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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
Cramer–Rao Bounds for the Estimation of Multipath
Parameters and Mobiles’ Positions in
Asynchronous DS-CDMA Systems
Cyril Botteron, Member, IEEE, Anders Høst-Madsen, Senior Member, IEEE, and Michel Fattouche, Member, IEEE
Abstract—Commercial applications for the location of subscribers of wireless services continue to expand. Consequently,
finding the Cramer–Rao lower bound (CRLB), which serves as
an optimality criterion for the location estimation problem, is of
interest. In this paper, we derive the deterministic CRLBs for the
estimation of the specular multipath parameters and the positions
of the mobiles in an asynchronous direct sequence code division
multiple access (DS-CDMA) system operating over specular multipath fading channels. We assume a multilateral radio location
system where the location estimates are obtained from some or all
of the estimated signal parameters at different clusters of antennas
of arbitrary geometry. Extension for unilateral and composite
radio location techniques is also discussed. As an application
example, we use numerical simulations to investigate the effects of
specular multipath and multiple access interference (MAI) on the
positioning accuracy for different radio location techniques.
Index Terms—Amplitude estimation, angle estimation, asynchronous DS-CDMA, Cramer–Rao lower bound, location,
position, specular multipath, time estimation.
I. INTRODUCTION
W
IRELESS localization using radio location systems has
become an important research area over the past few
years. A major application is personal safety, such as in the location-based emergency service (E-911), which is a requirement
for the wireless carriers in the United States [1]. Other applications include intelligent transportation systems, accident reporting, automatic billing, fraud detection, and other emerging
services [2], [3]. Radio location systems attempt to locate a mobile station (MS) by measuring the radio signals travelling between the MS and a set of fixed stations (FSs) of known coordinates. They can be classified as unilateral (or handset-based),
multilateral (or network-based), or composite [3]. In a unilateral
system, an MS forms an estimate of its own position based on
radio signals received from the FSs, and in a multilateral system,
an estimate of the MS’s location is based on a signal transmitted
by the MS and received at multiple FSs.
Manuscript received January 10, 2001; revised April 17, 2003. The associate
editor coordinating the review of this manuscript and approving it for publication was Prof. Randolph L. Moses.
C. Botteron was with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada. He is now with
the Institute of Microtechnology, University of Neuchâtel, CH-2000 Neuchâtel,
Switzerland (e-mail: [email protected]).
A. Høst-Madsen is with the Department of Electrical Engineering,
University of Hawaii at Manoa, Honolulu, HI 96822 USA (e-mail:
[email protected]).
M. Fattouche is with Cell-Loc Inc., Calgary, AB T2A 6T8 Canada (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TSP.2004.823490
In order to quantify the impact of different signal parameters, environment parameters, and to understand the relative
contribution to accuracy of different signal measurements, we
derive in this paper the CRLB for multilateral or network-based
radio location techniques. However, instead of assuming that
the signal measurements are (zero-mean) Gaussian distributed1
with known covariance matrix (see, e.g., [4]–[7]), we assume
for our CRLB derivation a specular multipath environment
and Gaussian distributed noise at the antenna receivers so that
we can express the CRLB directly in terms of the signal and
environment parameters. Extension of the CRLB for unilateral
or handset-based and composite techniques is also discussed.
Using the CRLB for the location estimation, we investigate
using numerical simulations the effects of the multiple access
interference (MAI) and specular multipath for a K-user asynchronous DS-CDMA system for which we also present the
deterministic and asymptotic CRLB for the joint estimation
of the signal parameters. Three different multipath scenarios,
where the number and relative clustering of the main reflectors
around the MS are varied, are also discussed.
This paper has been organized as follows. Section II describes the notations and some general model assumptions.
Section III presents the CRLB for the location estimation for
the mostly used radio location techniques. In Section IV, the
deterministic and asymptotic CRLBs for the joint estimation
of the time, bearing, and amplitude parameters in a K-user
asynchronous DS-CDMA system are derived. In Section V,
the effects of MAI and specular multipath on the relative
contribution to accuracy of different radio location techniques
are investigated. Finally, the conclusions are discussed in
Section VI.
II. RADIO LOCATION MODEL AND ASSUMPTIONS
We consider a specular multipath environment and a multilateral positioning system, where the location of a MS is estimated
based on the measurements of the signal transmitted by the MS
and received at FSs of known coordinates. Without loss of
. We consider the most comgenerality, we assume that
monly used signal parameter measurements,2 i.e, the angle of
1Because the signal measurements (such as time, bearings, etc.) cannot in
general be expressed as a linear function of the received signal samples at the
antenna receivers, their distribution should not be expected to be Gaussian, even
if the signal samples are statistically Gaussian distributed.
2For a more complete description of these different techniques, see, e.g.
[8]–[10], and the references therein.
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BOTTERON et al.: CRAMER–RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS
arrival (AOA), the time of arrival (TOA), the time difference of
arrival (TDOA) between multiple FSs, and the strength of arrival (SOA) measurements. These measurements can be used
individually or in combination (in which case, they are called
mixed measurements) to produce a position location. Note that
the model and theory developed in this paper can be easily extended to consider other measurements and positioning techniques (e.g., the extension of the model for unilateral and composite techniques is discussed in Section III-B).
For clarity, the generic assumptions used in this paper are
summarized and discussed below.
A1) The environment between the MS and the FS(s)
exhibits specular multipath due to a finite number
of dominant reflectors located in the far field of the
receiver(s).
A2) Every dominant path results from the superposition of
many component waves where delay spreads are much
smaller than the inverse bandwidth of the signal and the
dominant path delays.
A3) For short observation intervals, the signal parameters
(e.g., the number of dominant paths, their attenuation,
direction of arrival, and time-delay) can be assumed
constant.
A4) The transmitted signal is narrowband with respect to
the reciprocal of the excess delay across the receiving
antenna arrays (i.e., the signal bandwidth is assumed
to be very small compared to the carrier frequency).
A5) The noise present at the antenna receivers is zero-mean
Gaussian distributed, spatially independent between
the different FSs and the MS, and independent of the
signal and position parameters.
A6) In addition to the nonline of sight (NLOS) paths
coming from the main reflectors, a direct path always
exists between the MS and the FSs.
Assumptions A1–A4 are the conventional assumptions used
to model the direction and time estimation of the specular multipath arrivals from a narrowband source located in the far field
and in the same 2-D plane as an array of sensors (see, e.g,
[11]–[13]). The Gaussian noise assumption A5 is a basic and
mild assumption that is justified in practice, especially for the
outdoor channels. Note that only the noise between the different
FSs and the MS is assumed to be spatially independent. However, the noise processes are not assumed to be spatially independent between the different antenna receivers at one particular
FS, and the noise covariance matrix may be different from one
FS to another. Finally, assumption A6 is a little more restrictive but necessary to allow for an unbiased position estimator to
exist.
III. CRLB FOR THE LOCATION ESTIMATION
We define the following parameter vectors for the developments to come.
-dimensional complex vector,
where
observation vector
contains (in complex
baseband representation) all the received samples from all the antenna
receivers at the th FS.
863
: source-location
parameter vector
:
signal parameter vector
-dimensional real vector containing the positions in Cartesian
coordinates of all the main reflectors
and the MS(s) (i.e., the locations of
all the sources) plus any additional
nuisance parameters (such as the
noise parameters) necessary for
to completely parameterize the
probability density function (p.d.f.)
of the observation vector .
-dimensional real vector, where
contains the signal parameters
(e.g., AOA, TOA, SOA, ) that
characterize all of the specular multipath arrivals at the th FS plus
any additional nuisance parameter
(such as the noise parameters) that
to
is necessary for the vector
completely parameterize the p.d.f.
of the observation vector
at the th FS.
A. Derivation of the Location CRLB
Since the source-location parameters contained in form a
set of linearly independent parameters, the CRLB matrix for any
unbiased estimate of could be calculated, under standard regularity conditions and based on the above definitions and assumptions, using the standard CRLB formula as the inverse of the
, i.e., as (see, e.g., [14],
Fisher information matrix (FIM)
[15])
CRLB
(1)
denotes the expectation operator taken with respect
where
, and the derivatives are evaluated at the true values
to
of .
matrix of random samples
Assuming an
that consists of independent, -dimensional
random vectors normally distributed as
(see assumption A5), another method to calculate the elements
would be to use the standard CRLB formula
of the FIM
for the complex multivariate normal (MVN) model, i.e., (see,
e.g., [15])
tr
Re
(2)
However, unless a very simple system geometry [i.e., the position of the FSs and main reflectors with respect to the location
of the MS(s)] is assumed, deriving the CRLB for the location
of the sources using (1) or (2) is too difficult for the problem at
hand.
An alternative and simpler approach is to first calculate the
CRLB for the signal parameters of the sources at the FSs and
then use the following proposition
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
Proposition 1: Based on the above definitions and assumptions, define
TABLE I
CORRESPONDING PARAMETERS FOR MULTILATERAL
UNILATERAL RADIO LOCATION TECHNIQUES
AND
CRLB
..
.
CRLB
diag CRLB
CRLB
(3)
is the CRLB for the joint estimation of the
where CRLB
signal parameters of the sources at the th FS. Express the signal
parameters contained in
as a function of the location
as the functions
parameters contained in
for
or
with
. We then have
CRLB
where the elements of the matrix
(4)
are defined as
(5)
provided that the functions
do not affect the noise
parameters.
Proof: See Appendix A.
1) Discussion: We make the following observations.
• Proposition 1 looks like a standard transformation of parameters, as discussed in [14] and [15]. However, a transformation of parameters is defined for functions
with
, whereas here,
. Because of
the relationship between parameters, not all values of
are valid— is constrained to a submanifold of
, and
therefore is not a CRLB matrix (inverse of a Fisher
matrix).
do not need to be the
• The dimension of the vectors
. In other worlds, the CRLB for the
same for
location of the sources can be obtained using Proposition
1 for a different number of samples, number of antennas,
type of antennas, and noise covariance matrix at each of
the FS.
B. Extension for Unilateral and Composite Techniques
For an unilateral technique where the MS estimates its position based on the received signals coming from FSs, the observation vector at the MS will contain the superposition of the
multipath arrivals from FSs. Thus, it can be expressed as
(6)
where the received samples in
contains (in complex baseband representation) the contributions of the multipath signals
coming from the th FS, and is assumed Gaussian distributed
(see assumption A5) with covariance matrix
.
We observe that the above system model is similar to a -user
system model for multilateral radio location techniques where
a single FS is used to estimate the location of the MSs (see
Table I). Consequently, the CRLB for the signal parameters in
a -user system model for multilateral techniques can be applied for a single-user unilateral technique by using the correspondence of parameters given in Table I. In this case, the signal
parameter vector and the matrix
in Proposition 1 will
only contain the signal parameters of the sources estimated at
the MS and the CRLB for those signal parameters, respectively.
Finally, for a composite radio location technique where the received signals at the MS and the FSs are jointly used to estimate
the location of the MS, we deduce from Assumption A5 (i.e., assuming that the noise is spatially independent between the MS
and the FSs, which is reasonable in practice) that Proposition 1
can still be used. In this case, the signal parameter vector will
contain both the signal parameters estimated at the MS and at
will contain in block diagonal form the
the FSs, and
CRLB matrix for the estimation of the signal parameters at the
MS in addition to the CRLB matrices for the estimation of the
signal parameters at the FSs.
C. CRLB for the Location of a MS
FSs is large
If the number of specular multipaths at the
and we are only interested in bounding the positioning accuracy for the location of a MS (and not for the location of other
sources), using (3) –(5), which involves the inversion of a large
, is cumbersome. Furthermore, most position
matrix
estimators only use a subset of the signal parameters estimated
at FSs to estimate the location of an MS. For example, if we
assume a bearing-based radio location system, then only the estimated AOAs of the direct paths coming from the MS and reFSs will be used. In addition, if a radio location
ceived at
system only uses the estimated TOAs of the direct paths coming
from the MS and received at FSs and has no knowledge of
the time of transmission (TOT), then it will use these TOAs to
jointly estimate the unknown MS’s position with the unknown
MS’s TOT.
Thus, we now consider the case when some of the estimated
signal parameters are used to estimate the location of a MS.
Proposition 2: In addition to assumptions A1–A6, let us assume that the unbiased position estimator only uses the subset
of the jointly estimated signal parameters contained
, where
in to estimate the parameters contained in
contains all the parameters necessary for a function that
defines a continuous and bounded mapping from onto to
exist.
Then, the covariance matrix for any unbiased estimator is
bounded as follows:
cov
CRLB
diag CRLB
CRLB
where
are defined as
ments of the matrix
(7)
, and the ele-
(8)
BOTTERON et al.: CRAMER–RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS
provided that a) CRLB
exists
, and b)
is
exists).
positive definite (meaning
Proof: See Appendix B.
1) Discussion:
• Proposition 2 is very important since it allows the calculation
of the CRLB for the location of a MS, assuming an unbiased
estimator that only uses some of the estimated signal param.
eters without inverting the whole matrix
composing
can easily
• The CRLB matrices CRLB
by rebe obtained from the CRLB matrices CRLB
moving the rows and columns that correspond to the nui,
sance parameters contained in , where
i.e., the signal parameters that are not used to estimate the position parameter vector .
• Using the appropriate transformation matrix
and CRLB
, Proposition 2 can be used for any
matrices CRLB
positioning method (including multilateral, unilateral, and
composite radio location methods using mixed measurements), signal waveform, or system geometry.
that we would ob• It is worth noting that the CRLB for
tain by removing the rows and columns corresponding to the
from
rather than from
subset
would not be the same but correspond to assume that
is known by the positioning system. Using the
the subset
well-known fact that an augmentation of the nuisance parameters can only result in an increase of the corresponding
CRLB (see, e.g. [14]), we can write
CRLB
CRLB
CRLB
(9)
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the path loss between the MS and the th FS separated by a
as [16]
distance
dB
(10)
where
is the path loss at the reference distance , and
is the path loss exponent that indicates the rate at which the path
loss increases with distance. If the MS transmits with a fixed
dBm, we can thus express the power received at
power of
the th FS as
dBm
(11)
where
represents the total system losses (in decibels). On
the other hand, if the transmitted power is controlled by the th
serving FS and assuming perfect power control (p.c.), the received power at the th FS can be expressed as
dBm
(12)
where SNR (in decibels) and
(in decibel meters) are the
desired SNR (without antenna gain) and the received noise at
the th serving FS, respectively.
By defining the known 2-D position of the th FS in Cartesian
and the MS’s position as
coordinates as
, we can thus write the th signal parameter
as
, where
D. Expressions for the Transformation Matrix
We now give some analytical expressions for the transformain Proposition 2, assuming four different radio
tion matrix
location estimators that we categorize based on the type of direct path measurements they use to estimate the position of the
MS:
SOA-based: using the signal parameter vector
containing the estimated modulus of the direct paths’ fadings (we assume an estimator that has knowledge of the path loss model);
AOA-based: using the signal parameter vector
containing the estimated direct paths’ bearings;
TD-based: using the signal parameter vector
containing the estimated direct paths’ propagation-times obtained by subtracting the MS’s TOT
assumed to be known at the
FSs from the estimated
direct paths’ TOAs;
TOA-based or TDOA-based: using the signal parameter
containing
vector
the estimated direct paths’ TOAs. In this case, we assume
is also unknown and jointly estithat the MS’s TOT
mated with the location of the MS.
In order to find an analytical expression for the transformation
for a SOA-based radio location estimator, we write
matrix
and denotes the speed of light (in meters per second).
with respect to
By computing the derivatives of
[see (8)], we obtain the transformation matrices
,
,
, and
corresponding to the parameter
,
,
, and
, respectively, as
vectors
(13)
(14)
(15)
(16)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
where
is given assuming a fixed transmitted power
or perfect power control at the th FS by
or
, respectively, and
Note that for a mixed measurement vector
containing
any combination of the above measurements, the transformation
will simply be a combination of the columns of
matrix
and rows
(13)–(16) corresponding to the parameters in
corresponding to .
IV. EXPRESSIONS FOR CRLB
The CRLB derivation for direction and time estimation using
antenna arrays in a specular multipath environment can be found
in many publications. For example, assuming a single user and
no MAI, the conditional (sometimes also called deterministic)
CRLB for the estimation of the bearings and time delays was
derived in [11], [13], and [17], assuming unknown deterministic
path fading parameters. A compact CRLB derivation for parametric estimation of superimposed signals was also presented
in [18].
For a K-user asynchronous DS-CDMA communication
system where the MAI is explicitly modeled, the CRLB for the
time delays and path fadings treated as unknown deterministic
parameters was derived without bearing estimation in [19] and
[20]. However, the CRLB for the joint estimation of the path
amplitudes, bearing, and time-delays only appeared (without
proof) in [21]. Thus, we now briefly review the system model
in [21] and present the deterministic and asymptotic CRLB
for the joint estimation of the path amplitudes, bearing, and
time-delays in asynchronous DS-CDMA systems.
A. Asynchronous
where
and the complex noise vector
is defined similarly
.
denotes the number of dominant paths from
to
the th user impinging on the antenna array. The eleare
and
ments of
for
, where
and
,
is the th element of the steering vector
toward the AOA
, and
denotes the complex path
amplitude during the th symbol interval. The columns of
represent the shifted code sequences for the
dominant paths coming from the th user and impinging on the
antenna array. They are defined as
(18)
(19)
where
is the path delay, such that
is
, and
(i.e., we assume no
an integer, and
oversampling). The vector
and the permutation matrix
are defined as
Finally, by stacking the received vectors from all the
tennas in a single vector, we obtain
-User DS-CDMA System Model
Besides the generic assumptions A1–A6 (see Section II), we
assume that the complex path fading amplitudes can be considered constant over the duration of one symbol interval. The
users’ code waveforms are assumed to be rectangular and pe, where
is the chip period, and
riodic with period
is the processing gain. The modulation is BPSK, i.e., the th
user baseband signal is formed by pulse amplitude modulating
with a period of the -long
the data stream
as
.
code waveform
Using complex envelope representation and collecting the
received samples at the th FS during the th symbol interval
from the th element of a -element antenna array in a single
vector
, we can write3 (for more details, see, e.g., [19], [21],
[22])
an-
(20)
denotes the Kronecker product, and
where
defined similarly to
.
,
are
B. Deterministic CRLB for the DS-CDMA Signal Parameters
Based on the above signal model, we define the unknown
as
signal parameter vector
Re
Im
Im
Re
(21)
where
(17)
3For ease of notation, we often drop the FS subscript c unless it is necessary
to avoid confusion with the previous notations.
(22)
BOTTERON et al.: CRAMER–RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS
Assuming that the received noise at the antenna array is
, the
zero-mean complex circular Gaussian with variance
and conditioned on
log-likelihood function with respect to
the transmitted bits (practically, the transmitted bits are readily
available from the serving base stations) can be written as
and
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is defined as
diag
const.
By applying the CRLB formula (1) to the above log-likelitakes the
hood function, it can be shown that the FIM for
form4
..
..
.
.
(36)
We observe that the CRLB for the signal parameters belonging to the th user is independent of the received powers of
other users (this fact can be easily verified by noting
the
and using the same procedure as in
the diagonal form of
[19, Sec. IV]).
C. Extension for the SOA Estimation
(23)
where
(24)
crb
Re
Im
We now extend the previous derivation to obtain the deterministic CRLB for the joint estimation of the bearing, time, and
amplitude parameters using the following assumption:
A7) The real parameter vector of estimated path amplitudes
is obtained by averaging the estimated path amplitime intervals, i.e., as
tudes for the users over
(25)
Im
Re
Re
Im
(26)
Re
(28)
and the matrices
,
Re
(27)
, , and
are defined as
(29)
diag
(30)
Im
(37)
is defined in (22).
where
From (37), we can thus derive the CRLB for the joint estimation of the bearing, time, and amplitude parameters starting with
in (23) and using a transformation of
the FIM expression
parameters as
CRLB
(31)
CRLB
(38)
diag
(32)
The matrices
and
are defined similarly to
elements of these matrices are
. The
otherwise
is defined as
otherwise
(39)
where
Re
(34)
4Because
th element of the matrix
and can be written as
diag
(33)
otherwise
where the
Im
Re
Im
diag
and
denotes the element-wise (Schur-Hadamard) matrix
product.
(35)
the derivation of the deterministic CRLB assuming unknown deterministic path fading parameters for the above system model is somewhat
lengthly and follows the same steps as in, e.g., [19] and [23], a complete derivation is omitted.
D. Asymptotic CRLB for the DS-CDMA Signal Parameters
Proposition 3: Assuming that the transmitted data symbols
are i.i.d. with
and that the channel is static (with
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
multipath), the asymptotic CRLB for the joint estimation
of the complex path fading, bearing, and time-delay paramRe
eters contained in
Im
, where
, is given by
AsCRLB
CRLB
Re
Im
Re
Re
Im
Re
Im
Im
Re
Im
Re
Re
Re
Im
Re
Re
(40)
(41)
(42)
(43)
(44)
(45)
(46)
and
diag
matrices
and
otherwise
otherwise.
are de(47)
(48)
Proof: See Appendix C.
1) Discussion: We make the following observations.
• For a finite and sufficiently large observation interval
and a static multipath channel, we can approximate the
DS-CDMA CRLB expression with the asymptotic CRLB
as
CRLB
AsCRLB
V. NUMERICAL RESULTS
A. Effect of MAI on the Estimation of the Signal Parameters
for a K-User DS-CDMA System
where
the elements of the
fined as
• Because the asymptotic CRLB is not conditioned on the
transmitted symbol sequence (as opposed to the deterministc CRLB), it is much easier to calculate.
• If the MAI can be neglected or regarded as white Gaussian
noise or a single user is assumed, the conditional (deterministic) CRLB derived in [11] and [13] can be used instead of the asymptotic CRLB (40) to provide further insights into the effects of a pulse shaping filter and slowly
time-varying channel.
(49)
• The final expression for the asymptotic DS-CDMA CRLB
is similar to the conditional (deterministic) CRLB expressions derived in both [11, App. A] and [13, App. ]. The
expressions in [11] and [13], however, were derived under
quite different assumptions.
• The asymptotic CRLB for the th user is independent of
in the
the number of asynchronous DS-CDMA users
system.
We use numerical simulations to compare the deterministic
users is exK-user CRLB (38), where the MAI from the
plicitly modeled with the asymptotic CRLB [see (40) and (49)]
and the conditional single-user CRLB given in [11], [13], assuming a single-user transmitting a digital sequence modulated
by a raised cosine pulse with excess bandwidth 0.35 truncated
. The sensor array is an uniform linear array (ULA)
to
or
with half-wavelength element spacing composed of
three dipole sensors. Each user is assigned a different Gold code
. The number of multipath rays per
sequence of length
or 2. Their relative time-delays and bearings
user is
are randomly generated for every Monte Carlo simulation as
(in radians) and
(in
chip periods). The received path fading amplitudes have unit amplitude, and their phases are randomly gen(in radians). The
erated for every Monte Carlo trial as
SNR is calculated as the ratio of the received power of the first
dB.
ray to the noise variance, and the noise variance is
For each experiment, the CRLBs for the SOA, AOA, and TD parameters are averaged over the random phase of the amplitudes,
random bearings, and random time delays using 100 Monte
Carlo simulations.
1) Results: We considered two different experiments. In the
users and
first one (see Figs. 1 and 2), we simulated
plotted the standard deviation of the bearing and time-delay es) as a function of the obsertimates for the first user (i.e,
sensors. In the
vation interval and assuming a ULA of
, 5, 10,
second experiment (see Figs. 3–5), we simulated
15, 20, and 25 users and plotted the quotient of the standard deviation obtained using the K-user deterministic CRLB formula
with the conditional single-user CRLB formula as a function of
,
the number of users . The observation interval was
and the number of antenna elements and multipaths per user
to
and
to
,
was varied from
respectively.
We make the following observations.
• The conditional single-user CRLB gives similar results
as the K-user deterministic CRLB (see Figs. 1 and 2).
The smaller variance of the conditional single-user CRLB
can be attributed to the different assumptions used for
its derivation (e.g., the single user assumption, the pulse
shaping filter, and the assumption that a channel estimate
whose estimation noise is white and Gaussian with varialready exists (see [11] and [13]).
ance
BOTTERON et al.: CRAMER–RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS
869
Fig. 1. Standard deviation for ^ . K = 5 users, P = 3 antennas, Q = 1
path per user.
Fig. 3. Quotient of the standard deviation obtained using the det. K-user CRLB
formula with the cond. one-user CRLB formula. M = 50 symbols, P = 1
antenna, Q = 1 path per user.
Fig. 2. Standard deviation for ^ . K = 5 users, P = 3 antennas, Q = 1
path per user.
Fig. 4. Quotient of the standard deviation obtained using the det. K-user CRLB
formula with the cond. one-user CRLB formula. M = 50 symbols, P = 3
antennas, Q = 1 path per user.
• The increase of the variance when
users are assumed
relative to the variance when a single user is assumed is
about the same for the estimation of the bearing, time,
and amplitude parameters (see Figs. 3–5). Thus, we can
expect that an augmentation of the number of users will
similarly affect a radio location technique using any of
these measurements.
• A larger number of multipaths per user also results in an
increase of the variance for the estimation of the signal
parameters (see Figs. 4 and 5).
• The trends in performance previously observed in the literature as a function of the observation interval, numbers
of users, number of antennas, spreading sequence length,
etc. are also verified in our calculations of the CRLBs.
B. Influence of Specular Multipath on the Radio Location
Accuracy
We use numerical simulations to investigate the influence of
specular multipath on the achievable positioning accuracy for
Fig. 5. Quotient of the standard deviation obtained using the det. K-user CRLB
formula with the cond. one-user CRLB formula. M = 50 symbols, P = 3
antennas, Q = 2 paths per user.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
different radio location methods and environments. Since the effects of the number of DS-CDMA users on the estimation variance have already been discussed in Section V-A, we only consider a single user and use the conditional CRLB for the signal
parameters derived in [13]. Thus, the results we obtain are valid
for both a TDMA or a DS-CDMA system where the MAI can be
modeled as white Gaussian noise. As in Section V-A, we thus
assume a single user transmitting a digital sequence modulated
by a raised cosine pulse with excess bandwidth 0.35 truncated to
. No power control is assumed, the transmitted power
is normalized to 30 dBm, and the received noise variance at
dBm. The number of symbol intervals
the FSs is
and
, respecand code sequence length are
tively. The received power at the th FS is calculated using (10)
dB,
, and
and (11) with
dB (corresponding to an urban area, an FS’s antenna
MHz [16,
height of 30 m, and a carrier frequency of
Tab. III]). We assume that each FS is equipped with a six-dipole
interelement spacing.
uniform circular array with a
1) System Geometry and Distance Normalization: Because
different positioning methods may not depend on the system
geometry the same way, numerical results obtained for a given
system geometry may not be valid for another geometry. This
is particularly a problem when we want to compare different
positioning techniques since the comparison will only be valid
for the specific system geometry considered for the comparison.
In order to extend the comparison for any system geometry’s
size, we normalize all the distances in units. Thus, the results
we obtain can be generalized for any scaling of the system geometry using Proposition 4.
Proposition 4: If the system geometry’s distances are expressed in units, the radio location techniques discussed previously will be proportional to
CRLB
CRLB
CRLB
no power control (p.c.
perfect p.c.
no p.c.
perfect p.c.
no p.c.
perfect p.c.
where is the path loss exponent.
Proof: see Appendix D.
2) Results: The system geometry we consider is a
FSs centered in
five-hexagonal cell geometry with
each cell (see Fig. 6). For each experiment, the CRLB for the
estimation of the MS’s position was calculated for 500 Monte
Carlo trials, where in each trial, a new MS’s position and new
main reflectors positions were generated.
Table II shows the threshold for which the probability that
CRLB
is assuming an uniform distribution of
the MS’s position within cell
and a total of main reflectors uniformly distributed within the five hexagonal cells.
for which the probaTables III and IV show the threshold
is , assuming a uniform distribubility that CRLB
tion of the MS within cell
and a total of main reflectors
and
,
uniformly distributed within a circle of radius
respectively, and centered on the MS’s position.
Fig. 6. Five-hexagonal cells geometry. The Cartesian coordinates (
of the FSs are in
units.
D
b ,b
)
TABLE II
THRESHOLD
FOR WHICH
( CRLB( )
) = ASSUMING
MAIN REFLECTORS UNIFORMLY DISTRIBUTED WITHIN THE HEXAGONAL
CELLS. ( IS THE SEPARATION DISTANCE BETWEEN THE FSS IN
IS THE PULSE PERIOD IN MICROSECONDS)
KILOMETERS, AND
X
D
prob
w X
p
V
T
We make the following observations.
• From the normalization factors in Tables II–IV (see also
Proposition 4), we note that the relative contribution to accuracy of time measurements (i.e., TOA or TD) will not be
reduced as much as the contribution to accuracy of SOA
or AOA measurements when the cells’ size is increased.
As a consequence, we can infer that time measurements
will contribute more to the accuracy of radio location estimators than bearing measurements when the cells’ size
becomes larger than a given threshold. Furthermore, we
note that that this threshold will be function of the pulse’s
and the number of antennas
at the FSs (for
period
more details on the influence of the number of antennas
and pulse’s shaping filter, see [24] and [25]).
• Comparing the results in Tables II–IV, we note that for all
the radio location methods considered, the augmentation
of the number of main reflectors has a larger effect on the
accuracy of radio location estimators when the main reflectors are more closely scattered around the MS, which
is consistent with the observation that the delay spread
BOTTERON et al.: CRAMER–RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS
TABLE III
THRESHOLD X FOR WHICH prob( CRLB( )
X ) = p ASSUMING V
MAIN REFLECTORS UNIFORMLY DISTRIBUTED AROUND THE MS WITHIN A
CIRCLE OF RADIUS D=2. (D IS THE SEPARATION DISTANCE BETWEEN THE
FSS IN KILOMETERS, AND T IS THE PULSE PERIOD IN MICROSECONDS)
w and angle spread of the multipath arrivals at the FSs becomes smaller when the reflectors are more closely scattered around the MS, making the estimation of the signal
parameters more difficult.
• If we only consider the relative contribution to accuracy
of the time measurements with reference to the bearing
measurements and as a function of the number of main
reflectors for the three multipath scenarios we considered
(and regardless of the cells’ size and pulse’s period), we
make the interesting observation that the contribution to
accuracy of the time measurements relative to the bearing
measurements increases with the number of main reflectors (see Fig. 7). This increase is also more pronounced
when the main reflectors are more closely scattered around
the MS. This fact is consistent with the observation that
in the presence of closely time-spaced multipath arrivals,
it will be easier for a detector to detect the direct path’s
time of arrival (which will simply correspond in detecting
the first time of arrival) than detecting the direct path’s
AOA (which will be surrounded with the other multipath
AOAs).
• From the results in Tables II–IV, we note that the measurements that suffer the most from an augmentation of
the number of main reflectors are the SOA measurements.
This is despite the fact that for the derivation of the CRLB
for the SOA estimation, we assumed an unbiased estimator
that has perfect knowledge of the path loss model, which
is a very strong assumption that will not hold well in practice. We also note that the achievable positioning accuracy of the TOA measurements is not as good as for the
TD measurements. This can be easily explained since the
MS’s TOT is assumed to be known for the TD measurements and unknown for the TOA measurements.
VI. CONCLUSION
In this paper, we presented the CRLB for the estimation of
the specular multipath parameters in asynchronous DS-CDMA
systems and the CRLB for the estimation of the MS’s position
871
TABLE IV
THRESHOLD X FOR WHICH prob( CRLB( )
X ) = p ASSUMING V
MAIN REFLECTORS UNIFORMLY DISTRIBUTED AROUND THE MS WITHIN A
CIRCLE OF RADIUS D=5. (D IS THE SEPARATION DISTANCE BETWEEN THE
FSS IN KILOMETERS, AND T IS THE PULSE PERIOD IN MICROSECONDS)
w Fig. 7. Normalized quotient of the Threshold X for the AOA-based over the
TD-based positioning method as a function of the number of main reflectors for
a probability p = 0:67 and p = 0:90.
valid for multilateral, unilateral, and even composite radio location techniques. Because the CRLB for the MS’s position can be
expressed as a function of the CRLB for the estimation of the
signal parameters, it can be easily derived for many different
system models and provide interesting insights into the physics
of the localization problem. It is thus a valuable tool to compare different positioning methods, to assess the effects of different environment or system design parameters, or to evaluate
if a given cellular system can fulfill positioning requirements,
such as for the E-911 services.
One main limitation of our approach comes from the assumption that the CRLB matrices for the estimation of the
signal parameters at the FSs must exist. A second limitation
comes from the unbiasedness assumption, i.e., the CRLB for
the positions can be used as a benchmark or optimality criterion
for any unbiased radio location estimator. However, if the
estimator’s bias in not negligible compared with its variance,
then the CRLB will not provide a meaningful bound (note that
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
in [24] and [25], we considered another approach to relieve
these two limitations).
For the numerical simulations we considered, the MAI in a
K-user asynchronous DS-CDMA system was shown to affect
similarly the estimation of the amplitude, time, and delay parameters, which means that the contribution to accuracy of radio
location estimators using any of these measurements would be
similarly affected with the augmentation of the number of users
. We also investigated, using numerical results, the influence
of specular multipath on the contribution to accuracy of different
radio location estimators for a five-hexagonal cell geometry and
three multipath scenarios. In the first scenario, we assumed an
uniform distribution of the main reflectors within every cell, and
in the second and third scenarios, we assumed that the main reflectors were scattered around the MS within a circle of radius
and
, respectively, where denotes the distance between two FSs. As expected, the main reflectors had a greater
nuisance effect on the achievable accuracy in the last two scenarios. However, we also noticed that the time measurements
were not as much affected by an augmentation of the number of
main reflectors as were the bearing measurements. Thus, for a
radio location system using both time and delay measurements,
the contribution to accuracy of the time measurements relative
to the bearing measurements will augment with the number of
reflectors. That contribution will also be augmented when the
main reflectors are more closely scattered around the MS or
when the cells’ size is increased.
APPENDIX A
PROOF OF PROPOSITION 1
Based on assumption A5, i.e., assuming that the position and
noise parameters are independent, we rearrange the elements
,
of the source-location parameter vector as
where only contains the parameters for the noise covariance
and the parameters for the location parameters
matrix
in
. Thus, we can write, in block partition form, the FIM
expression given in (2) as
in (3) and using assumption A5, we rewrite
of the matrix
(4) as
(53)
where
diag CRLB
CRLB
(54)
diag CRLB
CRLB
(55)
(56)
(57)
In order to prove Proposition 1, we must prove that (4) is
correct. Note that this is equivalent to proving that (51) and
and
(52) can be expressed as
, respectively.
The first identity is easy to prove, since the noise parameare assumed independent of the signal paters in
rameters and statistically independent between different FSs.
Without loss of generality, we assume that the parameters in
are ordered as in
, i.e.,
. Thus,
can be
[see (54)], which is the same as
written as
, since
for
[see (56)].
in
To prove the second identity, we note that the vector
(52) can also be expressed as a function of the parameters in ,
. Thus, by applying the chain rule of differentiation
i.e., as
[26, Th. 9.15], we can write the derivatives contained in (52) as
(58)
(50)
where
tr
(51)
Re
(52)
Since the signal parameter vector must also contain the
same noise parameters as in (under the assumption that the
do not affect the noise parameters),
functions
we also rearrange the elements of
as
, where contains the noise’s parameters, and contains the signal parameters (plus any other
nuisance parameter, such as an MS’s TOT if it is assumed unknown at the FSs). Rearranging in a similar order the elements
where
, and denotes
the total number of elements in . Inserting the last equation
in (52) and noting that the vectors and only contain real
parameters, we obtain
Re
Re
Since
, we also have
. Thus, using
the fact that the noise at the FSs is Gaussian distributed, the
above expression can also be expressed in terms of the FIM
BOTTERON et al.: CRAMER–RAO BOUNDS FOR THE ESTIMATION OF MULTIPATH PARAMETERS
expression for the estimation of the signal parameters contained
in , i.e., as
873
In order to obtain a simple expression for the CRLB for
we rearrange the elements in the vector as
,
(63)
Re
(59)
Because the noise is also assumed statistically independent between different FSs, we can write
as
diag
. Finally, by noting that
[see (57)], we obtain
diag
, so that we can write the transformain block diagonal form as
where
tion matrix
(64)
are defined in (8).
where the elements of
By also rearranging the rows and the columns of the matrix
according to , we can express the FIM for [given as
the inverse of (62)] as
(60)
which completes the proof.
APPENDIX B
PROOF OF PROPOSITION 2
(65)
Since by assumption the position estimator only uses the
to estimate the posisubset of signal parameters
, the CRLB for
tion parameters contained in
can be obtained by assuming that the remaining parameters in
, which are not included in , are all nuisance parameters,
i.e., by assuming that they provide no information to estimate
and are thus independent of the parameters contained in .
Let us denote this subset of nuisance parameters as
, where
. Because these nuisance parameters are asno longer contains all the
sumed to be independent of ,
necessary parameters to completely parameterize the p.d.f. for
the observation vector . However, if we define a vector as
The CRLB for can now be obtained by taking the inverse of
the Schur complement corresponding to of (65), i.e., as
CRLB
(66)
(61)
then can be used to parameterize the p.d.f. for the observation
vector .5
Based on this new parametrization of the p.d.f for , we note
that Proposition 1 still holds and can thus be used to find the
CRLB matrix for (which also contains the CRLB for the parameters in ) as
CRLB
(62)
where
denotes the right lower block of a 2
2 block
partition matrix, and the last equality was obtained by noting
is the CRLB matrix for the estimation of
that
.
the signal parameters contained in
APPENDIX C
PROOF OF PROPOSITION 3
For a static channel, we have
write the asymptotic FIM for
where the elements of the matrix
.
Im
as
As
example, if we consider a location system that only uses the bearing parameters in to estimate the source-location parameters contained in , then will contain the remaining signal parameters in that are not used to estimate
(such as the time- and the strength-of-arrival parameters). Since these parameters are assumed independent of the location parameters in , they cannot be
expressed as a function of and, therefore, must also be added to so that completely parameterizes the p.d.f. for the observation vector .
w
w
Re
are defined as
5For
w
. Thus, we can
w
x
Re
Im
Re
Im
Re
Im
Re
Im
Re
Re
Im
Re
Re
Im
Re
Re
(67)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
where
Re
and
have the same
By noting that the matrices
, and proceeding in the same manner, it is easy
form as
to verify that the th block matrices of the other elements of
the asymptotic FIM can be written as in (42)–(46). Finally,
by noting that all the elements of the asymptotic FIM given
by (67) are block diagonal, we conclude that by rearranging
, we can write the asymptotic CRLB for
the elements in
the estimation of the signal parameters of the
users as
diag AsCRLB
AsCRLB
,
AsCRLB
where AsCRLB
is given in (40).
Im
Im
Re
Re
Im
Re
Im
APPENDIX D
PROOF OF PROPOSITION 4
Re
Re
Re
We start with the evaluation of
. From the block vertical
form of
[see (32)] and the block diagonal form of
, we rewrite
as
..
.
..
.
..
.
Thus, the
th block matrix of
is
. Because the transmitted bits
are assumed i.i.d. with
, it is easy to show that the
th element of
can be written as
First, we note from the FIM expression (23) that the CRLB
for the estimation of the bearing and time parameters is inversely
proportional to the square of the received power, while it is
independent of the received power for the estimation of the path
in (23) is
amplitudes [this is easily verified by noting that
, whereas
and are proportional
independent of
to
. Thus, by applying the formula for the inverse of a
,
partitioned matrix, we obtain that CRLB
Im
Re
Im
whereas CRLB Re
is independent of
. Finally, using (38), we obtain that
is also independent of the received powers].
CRLB
Second, we note from the path loss formulas (10)–(12) that
the received power for any multipath at any FS will be proportional to
only in the absence of power control, whereas it
will be independent of , assuming perfect power control.
Third, we note from (13)–(16) that the transformation matrices for the time measurements are independent of , whereas
the transformation matrix for the bearing measurements is inversely proportional to , and the one for the amplitude measurements is also inversely proportional to , assuming perfect
power control, and inversely proportional to
in the absence of power control (i.e., with fixed transmitted power).
Thus, inserting the above proportionalities in the CRLB formula for the estimation of the MS’s position [see (7)] concludes
the proof.
ACKNOWLEDGMENT
where
The authors are grateful to the anonymous reviewers for their
valuable comments.
Consequently,
is a -block diagonal matrix. By noting that the
th element of
is
and the
th element of
is
, we
as
can rewrite the th block matrix of
(68)
where
.
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875
Cyril Botteron (M’99) was born in Switzerland in
1969. He received the Dipl.-Ing. degree from the University of Applied Sciences, Le Locle, Switzerland,
in 1991 and the Ph.D. degree in electrical engineering
from the University of Calgary, Calgary, AB, Canada,
in 2003.
In 2000, he participated in the high-level design of
a network-based positioning system with Cell-Loc,
Inc., Calgary. Currently, he is a research scientist and
group leader with the Institute of Microtechnology,
University of Neuchâtel, Neuchâtel, Switzerland.
His research interests include statistical and discrete-time signal processing
techniques, RF systems design, and applications to wireless communications,
including ultra-wideband radio technology and global navigation satellite
systems.
Anders Høst-Madsen (M’95–SM’02) was born in
Denmark in 1966. He received the M.Sc. degree in
electrical engineering in 1990 and the Ph.D. degree
in mathematics in 1993, both from the Technical University of Denmark, Lyngby.
From 1993 to 1996, he was with Dantec Measurement Technology A/S, Copenhagen, Denmark,
from 1996 to 1998, he was an assistant professor
at Kwangju Institute of Science and Technology,
Kwangju, Korea, and from 1998 to 2000, he was an
assistant professor at the Department of Electrical
and Computer Engineering, University of Calgary, Calgary, AB, Canada,
and a staff scientist at TRLabs, Calgary. In 2001, he joined the Department
of Electrical Engineering, University of Hawaii at Manoa, Honolulu, as an
assistant professor. He was also a visitor at the Department of Mathematics,
University of California, Berkeley, in 1992. His research interests are in
statistical signal processing, information theory, and wireless communications,
including multiuser detection, equalization, and ad-hoc networks.
Dr. Høst-Madsen currently serves as Editor for multiuser communications for
the IEEE TRANSACTIONS ON COMMUNICATIONS.
Michel Fattouche (M’82) received the M.Sc. and
Ph.D. degrees in electrical engineering from the
University of Toronto, Toronto, ON, Canada, in
1982 and 1986, respectively.
He is chief technical officer of Cell-Loc, Calgary,
AB, Canada, and a tenured professor with the Department of Electrical and Computer Engineering,
University of Calgary, where he has taught and
conducted research since 1986. He is currently on a
leave of absence to allow him more time to dedicate
to technology development at Cell-Loc. He has been
affiliated with TR Labs, Calgary, since 1989, where he is currently an adjunct
professor, and he is on the board of directors of PsiNaptic Communications,
Calgary. He currently holds nine U.S. patents and has ten additional patents
pending. He has been published in a number of well-respected publications in
the field of digital wireless communications and has spoken at several industry
events.
Dr. Fattouche is a member of the Association of Professional Engineers, Geologists, and Geophysicists of Alberta. He was named Prairies Entrepreneur of
the Year 2000 for Communications and Technology as part of Ernst and Young’s
Entrepreneur of the Year (EOY) Program.