Kalman Filter

So far we have reviewed different approaches for dealing with the estimation of fixed

parameters. In tracking, however, parameters are assumed to evolve in time. To clearly

distinguish between fixed parameters and those that evolve in time, the latter ones are

often referred to as a state. The task in tracking is then to infer the state s[n] from

the measurements y[n] taken at time-step n. Obviously, we could implement tracking

by estimating states at every time-step individually, using some of the earlier discussed

methods. However, if an estimate ˆs[n ≠ 1] is available at time-step n, this estimate can

often be used as prior information for the estimation of ˆs[n]. When tracking the location

of a pedestrian, as an example, the change in location is clearly determined by the

pedestrian’s velocity. Thus, if we have obtained an estimate for the pedestrian’s location

and velocity at time-step n ≠ 1, we can already estimate the pedestrian’s location at

time-step n without taking any measurements. In order to exploit such knowledge, we

obviously need a good model for the evolution of the state. In Kalman filtering this

model is often referred to as the state transition and is given by

s[n] = F s[n ≠ 1] + w[n]

(2.21)

The state transition (2.21) consists of the the state transition matrix F and the zero-

mean driving noise w[n] with covariance matrix E[w[n]w

T

[m]] = ”

m≠n

R[n] where ”

i≠j

is the Kronecker delta function. Note that (2.21) is a linear state transition as required

in conventional Kalman filtering. Extensions for nonlinear models exist [91, p. 407].

However, in this thesis all state transitions will be of the above form.

In order to infer the state from the measurements, we have to find a model for the

measurement equation

y[n] = h(s[n]) + u[n]

(2.22)

where u[n] is the zero-mean measurement noise with a covariance equal to E[u[n]u

T

_{[m]] =}

”

m≠n

Q[n]. Note that we do not state the measurement equation in its most general

form (see [91, p. 407]). However, in contrast to the linear state transition we now

enable nonlinear relationships between the state and the measurements via the vector

function h. We allow this nonlinear relationship since all the measurements considered

in this thesis depend nonlinearly on the location that we track.

In cases where both the state transition as well as the measurement equations are

linear, i.e., y[n] = H[n]s[n] + u[n] it is possible to obtain state estimates using the

iterative Kalman filter. The Kalman filter is optimal in the MMSE sense if w[n] and

u[n] are jointly Gaussian. And if the Gaussian assumption does not hold, the Kalman

filter is still the optimal LMMSE estimator. However, as noted above the measurement

equations are often nonlinear. Thus, in practice we often use the so-called extended

Kalman filter (EKF) instead of the conventional Kalman filter. The main idea of the

EKF is to linearize the nonlinear parts of the models using the current state estimate.

As such, we generally cannot make any claims regarding optimality.

For each time-step n, the EKF consists of an a priori and an a posteriori estimation

stage. In the a priori stage, we estimate the state ˆs

≠

_{[n] and covariance of the state}

estimate P

≠

_{[n] at time-step n using all measurements up to but excluding y[n]. The}

estimation is given by

ˆs

≠

_{[n] = Fˆs}

+

_{[n ≠ 1]}

_(2.23)

P

≠

_{[n] = FP}

+

_{[n ≠ 1]F}

T

_{+ R[n].}

_(2.24)

In the a posteriori estimation stage, we estimate the state ˆs

+

_{[n] and covariance of the}

state estimate at time-step n using all measurements up to and including y[n]. The

2.2 Bayesian Estimation Theory

calculation in the a posteriori estimation stage is according to

K[n] = P

≠

_[n]H[n]

T

_(H[n]P

≠

_[n]H

T

_{[n] + Q[n])}

≠1

_(2.25)

ˆs

+

_{[n] = ˆs}

≠

_{[n] + K[n]}#y[n] ≠ h(ˆs

≠

_[n])$

_(2.26)

P

+

_{[n] = (I ≠ K[n]H[n])P}

≠

_[n]

_(2.27)

with the Jacobian matrix H[n] =

ˆh[n]

ˆs[n]

evaluated at ˆs

≠

[n]. Interestingly, the EKF

provides the possibility of state prediction. By executing the a priori estimation stage

N

times without the a posteriori stage, we are able to obtain a prediction ˆs

p

[n + N]

using only measurements up to time-step n.

Note that the EKF is not the only approach for nonlinear filtering. An overview

of other popular approaches such as unscented Kalman filters or particle filters can be

found, for example, in [91]. Note, moreover, that the CRB discussed in Section 2.1.2 does

not lower-bound the performance of Bayesian estimators such as the EKF. Sometimes,

we can learn certain general properties of the estimation problem by deriving the classical

CRB and benefit from the knowledge of these properties also in Bayesian estimation

(see Section 5.1 for an example). However, if we are interested in a lower performance

bound for, e.g., an EKF we have to refer to the so-called posterior CRB [101].

CHAPTER 3

Localization and Tracking in

Wireless Networks

T

he aim of localization or positioning is to estimate the location ¸ = [x, y]

T

of a

TN. In this thesis, we generally assume that this task is achieved by a set of K

collaborating ONs in a wireless network. In accordance with the discussion in Section 1.1,

we moreover assume that each of the ONs is equipped with some sort of a directional

antenna as depicted in Figure 3.1. In order to localize the TN, each ON k, k = 1 . . . K

makes a measurement

k

based on the TN signal and communicates this measurement

to the fusion center, where the measurements from all K ONs are combined into a TN

location estimate. In contrast to localization, we talk about location tracking when we

estimate the evolution of the TN location over time. In the following discussion, we

will mostly discuss localization. However, the majority of the topics discussed in this

chapter also apply to location tracking.

This chapter is organized as follows. In Section 3.1 we discuss important properties

of wireless networks, the ON devices, and the TN as well as their implications for

localization and location tracking. Thereafter, in Section 3.2 we discuss measurements

commonly used for localization in detail. Finally, in Section 3.3 we summarize the

Stansfield fusion algorithm.

3.1 Properties of Wireless Networks and Implications

for Localization and Location Tracking

In a practical wireless network, the choice of measurements for localization and location

tracking as well as the fusion mechanisms are confined by the network properties, the

ON devices and the nature of the TN. For time-based measurements the synchronization

Fig. 3.1. Considered localization system where the observing nodes (ONs) collaborate to estimate or

track the location of the target node (TN). ONs are assumed to be equipped with directional antennas.

of the network is of particular importance. In the best-case scenario, the clocks of the

ONs are perfectly synchronized with the TN clock. In that case the distance d

k

can be

extracted directly from the ToA estimates (see Section 3.2.3). However, already very

small synchronization errors significantly increase the ToA estimation error, which is

related to the relative clock offsets via the speed of light, c ¥ 3 ◊ 10

8

_{m/s. In many}

networks, a sufficiently accurate synchronization with the TN is not assumed to be

given. Thus, location estimates are often obtained by fusing time difference of arrival

(TDoA) estimates instead [5, 46]. However, utilizing TDoA estimates for localization

still requires synchronization of the ONs. Without any synchronization in the network,

the propagation time can be exploited using the so-called round-trip time of arrival

(RToA) [29,117]. In order to measure the RToA, an ON sends a packet to the TN. Once

the TN receives the packet, it responds by sending the packet back to the ON [117]. The

time between transmission and reception at the ON is then approximately equal to 2 d

k

/c.

However, RToA estimation obviously requires dedicated two-way localization signaling,

which may not be feasible in networks that are primarily designed for communication

purposes, such as the 5G network considered in [57] that also forms the basis for the

localization system proposed in [P7].

Apart from synchronization, the nature of the TN has important implications for

the applicability of time-based measurements. In PU localization, spectrum policy

enforcement, or military applications, the TN is generally non-cooperative [55,79,105–

107,109]. This means that both ToA and RToA estimates are not available for localization.

If, in addition, the network is unable to synchronize the ONs, TDoA estimates are also

unavailable. In these cases, localization is generally based on RSS [64, 109] or DoA

estimates [79, 105]. A problem arises when multiple non-cooperative TNs have to be

3.1 Properties of Wireless Networks and Implications for Localization and Location Tracking

localized simultaneously. Distinguishing between the TNs with RSS measurements only

is generally not possible. Thus, localization of multiple non-cooperative TNs is often

performed using DoA estimates [30,105]. However, in the fusion process DoAs estimates

have to be associated to the TNs. This is commonly done using Bayesian methods [10]

or by choosing the association that minimizes a given optimization criterion [30,105].

In general, the fusion process can be implemented in a central or distributed fashion.

In a classical cellular network architecture, a central fusion entity is readily available.

However, for CR networks, as an example, it is often assumed that fusion has to be

distributed among the CR devices [109]. Many centralized fusion algorithms such as the

Kalman filter, weighted centroid localization [14] and the Stansfield algorithm [92] exist

also in distributed versions [82,106,109] that achieve equal [82] or close to equal [106,109]

performance as their centralized counterparts.

The antenna type employed at individual ONs influences the way DoA estimates can

be obtained. Naturally, ONs need to be equipped with some sort of directional antenna in

order to make DoA estimation possible in the first place [117]. Most flexibility is achieved

when ONs are equipped with digitally controlled antenna arrays (DCAAs), which enables

array processing techniques such as the MUSIC DoA estimator [88]. However, DoA

estimators for other directional antennas exist as well. Later in Chapter 4, for example,

we develop DoA estimators for the group of sectorized antennas. In fact these estimators

could also be used with DCAAs if a less complex DoA estimation is desired. Low

complexity might be required if the ONs are mobile devices with limited battery and/or

computational resources. Such a scenario arises, e.g., in CR networks where the SU

devices are involved in the PU localization [79, 105–107, 109] or in networks, where

the UNs assist in the localization of another UN by means of device-to-device (D2D)

communication [27]. In some cases it may even be required to select only a subset of all

available ONs in order to reduce the overall energy consumption in the network. The

selection of ONs employing DoA estimation, as an example, has been studied in [53,55],

taking into account that certain ONs-TN geometries produce better location estimates

than others.

Note that the above discussion serves as an overview of topics that have been

important in the making of this thesis. Therefore, it is not complete and the referenced

literature by no means exhaustive. In general, it is also important to note that the

discussed properties and implications are interdependent. In a network where the ONs

are not synchronized, it could be possible to obtain synchronization by estimating the

clock offsets of the ONs. However, this would probably increase the burden on the ON

devices, which may again be prohibitive for resource limited ONs. Similarly, it was

shown that it is possible to employ array processing techniques with certain sectorized

antennas by letting the TN repetitively send the same signal [75,96,103]. Again, however,

this is only an option if dedicated localization signaling is possible, which also always

implies a cooperative TN. A simplified summary of the discussion in this section can be

found in Table 3.1.

In document Directional Antenna System-Based DoA/RSS Estimation, Localization and Tracking in Future Wireless Networks: Algorithms and Performance Analysis (Page 30-37)