Accuracy Bounds for Array-Based Positioning in Dense Multipath Channels

Abstract

The accuracy of radio-based positioning systems will be limited by multipath interference in realistic application scenarios. This paper derives closed-form expressions for the Cramér⁻Rao lower bound (CRLB) on the achievable time-of-arrival (ToA) and angle-of-arrival (AoA) estimation-error variances, considering the presence of multipath radio channels, and extends these results to position estimation. The derivations are based on channel models comprising deterministic, specular multipath components as well as stochastic, diffuse/dense multipath. The derived CRLBs thus allow an evaluation of the influence of channel parameters, the geometric configuration of the environment, and system parameters such as signal bandwidth and array geometry. Our results quantify how the ToA and AoA accuracies decrease when the signal bandwidth is reduced, because more multipath will then interfere with the useful LoS component. Antenna arrays can (partly) compensate this performance loss, exploiting diversity among the multipath interference. For example, the AoA accuracy with a 16-element linear array at 1 MHz bandwidth is similar to a two-element array at 1 GHz , in the magnitude order of one degree. The ToA accuracy, on the other hand, still scales by a factor of 100 from the cm-regime to the m-regime because of the dominating influence of the signal bandwidth. The position error bound shows the relationship between the range and angle information under realistic indoor channel conditions and their different scaling behaviors as a function of the anchor⁻agent placement. Specular multipath components have a maximum detrimental influence near the walls. It is shown for an L-shaped room that a fairly even distribution of the position error bound can be achieved throughout the environment, using two anchors equipped with 2 × 2 -array antennas. The accuracy limit due to multipath increases from the 1⁻10-cm-range at 1 GHz bandwidth to the 0.5⁻1-m-range at 100 MHz .

Keywords: AoA; CRLB; ToA; antenna-array signal processing; indoor positioning; wireless positioning.

Figure 1

Geometry of the array at the anchor position ${\mathbf{q}}^{(\ell )}$ (reference point) with positions of the mth array element ${\mathbf{q}}_m^{(\ell )}$, described by the angle ${\psi }_m^{(\ell )}$ and the distance $d_m^{(\ell )}$ of element m from the reference point ${\mathbf{q}}^{(\ell )}$. ${\varphi }^{(\ell )}$ is the AoA and ${\tau }^{(\ell )}$ the ToA from an agent at a position $\mathbf{p}={\left[x,y\right]}^{\mathrm{T}}$.

Figure 2

REB and AEB parameters over different pulse bandwidths $1/T_p$: effective SINR for the delay estimation ${\widetilde{\mathrm{SINR}}}_{\tau }$, effective SINR for the angle estimation ${\widetilde{\mathrm{SINR}}}_{\varphi }$, SINR, whitening gain for the delay estimation ${\gamma }_{\tau }$, whitening gain for the angle estimation ${\gamma }_{\varphi }$, and loss factor ${sin}^2(\xi )$.

Figure 3

Ranging error bound (REB) and angulation error bound (AEB) over different pulse bandwidths $1/T_p$, and performance results for matched filter estimator (MFE) and maximum likelihood estimator (MLE). Simulations were performed using $N_{\mathrm{r}}=\mathrm{10,000}$ realizations for ULAs with $M=2$ (dashed lines) and $M=16$ (solid lines) elements and inter-element spacing of $\lambda /2$; the resulting array aperture is $(M-1)\lambda /2$.

Figure 4

PEB as stated in Equation (41) for two anchors ${\mathbf{q}}^{(1)}$ and ${\mathbf{q}}^{(2)}$ equipped with antenna arrays, visualizing the influence of the different parts attributable to ranging (in direction of the agent) and angulation (perpendicular to the agent direction) that contribute to the positioning accuracy of an agent $\mathbf{p}$.

Figure 5

Floorplan of the synthetic environment used for the PEB evaluation with two anchors A1 (⊗) and A2 (⊗) at positions ${\mathbf{q}}^{(1)}={[10,7]}^{\mathrm{T}}$ and ${\mathbf{q}}^{(2)}={[2,1]}^{\mathrm{T}}$. The specular components were modeled via reflections at the walls. We modeled reflections up to second order, i.e., two wall interactions.

Figure 6

PEB ellipses (1-fold) for LoS-only positioning using a $2\times 2$ array at anchors A1 (⊗) and A2 (⊗) for pulse durations $T_p=\{1,10\}\mathrm{ns}$. Ellipses for range- and angle-only positioning are only shown where two anchors are visible. Underlying colors indicate the total PEB.

Figure 7

PEB ellipses (1-fold) for AWGN-plus-DM with path-overlap and $T_p=10\mathrm{ns}$ comparing the full PEB using anchors A1 (⊗) and A2 (⊗) and single anchor positioning using solely A1 (⊗). Underlying colors show the same PEB as the ellipses.

Figure 8

PEB ellipses (1/4-fold) for AWGN-plus-DM with path-overlap and $T_p=10\mathrm{ns}$ comparing angle-only and range-only positioning (both anchors are needed). Underlying colors show the same PEB as the ellipses.