## 3 Proposed SAGE Estimation Algorithm
\begin{split}&\mathrm{U}\Big{(}\Big{[}\mathrm{H}_{1}^{(k)}, \mathrm{H}_{2}^{(k)}\Big{]}\Big{|}\Big{[}\widehat{\mathrm{H}}_{1}^{(k)}(\zeta ),\widehat{\mathrm{H}}_{2}^{(k)}(\zeta)\Big{]}\Big{)}=\\ &\mathrm{E}\Big{[}\log\mathrm{Pr}\Big{(}\mathbf{z}_{1}^{(k)}, \mathbf{z}_{2}^{(k)}\Big{|}\mathrm{S}^{(k)},\mathrm{H}_{1}^{(k)},\mathrm{H}_{ 2}^{(k)}\Big{)}\Big{|}\mathbf{z}_{1}^{(k)},\\ &\mathbf{z}_{2}^{(k)},\widehat{\mathrm{H}}_{1}^{(k)}(\zeta), \widehat{\mathrm{H}}_{2}^{(k)}(\zeta)\Big{]}\end{split} \] (27) \[=\int_{\mathrm{S}^{(k)}}\log\mathrm{Pr}\Big{(}\mathbf{z}_{1}^{(k)}, \mathbf{z}_{2}^{(k)}\Big{|}\mathrm{S}^{(k)},\mathrm{H}_{1}^{(k)},\mathrm{H}_{ 2}^{(k)}\Big{)}\times\] \[\mathrm{Pr}\Big{(}\mathrm{S}^{(k)}\Big{|}z_{1}^{(k)},\mathbf{z}_{2}^{(k)}, \widehat{\mathrm{H}}_{1}^{(k)}(\zeta),\widehat{\mathrm{H}}_{2}^{(k)}(\zeta) \Big{)}d\mathrm{S}^{(k)},\] where the expectation is executed on the broadcast matrix \(\mathrm{S}^{(k)}\). * By substituting (21), (22), and (23) into (27) and eliminating the unnecessary elements, we obtain \[ \begin{split}&\mathrm{U}\Big{(}\Big{[}\mathrm{H}_{1}^{(k)}, \mathrm{H}_{2}^{(k)}\Big{]}\Big{|}\Big{[}\widehat{\mathrm{H}}_{1}^{(k)}(\zeta ),\widehat{\mathrm{H}}_{2}^{(k)}(\zeta)\Big{]}\Big{)}\propto\\ & 2\Re\Big{(}z_{1}^{(k)}\mathbf{B}_{1}\mathrm{H}_{1}^{(k)}\Big{)}- \mathrm{H}_{1}^{(k)\ddagger}\mathbf{B}_{2}\mathrm{H}_{1}^{(k)}\\ &+2\Re\Big{(}z_{2}^{(k)\ddagger}\mathbf{B}_{3}\mathrm{H}_{2}^{(k )}\Big{)}-\mathrm{H}_{2}^{(k)\ddagger}\mathbf{B}_{4}\mathrm{H}_{2}^{(k)},\end{split} \] (28) where the superscript \(\ddagger\) is the vector-transpose conjugate, and \[\mathbf{B}_{1}=\int\mathrm{S}^{(k)}\mathrm{Pr}\Big{(}\mathrm{S}^{(k)}\Big{|} \mathbf{z}_{1}^{(k)},\widehat{\mathrm{H}}_{1}^{(k)}(\zeta)\Big{)}d\mathrm{S}^{ (k)},\] (29a) \[\mathbf{B}_{2}=\int\mathrm{S}^{(k)}\mathrm{I}\mathrm{S}^{(k)}\mathrm{Pr}\Big{(} \mathrm{S}^{(k)}\Big{|}\mathbf{z}_{1}^{(k)},\widehat{\mathrm{H}}_{1}^{(k)}( \zeta)\Big{)}d\mathrm{S}^{(k)},\] (29b) \[\mathbf{B}_{3}=\int\mathrm{S}^{(k)}\mathrm{Pr}\Big{(}\mathrm{S}^{(k)}\Big{|} \mathbf{z}_{2}^{(k)},\widehat{\mathrm{H}}_{2}^{(k)}(\zeta)\Big{)}d\mathrm{S}^{ (k)},\] (29c) \[\mathbf{B}_{4}=\int\mathrm{S}^{(k)\ddagger}\mathrm{S}^{(k)}\mathrm{Pr}\Big{(} \mathrm{S}^{(k)}\Big{|}\mathbf{z}_{2}^{(k)},\widehat{\mathrm{H}}_{2}^{(k)}( \zeta)\Big{)}d\mathrm{S}^{(k)},\] (29d)
* Hence, the M-step is derived as \[ \begin{split}\Big{[}\widehat{\mathrm{H}}_{1}^{(k)}(\zeta+1), \widehat{\mathrm{H}}_{2}^{(k)}(\zeta+1)\Big{]}=\arg\max_{\mathrm{H}_{1}^{(k)}, \mathrm{H}_{2}^{(k)}}\\ \mathrm{U}\Big{(}\Big{[}\mathrm{H}_{1}^{(k)},\mathrm{H}_{2}^{(k)} \Big{]}\Big{|}\Big{[}\widehat{\mathrm{H}}_{1}^{(k)}(\zeta),\widehat{\mathrm{H}}_ {2}^{(k)}(\zeta)\Big{]}\Big{)}.\end{split} \] (30)When we zero out the derivative of (28) with regard to \(\mathrm{H}_{1}^{(k)}\) and \(\mathrm{H}_{2}^{(k)}\), respectively, we are able to refine the previous estimates of \(\mathrm{H}_{1}^{(k)}\) and \(\mathrm{H}_{2}^{(k)}\) as  
\[\widehat{\mathrm{H}}_{1}^{(k)}(\zeta+1)=\mathbf{B}_{2}^{-1}\mathbf{B}_{1} \mathbf{z}_{1}^{(k)}, \tag{31a}\] \[\widehat{\mathrm{H}}_{2}^{(k)}(\zeta+1)=\mathbf{B}_{4}^{-1}\mathbf{B}_{3} \mathbf{z}_{2}^{(k)}. \tag{31b}\]  
**4. Practical Considerations**  
The following are a number of realistic remarks that need to be taken into account. * The concern that arises is how one would proceed with actually determining the matrices of \(\mathbf{B}_{1}\), \(\mathbf{B}_{2}\), \(\mathbf{B}_{3}\), and \(\mathbf{B}_{4}\) in the real world. Implications from (4) are that \[ \begin{split} s_{S}^{(k)}(n_{1})=\sum_{n_{2}=-\
u}^{N-1}\mathrm{E} \Big{[}\tilde{x}_{S}^{(k)}(n_{2})\Big{]}\exp(j2\pi n_{1}n_{2}/N).\end{split} \] (32) Using (1), (2), and (3), \(\mathrm{E}\Big{[}\tilde{x}_{S}^{(k)}(n_{2})\Big{]}\) is expressed as \[