Signal Level Mitigation

Empirical Estimation of Cross-Measurement Correlations

This technique relies on the intuition that the knowledge of cross-correlations between the components of the measurement vector provides relevant information to CLoc [103].

Recalling that, although the x-to-y correlation in GNSS position is commonly assumed to be null, the cross-correlations between links’ fading measurements are accounted in the 4-D shadowing map and can be determined. More particularly, an “ego” vehicle can infer from its “ego” position and the constellation of its “virtual anchors” the correlations between links’ fading measurements. From the aforementioned 4-D correlated shadowing model, we therefore derive the cross-correlation between two separate links a= (i → j) and b= (l→m) as follows:

where kx_i−x_lk and kx_j −x_mk are the Euclidean distances between the transmitters i and l and between the receivers j and m, respectively.

For illustration, we consider a simplified example where the “ego” car i moving at speedvi collects three asynchronous RSSI readings with respect to the three neighbors 1, 2, and 3 during the time interval ∆T (e.g., every 100 ms or equivalently, at the fusion rate of 10 Hz). The covariance matrix for the shadowing experienced over these three links is thus inferred from (4.4) as

with

R_Sh(j, l→i) =σ_Sh² exp

−kx_j−xlk+vi|t_j−tl| d^Sh_cor log 2

, j, l∈ {1,2,3}, (4.6)

wheret_j and t_l represent the time instants at which vehicleireceives the CAMs from its neighbors j andl, respectively.

Note that (4.6) is deduced after applying (4.4) to a pair of links that has a common end point (i.e., “ego” vehiclei). As vehicleicollects data while moving, cross-link correlation depends on the traveling distance between two corresponding CAMs. Hence, this distance varies from one pair of links to the others. In practice, the true positions (e.g., xj, xl

in (4.4)) cannot be perfectly known. Accordingly, a possible and reasonable approximation Rb_Sh(j, l → i), j, l ∈ {1,2,3} leading to Rb_Sh(1,2,3 → i) can be estimated as a function of the estimated positionsxb_j,xb_l, j, l ∈ {1,2,3}, which are included in/derived from the received CAM payloads. In practice, when the “ego” vehicle has more reference neighbors, the generalization is straightforward.

Differential Measurements

In the literature, there exists a couple of techniques to deal with correlated/colored ob-servation noise. One first approach is to augment the state with the obob-servation noise components [86, 101]. However, this causes a singular measurement noise covariance, which often results in numerical problems [86]. Hence, we concentrate in our work on the second option, referred to as differential measurement (DM). As suggested by its name, the key idea is to whiten the noise by subtracting the correlated part. This problem is solved by building a noise prediction model (from its correlation properties). Being both characterized by the exponential ACF, GNSS residual error and shadowing can be pre-dicted by a Gauss–Markov model. In addition, the most dominant mobility pattern in the vehicular context is platooning-like when vehicles move in groups (coordinated or not).

Accordingly, their velocities become highly correlated and thus, the memory levels in the prediction model are almost time-invariant in first approximation². For the GNSSx- and

2The technique is not limited to highly correlated mobility. In a general case, the memory levels become time-variant i.e., depending on the last known speeds of the participants, leading to prediction noises that are statistically independent but not identically distributed (i.e., varying standard deviation).

y-residual errorsn^x_i,k and n^y_i,k respectively, this yields

n^x_i,k =λ^x_GNSSn^x_i,k−1+en^x_i,k, n^y_i,k =λ^y_GNSSn^y_i,k−1+ne^y_i,k, (4.7)

and for the shadow fading of the link (j→i), denoted by sj→i,k, this leads to

sj→i,k =λ_Shsj→i,k−1+esj→i,k, (4.8)

where en^x_i,k, en^y_i,k, and esj→i,k are zero mean white Gaussian processes with small variances of (1−(λ^x_GNSS)²)(σ_GNSS^x )², (1−(λ^y_GNSS)²)(σ_GNSS^y )², and (1−λ²_Sh)σ²_Sh, respectively.

The memory levels are λ^x_GNSS ≈ exp (−v_i∆T /d^x_cor), λ^y_GNSS ≈ exp (−v_i∆T /d^ycor), and λ_Sh ≈ exp −(v_i+v_j)∆T /d^Sh_cor

≈ exp −2v_i∆T /d^Sh_cor3

, where ∆T is the measurement sampling period, v_j andv_i the asymptotic mean speeds of the Txj and the Rx i, respec-tively. In the time interval ∆T till the next fusion time k, the “ego” car i communicates with its set N_→i,k of “virtual anchors” whose cardinality |N_→i,k| is denoted by ¯N¯_i,k for simpler notations. Hence, the prediction model in the vector form is

n_i,k =λni,k−1+ne_i,k, (4.9)

whereλ= diag(λ^x_GNSS, λ^y_GNSS, . . . , λSh, . . .),λ:R^N¯¯i,k+2→R^N¯¯i,k+2 represents the diagonal memory matrix, ni,k = (n^x_i,k, n^y_i,k, . . . , sj→i,k, . . .)^† ∈ R^N¯¯i,k+2 represents the observation noise vector, and finally, en_i,k = (en^x_i,k,en^y_i,k, . . . ,sej→i,k, . . .)^†∈R^N¯¯i,k+2 is the whitened noise vector.

Now, the so-called auxiliary measurementez_i,k can be expressed as

ez_i,k =z_i,k −λzi,k−1 =h(Xe _i,k,XS→i,k) +ne_i,k (4.10) with

h(Xe _i,k,XS→i,k) =h(X_i,k,XS→i,k)−λh(Xi,k−1,XS_k→i,k−1) and

nei,k =ni,k−λni,k−1,

3We consider here the fusion/filter rate equal to the GNSS rate i.e., 1/∆T, therefore, only vehicles that send CAMs at this rate (or higher) can become “virtual anchors”. If so, the time interval between two consecutive received CAMs/RSSI readings is more or less ∆T due to random CAM generation time and/or congestion control.

where X_i,k ∈R^nx, X_S→i,k ∈ R^N¯¯i,k×nx are the state vector of “ego” vehicle i and the ag-gregated state vector of its cooperative neighbors as “virtual anchors” (i.e., the setN_→i,k) respectively,n_x the dimension of the state vector X_i,k,z_i,k = (x_i,k, y_i,k, . . . , z_j→i,k^RSSI, . . .)^†∈ R^N¯¯i,k+2 the aggregated measurement vector, eh : R^nx ×R^N¯¯i,k×nx → R^N¯¯i,k+2 the corre-sponding model for the new measurement vector ez_i,k ∈ R

¯¯

Ni,k+2, and en_i,k ∈ R

¯¯

Ni,k+2 the prediction noise vector, which is assumed white with a diagonal covariance matrix but cross-correlated with the process noise [86, 101], although this cross-correlation can be neglected at the price of marginal accuracy degradation [101].

Accordingly, our new equivalent observation model can now be written in the same form as (4.10). Note that contrarily to our proposal, the initial differential measurement technique relies on a new measurementez_i,k =z_i,k+1−λz_i,k, which uses the future mea-surementz_i,k+1. This technique is somehow equivalent to 1-lag smoothing [86], thus likely yielding better accuracy gains. Nevertheless, it is inappropriate for real-time tracking in high-mobility contexts such as VANETs.

In addition, in realistic settings, the use of random CAM transmissions introduces specific challenges that should be accounted carefully. Even in case of periodic CAMs, the transmissions are still random due to a so-called CAM generation time between the instant when CAM generation is triggered and the instant when the CAM is delivered to the networking transport layer [19], as illustrated in Figure 4.4. Assume that the CAMs are triggered right after estimating the position, it is possible that the CAM is transmitted and thus received too late with respect to the “ego” estimation time, causing 1) a lack of up-to-date CAMs (e.g., time window k−1 in Figure 4.4) and 2) redundant CAMs afterwards (e.g., time window k in Figure 4.4). In the former subcase, the solution is to simply exclude this neighbor j from the list of “virtual anchors” since there is no RSSI measurement with respect toj available at the estimation time (i.e.,ti,k−1). In the latter subcase, it is reasonable to retain the latest CAM and to drop the oldest CAMs (e.g., the late CAM in Figure 4.4). We observe that this scenario usually occurs as a result of late CAMs. Since there was no observation associated with j at time ti,k−1, the DM can not be performed at time t_i,k. In other words, a late CAM can prevent its transmitter from becoming a “virtual anchor” up to two consecutive “ego” estimates when adopting the DM technique.

𝑿𝑗,𝑘−2 global window 𝑘 − 1 global window 𝑘 global window 𝑘 − 2

CAM gen. time CAM gen. time

Figure 4.4: Impacts of asynchronous position estimates and CAM transmissions on the information fusion.