Performance Analysis of a Novel Antenna Array Calibration Approach for Direction Finding Systems

Transcription

1 EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS Eur. Trans. Teecomms. 2012; 00:1 14 RESEARCH ARTICLE Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra 1, A. M. Toneo 1 1 Dipartimento di Ingegneria Eettrica, Gestionae e Meccanica, University of Udine, via dee Scienze, 208, Udine, Itay ABSTRACT This paper addresses the probem of the direction of arriva (DoA) estimation in the presence of an uncaibrated array, i.e., when phase offset, carrier frequency offset, and phase noise occur among the array signas. The method that we propose consists in the use of a particuar antenna array which comprises the eements for the DoA estimation and a common antenna which is shared with a spitter among the receivers. The signa that impinges on the common antenna is acquired by every receiver and aows the estimation of the phase ambiguity. After the caibration step is performed, anaog switches seect the array eements for the DoA estimation. A method which enabes the phase ambiguity estimation is presented, and severa simuated performance resuts are iustrated. Furthermore, we show the effect on caibration due to the presence of different carrier frequency offsets among the receivers, and we propose an approach for their compensation. The presence of phase noise is aso considered, and its effect is anayzed. Copyright c 2012 John Wiey & Sons, Ltd. Correspondence Dipartimento di Ingegneria Eettrica, Gestionae e Meccanica, University of Udine, via dee Scienze, 208, Udine, Itay. E- mai: toneo@uniud.it 1. INTRODUCTION The estimation of the direction of arriva (DoA) in wireess radio systems with the use of antenna arrays has attracted considerabe interest. The DoA estimation can be expoited for positioning and tracking of wireess nodes, for navigation, as we as for supporting context aware communication services in ceuar, vehicuar and wireess sensor networks. The fundamenta aspects of pane wave propagation and the properties of panar arrays that aow DoA estimation have been discussed in [1]. It has been shown that the deays of the signas in individua array eements are a function of the DoA. Over the years severa improved approaches have been proposed as the agorithms based on the maximum ikeihood paradigm [2], and the so caed subspace-based methods [3, 4]. An important aspect in DoA estimation is the array caibration. In fact, gain and phase mismatches among the sensors, and the unknown position of the sensors introduce ambiguities in the array manifod which may resut in poor DoA estimation. These mismatches are caused by the fact that each receiver consists of a different integrated circuit (IC) that manifests, for instance, its own phase offset (since the RF oca osciator is internay impemented). The compensation of these impairments is typicay done with the aid of an auxiiary reference signa. The array precaibration in [5] and [6] is done with the injection (directy to the receiver inputs and by using coupers, respectivey) of a oca reference signa, which requires an additiona radio frequency (RF) osciator. In [7] and [8], instead, the presence of a reference signa from a known direction is expoited. In particuar, in [7] three caibration methods are proposed and compared, whie in [8], the signa energy knowedge is used to estimate the ampitude and phase errors directy from the signa covariance matrix, without impementing an eigen-decomposition. Copyright c 2012 John Wiey & Sons, Ltd. 1 [Version: 2010/06/21 v2.00]

2 Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra, A. M. Toneo Other approaches that do not require any auxiiary sources can aso be found in the iterature. The method in [9] reaizes array phase caibration under the assumption of a arge number of sensors, which is not aways reaistic. The signa correation matrix structure is expoited in [10] to compensate the gain and phase differences among sensors. However, the correation matrix estimation and its severa eaborations impy high compexity. A particuar appication of the signa subspace method, combined with a QR-based Gauss-Newton agorithm, works we ony for sma phase, gain, and sensor ocations perturbations, as shown in [11]. Even in this case the computationa compexity is high. A genetic approach is expoited instead in [12] for array gain and phase caibration. More recenty, the instrumenta sensors (ISM) method has been derived in [13], whie the approach based on the combination of ISM and the estimation signa parameter via a rotationa invariant technique (ESPRIT) has been proposed in [14]. The atter requires the use of two we caibrated receiver paths, i.e., from the antenna to the baseband output, in order to estimate the phase and gain uncertainties through eigen-decomposition. In [15], a bind caibration agorithm based on the independent component anaysis (ICA) is presented. The ICA technique is used to resove the presence of mutipe sources in order to produce a set of reference data that can be expoited to iterativey obtain the caibration matrix. Hovewer, the ICA needs the received signas to be independent and nongaussian. The method invoves an eigen-decomposition which increases compexity. There are other impairments introduced by both the RF and acquisition (baseband) hardware stages that cannot be negected. In particuar, we have observed the presence of different phase noise and carrier frequency offsets among the receivers, through the experimenta deveopment of a hardware test bed that depoys a state-ofthe-art muti channe direct-conversion receiver foowed by an acquisition board based on the Lyrtech patform [16]. These effects are caused by the fact that each receiver consists of a different IC. Athough these hardware impairments are usuay considered in the context of data transmission systems [17, 18, 19], they are often negected in the context of DoA estimation with antenna arrays. In this paper, we consider the DoA estimation probem when frequency and phase mismatches occur. In particuar, the contribution of this work is fourfod: we propose a nove caibration technique which reies on a particuar array configuration. we derive a simpe method for the phase difference estimation (based on a signa correation) that can be used with the 2D DoA estimator in [16]. we show how carrier frequency offsets can degrade the phase estimation. A compensation agorithm is proposed in order to mitigate this degradation. we anayze the performance of the proposed method in the presence of phase noise when it is used with the DoA estimator in [16]. As it wi be shown, the caibration procedure does not depend on the specific transmitted signa and it works for a dispersive channe as we. Our method requires the use of ony one additiona antenna eement that is shared with a spitter among the receivers. This yieds a simper architecture compared to the approach in [14] that requires two we caibrated receiver paths, or to the method in [5] that requires an auxiiary RF oca osciator. As it wi be expained in the foowing, caibration is achieved via a signa correation operation and not via any eigenvaue decomposition which is more compex. In this way, the carrier frequency offset contributions are we compensated since their differences are imited. The paper is organized as foows. In Section 2, we describe a typica DoA estimation system mode affected by non-ideaities. The non-ideaities modes are reported in Section 3. In Section 4, the proposed caibration procedure is expained, as we as the derivation of the methods which aow the phase and frequency ambiguities estimation and compensation. Some practica considerations are discussed in Section 5. We study the mean-square-error performance of the parameter estimation in Section 6. Finay, concusions are given in Section DOA ESTIMATION SYSTEM MODEL Let us assume a radio system consisting of the foowing components: i) a receiver equipped with a generic antenna array of M sensors, ii) a source that emits a baseband signa x(t) that is upconverted to the RF f 0,RF to obtain x(t)e j2πf 0,RF t, iii) a singe-input-mutipe-output (SIMO) propagation channe whose baseband impuse 2 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd.

3 D. Inserra, A. M. Toneo Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems response can be expressed as h (i) (t) = L =1 ( α (i) δ(t τ )e j ψ (i) +2πf 0,RF τ ), (1) where i {1,.., M} is the antenna index, α (i) represents the compex ampitude of the -th ray, τ is its deay, and ψ (i) is a phase contribution due to the DoA. We aso consider a direct-conversion receiver architecture as depicted in Fig. 1 for the i-th sensor. expressed as β (i) (nt ) = 2πf (i) nt + ϕ (i) (nt ) + Φ (i). (3) In this paper we assume that the compex gain α (i) is due to the propagation oss. Finay, w (i) (nt ) is the background noise contribution. As it can be observed from (2), the presence of the phase term β (i) (nt ) makes the DoA estimation chaenging. In fact, without oss of generaity, if a ineary equispaced (LES) antenna array is considered and a channe mode with a singe ine of sight (LOS) path, i.e., L = 1 and α (i) 1 = 1, i {1,.., M} is assumed, we can write (2) in matrix form as r = xa(φ) + w (4) Figure 1. Receiver architecture for each channe (antenna) of our testbed. As it can be observed, the signa captured by the antenna is fitered by a band seect fiter (BSF), ampified with a ow-noise ampifier (LNA), and downconverted by two mixers fed by two 90 -deayed carriers. The downconverted signas are fitered with a channe seect fiter (CSF) that attenuates the interferers and DC offset. Finay, a variabe gain ampifier (VGA) and an antiaiasing fiter (AAF) render the signas suitabe to be acquired by the ADCs. The carrier frequency is generated by a phase-ocked oop (PLL) from a ow-frequency reference osciator shared by every receiver. Hence, the received RF signas are downconverted using a oca osciator with frequency f (i) LO = f 0,RF f (i), where f (i) represents the carrier frequency offset. If we sampe the baseband signas with period T, the sequence of compex sampes r (i) (nt ) associated to the i-th sensor can be written as r (i) (nt ) = L e j ( α (i) x(nt τ )e j2πf 0,RF τ =1 β (i) (nt ) ψ (i) ) + w (i) (nt ), where β (i) (nt ) incudes the phase uncertaintes due to the carrier frequency offset f (i), the phase noise process ϕ (i) (nt ), and the phase offset Φ (i). For the sensor i, it is (2) where r = [r (1) (nt ), r (2) (nt ),.., r (M) (nt )] T, x = x(nt τ)e j2πf 0,RF τ, w = [w (1) (nt ), w (2) (nt ),.., w (M) (nt )] T, and the array manifod a(φ) (that is usuay considered to estimate the DoA) at the time nt can be written as a(φ) = [e jβ(1) (nt ), e j2π d sin(φ)+jβ (2) (nt ),...., e j2π d (M 1) sin(φ)+jβ (M) (nt ) ] T. This representation does not have the famiiar Vandermonde structure due to the presence of the phase ambiguity β (i) (nt ), i {1,.., M}. Therefore, the term β (i) (nt ) has to be estimated and compensated to caibrate the array manifod. This procedure is not feasibe from (2) since neither the channe nor the DoA is known. (5) The array caibration method that we propose is based on a particuar array configuration that requires the use of an additiona antenna eement that is shared by a the receivers and it acts as a reference. In this way, the DoA is forced to be φ = 0 and the array manifod reads a(0) = [e jβ(1) (nt ), e jβ(2) (nt ),.., e jβ(m) (nt ) ] T. (6) Thus, we are abe to estimate the array manifod from (4) at east within an arbitrary rotation factor, aso when mutipath propagation occurs. This is because it is not important the shape of the reference signa to obtain the phase φ = 0 but ony that the same signa is aquired by a the receivers. Athough in this paper we deveop Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd. 3

4 Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra, A. M. Toneo a ow compexity compensator for the uncertainties β (i) (nt ) β (i+1) (nt ), i {1,.., M 1} so that it can be used by the DoA estimator in [16], the proposed array configuration can be used together with other DoA agorithms and other array geometries. the common sensor that is shared with a spitter among the receivers. It foows that the caibration step and the DoA estimaton step are time mutipexed. 3. HARDWARE IMPAIRMENT MODELS The hardware impairments that we take into account in this work, i.e., phase noise, phase offset, and carrier frequency offset, have been characterized and modeed in [16]. We briefy report herein the main resuts Phase Noise We have used a parametric approach as in [20], and the phase noise power spectrum parameters are reported in [16]. Statisticay, the phase noise has a norma distribution with zero mean and measured standard deviation σ P N = 1 deg, and it is uncorreated among the antenna eements, i.e., E{φ (i) (nt )φ (j) (nt )} = σ P N δ i,j. Furthermore, it is sowy time variant Phase Offset The phase offsets Φ (i), i {1,.., M} can be assumed uniformy distributed in the range [0, 2π] and constant during the observation window Carrier Frequency Offset We assume a Gaussian distribution for the carrier frequency offset f (i). The frequency offsets are considered independent among the receivers, with identica mean µ CF O = 15 khz and standard deviation σ CF O = 100 Hz. 4. CALIBRATION METHOD DESCRIPTION Figure 2. Exampe of a 2 eement antenna array which can be used to impement our caibration method. In this way, the resuting channe impuse response when the switches seect the common antenna becomes h C (t) = L ˆα δ(t ˆτ )e j2πf 0,RF ˆτ, (7) =1 where ˆα and ˆτ are the compex ampitude and the propagation deay, respectivey, for the -th tap of the channe from the transmitter to the common antenna. The i-th acquired caibration signa reads where r (i) C (nt ) = A(nT )ej(2πf (i) nt +ϕ (i) (nt )+Φ (i) ) A(nT ) = + w (i) (nt ), i {1,.., M}, (8) L ˆα x(nt ˆτ )e j2πf 0,RF ˆτ. (9) =1 As it can be observed, the caibration signa r (i) C (nt ) does not depend on the direction of arriva. The caibration technique that we now describe does not rey on the knowedge of the transmitted signa and is aso independent of the channe and the DoA. It requires the use of a particuar antenna array as shown in Fig. 2. Each receiver is connected to an anaog switch that can aternativey seect the sensor for the DoA estimation or 4.1. Frequency and Phase Offset Caibration From (8), we can not directy estimate the phase offset Φ (i) because of the presence of both the compex channe contribution A(nT ) and the carrier frequency offset. To proceed, we propose to correate the antenna signas as 4 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd.

5 D. Inserra, A. M. Toneo Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems foows Λ (i) (nt ) = r (i) C (nt ) r(i+1) C (nt ) = A(nT ) 2 e j(2π f (i) nt + ϕ (i) (nt )+ Φ (i) ) + W (i) (nt ), i {1,.., M 1}, (10) where W (i) (nt ) is the resutant noise term, f (i) = f (i) f (i+1), ϕ (i) (nt ) = ϕ (i) (nt ) ϕ (i+1) (nt ), and Φ (i) = Φ (i) Φ (i+1). It can be observed that the channe contribution appears as a rea gain factor so it does not contribute in the argument of the exponentia. Furthermore, the phase offset term now appears as the difference of two absoute phase offsets. This fact, however, does not affect the fina DoA estimation since the DoA estimator that we herein impement uses the same correation operation in (10) as it wi be shown beow. Thus, it is aso impaired by Φ (i). Now, the residua carrier frequency offset f (i) is Gaussian with zero mean and standard deviation 2σCF O. Depending on whether the residua carrier frequency offset f (i) is negigibe or not, we propose two different estimators of Φ (i). offsets between the eement pairs can be estimated as Φ ˆ (i) = 1 N AV G 1 { Λ (i) (nt )e j2π N AV G i {1,.., M 1}, f ˆ(i) nt }, (12) where we have used N AV G snapshots of the signa Λ (i). It shoud be noted that the phase noise affects the estimation of Φ (i) (nt ) through the term ϕ (i) (nt ). If N AV G + N DOA is not too arge we have that ϕ (i) (nt ) ϕ (i) ((n + N AV G + N DOA )T ). In such a case the phase noise does not affect the estimation of Φ (i) (nt ). In the foowing, we refer to Φ ˆ (i) simpy as phase offset Equa Carrier Frequency Offset If we can assume that the differences of the carrier frequency offsets among the antennas are negigibe, i.e., f (i) 0, i {1,.., M 1}, the signa in (10) reads Λ (i) (nt ) = A(nT ) 2 e j( ϕ(i) (nt )+ Φ (i) ) + W (i) (nt ), i {1,.., M 1}. (13) We can now estimate the phase offsets between the eement pairs simpy as Different Carrier Frequency Offset The difference of the carrier frequency offset f (i) among the antenna eements can be estimated as ˆ f (i) = N CF O 1 1 { } Λ (i) (nt + T ) Λ (i) (nt ), 2πN CF O T i {1,.., M 1}, where we have used N CF O (11) sampes of the caibration signas in (8). It shoud be noted that since subsequent phase noise sampes are highy correated, we have that ϕ (i) (nt ) ϕ (i) ((n + 1)T ), so that the carrier frequency offset estimate is not affected by the phase noise. The carrier frequency offset is compensated using (11). It foows that the istantaneous difference of the phase Φ ˆ (i) = 1 N AV G 1 { } Λ (i) (nt ), N AV G i {1,.., M 1}, (14) without performing the carrier frequency offset compensation as it is differenty done in (12). Figure 3. Tempora order of the caibration agorithm operations. In summary, the caibration agorithm consists of the foowing steps (as aso shown in Fig. 3): Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd. 5

6 Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra, A. M. Toneo If we need to compensate the carrier frequency offset, we acquire N CF O sampes of r (i) C (nt ) to firsty compute Λ (i) (nt ) in (10) and then to estimate the carrier frequency offset by using (11). We acquire N AV G sampes to estimate the phase offset Φ (i) (nt ). This procedure can be done with (14) if the carrier frequency offset is negigibe. Otherwise, the phase offset estimation is done with (12) which takes into account the compensation of the carrier frequency offset. After the phase offset Φ (i) (nt ) compensation, we can estimate the DoA. As aready mentioned, this agorithm is suitabe for use with the DoA estimator in [16] that is briefy described beow. In Section 6, we compare the performance of the estimator (14) that does not compensate the carrier frequency offset (in the foowing we denote it as method NOCOMP) with the performance of the estimator (12) that compensates the carrier frequency offset (we denote it as method COMP) D DoA Estimator a singe tap LOS channe, the signa in (2) becomes r (a,i) (nt ) = e j(β(a,i) (nt ) ψ (a,i) ) (15) + w (a,i) (nt ), i {1,.., M}, where the index a represents the eements of a subarray x, y, or z. Furthermore, β (a,i) (nt ) = 2πf (a,i) nt 2πf 0,RF τ (a) Φ (a,i) ϕ (a,i) (nt ), (16) and ψ (a,i) is the term due to the 2D DoA that reads 2π d (i 1) cos(φ) sin(ϑ), a = x ψ (a,i) = 2π d (i 1) sin(φ) sin(ϑ), a = y 2π d (i 1) cos(ϑ), a = z. (17) We now correate pairs of antenna eement signas obtaining z (a,i) (nt ) = r (a,i) (nt ) r (a,i+1) (nt ) = e ( ˆψ (a,i) Φ (i) ) + ŵ (a,i) (nt ), (18) i {1,.., M}, where 2π d cos(φ) sin(ϑ), a = x ˆψ (a,i) = 2π d sin(φ) sin(ϑ), a = y 2π d cos(ϑ), a = z, (19) and we have assumed the carrier frequency offset to be identica on the same subarray, that is, f (a,i) = 0, i {1,.., M 1}. It shoud be noted that the term Φ (a,i) has to be compensated to estimate the DoA. At this point, we can average in time and in space the sampes z (a,i) (nt ) to increase the immunity to noise, after the compensation of the phase offset, as foows: Figure 4. 3D L-shaped array configuration. We consider a 3D L-shaped array (Fig. 4) that has three ineary equispaced arrays dispaced aong the three axes, x, y, and z. Then, with a singe tone transmitted signa and N DOA 1 z (a) 1 = z (a,i) (nt )e j Φ ˆ (a,i), N DOA (M 1) (20) where Φ ˆ (a,i) is the phase offset contribution that we have estimated with (14) or (12). 6 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd.

7 D. Inserra, A. M. Toneo Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems Finay, we can determine the 2D DoA estimates as ˆΦ = arctan 2 ( z (y), z (x)) ) ˆϑ = arctan 2 ( ( (x)) 2 + ( z (y) ) 2, z z, (21) where arctan 2(y, x) is the arctangent function defined within [0, 2π). 5. IMPLEMENTATION ISSUES Figure 5. Exampe on how to reaize a mutiport spitter. The particuar configuration of the array for the caibration requires further attention, particuary in reference to its practica impementation. In fact, as expained in Section 4, we have to connect the common antenna to every receiver s input (in particuar to one input of the anaog switch as in Fig. 2). This connection can not be simpy done with a short circuit because of the introduction of impedance mismatches into the signa path. Hence, it is necessary to use a mutiport microwave power divider [21] which equay spits the input signa among the receivers and which must not introduce phase differences among the outputs. It shoud be noted that since the signa that passes through the mutiport power divider channes is the same, i.e., the RF signa at frequency f 0,RF, having paths with equa engths guarantees the phase equaization among the mutiport spitter paths. In this way, the caibration signa mode in (8) hods true. Off the shef devices are avaiabe for this purpose. Aternativey, a mutiport spitter can be reaized by using two output ports dividers and appying other dividers to those ports as shown in Fig. 5. The proposed mutiport spitter can be used with any number of array eements M. In fact, even if it has a tota number of outputs which is a power of two, appying a 50Ω oad into the unused port, the correct operation of the device is not affected. Another important practica aspect concerns the ength of the cabes that connect the antennas of the DoA estimation array with the RF front-end inputs. In fact, the phase offset Φ (i) in (3) does not incude the phase differences among the channes due to different path/cabe engths. If the connection cabes have different ength, then the signa in (2) becomes r (i) (nt ) = L e j ( α (i) x(nt τ )e j2πf 0,RF τ =1 β (i) (nt )+ζ (i) ψ (i) ) + w (i) (nt ), (22) where ζ (i) = 2πf (i) 0,RF c 0 is the phase contribution due to the propagation in the cabe of ength (i) (it shoud be noted that we have assumed x(t τ (i) c 0 ) x(t τ ) since the signa x(t) is narrowband). Since the phase differences due to this impairment are time invariant, it is reasonabe to assume an offine precaibration procedure that estimates ζ (i). Aternativey, an accurate path/cabe engths design that gives ζ (i) = ζ, i {1,.., M} can aso be done. 6. PERFORMANCE ANALYSIS This section reports the performance anaysis of the proposed caibration method. In particuar, we anayze the root mean-squared error (RMSE) in the estimation of the phase offset Φ (i) in (14) or in (12) defined as 1 RMSE = E M 1 M 1 i=1 2 Φ (i) Φ ˆ (i), (23) Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd. 7

8 Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra, A. M. Toneo and the aggregate RMSE of the 2D DoA estimation, defined as { ( RMSE DoA = E φ ˆφ ) 2 ( + ϑ ˆϑ ) } 2, (24) where φ and ˆφ are the azimuth and its estimate, respectivey, whie ϑ and ˆϑ are the eevation and its estimate, respectivey (in the simuation we have used φ = 30 and ϑ = 50 ). The parameters of the 2D DoA estimator used in the numerica exampes are f 0,RF = 5.8 GHz, and d = 2. The tota number of eements becomes 3M 2 = 4, where M = 2 is the number of eements per subarray. Finay, we assume 1 = 50 MHz. T As far as the channe mode is concerned, we have considered two modes: the first is a singe tap LOS channe (L = 1) with deterministic ampitude, whie the second mode is a two rays channe (L = 2) with the second ray that exhibits temporay and spatiay uncorreated Rayeigh faded ampitude and uniform phase. We define the factor γ as the ratio between the power of the LOS tap, M LOS, and the power of the second tap. Furthermore, we define the signa to noise ratio SNR as noise) Φ ˆ (i) = 1 N AV G 1 { A(nT ) 2 e j Φ(i) N AV G } e j2π f (i) nt + W (i) (nt ) where W (i) Φ Φ (i) + ε (i) 1 + W (i), i {1,.., M 1}, Φ is the noise that affects the estimation, and ε (i) 1 = 1 N AV G 1 2π f (i) nt, N AV G = π f (i) (N AV G 1)T (26) (27) is the rotation factor due to the carrier frequency offset. From (27), and since f (i) = f (i) f (i+1), i {1,.., M 1} has zero mean and standard deviation 2σCF O, we obtain the standard deviation of ε (i) 1 as σ ε1 = 2πσ CF O (N AV G 1)T. (28) In Fig. 6, we report σ ε1. SNR = M LOS, (25) σw 2 where we have considered a transmitted signa with unitary 0.5 power, whie σ 2 W is the variance of the background noise σ CFO =100 Hz w (i) (nt ) that we have considered equa for a receivers. We have set the parameters of the carrier frequency offset mode with the defaut vaues in [16], i.e., µ CF O = 15 khz and σ CF O = 100 Hz, as aso reported in Section 3. The carrier frequency offset estimation with (11) has been done assuming N CF O = Finay, the standard deviation of the phase noise is assumed σ P N = 1 deg. σ ε1 [deg] =100, σ ε1 = =1000, σ ε1 = Figure 6. Standard deviation of ε 1 as a function of N AV G Theoretica Considerations Carrier Frequency Offset Effects When we consider the NOCOMP method but the frequency offset differences among the antennas are not negigibe, (14) can be written as (negecting the phase As it can be deducted from (26), the rotation factor ε (i) 1, i {1,.., M 1} imits the performance of the estimator introducing a phase error which is proportiona to the number of sampes N AV G used in the averaging operation. We wi aso observe in the simuation resuts 8 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd.

9 D. Inserra, A. M. Toneo Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems that the vaues of σ ε1 in Fig. 6 are the vaues of the foor exhibited by the RMSE curves evauated as in (23). To compensate this effect we have proposed the method COMP that comprises a carrier frequency offset caibration. By appying this caibration, (12) can be rewritten as (26), where the rotation factor (that now we denote with ε (i) 2 ) can be expressed as ε (i) 2 = π( f (i) ˆ f (i) )(N AV G 1)T. (29) RMSE [Hz] µ CFO =15 khz σ CFO =100 Hz N CFO =1000 CFO Estimator performance with σ PN =0 deg The standard deviation of the rotation factor ε (i) 2 is 10 0 Theoretica imit RCRB σ ε2 = πσ ˆ f (N AV G 1)T, (30) where σ f ˆ is the RMSE of the carrier frequency offset estimator which is defined as σ ˆ f = E { f (i) ˆ f (i) 2 }. (31) SNR [db] Figure 7. RMSE of the carrier frequency offset estimator in (11) as a function of SNR, with the presence of carrier frequency offset, and L = 1. Hence, comparing (28) with (30), we can assert that the method COMP works better than the NOCOMP method if σ ˆ f < 2σ CF O. In order to determine a ower bound for the carrier frequency offset estimator performance, we have evauated the root of the Cramer-Rao Bound (RCRB) for the estimation of the carrier frequency offset from (10). Its derivation is reported in Appendix A. It reads σ f ˆ 3 (32) (2πT ) 2 γ Λ N CF O (N CF O 1)(2N CF O 1), where γ Λ is the signa-to-noise ratio obtained from the signa in (10). It is easy to prove that γ Λ is 3 db ess than the SNR in (25). In Fig. 7 we report the RMSE for the carrier frequency offset estimator in (11) as a function of SNR, using N CF O = It shoud be noted that the carrier frequency offset estimation in (11) yieds a RMSE smaer than 2σ CF O = Hz for SNRs arger than approximatey 38 db. It foows that the estimation and compensation of f (i) is beneficia for SNRs arger than approximatey 38 db Phase Noise Effects If we consider the COMP method and we do not negect the phase noise contribution, (11) can be approximated as ˆ f (i) f (i) + ξ (i) + W (i) f (33) where W (i) f is the noise that affects the carrier frequency offset estimation, and ξ (i) = 1 ( ) (34) ϕ (i) (N CF O T ) ϕ (i) (0) 2πN CF OT is a frequency shift due to the fact that the phase noise sampes are not perfecty correated. This frequency shift wi determine an error foor in the RMSE of the carrier frequency offset estimation, as aready observed for the rotation factor ε (i) 1. From (34), we can obtain the standard deviation of ξ (i) as 2σP 2 N σ ξ = r(i) P N (N CF OT ), (35) 2πNCF O T where r (i) P N (N CF OT ) = E{ ϕ (i) (nt + N CF O T ) ϕ (i) (nt )} is the correation of the sampes ϕ (i) (nt ) evauated in N CF O T. Then, if the phase noise sampes were perfecty correated after N CF O sampe periods, we woud obtain σ ξ = 0. On the other hand, if the phase noise sampes are uncorreated, with N CF O = 1000 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd. 9

10 Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra, A. M. Toneo and σ P N = 1 deg, we wi obtain σ ξ = Hz. Hence, in this case the carrier frequency offset compensation does not improve the performance since the effect of the phase noise is more detrimenta than the carrier frequency offset. In practica situations, the vaue of N CF O can be chosen so that a tradeoff is made between the decrease of the correation r (i) P N (NCF OT ) and the overa decrease of σ ξ computed as in (35) Simuation Resuts Carrier Frequency Offset Effects In Fig. 8, the RMSE of the phase offset is shown as a function of the SNR evauated by using the two methods described in Section 4, with N AV G = {1, 10, 100, 1000} and L = 1. In this case we have not considered the effect of the phase noise (σ P N = 0 deg). RMSE [deg] =1000 µ CFO =15 khz σ CFO =100 Hz N CFO =1000 σ PN =0 deg L=1 =100 soid ines: Method NOCOMP dash dot ines: Method COMP =10 = SNR [db] Figure 8. RMSE of the phase offset as a function of SNR and N AV G, with the presence of carrier frequency offset, and L = 1. Now, et us consider the curves generated using the method NOCOMP. As we can observe, the increase of the SNR eads to better performance. Furthermore, by increasing N AV G the RMSE decreases. However, for high SNR vaues the performance reaches a foor and this gets arger as the N AV G increases. This is due to the presence of different carrier frequency offsets among the array eements whose effect becomes dominant at high SNRs. In particuar, with the considered parameters, we reach the vaues RMSE 0.05 deg with N AV G = 100, and RMSE 0.5 deg with N AV G = 1000 (aso for the curves with N AV G = 1 and N AV G = 10 a foor in the performance occurs, but this is not visibe with the considered SNR vaues), that are exacty the vaues of σ ε1 obtained with (28) and reported in Fig. 6. Furthermore, in Fig. 8 we can observe that the curves obtained with the use of (12) do not experience a foor, as expained before. However, the RMSEs evauated with N AV G = {100, 1000} are arger than the ones obtained with the method NOCOMP up to the SNR vaues ower than 38 db, as previousy anayzed. In Fig. 9 we show the aggregate RMSE of the 2D DoA estimator in [16] when both the proposed caibration methods are used, with N AV G = {1, 10, 100, 1000}, N DOA = 1, and L = 1. RMSE DOA [deg] µ CFO =15 khz σ CFO =100 Hz N CFO =1000 4dB Method {NOCOMP,COMP} ={1,10,100,1000} σ PN =0 deg Method NOCOMP =1000 Method NOCOMP =100 No phase offset case SNR [db] Figure 9. Aggregate RMSE of the 2D DoA estimator as a function of SNR and N AV G, with the presence of carrier frequency offset, and L = 1. The COMP and NOCOMP methods have identica performance with N AV G = 1. It shoud aso be noted that a the performance curves are concentrated into a range of 4 db. The foor that occurs in Fig. 9 by using the method NOCOMP with N AV G = {100, 1000} is due to the carrier frequency offset that is not compensated. In Fig. 10, we consider a mutipath channe with L = 2 and γ = {10, 20, 30} db, and we show the RMSE of the phase offset as a function of the SNR, with N AV G = {1, 10, 100, 1000}. It shoud be noted that the performance of the caibration method is not infuenced by the presence of mutipath since the curve with L = 1 and L=1 the ones with L = 2 are practicay overapped. 10 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd.

11 D. Inserra, A. M. Toneo Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems σ PN =0 deg =1 phase noise on the performance of the 2D DoA estimator when we appy the caibration method proposed in this paper, with N AV G = {10, 100} and N DOA = RMSE [deg] 10 1 Method {NOCOMP,COMP} L={1,2} 10 1 With Phase Noise =100 σ PN =1 deg L=1 Method COMP 10 2 µ CFO =15 khz σ CFO =100 Hz N CFO = SNR [db] Figure 10. RMSE of the phase offset as a function of SNR and N AV G, with the presence of carrier frequency offset, with L = 2. RMSE DOA [deg] With Phase Noise =10 µ CFO =15 khz σ CFO =100 Hz N CFO =1000 Method NOCOMP Without Phase Noise σ PN =0 deg, = SNR [db] Phase Noise Effects In Fig. 11, we show the effect of the phase noise to the performance of the 2D DoA estimator when we appy the method NOCOMP. We have considered the phase noise mode from the measurements described in Section 3, N AV G = {1, 10}, and N DOA = 1, whie we have negected the carrier frequency offset, and L = 1. RMSE DOA [deg] σ CFO =0 Hz With Phase Noise σ PN =1 deg =1 Method NOCOMP L=1 Without Phase Noise σ PN =0 deg = SNR [db] Figure 11. Aggregate RMSE of the 2D DoA as a function of SNR and N AV G, with the presence of phase noise, and L = 1. As we can observe, the error foor increases with the increase of N AV G. In Fig. 12, we show the effect of the Figure 12. Aggregate RMSE of the 2D DoA as a function of SNR, with N AV G = {10, 100}, both the presence of carrier frequency offset and phase noise, and L = 1. Considering the curves associated to the method COMP, we can observe that the phase noise sti imits the performance. It shoud be noted that with the vaues of phase noise and carrier frequency offset considered in this paper, the two methods reach very simiar performance. Assuming N AV G = 100, the RMSE DOA foor is as sma as 0.4 deg. The error foor can be owered with an opportuna choice of N CF O. 7. CONCLUSION In this paper we have considered the DoA estimation probem in the presence of an uncaibrated mutipe antenna system, i.e., when phase offset, carrier frequency offset, and phase noise occur among the array signas. We have proposed a nove technique for the caibration of the antenna array system that reies on the use of a particuar antenna array which aows the estimation of the phase offset differences of the array. The array comprises a common antenna which brings the same signa to each receiver. After the caibration step is performed, anaog switches can seect the antennas for the DoA estimation. Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd. 11

12 Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra, A. M. Toneo We have shown that the caibration method performance does not degrade in the case of a mutipath channe. This caibration technique works with either equa carrier frequency offsets or different carrier frequency offsets among the receivers. To better address the atter case, we have proposed another method which firsty estimates the carrier frequency offset differences and then it performs compensation. The anaisys has reveaed that the method without carrier frequency offset compensation works better at ow SNRs whie at high SNRs it exhibits an error foor. Consequenty, at high SNRs it is beneficia to perform carrier frequency offset compensation. Another important issue is the effect of the phase noise. The phase offset compensation method can mitigate the effects of phase noise depending on its spectrum, i.e., if it is sowy time variant. A. CRAMER-RAO BOUND FOR THE CARRIER FREQUENCY OFFSET ESTIMATOR We have evauated herein the Cramer-Rao bound for the carrier frequency offset estimation from the signa in (10), negecting both the phase offset and the phase noise, and under the assumption that we do not know either the statistics of f (i) or the statistics of A(nT ). Furthermore, for the sake of simpicity, in the notation we do not report the apex (i). The unknown parameters are the carrier frequency offset f, and the vaues of the signa ampitude A(nT ) 2, n {0,.., N CF O 1}. Then, the vector of the unknown parameters can be defined as θ = [ f, A(0) 2,.., A((N CF O 1)T ) 2 ]. At this point we can define the probabiity density function of the vector Λ = [Λ(0),.., Λ((N CF O 1)T )] conditioned by the unknown parameters θ as N CF O 1 1 p(λ θ) = πσ W { 1 } exp Λ(nT ) A(nT ) 2 e j2π fnt 2 σ 2 W (36) and the og-ikeihood function L(θ) obtained by appying the ogarithm on p(λ θ) and negecting the constant terms, L(θ) = 1 N CF O 1 Λ(nT ) A(nT ) 2 e j2π fnt σw 2 + Λ(nT ) A(nT ) 2 e j2π fnt A(nT ) 4. (37) After cacuating the derivatives and evauating their expectation we obtain { } 2 L(θ) E = 0, i j, (38) θ i θ j which aows evauating the Cramer-Rao bound as CRB( f) = E 1 { 2 L(θ) θ (2πT ) 2 γ Λ N CF O (N CF O 1)(2N CF O 1), } (39) where γ Λ is the approximated signa to noise ratio of the signa in (10) (we have approximated A(nT ) 4 as a constant). We have aso used N 1 n2 = N(N 1)(2N 1). 6 ACKNOWLEDGMENT D. Inserra acknowedges that part of the work herein presented was funded by CNIT, Consorzio Nazionae Interuniversitario per e Teecomunicazioni, with a doctora research grant. REFERENCES 1. Amitay N, Pecina RG, Wu CP. Radiation properties of arge panar arrays. Technica Report, Be Teephone Lab. Inc., Whippany, New Jersey Swindehurst AL. Maximum ikeihood DOA estimation and detection without eigendecomposition. Proc. IEEE Internationa Conference on Acoustic, Speech and Signa Processing (ICASSP 92), 1992; Schmidt RO. Mutipe emitter ocation and signa parameter estimation. IEEE Transactions on Antennas and Propagation Mar 1986; AP-34: Roy R, Kaiath T. ESPRIT - estimation of signa parameters via rotationa invariance. IEEE 12 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd.

13 D. Inserra, A. M. Toneo Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems Transactions on Acoustics, Speech and Signa Processing Ju 1989; 37(7): Dandekar KR, Ling H, Xu G. Smart antenna array caibration procedure incuding ampitude and phase mismatch and mutua couping effects. Proc. IEEE Internationa Conference on Persona Wireess Communications (ICPWC 2000), 2000; Cooper T, McCormack J, Farre R, Badwin G. Toward scaabe, automated tower-top phased array caibration. Proc. of the IEEE 65th Vehicuar Technoogy Conference (VTC 2007 Spring), Dubin, Ireand, Cherntanomwong P, Takada J, Tsuji H, Miura R. Array caibration using measured data for precise ange-of-arriva estimation. Proc. of WPMC 2005, Aaborg, Denmark, Zeyang D, Yuming D. Fast active caibration for uniform inear array with ampitude and phase errors. Proc. of the 2009 Internationa Workshop on Information Security and Appication (IWISA 2009), Qingdao, China, Wyie MP, Roy S, Messer H. Joint DOA estimation and phase caibration of inear equispaced (LES) arrays. IEEE Transactions on Signa Processing Dec 1994; 42: Pauraj A, Kaiath T. Direction of arriva estimation by eigenstructure methods with unknown sensor gain and phase. Proc. IEEE Internationa Conference on Acoustics, Speech and Signa Processing (ICASSP 88), New York City, 1988; Wang C, Cadzow JA. Direction-finding with sensor gain, phase and ocation uncertainty. Proc. IEEE Internationa Conference on Acoustics, Speech and Signa Processing (ICASSP 91), 1991; Xu Q, Tao HH, Liao GS. Array gain and phase error correction based on GA. Systems Engineering and Eectronics 2006; 28(5): Wang BH, Wang YL, Chen H. Array caibration of anguary dependent gain and phase uncertainties with instrumenta sensors. Proc. of IEEE Internationa Symposium on Phased Array Systems and Technoogy, Boston, Massachusetts, USA, Bing L, Guisheng L. Method for array gain and phase uncertainties caibration based on ISM and ESPRIT. Journa of Systems Engineering and Eectronics 2009; 20(2): Shimada Y, Yamada H, Yamaguchi Y. Bind array caibration technique using ICA. Proc. of ISAP 2007, Niigata, Japan, Inserra D, Toneo AM. Characterization of hardware impairments in mutipe antenna systems for DoA estimation. Journa of Eectrica and Computer Engineering Proakis JG. Digita Communications. 3rd edn., McGraw Hi: New York, Meyr H, Moenecaey M, Fetche SA. Digita Communication Receivers: Synchronization, Channe Estimation, and Signa Processing. 2nd edn., Wiey- Interscience: New York, Lindoff B, Mam P. BER performance anaysis of a direct conversion receiver. IEEE Transactions on Communications May 2002; 50: Robertson P, Kaiser S. Anaysis of the effect of phasenoise in orthogona frequency division mutipex (OFDM) system. Proc. IEEE ICC 95, Seatte, Pozar DM. Microwave Engineering. John Wiey and Sons, AUTHORS BIOGRAPHIES Daniee Inserra Daniee Inserra received the BSc Degree (2007) and the MSc Degree (2009) in eectrica engineering both from the University of Udine. Currenty he is a member of the wireess and power ine communications aboratory, WiPLi Lab. Since 2010, he has been a Ph.D. student in teecommunications. His research interests are on infomobiity, wireess communications, radio positioning and ocaization techniques, hardware/software co-design. Andrea M. Toneo Andrea M. Toneo received the Doctor of Engineering degree in eectronics (summa cum aude), and the Doctor of Research degree in eectronics and teecommunications, both from the University of Padova, Itay. From 1997 to 2002, he worked with Be Labs - Lucent Technoogies, first as a Member of Technica Staff, then he was promoted to Technica Manager and he was appointed Managing Director of Be Labs Itay. In 2003, he joined the University of Udine, Itay, where he is an Aggregate Professor. Dr. Toneo received severa awards among Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd. 13

14 Performance Anaysis of a Nove Antenna Array Caibration Approach for Direction Finding Systems D. Inserra, A. M. Toneo which the Distinguished Visiting Feowship from the Roya Academy of Engineering, UK, in 2010, and the IEEE VTS Distinguished Lecturer Award He was the TPC Co-chair of IEEE ISPLC 2007, the Chair of the Workshop on Power Line Communications 2009, and the Genera Chair of IEEE ISPLC He serves as an Associate Editor for the IEEE Transactions on Vehicuar Technoogy and for the IEEE Transactions on Communications. He is the Vice-chair of the IEEE Communications Society Technica Committee on Power Line Communications. 14 Eur. Trans. Teecomms. 2012; 00:1 14 c 2012 John Wiey & Sons, Ltd.