Tensor-Based Subspace Tracking for Time-Delay Estimation in GNSS Multi-Antenna Receivers

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. 2019 Nov 20;19(23):5076. doi: 10.3390/s19235076

Algorithm 1: PAST–HOSVD time-delay estimation.

Initialization.
- ${\hat{U}}_{s}^{(1)} (0) = I_{2 \times L}$ , ${\hat{U}}_{s}^{(3)} (0) = I_{M_{s} \times L}$ ; ${\hat{U}}_{s}^{(4)} (0) = I_{L_{s} \times L}$
- ${\hat{C}}_{yy}^{(1)} (0) = {\hat{C}}_{yy}^{(3)} (0) = {\hat{C}}_{yy}^{(4)} (0) = I_{L}$
Section: Tensor-based PAST [21].
- for $t = 0, 1, \dots$ do:
  - % – Tracking the signal subspace related to the dimension K.
  - $Y^{(1)} (t + 1) = {\hat{U}}_{s}^{(1)}^{H} (t) {[Z_{FBA + ESPS} (r)]}_{(1)} (t + 1)$
  - $C_{yy}^{(1)} (t + 1) = β C_{yy}^{(1)} (t) + Y^{(1)} (t + 1) Y^{(1)} H (t + 1)$
  - $G^{(1)} (t + 1) = {C_{yy}^{(1)}}^{- 1} (t + 1) Y^{(1)} (t + 1)$
  - $E^{(1)} (t + 1) = {[Z_{FBA + ESPS} (r)]}_{(1)} (t + 1) - {\hat{U}}_{s}^{(1)} (t) Y^{(1)} (t + 1)$
  - ${\hat{U}}_{s}^{(1)} (t + 1) = {\hat{U}}_{s}^{(1)} (t) + E^{(1)} (t + 1) {G^{(1)}}^{H} (t + 1)$
  - % – Tracking the signal subspace related to the dimension $M_{s}$ .
  - $Y^{(3)} (t + 1) = {\hat{U}}_{s}^{(3)}^{H} (t) {[Z_{FBA + ESPS} (r)]}_{(3)} (t + 1)$
  - $C_{yy}^{(3)} (t + 1) = β C_{yy}^{(3)} (t) + Y^{(3)} (t + 1) Y^{(3)} H (t + 1)$
  - $G^{(3)} (t + 1) = {C_{yy}^{(3)}}^{- 1} (t + 1) Y^{(3)} (t + 1)$
  - $E^{(3)} (t + 1) = {[Z_{FBA + ESPS} (r)]}_{(3)} (t + 1) - {\hat{U}}_{s}^{(3)} (t) Y^{(3)} (t + 1)$
  - ${\hat{U}}_{s}^{(3)} (t + 1) = {\hat{U}}_{s}^{(3)} (t) + E^{(3)} (t + 1) {G^{(3)}}^{H} (t + 1)$
  - % – Tracking the signal subspace related to the dimension $L_{s}$ .
  - $Y^{(4)} (t + 1) = {\hat{U}}_{s}^{(4)}^{H} (t) {[Z_{FBA + ESPS} (r)]}_{(4)} (t + 1)$
  - $C_{yy}^{(4)} (t + 1) = β C_{yy}^{(4)} (t) + Y^{(4)} (t + 1) Y^{(4)} H (t + 1)$
  - $G^{(4)} (t + 1) = {C_{yy}^{(4)}}^{- 1} (t + 1) Y^{(4)} (t + 1)$
  - $E^{(4)} (t + 1) = {[Z_{FBA + ESPS} (r)]}_{(4)} (t + 1) - {\hat{U}}_{s}^{(4)} (t) Y^{(4)} (t + 1)$
  - ${\hat{U}}_{s}^{(4)} (t + 1) = {\hat{U}}_{s}^{(4)} (t) + E^{(4)} (t + 1) {G^{(4)}}^{H} (t + 1)$
Eigenfiltering section [10].
- Select the dominant singular vectors of each ${\hat{U}}_{s} (t + 1)$ previously estimated. Then apply the filtering process detailed in (10).
- $q_{FBA + ESPS} (r) = (\begin{matrix} Z_{FBA + ESPS} (r) \times_{1} {({\hat{u}}^{(1)})}^{H} \times_{3} {({\hat{u}}^{(3)})}^{H} \times_{4} {({\hat{u}}^{(4)})}^{H} \end{matrix}) Σ V^{H} \in C^{Q \times 1}$
- end