Effects of Angle-of-Arrival Estimation Errors, Angular ... - IEEE Xplore

Effects of Angle-of-Arrival Estimation Errors, Angular Spread and Antenna Beamwidth on the Performance of Reconfigurable SISO Systems Vida Vakilian and Jean-François Frigon

Sébastien Roy

Dept. of Electrical Engineering ´ Ecole Polytechnique de Montréal Montréal, QC, Canada Emails: {vida.vakilian,j-f.frigon}@polymtl.ca

Dept. of Elec. and Comp. Engineering Université Laval Québec, QC, Canada Email: [email protected]

Abstract—This paper investigates the average bit error rate (BER) performance of reconfigurable single-input single-output (RE-SISO) antenna systems in the clustered channel model. The main focus of the paper is to evaluate the impacts of angle of arrival (AoA) estimation errors, angular spread and antenna beamwidth on the performance of a reconfigurable system. In particular, the BER for the RE-SISO systems is presented under perfect AoA knowledge at the receiver. Then, an analytical expression for the BER of the system is derived in the presence of AoA estimation errors. Along with this, the effects of angular spread and antenna beamwidth on the system performance are examined for both perfect and imperfect AoA estimation scenarios. The results indicate that the effect of AoA estimation error is more significant when the cluster angular spread is small. The same conclusion can be made for the antenna beamwidth.

I. I NTRODUCTION Reconfigurable antennas have received significant attention in recent years due to their potential applications in various wireless communication systems [1]–[3]. One of the attractive features of these antennas is capability to dynamically change their radiation patterns which have been traditionally assumed fixed. Several reconfigurable antenna architectures have been reported in the literature to provide radiation pattern diversity [4]–[7]. Technically, most of these antennas are offering pattern diversity based on altering their physical configuration, achieving either by using Microelectromechanical (MEM) switches or active devices such as diodes or field-effect transistors (FETs). Having this property enables the system to steer the radiation patterns toward the desired users, promising higher data rates and better quality of service (QoS). For the system to achieve higher data rates and better QoS, knowledge of the main angle of arrival (AoA) plays a vital role. The angle of arrival of the signals can be estimated through well-known estimation algorithms, such as the multiple signal classification (MUSIC) algorithm, beamspace MUSIC, MinNorm Linear Prediction and the estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm. For a single reconfigurable antenna system, AoA estimation has been intensively studied in recent years [8]–[11]. One of the proposed AoA estimation algorithms for this type of

antenna is based on the popular beamspace MUSIC algorithm [12]–[14]. In these approaches, the AoA is generally estimated based on forming the covariance matrix that is computed using received signals from different beams, and the AoA is estimated by decomposing the covariance matrix into signal and noise subspaces using an eigenvalue decomposition technique. In reconfigurable antenna systems, AoA estimation errors can significantly affect the performance of the system. The impact of AoA estimation errors on the outage probability of a wireless system have been investigated in [15]. In this reference, the authors have used an antenna array beamformer to examine the impact of beamforming impairments, such as direction-of-arrival (DoA) estimation errors, signal spatial spreads, antenna array perturbation, and mutual coupling. It was then demonstrated that the outage probabilities increase due to the AoA estimation errors. However, they consider the impact of AoA estimation errors on the outage probability using a uniform linear array (ULA). In this paper, we analyze the effect of AoA estimation errors on the BER performance of systems, employing a single reconfigurable antenna at the receiver and an omni-directional antenna at the transmitter. As a first step, we assume perfect knowledge of the angle of arrival at the receiver. Then, we compute the average BER performance of the system based on this information. Next, we investigate the BER of the system when the AoA is estimated with an error. Moreover, we evaluate the effect of angular spread and antenna beamwidth on the system performance. We use the practical clustered channel model, validated in [16], which currently is being utilized in different wireless standards such as the IEEE 802.11n Technical Group (TG) [17] and 3GPP Technical Specification Group (TSG) [18]. In this model, groups of scatterers are modeled as clusters around transmit and receive antennas. The paper is organized as follows. In Section II, we present the cluster channel model and also the signal model for a reconfigurable SISO system for the case of perfect AoA knowledge. Then, in Section III, the average BER performance is computed for a reconfigurable SISO system under imperfect

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978-1-4577-0253-2/11/$26.00 ©2011 IEEE

can be written as

Cluster n Radiation pattern

T Err

gr (T )

Multipath m

y Reconfigurable antenna

Mean AoA of the cluster

Tp

Omni-directional antenna

Mean AoD of the cluster

II. S YSTEM D ESCRIPTION In this section, we first review the clustered channel model for the reconfigurable SISO system and then present a signal model for this system in the case of perfect angle of arrival estimation. A. Clustered Channel Model In this paper, we consider a clustered channel model to characterize the wireless channel of a SISO system equipped with an omni-directional antenna at the transmitter, and a reconfigurable antenna at the receiver as shown in Fig. 1. Let hn (t) be the channel between transmitter and receiver for the nth cluster where each cluster has M multipaths. Then, the impulse response for the nth cluster can be given by [19]

= =

M 1 n )αn (t)δ(t − τ ) √ gr (θm n m M m=1 M 1 n n √ gr (θAOA + ϑnm )αm (t)δ(t − τn ) M i=1 M

hnm (t)δ(t − τn ),

hm x + z

(2)

m=1

AoA estimation at the receiver. The simulation and theoretical results are presented in Section IV. Finally, conclusions are drawn in Section V.

=

M

=

Fig. 1: System model for the RE-SISO system with a reconfigurable antenna at the receiver

hn (t)

= hx + z M 1 gr (θm )αm x + z = √ M m=1

(1)

m=1 n (t) is the where M is the number of paths per cluster, αm th n complex gain of the m multipath (the αm (t) are zero mean unit variance independent identically-distributed (i.i.d.) n ) is the reconfigurable complex random variables), and gr (θm n n n + ϑnm where θAoA antenna gain at the direction θm = θAoA th is the mean angle of arrival (AoA) of the n cluster and ϑnm is the deviation of the paths from mean AoA. The ϑnm are modeled as i.i.d. Gaussian random variables, with zero 2 n . Furthermore, the αm (t) and ϑnm are mean and variance σAoA independent.

where y is the received signal, x is the transmitted signal, z is a zero mean complex additive white Gaussian noise (AWGN) with unit variance, and hm = √1M gr (θm )αm are i.i.d. complex random variables. A notable reconfigurable antenna structure which can perform a continuous beam scanning from backfire to endfire is the composite right/left-handed (CRLH) leaky-wave (LW) antenna. The radiation pattern of the CRLH LW antenna over the azimuth plane can be derived via the array factor approach [20] gr (θ) =

I0 e−α(n−1)d+j(n−1)ko d[sin(θ)−sin(θp )]

(3)

n=1

where Nc is the number of cells in the CRLH LW antenna structure (proportional to the antenna length), α is the leakage factor, d is the period of the structure, k0 is the free space wavenumber, and θp is the pointing angle of the radiation pattern. To simplify the theoretical analysis, the CRLH-LW antenna radiation pattern gr (θ) was approximated by a parabolic function which can be expressed as [21] gr (θ) =

2π 100.1A(θ) , B3dB

(4)

2 θ−θ where A(θ) = −η B3dBp in dB, η is a constant (set to 12 in [21]), B3dB is the 3dB reconfigurable antenna beamwidth in radians, and θp is the antenna pointing angle. For the ideal case considered in this section we have θp = θAoA . M Since h = m=1 hm is a sum of independent random variables, based on the the Central Limit Theorem (CLT) for a large number of multipaths, it can reasonably be modeled as a zero-mean Gaussian random variable with variance σh2 = M var[hm ], where var[hm ] can be computed as follows var[hm ]

= =

1 (E[( gr (θm )αm )2 ] − E[ gr (θm )αm ]2 ) M 1 2π √ (E[gr (θm )]) = , (5) M M B3dB 2cσ 2 + 1 0.1ησ 2

AoA where c = ln 10 is a constant and σ 2 = (see 2 B3dB Appendix-I). Using (5), the variance of h can be defined as follows:

σh2

B. Signal Model In this paper, we consider a time-invariant single cluster channel model and therefore, using (1), the received signal

Nc

=

2π 2 0.2cησAoA

+ B3dB

(6)

Note that the variance of the channel coefficient is a function of the variance of the AoA and the antenna beamwidth.

516

10

Now, let us define the average error probability for a BPSK modulation with respect to the channel statistics as follows 2|h|2 SNR (7) p¯e = Eh Q

10

Average Bit Error Rate

where SNR is the average received signal-to-noise ratio, Q(x) denotes the Gaussian-Q function Q(x) = ∞ √1 exp(−t2 /2)dt and |h|2 is exponentially distributed. 2π x Therefore, direct integration of (7) yields

σh2 SNR 1 1− (8) p¯e = 2 1 + σh2 SNR

In this section, we compute the average BER of the reconfigurable SISO system under imperfect AoA estimation. In this respect, we first derive the variance of the channel in this scenario and then calculate the average BER based on the computed channel variance. Now, taking AoA estimation errors into consideration, (9) can be written as y

= hx + z M 1 gr (θm )αm x + z = √ M m=1 =

M

hm x + z

(9)

m=1

where h is the channel coefficient when the AoA is estimated with a fixed error of θErr and gr (θ) is the antenna gain which can be written as ˆ 2 θ−θ −0.1η B p 2π 3dB gr (θ) = 10 (10) B3dB where θˆp = θAoA + θErr . By expanding θˆp in the above expression, we get θ−(θAoA +θErr ) 2 2π −0.1η B3dB 10 gr (θ) = B3dB = βgr (θ), (11) θErr 2 (θ−θAoA )θErr −0.1η

B

−2

2

B 3dB 3dB where β = 10 . Therefore, the variance of hm can be computed as follows, 1 (E[( gr (θm )αm )2 ] − E[ gr (θm )αm ]2 ) var[ hm ] = M 1 (E[gr (θm )]) = M μ 2 ( ) 2 σ2 − μ2 2π 2σ 4(ln 10+1/2σ 2 ) √ = , (12) e M B3dB 2cσ 2 + 1

√

Err and σ 2 = where μ = − 0.1ηθ B3dB variance of h can be defined as

σ2 h

2 0.1ησAoA . 2 B3dB

= M var[ hm ]

Therefore, the

(13)

-1

T h eory(O m ni-N t=1, Reconfig-N r=1),

θErr = 0°


θErr = 10°


θErr = 15°


θErr = 20°

iSmulation (O m ni-N t=1, Reconfig-N r=1),

θErr = 0°


θErr = 10°


θErr = 15°


θErr = 20°

O m ni-N t=1, O m ni-N r=1

10

10

III. BER A NALYSIS FOR A R ECONFIGURABLE SISO S YSTEM W ITH AOA E STIMATION E RROR

0

10

-2

-3

-4

0

2

4

6

8

10 N S R, dB

12

14

16

18

20

Fig. 2: Effect of AoA estimation errors on the average bit error rate Then, by using (8) the average BER for BPSK modulation is given by ⎛ ⎞ σ2 SNR 1⎝ h ⎠ 1− (14) p¯e = 2 1 + σ 2 SNR h IV. S IMULATION R ESULTS In this section, we perform several simulations to investigate the impact of AoA estimation errors on the performance of reconfigurable SISO systems. We also study the effect of angular spread and antenna beamwidth on the system performance. Fig. 2 illustrates the BER versus signal to noise ratio (SNR). In this simulation, we set the antenna beamwidth at B3dB = 20◦ and angular spread at σAoA = 5◦ . From this figure, we can observe that the theoretical results exactly match the Monte-Carlo simulations, validating the analysis. It also can be observed that the BER increases as the AoA estimation error increases. It is interesting to note that even though the AoA estimation error is as large as the half beamwidth of the radiation pattern, the BER performance of the reconfigurable systems still outperform a traditional system with a fixed omnidirectional antenna. Fig. 3 depicts the BER performance of the RE-SISO system versus AS for different AoA errors and fixed antenna beamwidth. We observe that in the case of perfect AoA estimation (θErr = 0), the average BER increases as the angular spread increases. This is expected since as the anuglar spread increases we are receiving fewer multipaths in the middle of the radiation pattern when the gain is the largest. On the other hand, in the presense of significant estimation error, the average BER initially decreases as a function of the angular spread. This is due to the fact that the number of multipaths impinging in the misaligned radiation pattern increases as a function of the angular spread. Fig. 4 also illustrates the impact of angle of arrival estimation error on the performance of the reconfigurable antenna

517

10

10

-2

10

B3dB = 20°

θErr = 10° , B3dB = 20°

B3dB = 30°

θErr = 15° , B3dB = 20°

B3dB = 40°

θErr = 20° , B3dB = 20°

B3dB = 50°

-3

6

8 10 Angular p Sread, Degree

12

10

14

Fig. 3: Effect of AS on the average bit error rate with different amounts of AoA estimation errors 10

0

5

10 AoA Estimation Error, Degree

15

20

Fig. 5: Effect of antenna beamwidth on the average bit error rate for different amounts of AoA estimation errors

Angular spread = 12° , B3dB = 20° Average Bit Error Rate

-4

the higher received signal energy as compared to a narrow misaligned beam which will miss most of the energy.

Angular spread = 8° , B3dB = 20°

10

-3

-2

Angular spread = 4° , B3dB = 20°

10

10

-4

4

-2

θErr = 0° , B3dB = 20°



10

V. C ONCLUSION

-3

-4

0

5

10 AoA Estimation Error, Degree

15

20

Fig. 4: Effect of AoA estimation errors on the average bit error rate for different values of angular spread (AS) system. As shown in the figure, for all AS values, the performance loss increases as the AoA errors increase. It can be observed that the BER performance of the system with smaller AS is better than the BER of the system with greater AS in the case of small AoA error. This is due to the fact that the antenna receives almost all the multipaths from scatterers near to its maximum gain when the AS is small. However, as the figure illustrates, the situation is reversed for large AoA error since most of the multuipaths are tightly clustered and imping on the side of the misaligned radiation pattern. Therefore, it is important to improve the angle of arrival estimation error for channel with small angular spread. The impact of antenna beamwidth is examined in Fig. 5 when the angular spread is fixed at 5◦ and the AoA estimation error is varied from 0 to 20 degrees. As expected, for small values of AoA estimation error, the most directive antenna yields the lowest BER, profiting directly from the higher antenna gain. However, for large AoA error, the antenna with wider beamwidth has a better BER performance due to

In this paper, we analyzed the BER performance of a reconfigurable SISO system employing an omnidirectional antenna at the transmitter and a reconfigurable antenna at the receiver. The impact of different parameters, including the AoA estimation error, angular spread and antenna beamwidth, on the BER of the reconfigurable SISO system were examined. Simulation results showed that the BER of the RE-SISO system with fixed antenna beamwidth and angular spread increases due to error in AoA estimation. Moreover, it was shown that with small AoA estimation error, the system with smaller angular spread has a better performance than that with larger angular spread. However, for large AoA estimation error, this relationship is reversed. We also examined the BER performance of the system for different values of antenna beamwidth. While the BER of the system with narrower beamwidth has a better performance in small AoA estimation error, it is observed that in large AoA estimation error the system with wider beamwidth outperforms. These results can be used to design appropriate angle of arrival estimation algorithm for communicaiton systems with reconfigurable antennas. VI. A PPENDIX -I For θp = θAoA , we have from (4) that gr (θm ) = gr (θAoA + ϑm ) =

2π −0.1η( Bϑm )2 3dB 10 , B3dB

(15)

2 where √ ϑm ∼mN (0, σAoA ). Applying the change of variable we then have y = 0.1η Bϑ3dB

gr (y)

518

=

2 2π 10−y , B3dB

(16)

0.1ησ 2

where y ∼ N (0, σ 2 ) and σ 2 = B 2 AoA . Thus, the expected 3dB value of gr (y) can be written as ∞ gr (y)pY (y)dy E[gr (y)] = −∞ ∞ 2 2 2 1 2π = √ 10−y e−y /2σ dy 2πσ −∞ B3dB 2π = B3dB σ 2 ln 10 + 1/σ 2 2π √ = . (17) B3dB 2cσ 2 + 1

[18] “3GPP Technical Specification Group Spatial channel model SCM-134 text V6.0,” Spatial Channel Model AHG (Combined Ad-Hoc from 3GPP and 3GPP2), Apr. 2003. [19] D. Tse and P. Viswanath, Fundamentals of wireless communication. Cambridge Univ Pr, 2005. [20] C. Caloz and T. Itoh, “Array factor approach of leaky-wave antennas and application to 1-D/2-D composite right/left-handed (CRLH) structures,” IEEE Microwave and wireless components letters, vol. 14, no. 6, pp. 274–276, 2004. [21] 3GPP, “Spatial channel model for multiple input multiple output (MIMO) simulations,” Tech. Rep., Sept. 2003.

R EFERENCES [1] D. Piazza, N. Kirsch, A. Forenza, R. Heath, and K. Dandekar, “Design and evaluation of a reconfigurable antenna array for MIMO systems,” IEEE Trans. Antennas Propagat., vol. 56, pp. 869–881, March 2008. [2] D. Piazza and K. Dandekar, “Reconfigurable antenna solution for MIMO-OFDM systems,” Electronics Letters, vol. 42, pp. 446–447, Aug. 2006. [3] J. Frigon, C. Caloz, and Y. Zhao, “Dynamic radiation pattern diversity (DRPD) MIMO using CRLH leaky-wave antennas,” in Radio and Wireless Symposium, 2008 IEEE, 2008, pp. 635–638. [4] G. Huff and J. Bernhard, “Integration of packaged RF MEMS switches with radiation pattern reconfigurable square spiral microstrip antennas,” IEEE Trans. Antennas Propagat., vol. 54, pp. 464–469, Feb. 2006. [5] J. Cheng, M. Hashiguchi, K. Iigusa, and T. Ohira, “Electronically steerable parasitic array radiator antenna for omni-and sector pattern forming applications to wireless ad hoc networks,” in Microwaves, Antennas and Propagation, IEE Proceedings-, vol. 150, Apr. 2003, pp. 203–208. [6] W. Weedon, W. Payne, and G. Rebeiz, “MEMS-switched reconfigurable antennas,” in Antennas and Propagation Society International Symposium, 2001. IEEE, vol. 3, 2001, pp. 654–657. [7] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” Microwave Theory and Techniques, IEEE Transactions on, vol. 52, pp. 2678–2690, Dec. 2004. [8] E. Taillefer, W. Nomura, J. Cheng, M. Taromaru, Y. Watanabe, and T. Ohira, “Enhanced Reactance-Domain ESPRIT Algorithm Employing Multiple Beams and Translational-Invariance Soft Selection for Direction-of-Arrival Estimation in the Full Azimuth,” IEEE Trans. Antennas Propagat., vol. 56, pp. 2514–2526, Aug. 2008. [9] C. Plapous, J. Cheng, E. Taillefer, A. Hirata, and T. Ohira, “Reactance domain MUSIC algorithm for electronically steerable parasitic array radiator,” IEEE Trans. Antennas Propagat., vol. 52, p. 3257, Dec. 2004. [10] E. Taillefer, C. Plapous, J. Cheng, K. Iigusa, and T. Ohira, “Reactancedomain MUSIC for ESPAR antennas (experiment),” 2003 IEEE Wireless Communications and Networking, 2003. WCNC 2003, pp. 98–102, 2003. [11] E. Taillefer, A. Hirata, and T. Ohira, “Direction-of-arrival estimation using radiation power pattern with an ESPAR antenna,” IEEE Trans. Antennas Propagat., vol. 53, pp. 678–684, Feb. 2005. [12] X. Xu and K. Buckley, “Statistical performance comparison of MUSIC in element-space and beam-space,” in Acoustics, Speech, and Signal Processing, 1989. ICASSP-89., 1989 International Conference on, pp. 2124–2127. [13] P. Stoica and A. Nehorai, “Comparative performance study of elementspace and beam-space MUSIC estimators,” Circuits, systems, and signal processing, vol. 10, pp. 285–292, March 1991. [14] Y. Yang, “Studies on beamforming and beamspace high resolution bearing estimation techniques in sonar systems,” Ph.D. dissertation, Ph. D Thesis, Northwestern Polytechnical University, 2002. [15] H. Li, Y. Yao, and J. Yu, “Outage probabilities of wireless systems with imperfect beamforming,” Vehicular Technology, IEEE Transactions on, vol. 55, pp. 1503–1515, May 2006. [16] A. Saleh and R. Valenzuela, “A statistical model for indoor multipath propagation,” Selected Areas in Communications, IEEE Journal on, vol. 5, pp. 128–137, Feb. 1987. [17] V. Erceg, “TGn channel models,” IEEE 802.11-03/940r4, May 2004.

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