Direction-of-Arrival Estimation in a Clustered Channel Model Vida Vakilian and Jean-Franc¸ois Frigon
S´ebastien Roy
Dept. of Electrical Engineering ´ Ecole Polytechnique de Montr´eal Montr´eal, QC, Canada Emails: {vida.vakilian,j-f.frigon}@polymtl.ca
Dept. of Elec. and Comp. Engineering Universit´e Laval Qu´ebec, QC, Canada Email:
[email protected]
Abstract—- In this paper, we study the performance of the MUltiple SIgnal Classification (MUSIC) direction-of-arrival (DoA) estimation algorithm in a clustered channel model. We examine the effect of angular spread on the estimation accuracy of the MUSIC algorithm. Simulation result shows that the performance of the MUSIC algorithm degrades as the angular spread increases. Furthermore, an experimental study is performed in an anechoic chamber to evaluate the performance of DoA estimation using a single reconfigurable antenna element known as Composite Right/Left Handed (CRLH) Leaky-Wave (LW) antenna that offers a high level of directivity with electronically controllable radiation patterns. Experimental results show that the DoA estimation technique succeeded in estimating the DoA using a single antenna element.
I. I NTRODUCTION The area of DoA estimation has received a considerable amount of attention in the literature due to its numerous applications in communication, radar, sonar and biomedical engineering [1]–[3]. Several DoA estimation algorithms have been developed that in general can be categorized into two classes, namely conventional techniques and subspace-based techniques [4]. Conventional approaches include methods such as delay-and-sum and minimum variance distortionless look (MVDL) methods. These methods work based on the concept of beamforming that means they estimate the DoA by measuring the signal power in all possible directions and selecting the direction that produces maximum power as the DoA of the desired signal [2], [5]. Although these methods are simple in concept, they need a large number of elements to achieve high resolution performance and may fail to estimate the DoA when the signals are highly correlated, such as in multipath propagation environments. The other class of DoA estimation algorithms are subspacebased methods that exploit the orthogonality between the noise and signal subspaces to estimate the DoA. One of the most popular subspace-based DoA estimation techniques is the MUltiple SIgnal Classification (MUSIC) algorithm [6]. The MUSIC algorithm provides high resolution estimation and can be used with a variety of array geometries. Many studies have been conducted on the performance evaluation of MUSIC algorithm in the presence of point sources without considering any local scattering around the sources. However, in practice and applications such as mobile communications, the effect
of multipath propagation must be taken into account. In this paper, we study the performance of the MUSIC algorithm in the multipath propagation environment. To do so, we consider a clustered channel model where the propagation paths with the same angle and delay are considered as a cluster. There are several cluster-based channel models such as the 3GPP Spatial Channel Model (SCM), WINNER II, and the IEEE 802.11 TGn channel model. We focus herein on DoA estimation problem using the MUSIC algorithm in a multipath environment based on the SCM channel model. This model is appropriate and realistic for assessing DoA methods because it models the geographical position of scatterers according to clusters, an approach which has been validated through measurements in [7], [8]. By using this channel model, we are able to study the effects of different real-world parameters on the performance of the DoA estimation algorithms. The rest of this paper is organized as follows. Section II introduces the spatial channel model used in this work. Section III presents the signal model and Section IV describes the MUSIC algorithm in the cluster-based channel. The simulation results are presented in Section V. The measurement setup and experimental results are presented in Section VI. Finally, conclusions are drawn in Section VII. II. S PATIAL C HANNEL M ODEL The Spatial channel model (SCM) is a statistical model developed by the 3rd Generation Partnership Project (3GPP) for three propagation environments including urban microcell, urban macro-cell and suburban macro-cell [9]. In this model, the propagation paths with the same angle and delay are considered as a cluster. Moreover, this model takes into account the impact of several physical parameters of wireless channels such as direction-of-arrival (DoA), directionof-departure (DoD), path power, antenna radiation patterns, angular and delay spread. The channel coefficient between a single transmit antenna and receiver antenna j for the l-th cluster, for l ∈ {0, 1, · · · , L − 1}, is given by
978-1-4673-0859-5/12/$31.00 ©2012 IEEE 313
√
hj (l)
M √ Pl ∑ m αl g t (θlm ) M m=1 √ ( ) m gjr (ϕm × l ) exp jk0 dr (j − 1) sin(ϕl ) , (1)
=
received signal at time t at the j-th receive antenna is given as
First cluster m 0, DoA
sub-path m
Rx Antenna Array
yj (t) =
m 0
m 0
1
m 1, DoA
yj = hj ∗ x + nj , for j = 1, 2, · · · , Nr
m 1
1
m 1, DoD
The spatial channel model for two clusters scenario.
where Pl is the power of the l-th cluster which is normalized so that the total average power for all clusters is equal to one, M is the number of unresolvable multipaths per cluster that have similar characteristics, k0 = 2π/λ is the free space wavenumber, where λ is the free-space wavelength, dr is the antenna spacing between two elements at the receiver side, αlm is the complex gain of the m-th multipath of the l-th path (the αlm are zero mean unit variance independent identicallydistributed (i.i.d) complex random variables), g t (θlm ) is the gain of transmit antenna, and gjr (ϕm l ) is the gain of j-th receive antenna. θlm and ϕm l are the DoD and DoA for the m-th multipath of the l-th cluster, respectively, and can be given by θlm ϕm l
= =
θl + ϑm l,DoD , ϕl + ϑm l,DoA ,
(2) (3)
where θl and ϕl are the mean DoD and the mean DoA of m the l-th cluster, respectively. The ϑm l,DoD and ϑl,DoA are the deviation of the paths from mean DoD and DoA, respecm tively. The ϑm l,DoD and ϑl,DoA are modeled as i.i.d. Gaussian random variables, with zero mean and variance σθ2 and σϕ2 , respectively. The channel impulse response between a transmit antenna and receive antenna j can be modeled as hj (τ ) =
(26)
where Second cluster
Fig. 1.
(25)
where τl are ordered so that τL−1 > · · · > τ1 > τ0 = 0. For simplicity, we assume that τl = lTs , where Ts is the symbol period. After K time slots, the received signal by j-th antenna element can be given by
m 1
Tx
hj (l)x(t − τl ) + nj (t),
l=0
0
m 0, DoD
0
L−1 ∑
L−1 ∑
hj (l)δ(τ − τl ),
hj = [hj (0), hj (1), · · · , hj (L − 1)], x = [x(1), x(2), · · · , x(K)],
(27) (28)
nj = [nj (0), nj (1), · · · , nj (K + L − 1)],
(29)
where ∗ is the convolution operator, hj ∈ C 1×L is the channel vector, x is the transmitted signal vector with independent symbols and the total transmit power of E{xxH } = σx2 , where E[.] denotes the expectation, and nj are modeled as independent Gaussian random variables with zero-mean and variance σn2 . In other words, the overall received signal over Nr antennas, Y ∈ C Nr ×(K+L−1) can be represented in matrix format as follows, Y = HX + N,
(30)
where H = [hT1 , hT2 , · · · , hTNr ]T ∈ C Nr ×L , N= and
[nT1 , nT2 , · · ·
, nTNr ]T
x(1) x(2) · · · x(K) x(1) x(2) ··· 0 0 0 x(1) x(2) X= . .. .. .. . . . . . 0 ··· 0 0
∈C
(31)
Nr ×(K+L−1)
,
(32)
0 x(K)
0 0
··· ···
··· .. .
x(K) .. .
··· .. .
x(1)
x(2)
0 0
0 .. . · · · x(K) (33)
IV. MUSIC A LGORITHM IN THE CLUSTER - BASED (24)
l=0
where τl is the l-th cluster delay, and hj (l) is the complex gain of the l-th cluster defined in (1). III. S IGNAL MODEL Consider a uniform linear array (ULA) composed of Nr omnidirectional elements, which is receiving a signal from a single scattered source, and a single omnidirectional element antenna at the transmitter. Moreover, assume L clusters, with each cluster consisting of M subpaths and maximum cluster delay is less than one symbol period. In this scenario, the
CHANNEL MODEL
In this section, we describe the general concept of the MUSIC algorithm in a cluster-based channel model. The Music algorithm proposed by Schmidt [6] uses the orthogonality property between the signal subspace and noise subspace of the spatial covariance matrix, to estimate the DoAs of the impinging signals. The algorithm is comprised of 3 main steps, described below. Step 1: The covariance matrix of the observed signal is calculated. Note that for simplicity, we omit the time index t for the remainder of this section. The received covariance matrix can be computed as follows:
314
.
TABLE I S IMULATION PARAMETERS FOR SCM C ANNEL M ODEL
40
35
Value 2.4 GHz 0.5λ 0-20 dB 10,000 3 20 (0, 0.26, 0.52) µs (0.46, 0.38, 0.16) 2◦ η(0, σθ2 ) 2◦ 2) η(0, σϕ
q
I = -50 1
30 q
I = 40 Power Spectrum (dB)
Parameters Carrier Frequency fc Rx antenna spacing SNR ρ No. of channel realization No. of Paths Clusters (L) No. of Sub-paths per cluster (M ) Cluster delay (τ0 ,τ1 ,τ2 ) Cluster power (P0 ,P1 ,P2 ) AS at Tx (σθ ) Tx Per-path DoD Distribution AS at Rx (σϕ ) Rx Per-path DoA Distribution
3
25 q
I = -10 2
20
15
10
5
0 -100
-80
-60
-40
-20
7
0 20 Angle (Degree)
40
60
80
100
Fig. 3. The MUSIC spectrum for the estimation of ϕ0 = −50◦ , ϕ1 = −10◦ , ϕ2 = 40◦ and SNR=20 dB
6.5
6
10 RMSE (Degree)
5.5
9 5
8
RMSE (Degree)
4.5
4
3.5
3
0
2
4
6
8
10 SNR (dB)
12
14
16
18
20
7
6
5
Fig. 2. RMSE of DoA estimation versus SNR. Nr = 4, Nt = 1, and σϕ = 2 ◦ .
4
3
2
3
4
5 V (Degree)
6
7
8
I
RY
Fig. 4. RMSE of DoA estimation versus the angular spread. Nr = 4, Nt = 1, ϕ0 = 30◦ , and SNR =20 dB.
= E[Y.YH ], [ ] = E (HX + N)(HX + N)H , = σx2 RH + RN ,
where (.)H denotes the conjugate transpose, RH = E{HHH }, and RN = E{NNH } are the channel and noise covariance matrix, respectively. Step 2: The eigenvalues and eigenvectors of the received covariance matrix RY are determined. Step 3: The MUSIC pseudo-spectrum is computed as follows: PM U SIC (ϕ) =
aH (ϕ)a(ϕ) , aH (ϕ)Qn Qn H a(ϕ)
V. S IMULATION R ESULTS
(34)
(35)
where Qn is the noise eigenvector and a(ϕ) = [1, ejk0 dr sin(ϕ) , · · · , ejk0 dr (Nr −1) sin(ϕ) ]T is the steering vector. Note that since the noise eigenvectors Qn are orthogonal to the signal steering vectors a(ϕ), the denominator becomes zero when ϕ is a signal direction. Therefore, the estimated DoA is the peak value in the pseudo-spectrum.
In this section, we evaluate the performance of the MUSIC algorithm in the SCM channel model. The simulation results are obtained by averaging over 10,000 independent channel realizations. Also, the important parameters used in the simulations are given in Table I [9]. In these simulations, we consider a ULA composed of 4 sensors at the receiver and a single antenna at the transmitter. Fig. 2 shows the root-mean-square error (RMSE) of DoA estimation for different values of SNR at a DoA of 30◦ . It is observed that the RMSE of DoA estimation improves, as the SNR increases. Moreover, the MUSIC power spectrum is plotted in Fig. 3 for the case of receiving signals from three clusters with mean DoAs of −50, −10, 40◦ . Fig. 4 depicts the RMSE of DoA estimation versus the angular spread σϕ . As shown in this figure, as the angular spread increases, the performance of the DoA estimation algorithm degrades. This happens as a result of increase in the spread of multipath components around the mean DoA of the cluster.
315
TABLE II E XPERIMENTAL PARAMETERS Parameters RF frequency Modulation Tx antenna Rx antenna
Horn Antenna
CRLH-LW Antenna
Value 2.4 GHz QAM Horn antenna CRLH-LW antenna
! 30
VI. M EASUREMENT SETUP AND E XPERIMENTAL R ESULTS
VII. C ONCLUSION In this paper, we investigated the performance of the MUSIC DoA estimation algorithm in the cluster-based channel model, called Spatial Channel Model (SCM). We also studied the impact of angular spread on the performance of the MUSIC algorithm in the SCM model. Moreover, an experimental study was performed as a preliminary evaluation of DoA estimation algorithm using a single CRLH-LW antenna element. The experiment was conducted inside an anechoic chamber. The results show that the DoA was estimated accurately using a single element antenna. Related theoretical analysis, algorithm development and experiments will be reported in future papers. R EFERENCES
RF front-end Q D/A
RF front-end Control Unit
I D/A
I
Q
A/D
SIGNAL GENERATOR
A/D
MUSIC Algorithm
Fig. 5. The measurement setup for one-source DoA estimation in an anechoic chamber. 35
30
25 Power Spectrum (dB)
In this section, we present the DoA estimation experiment in an anechoic chamber. The measurement setup is illustrated in Fig. 5, where the antenna at the transmitter is the horn and at the receiver is a single element composite right/left-handed (CRLH) leaky-wave (LW) which provides electronically controllable radiation patterns. Furthermore, we use the Lyrtech MIMO advanced development system which is a baseband-to-RF solution and consists of several signal processing platforms, such as the VHS-ADC for analog-todigital conversion (ADC), the VHS-DAC for digital-to-analog conversion (DAC), the SignalMaster Quad for baseband signal processing, and the Quad Dual Band RF Transceiver for up and down conversion . At the transmitter, the data is first generated using an Agilent signal generator, and then the resulting baseband signal is split into in-phase (I) and quadrature (Q) components. These components are then converted to analog using the Lyrtech VHS-DAC. Next, using Quad Dual Band RF Transceiver which is a four-channel RF analog front-end, the analog signal is upconverted to the carrier frequency of 2.4 GHz and transmitted over the air. At the receiver, the signal is captured by the CRLH-LW antenna at different scanning steps and sent to the RF front-end to down-convert the RF signal to baseband. Then, the baseband analog signal is sampled by the Lyrtech VHS-ADC. The resulting sampled streams are stored in real time on memory boards and transferred offline to a PC for estimating the DoA. The experimental parameters are shown in Table II. Fig. 6 shows the MUSIC spectrum obtained using a single CRLH-LW antenna at SNR = 20 dB in an anechoic chamber.
Antenna output
20
15
10
5
0 -100
-80
-60
-40
-20
0 20 Angle (Degree)
40
60
80
100
Fig. 6. Experimental result for one signal arriving from the 30◦ direction at SNR=20 dB
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