TIME A DOMAIN METHOD FOR JOINT ESTIMATION OF TIME DELAYS, DOPPLER SHIFTS AND SPATIAL SIGNATURES
Andreas Jakobsson
A. Lee Swindlehurst Dept. of Elec. & Comp. Eng. Brigham Young University Provo, UT 84602, USA.
2. PROBLEM FORMULATION Suppose an antenna receives several scaled, time-delayed, and Doppler-shifted copies of a known transmitted baseband signal. The received signals could, for instance, be the echoes from a pulse transmitted by an active sonar, or they could result from a training sequence sent over a multipath communication channel. In either case, we may model the received signal, , as a delayed and time scaled version of the transmitted signal, , (see, e.g., [9]):
Systems and Control Group Box 27, SE-751 03 Uppsala Sweden.
ABSTRACT This paper presents an efcient algorithm for estimating the time delays, Doppler shifts and spatial signatures of a known waveform received via several distinct paths by an array of antennas. The algorithm is based on the Signal Subspace Fitting approach. Unlike the approach recently presented in [1], this paper uses a more accurate time domain model for the problem. Simulation examples are included to illustrate the algorithms performance relative to the Cram r-Rao bound (which is also derived). e
(1)
where and dened as 1. INTRODUCTION The problem of estimating the time delays, Doppler shifts and spatial signatures of a known signal is a central problem in many elds including radar, sonar and communications. The recent literature considers primarily the case of multiple backscatters of non-moving reectors (see, e.g., [2, 3, 4, 5]). However, in the applications mentioned the scatterers are often rapidly moving, and taking the Doppler shift into account provides more accurate time delay estimates, as well as information about the position and relative motion of the reecting objects. This more involved problem has attracted relatively little attention in the literature (see, e.g., [1, 6, 7, 8]). In [1] a commonly used approximate Doppler shift model is used, and the time delays are estimated by transforming this data model to the frequency domain. The model used in this paper avoids using this approximate Doppler model, and consequently yields more accurate results.
This work was supported in part by the Swedish Institute, the Royal Swedish Academy of Sciences, and by the Ofce of Naval Research under grant N00014-96-1-0934. 1 Andreas Jakobsson is currently on leave at Dept. of Elec. & Comp. Engineering, Brigham Young University, Provo, UT 84602.
and , and are respectively the propagation speed, the range at and the range rate. Given an array of antennas, the output of the array can be expressed in vector form as
8 6 973 5 4 ) " 2 3 $ 1"
where
Q R C P C I C G F D H7EC A B @6
and and represent the number of different multipath arrivals and the spatial signature of the :th arrival, respectively. The additive noise vector, , is assumed to be a zero mean temporally and spatially white noise process with covariance . The standard narrowband assumption is employed here; i.e., the propagation time of the signal across the array is assumed to be much less than the reciprocal of the signal bandwidth. To simplify the problem, we do not use an explicit parameterization of the spatial response
T 98 X V YWU C G S
& ) " ! 0(
& $ " %#!
'
are the time-delay and Doppler parameters
(2) (3)
&
(4)
(5)
where and jugate transpose,
z
denote the trace operator and the conis the orthogonal projection matrix
Q F ( P X
(6)
is the matrix whose columns are the left singular vectors corresponding to the largest singular values of , and is a diagonal weighting matrix. In the simulations presented later, we use the stochastic ML weighting (21)
S r V X V U Q F S
(15)
w
d
w } y u
F
f
w
w @ 3 f
In this section, we present a computationally efcient Signal Subspace Fitting (SSF) algorithm that approximates the maximum likelihood estimator of , , and . The estimator is derived similarly to the SSF estimator presented in [1] and is thus presented in a somewhat condensed form. The
i f
Inserting (27) into the cost function leads to the following criteria for estimating : (28)
d
F
i
3. SIGNAL SUBSPACE FITTING
where and denote the real part and the SchurHadamard product, respectively. Then, minimization of (19) with respect to yields (see [1] for more details)
s
d qa
and where, for example, diag( ) is a diagonal matrix with the elements of the vector along its diagonal.
i
Q n
V V
y w Ixu
(18)
F YxFw
diag
Fw G g
n
(16) (17)
where denotes the pseudo-inverse. Also, let be the vector formed from the real part of the diagonal elements of , and dene
V n F
d
diag
(14)
w C a p
V xV
(13)
Q t
xt
s
V
F
V F F xF
(12)
mfpq h
S e S
@
(11)
and let
be the
blocks of the matrix (24)
X
h
(10)
Q
F
where denotes transpose, same way as , and
d d YYd F g r f
is formed from
w
in the
X
i ! 00S
and suppose that Introduce
F n
and that
has full column rank.
h
(9)
e8
i ! 0S
Q
g
g
(
h
C p S h a3q
w
i S d d YYd C F g d dYd C F g dYdd F l d d Yd F %g k
h
h A j w h A
h A k
i A G
A
w
d d YYd
d d Yd
d d Yd
p yq w 5
. . .
(8)
V U
r
S
4
r
Assuming that is an column vector, and that a total of snapshots are collected from the array at time instances , the data may be arranged in matrix form as
g e hf4 F x5 w p qQ tuv d d Yd s Q F i r
where is a diagonal matrix formed from the largest squared singular values of , and is a consistent estimate of the noise variance (obtained, for example, as the average of the smallest non-zero squared singular values of ). The Doppler parameters can be explicitly estimated using SSF, but only for the case where , which is not a serious restriction in most cases [1]. Let (22)
b S dc c@ b
where
6
(7)
Q
pt
}|~ z y w u s xt }{xvtQ f rpn q o o |~ z d |~ d xw u
in terms of directions of arrival (DOA), but instead treat the elements of as deterministic parameters to be estimated. This allows us to consider a cluster of coherent arrivals that share a given time delay and Doppler shift, without the necessity of estimating the number of such arrivals nor their individual DOAs and amplitudes. In addition, this assumption eliminates the need for an accurately calibrated array. Under the assumption that the Doppler shifts are small, it is possible to simplify the dependence of (5) on the Doppler parameters neglecting by the higher order terms in a Taylor series expansion of :
S C C P7 C aI C C ` @ C C I C G
SSF estimates of the delays and Doppler shifts can be found by minimizing [10, 11, 12] (19)
(20)
Q P0 F %g j F g
s w d i C j C k
(23)
(25) (26)
(27)
Thus the SSF algorithm is implemented by rst performing the -dimensional search over in (28), and by then solving for using (27). Note that the computation required to evaluate the SSF criterion can be signicantly simplied by performing the trace calculation in (28) as
d y u t w t f xt w u y i S
can be seen from the gure, the SSF estimates are found to be efcient at about SNR = 0 dB. For SNRs below 0 dB the Doppler-shift estimate was found to be biased towards zero, which is the reason why its rMSE is actually below the CRB in this region.
(29)
10
1
It should be noted here that the algorithm requires an initial estimate of the multipath parameters. In the simulations presented in the next section we have used an initial Doppler estimate of zero, and a fast ESPRIT-based estimator presented in [5] for estimating time-delays in cases when the Doppler offset is zero. Our empirical results indicate that this approach still gives reasonable time delay estimates even when the Doppler is non-zero but small. The fact that the algorithm yields the desired estimates in closed form (i.e., without search) makes it an attractive alternative for initialization. 4. SIMULATION RESULTS In this section we study the performance of the estimator as the signal-to-noise ratio (SNR) varies. Simulation data was generated using (4) for two multipath signals ( ) with time-delays and Doppler shifts . The data was corrupted by spatially and temporally white circular Gaussian noise with zero mean and standard deviation . The two columns of the signature matrix, , were given by the array response of a 5-element, half-wavelength spaced uniform linear array with DOAs and . The signal sequence was chosen to be the raised cosine pulse sinc
! S d w d 2 U g @ w 2 d 2 2 f 2 ! 2 g d 2 2 2 i
10
0
SSF ESPRIT CRB
10
1
rMSE
10
2
10
3
10
4
10 10
5
0
10
20
30 SNR
40
50
60
70
Figure 1: The rMSE of the rst time-delay estimate, , for the SSF and the ESPRIT-based estimators as a function of the SNR.
10
0
SSF CRB
10
1
Figure 2: The rMSE of the rst Doppler-shift estimate, for the SSF estimator as a function of the SNR.
F
For the simulations presented here, , and samples are assumed to be taken from the array. The root mean squared error (rMSE) of the time-delay and Doppler estimates were calculated for the algorithm based on 200 Monte Carlo trials for various SNR values. The resulting rMSE for the rst time delay estimate (the second time-delay estimate behaves similarly) are plotted in Figure 1, together with the appropriate Cram r-Rao Bound e (CRB). See Appendix A for a derivation of the CRB. As can be seen from the gure, the SSF algorithm (dashed line) achieves the CRB (solid line) at about SNR = 10 dB. The excess error for the ESPRIT algorithm (dotted line) is of course due to the fact that it assumes the Doppler is zero when estimating the time-delays; thus it yields biased time delay estimates. Figure 2 shows the rMSE for the rst Doppler shift estimate with the corresponding CRB as a function of SNR. As
g g i
10
rMSE
2
10
3
10
4
10 10
5
0
10
20
30 SNR
40
50
60
F
2
V
70
,
A. THE CRAMER-RAO BOUND In this Appendix, we derive the CRB for the current estimation problem. Specically, we derive the elements of the Fisher information matrix, , whose inverse yields the CRB. By denition (see, e.g., [13], Appendix B): E
b C b V b W r 7 { s X C X W
[2] J. Li, B. Halder, P. Stoica, and M. Viberg, Computationally Efcient Angle Estimation for Signals with Known Waveforms, IEEE Trans. on Signal Processing, 43(9):21542163, Sept. 1995. [3] A. van der Veen, M. Vanderveen, and A. Paulraj, J...