TOA estimator

Signal Processing 84 (2004) 1359 – 1365

www.elsevier.com/locate/sigpro

Performance analysis of joint DOA/TOA estimator F. Mrabtia , M. Elhajjamib , M. Zouaka; b;∗ a UFR,

Automatique et Analyse des Systemes, Faculte des sciences Fes-Dhar El Mehraz, BP 1796, Fes, Atlas 30000, Morocco of Sidi Mohammed Ben Abdellah, UFR, Laboratory Signaux, Systemes et Composants, Faculte des Sciences et Techniques, Fes, BP 2202, Fes, VN 30000, Morocco

b University

Received 20 June 2003; received in revised form 21 April 2004

Abstract In many applications such as radar and mobile communication, the multipath propagation e5ects are described as a sum of contributions of a large number of wavefronts that arrives at the sensor array in clusters of rays, distributed around a nominal direction of the signal sources. Based on this observation and on the work of Bengtsson and Ottersten (Proceeding of Norsig-98, IEEE Nordic Signal Processing Symposium, April 1998), this paper jointly estimate the directions of arrival and the times of arrival of scattered sources. A theoretical performance analysis is given in terms of asymptotic error variance and illustrated by a simulation study. ? 2004 Elsevier B.V. All rights reserved. Keywords: Sensor array processing; Parameter estimation; Mobile communication; Scattered sources

1. Introduction For several years, many algorithms have been proposed to resolve the problem of signal parameter estimation in sensor array processing. In fact, the signal can be separated at the array based on the knowledge of their spatial and/or temporal “signatures”. Traditionally, the studies assume that the received signals originate from far-
Corresponding author. University of Sidi Mohammed Ben Abdellah, UFR, Laboratory Signaux, SystAemes et Composants, FacultBe des Sciences et Techniques, FAes, BP 2202, FAes, VN 30000, Morocco. E-mail addresses: f [email protected] (F. Mrabti), [email protected] (M. Elhajjami), mohcine [email protected] (M. Zouak).

from a
0165-1684/$ - see front matter ? 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.05.010

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F. Mrabti et al. / Signal Processing 84 (2004) 1359 – 1365

technique, MUSIC, is used for channel parameters estimation. This paper is organised as follows: In Section 2, a modelling framework is presented for the space-time processing. The algorithm MUSIC, to estimate the parameters channel, is investigated in Section 3 followed by a MontBe Carlo simulations in Section 4. The paper is concluded in Section 5.

θ2

2. Data model In an urban area, the signal received by the base station from a mobile terminal can be considered as resulting mainly from reKections of the radiowaves. None of these di5erent signal paths will dominate, unless there is free sight between the mobile and the base station. The source can be modelled as a number of scatterers spreading out on a circle surrounding it (Fig. 1), [1,4]. Assume a large number of wavefronts impinging on a uniform linear array (ULA), emanating from

θ1

mobile

Fig. 2. Approximation by two points sources.

θBW

θ

Base station

Fig. 1. Model of base station received scattered source.

reKections close to the source [1,2,10]. The authors of [3], under the assumption that the time delays is modelled as phase shifts, approximate each cluster by two point sources at 1 and 2 , (Fig. 2). This approximation permits us to use the formulation of the DOA estimation problem which requires that the number of impinging signals is less than the number of sensors in the array. This approximation can be justi
F. Mrabti et al. / Signal Processing 84 (2004) 1359 – 1365

However, since: 
2m 61 − 1 6 sin M −1 ⇒  − 12 BW 6 m 6  + 12 BW :

white gaussian noise with the same dimension of the vector Y. (2.2)

We can represent the spreading of m by the extrema of the circle’s diagonal, therefore, we consider the scatterers as the two extremes of the interval and the approximation of two point sources [3] are justi
and

2 =  + 12 BW :

(2.3)

The statistical spatio-temporal ultra high frequency (UHF) signal model is then a sum of the two rays at 1 and 2 . The signal received by the N -element antenna array can be collected in a vector (N × 1): 2  y(t) =

i a(i )s(t − i ) + b(t) (2.4) i=1

with s(t) is the base band transmitted signal, i (i=1; 2) is the group delay. With
 d (2.5) an (i ) = exp 2j (n − 1)sin(i )  the nth elements of (N × 1) vector a(i ), the array response vector, spaced by a distance d;  is the wavelength, and

i = i exp(j’i )

(2.6)

is a random complex gain factor, assumed to be independent from snapshot to snapshot as from ray to ray, where i is Rayleigh distributed and ’i is uniform [0; 2] distributed. b(t) is a white gaussian noise. Oversampling y(k) at K times gives a spatio-temporal vector of (NK × 1) observations: 2  Y(k) =

i C(i ; i ) + B(k) (2.7) i=1

3. 2D-Music algorithm Based on the parametric modelling of Y(k) in Eq. (2.7), the empirical covariance matrix Rˆ y can be expressed as P 1 Rˆ y = Y(p)YH (p); P

(3.1)

p=1

where ( )H denote the complex conjugate transpose operation and P the number of snapshots. The matrix Rˆ y can be decomposed into two disjoint orthogonal subspaces, signal and noise subspace, which are spanned by the respective signal and noise ˆ eigenvectors, Sˆ and G. Let 1 ¿ 2 ¿ · · · ¿ NK denote the eigenvalues of Rˆ y . The rank of the signal subspace Sˆ is M. Denote the unit-norm eigenvectors associated with 1 : : : M by s1 : : : sM and those corresponding to M +1 : : : NK by g1 : : : gNK−M . Also de
(3.2)

Gˆ = [g1 : : : ; gNK−M ]:

(3.3)

In the MUSIC algorithm, the DOAs and TOAs are jointly estimated by searching one by one for values ˆ of  and that make c(; ) nearly orthogonal to G. The MUSIC cost function is de
C(; )H Gˆ Gˆ H C(; ) C(; )H C(; )

(3.4)

and the minima of f(; ) are taken to be the estimates of the DOAs and TOAs of the point sources. 4. Performance analysis

with C(i ; i ) = a(i ) ⊗ s( i );

(2.8)

Y(k) = [yT (k); : : : ; yT (k − K + 1)]T ;

(2.9)

s( i ) = [s(k − i ); : : : ; s(k − K + 1 − i )]T :

1361

(2.10)

s( i ) is the (K × 1) collected vector of the transmitted signal, ⊗ is a Kronecker product and B(k) is a

Let us
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F. Mrabti et al. / Signal Processing 84 (2004) 1359 – 1365

In this section, we derive the asymptotic variances of the errors of the estimates obtained by the 2D-MUSIC algorithm. The proposed analysis is asymptotic, i.e., the analytical formulas derived are correct up to 0 (P −1 ). Given that the estimated parameter vector ˆ i = (ˆi ; ˆi ); (i = 1; 2) minimizes the cost function (3.4) it is also solution of:  ( ˆ i ) = 0: fMU

(4.1)

A Taylor series expansion of (4.1) around

i

gives

  = fMU ( i ) + fMU ( i )( ˆ i −

i)

+ ::::

(4.2)

fMU ( i ) = f( i )=r( i );

(4.3)

where f( i ) = C( i )H Gˆ Gˆ H C( i )

(4.4)

and r( i ) = C( i )H C( i ):

(4.5)

We observe that r( i ) is related to the numerator, and the estimates obtained by minimizing (4.3) for any function r( i ) have the same asymptotic (for large P) distribution as the f( i ) function. A theorem and a proof are given in [7] for a scalar. This theorem can extend to a vector parameter of dimension 2. Theorem 1. Assume that the function r( i ) satis=es the regularity condition r( i ) = 0, but is otherwise arbitrary. Then the estimates minimizing f( i ) and fMU ( i ) have the same asymptotic distribution. Proof.

f ( i ) f ( i ) ˆ + ( i− r( i ) r( i )

i)

i );

(4.7)

where the neglected terms go to zero faster than ( ˆ i − i ), when P tends to in




f ( i )r( i ) − f( i )r ( i ) ; r2( i )

( ˆi −

Gr =

i)

i)

≈0

≈ −H · Gr;

(4.8)

(4.6a)

[f ( i )r( i ) − f( i )r  ( i )]r 2 ( i ) r4( i ) [f ( i )r( i ) − f( i )r  ( i )]2r( i )r  ( i ) : r4( i ) (4.6b)

(4.9)

@f( i )=@i

@f( i )=@ i  H H @C( i )    C ( i )Gˆ Gˆ @i = 2 Re    C H ( i )Gˆ Gˆ H @C( i ) @ i

(4.10)

and H = 2 · Re   @C( i ) @C( i ) @C( i ) @C( i ) −1 EN ENH EN ENH  @i @i @i @ i   ; × @C( )  i H @C( i ) @C( i ) H @C( i ) EN E N EN E N @ i @i @ i @ i (4.11) H is the inverse of the limiting Hessian and Gr the gradient. Next, note that (see Refs. [7,9]): ˆ H C Gˆ Gˆ H ≈ CSS H GG

 ( i) fMU

−

≈

where 

The MUSIC estimator can be written as

=

  0 ≈ fMU ( i ) + fMU ( i )( ˆ i −

Consequently

 ( ˆi) 0 = fMU

 ( i) = fMU

Since f( i ) = 0(1=P) and f ( i ) = 0(1=P), it follows from (4.2) and (4.6) that

(4.12)

Inserting (4.12) in the expression (4.10) for Gr, we get   NK−M  H @C H g · CSS gˆk     k=1 k @i   (4.13) Gr =  ;   NK−M  @C   gkH · CSS H gˆk @ i k=1

where NK is the dimension of the observation vector ˆ and gˆk is a kth vector of G.

F. Mrabti et al. / Signal Processing 84 (2004) 1359 – 1365

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0.06

EQM des DAs

CRB MUSIC 0.04

0.02

0

-0.02 0

5

10

(a)

15

20

25

30

SNR

EQM des TAs

0.2 CRB MUSIC

0.15 0.1 0.05 0 -0.05 0

5

10

(b)

15

20

25

30

SNR

Fig. 3. MSE of estimated parameters compares to the CRB versus the SNR: (a) MSE of DOA; (b) MSE of TOA.

It can be shown easily that the theoretical variance of the 2D-MUSIC estimator is given by 2 E((Ti )2 ) = Re 2P 
H   H @C   2 H H @C   H (C UC) GG    11  @ @   i i     
  H H @C @C H H ; +2H11 H12 (C UC) GG  @i @ i      
H   H     @C   2 H H @C  +H12  (C UC) GG @ i @ i (4.14) 2  E((T i )2 ) = Re 2P 
H   @C H @C   2 H H   H (C UC) GG   22   @ @   i i    
H   H @C @C H H ; +2H22 H21 (C UC) GG  @i @ i      
H       @C H @C   2  +H21  (C H UC) GG H @i @i (4.15)

where [7–9] U=

M  k=1

(2

k sk skH − k ) 2

(4.16)

and (k ; sk ) are the kth associated eigenvalue and eigenvector of the covariance matrix.

5. Simulation In this simulation, we consider a 5-element ULA with half-wavelength element spacing. The DOAs and TOAs, characterizing the two point sources, are (10◦ , 1.5Te) and (20◦ , 2.1Te). Te is the sampling period. The transmitted signal is a BPSK and the channel outputs were corrupted by additive Gaussian white noise of variance 2 . The temporal duration is K = 10 samples. The number of snapshots considered is P = 100. We limit our study for two points sources, which is a particular case of the scattered sources.

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F. Mrabti et al. / Signal Processing 84 (2004) 1359 – 1365 0.04 CRB MUSIC

MSE of DOAs

0.03 0.02 0.01 0 -0.01 50

100

150

200

250

300

(a)

350

400

450

500

550

600

number of snapshots 0.2 CRB MUSIC

MSE TOAs

0.15 0.1 0.05 0 50

(b)

100

150

200

250

300

350

400

450

500

550

600

number of snapshots

Fig. 4. MSE of estimated parameters compared to the CRB versus the number of snapshots: (a) MSE of DOA; (b) MSE of TOA.

Computer simulations were carried out to test the performance of the estimator in terms of the mean-squared error (MSE). The MSE are computed by the average squared error over 50 experiments and ploted versus the signal to-noise ration (SNR) and the number of snapshots. Figs. 3 and 4 compares the MSE of DOA and TOA estimation using the MUSIC and the CramBer–Rao bound (CRB). The CRB has been calculated as in [6,10]. We see that the MSE converges to the CRB as the number of snapshots is increased resulting from a better estimation of the covariance matrix.

6. Conclusion Herein, we investigate the problem of jointly estimating the direction of arrival and time of arrival of local scatterers, approximated by two points sources. We presented an asymptotic accuracy analysis of the 2D-MUSIC estimator and an explicit expres-

sion, for its theoretical error variance have been provided.

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F. Mrabti et al. / Signal Processing 84 (2004) 1359 – 1365 [7] P. StoVWca, A. Nehorai, MUSIC, maximum likelihood, and Cramer–Rao bound, IEEE Trans. Acoust. Speech Signal Process. 37 (5) (May 1989) 720–741. [8] P. Stoica, A. Nehorai, MUSIC, maximum likelihood, and Cramer–Rao bound: further results and comparisons, IEEE Trans. Acoust. Speech Signal Process. 38 (12) (December 1990) 2140–2150.

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[9] P. Stoica, T. Soderstrom, Statistical analysis of MUSIC and subspace rotation estimates of sinusoidal frequencies, IEEE Trans. Signal Process. 39 (8) (August 1991) 1836–1847. [10] T. Trump, B. Ottersten, Estimation of nominal direction of arrival and angular spread using an array of sensors, Signal Processing 50 (8) (1996) 57–69.