## 3 Proposed SAGE Estimation Algorithm
The basic idea is to rotate between executing an expectation (E) step, which constructs a function for the expected log-likelihood assessed using the latest assessment for the variables, and a maximization (M) step, which evaluates variables optimizing the expected log-likelihood determined on the E step. During the following E step, these variable-predictions are utilized to calculate the distribution of the observed variables. This solution is particularly beneficial when the M-step is simpler than optimizing the initial likelihood. A conventional expectation maximization algorithm's concurrent updating needs too-informative complete-data spaces, which leads to sluggish convergence [64]. In general, an expectation maximization process's convergence speed decreases as its full-data space's Fisher information increases [65]. Hence, a significant number of iterations is necessary to achieve a satisfactory performance level. [PERSON] developed the SAGE approach in 1994 to speed up the convergence time of the expectation maximization process for multidimensional parameter estimation. As compared to the standard expectation maximization process, the SAGE algorithm differs in that it successively updates groups of parameters rather than all of them at once. It was shown in [66] that the SAGE process converges more quickly than the expectation maximization procedure while preserving the benefits of computational simplicity and stability. In recent years, SAGE has been used in a great number of studies including the estimate of the direction-of-arrival, and channel estimation and synchronization for upstream scenarios [67; 68]. Theoretical details of the SAGE procedure devised for calculating the parameters under consideration are described as follows. We generate \(K\) separate subgroups, none of which are identical to one another; \(\left\{\mathrm{H}_{1}^{(0)},\mathrm{H}_{2}^{(0)}\right\}\), \(\cdots\), \(\left\{\mathrm{H}_{1}^{(K-1)},\mathrm{H}_{2}^{(K-1)}\right\}\). Alterations to the parameters are made on a one-by-one basis across all of the users. Each round is made up of \(K\) cycles in which a user's subgroup is modified. Given the early guesses, \(\left\{\widehat{\mathrm{H}}_{1}^{(0)}(0),\widehat{\mathrm{H}}_{2}^{(0)}(0)\right\}\), \(\cdots\), \(\left\{\widehat{\mathrm{H}}_{1}^{(K-1)}(0),\widehat{\mathrm{H}}_{2}^{(K-1)}(0)\right\}\), the following steps compose the \((\zeta+1)\) round. * During the \(k\)th loop, the user \(k\) subgroup \(\left\{\mathrm{H}_{1}^{(k)},\mathrm{H}_{2}^{(k)}\right\}\) is updated, while the subsets of all other users remain unchanged. * By conducting time domain multiple access interference cancellation, we are able to retrieve the received signal for user \(k\), denoted by \(z^{(k)}\), by eliminating the contributions from all users except user \(k\). We can formally express \[\mathbf{z}_{1}^{(k)}=\overline{\mathbf{y}}_{D}^{(1)}-\sum_{k^{\prime}=0\
ot{k}^{ \prime}\
eq k}^{K-1}\mathrm{A}^{(K^{\prime})}\widehat{\mathrm{H}}_{1}^{(k)}( \zeta),\] (24) \[\mathbf{z}_{2}^{(k)}=\overline{\mathbf{y}}_{D}^{(2)}-\sum_{k^{\prime}=0\
ot{k} ^{\prime}\
eq k}^{K-1}\mathrm{A}^{(K^{\prime})}\widehat{\mathrm{H}}_{2}^{(k)}( \zeta),\] (25) where \[\mathrm{A}^{(K^{\prime})}=\int_{\mathrm{S}^{(k^{\prime})}}\mathrm{S}^{(k^{ \prime})}\mathrm{Pr}\Big{(}\mathrm{S}^{(k^{\prime})}\Big{|}\overline{\mathbf{y }}_{D}^{(1)},\overline{\mathbf{y}}_{D}^{(2)},\widehat{\mathrm{H}}_{1}^{(k^{ \prime})}(\zeta),\widehat{\mathrm{H}}_{2}^{(k^{\prime})}(\zeta)\Big{)}d \mathrm{S}^{(k^{\prime})}.\] (26)
* The E-step measures the expected logarithmic likelihood of the observed signal given the conditional distribution of the unobserved data, based on what we know about the unknown parameters so far. This is typically expressed as \[