Expectation–Maximization-Based Simultaneous Localization and Mapping for Millimeter-Wave Communication Systems

. 2022 Sep 14;22(18):6941. doi: 10.3390/s22186941

Algorithm 1: Stochastic Approximation EM-based AP positioning.

Input: Observed ADOA vectors

{\tilde{θ} (p_{n})}_{n = 1}^{N}

at a large number of random and unknown positions

1:
Initialization: ${\hat{a}}_{3 : L}^{0}, {\hat{a}}_{1} = [0 0], {\hat{a}}_{2} = [1 0], t = 0$
2:
repeat
3:
$t \leftarrow t + 1$

E-step in t-th iteration:
4:
Select M MT positions ${{\bar{p}}_{m}}_{m = 1}^{M}$ randomly
5:
Generate $θ ({\hat{a}}_{3 : L}^{t - 1}, {\bar{p}}_{m})$ based on Equation (8)
6:
Obtain the position samples ${{\hat{p}}_{n}^{t}}_{n = 1}^{N}$ for observed ADOA vectors ${\tilde{θ} (p_{n})}_{n = 1}^{N}$ according to Equation (12)

M-step in t-th iteration:
7:
update ${\hat{a}}_{3 : L}^{t}$ based on ${{\hat{p}}_{n}^{t}}_{n = 1}^{N}$ according to Equation (14)
8:
until $\sum_{l = 3}^{L} {∥{\hat{a}}_{l}^{t} - {\hat{a}}_{l}^{t - 1}∥}^{2} / {∥{\hat{a}}_{3 : L}^{t}∥}^{2} < ε$ or $t > M a x I T$

Output: the AP position estimates

{\hat{a}}_{3 : L}^{t}

and

{\hat{a}}_{1 : 2}