Sistemas de representación

For notational convenience the rv in the subscript of the density fV(v) is dropped,

i.e., fV(v) = f (v). A capital letter in the argument of f (·) denotes an observed time

sequence, i.e., Yk_{={y(i), i = 1, . . . , k}.}

2.2.1.1 Nonlinear System Model

Consider that the UE is moving on a 2D plane surrounded by M FTs. The parameters of interest are the position and velocity of the UE at each time step k contained in the state vector x(k) = [x(k) y(k) ˙x(k) ˙y(k)]T

, where ˙x(k) and ˙y(k) denote the derivatives of x(k) and y(k) with respect to time, corresponding to the velocity in x- and y- direction. The following discrete-time stochastic model describes the movement of the UE, i.e.,

x(k) = a(x(k_{− 1)) + ω(k − 1), k = 1, 2, . . . , K,} (2.32) where a(_{·) is a known linear or nonlinear function describing the movement of the} UE from time step k − 1 to k, and vector ω(k − 1) is white Gaussian process noise with covariance matrix Q(k), k = 1, . . . , K, describing the uncertainty one has about the motion model. Thus, x(k) is a first-order Markov process meaning that x(k) only depends on x(k − 1) and ω(k − 1). Consequently, the condition f (x(k)_{|x(k − 1), x(k − 2), . . . , x(0)) = f(x(k)|x(k − 1)) holds, where f(x(k)|x(k − 1))} is the pdf of the process noise. Various motion models with different a(·) to describe different dynamics exist in the literature [85].

Since the UE is moving, the nonlinear model from (2.1) becomes time-dependent, i.e., y(k) = h(x(k)) + v(k), k = 1, 2, . . . , K. (2.33) where h(_{·) describes the nonlinear relationship between the position of the UE} and the FTs as explained in Section 2.1.1. The measurement noise sequence v(k) = [v1(k), v2(k), . . . , vM(k)]T is assumed white over time and FTs, and each el-

ement vm(k) is distributed according to (2.2). The measurement covariance matrix

R(k) = E{(v(k) − E{v(k)})(v(k) − E{v(k)})T

} can be time-variant. For the signal parameters TOA, AOA and RSS and a fixed time step k, given that v(k) is iid among FTs, R(k) = σ2

VIM = (σ2G+ εση2)IM, where IM is the M × M identity matrix. Note

that R(k) is not a diagonal matrix for TDOA positioning. This is because a reference FT is chosen and the time differences to all FTs are calculated leading to correlations among the different pairs. Note that v(k) and ω(k) are mutually independent for all k.

π11 π12 π22

π21

V _{∼ N (0, σ}2

G) V ∼ H

LOS NLOS

Figure 2.5: Markov chain for modeling LOS/NLOS occurrences.

2.2.1.2 Jump-Markov Nonlinear Model

Unless the UE is maneuvering quickly, which is beyond the scope of this work, the whiteness assumption for the process noise in (2.32) is reasonable. In contrast, the assumption for a white process v(k) (2.33) may not hold in reality since the wireless channel for moving objects highly depends on the environments. Thus, the perturba- tions due to the NLOS effect can undergo switching over time, meaning at time step k− 1 an FT can be in LOS and at time step k it changes to NLOS or vice versa. Imag- ine that a UE is driving alongside an FT shadowed by a high-rise building. During this time, consecutive measurements are in NLOS condition. When the building is passed, the condition of the channel abruptly changes to LOS. The opposite, i.e., consecutive measurements are in LOS condition, is true for a UE traveling on a flat plane where no obstacles prevent the signals to arrive at the FT via the direct path. In this case a sudden change of the channel to NLOS occurs when the UE passes an obstacle. To model these time dependencies and sudden changes, for each FT we use a first-order time-homogeneous Markov chain [25], depicted in Figure 2.5, consisting of r = 2 states. Note that a process_{M(k) is called Markov chain (MC) when the future of the process} only depends on the present and not on the past. In the following M(k) is called the mode variable, assumed to be among the r possible modes, _{M(k) ∈ {M}j}rj=1, where

M1 is assigned to the event “LOS” and M2 is assigned to the event “NLOS”. The

transition probabilities πij ≥ 0 denote the conditional probability for changing to state

Mj at time step k given that state Mi is in effect at k− 1,

and the transition probability matrix (TPM) of the MC depicted in Figure 2.5 is Π =  π11 π12 π21 π22  , (2.35)

where P2_j=1πij = 1, i = 1, 2. Given any initial probabilities, meaning the proba-

bilities that the MC at time step k = 1 is in LOS, i.e., Pr_{{M(1) = M}1} = π1 and

Pr_{{M(1) = M2} = 1 − π1}, where 0≤ π1 ≤ 1, then the MC converges to a stationary

distribution when k increases [25], lim k→∞Pr{M(k) = M1} = π21 π12+ π21 = 1− ε, (2.36) lim k→∞Pr{M(k) = M2} = π12 π12+ π21 = ε. (2.37)

However, considering M FTs, there are r = 2M _{different noise constellations or modes,}

meaning the MC consists of 2M _{different states. Assuming that the LOS/NLOS transi-}

tions among different FTs are independent we can calculate the TPM for the augmented MC (meaning the whole system with M FTs) using the Kronecker product (_{⊗), i.e.,}

T = Π1 ⊗ Π2 ⊗, . . . , ⊗ ΠM, (2.38)

where Πm is the TPM (2.35) of the m-th FT and T with dimensions r× r is the TPM

of the augmented MC. The elements of T are the transitions probabilities between the r different states. Since (2.1) is now time-dependent, the time-varying LOS/NLOS occurrences are modeled as a jump-nonlinear system [5] with discrete mode variable M(k) ∈ {Mj}r=2

M

j=1 describing the mode the system is in at time step k,

y(k) = h(x(k)) + v(k,_M(k)) k = 1, 2, . . . , K, (2.39) Such systems are called hybrid because they consist of continuous components, such as the state vector x(k), and the discrete mode variable_{M(k). Note that, v(k, M(k))} and ω(k) in (2.32) and (2.39) are assumed mutually independent. To model maneuver, the mode variable _{M(k) can also be incorporated into the process noise in (2.32).} Here, we only assume v(k,_{M(k)) to be mode-dependent to model NLOS effects with} R(k) = E_{{(v(k, M(k)) − E{v(k, M(k))})(v(k, M(k)) − E{v(k, M(k))})}T

}. Thus, the measurement covariance matrix is strongly time-dependent and changes between consecutive time steps due to LOS/NLOS switching according to the augmented MC in (2.38). Note that for TOA, AOA, RSS the diagonal elements of R(k) at a fixed time step are equal either to σ2

G or to σ2G+ ση2 and do not depend on ε anymore. As

in Section 2.2.1.1 the structure of R(k) is non-diagonal for TDOA. Note also that the process noise covariance Q(k) is assumed as in Section 2.2.1.1. The algorithms for tracking a UE, developed in Chapter 4 and Chapter 5, are tested under both, the nonlinear measurement model in (2.33) and the jump-nonlinear model in (2.39).