.edu and balanis@asu.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2008.924764 strong co-channel interferers as long as the signals are from different directions. It has shown that the adaptive array has great beamforming performance [7]–[10], which can greatly mitigate the effects of co-channel interference. Thus, the adap- tive array system can significantly improve the system capacity and range of wireless systems. In order to investigate the system performance of adaptive ar- rays in wireless communications, several spatial channel models have been proposed [11], [12]. Meanwhile, the performances of adaptive arrays in cellular systems, such as the code division multiple access (CDMA), are examined in [13]–[18]. The influ- ence of the array geometry on adaptive beamforming has been investigated in [19], [20]. However, the influence of antenna ge- ometries on BER performance of adaptive arrays has not yet been considered, and it is of interest. This paper presents and compares the BER performances of different antenna array geometries. The uniform circular array (UCA) and uniform rectangular array (URA) with the trellis coded modulation (TCM) and different adaptive algorithms are implemented to determine the influence of array geometries on the system. The uniform linear array is not addressed because it is an 1D geometry, and it should not be compared with 2D structures. This paper begins with a system with binary-phase-shift- keying (BPSK) modulation and investigates the BER of the UCA in an AWGN channel. The system is then extended to include TCM and to compare the system performances of the UCA, URA, and UCA with a center element (UCA-CE). The influence of array geometries on the BER is examined in well-separated-signals and in random-direction-signals envi- ronments. The results indicate that, in some cases, the URA has the best BER. However, the UCA has better average perfor- mance because the UCA has a symmetrical geometry that treats the azimuth plane equally. In addition, both the least mean square (LMS) and recursive least square (RLS) algorithms are implemented to investigate the convergence of the BER. The RLS converges much faster than the LMS, and thus it achieves much better performance than the LMS with a short training sequence. The system model of a wireless system with an adaptive array is presented in Section II, and the instantaneous output SINR and BER of the system are evaluated. System configurations, including adaptive algorithms and TCM, are briefly reviewed in Section III. In Section IV the gain of TCM in adaptive arrays is verified. Simulated BERs of URA, UCA and UCA-CE, with the LMS and RLS in different signal environments, are compared in Section V. Conclusions are included in Section VI. 0018-926X/$25.00 © 2008 IEEE
  2. 2. 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.
  3. 3. 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
  4. 4. 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. 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
  6. 6. 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. 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
  8. 8. 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. 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. REFERENCES [1] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cam- bridge Univ. Press, 2005. [2] G. L. Stuber, Principle of Mobile Communication, 2nd ed. Boston, MA: Kluwer , 2001. [3] B. Sklar, Digital Communications Fundamentals and Applications, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1995. [4] R. Kohno, “Spatial and temporal communication theory using adaptive antenna array,” IEEE Pers. Commun. Mag., vol. 5, no. 1, pp. 28–35, Feb. 1998. [5] A. Paulraj and C. B. Papadias, “Space-time processing for wireless communications,” IEEE Pers. Commun. Mag., vol. 14, no. 5, pp. 49–83, Nov. 1997. [6] G. V. Tsoulos, “Smart antennas for mobile communication systems,” Elect. Commun. Eng. J., vol. 11, no. 2, pp. 84–94, Apr. 1999. [7] B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode, “Adaptive antenna systems,” Proc. IEEE, vol. 55, pp. 2143–2159, Aug. 1967. [8] S. P. Applebaum, “Adaptive arrays,” IEEE Trans. Antennas Propag., vol. AP-24, pp. 585–598, Sep. 1976. [9] R. O. Schmidt, “Multiple emitter location and signal parameter estima- tion,” IEEE Trans. Antennas Propag., vol. AP-34, pp. 276–280, Mar. 1986. [10] H. Krim and M. Viberg, “Two decades of array signal processing research: The parametric approach,” IEEE Signal Process. Mag., pp. 67–94, Jul. 1996. [11] R. Ertel, P. Cardieri, K. W. Sowerby, T. Rappaport, and J. H. Reed, “Overview of spatial channel models for antenna array communication systems,” IEEE Pers. Commun. Mag., vol. 5, no. 1, pp. 10–22, Feb. 1998. [12] Y. Mohasseb and M. P. Fitz, “A 3-D spatio-temporal simulation model for wireless channels,” IEEE J. Select. Areas Commun., vol. 20, pp. 1193–1203, Aug. 2002. [13] Z. J. Hass, J. H. Winters, and D. S. Johnson, “Simulation results of the capacity of celluar systems,” IEEE Trans. Veh. Technol., vol. 46, no. 4, pp. 805–817, Nov. 1997. [14] J. H. Winters and J. S. Salz, “Upper bounds on the bit-error rate of optimum combining in wireless systems,” IEEE Trans. Commun., vol. 46, pp. 1619–1624, Dec. 1998. [15] J. S. Thompson, P. M. Grant, and B. Mulgrew, “Smart antenna arrays for CDMA systems,” IEEE Pers. Commun. Mag., vol. 3, no. 5, pp. 16–25, Oct. 1996. [16] J. C. Liberti and T. S. Rappaport, “Analytical results for capacity im- provements in CDMA,” IEEE Trans. Veh. Technol., vol. 43, no. 3, pp. 680–690, Aug. 1994. [17] A. F. Naguib, A. Paulraj, and T. Kailath, “Capacity improvement with base-station antenna arrays in cellular CDMA,” IEEE Trans. Veh. Technol., vol. 43, no. 3, pp. 691–698, Aug. 1994. [18] G. V. Tsoulos, M. A. Beach, and S. C. Swales, “Adaptive antennas for third generation DS-CDMA cellular systems,” in IEEE Vehicular Technology Conf., Chicago, IL, Jul. 1995, vol. 1, pp. 45–49. [19] P. J. Bevelacqua and C. A. Balanis, “Optimizing antenna array geom- etry for interference suprression,” IEEE Trans. Antennas Propag., vol. 55, pp. 637–641, Mar. 2007. [20] P. Ioannides and C. A. Balanis, “Uniform circular and rectangular ar- rays for adaptive beamforming applications,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 351–354, 2005. [21] G. Ungerboeck, “Channel coding with multi-level/phase signals,” IEEE Trans. Inf. Theory, vol. 28, no. 1, pp. 55–67, Jan. 1982. [22] A. J. Viterbi, “Error bounds for convolutional codes and an asymp- totically optimum decoding algorithm,” IEEE Trans. Inf. Theory, vol. IT13, pp. 260–269, Apr. 1967. [23] Z. Huang and C. A. Balanis, “The MMSE algorithm and mutual cou- pling in adaptive array,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1292–1296, May 2008. [24] S. Haykin, Adaptive Filter Theory. Upper Saddle River, NJ: Prentice Hall, 2002. 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).

×