Table 4 Four designs of precoder F and decoder G subject to either ATP or IAP constraint

PMMSE

$\underset{\mathbf{G},\mathbf{F}}{min}E\left\{{∥\mathbf{Gy}-\mathbf{s}∥}_F^2\right\}=\underset{\mathbf{G},\mathbf{F}}{min}\mathit{tr}\left\{{\sigma }_{\mathbf{s}}^2\left(\mathbf{GHF}-{\mathbf{I}}_m\right){\left(\mathbf{GHF}-{\mathbf{I}}_m\right)}^{*}+\beta \mathbf{G}{\mathbf{G}}^{*}\right\}$

ATP (9a)

IAP (11a)

PMMI

$\underset{\mathbf{G},\mathbf{F}}{max}log\frac{\left|\mathbf{G}\left({\sigma }_{\mathbf{s}}^2\mathbf{HF}{\mathbf{F}}^{*}{\mathbf{H}}^{*}+\beta {\mathbf{I}}_r\right){\mathbf{G}}^{*}\right|}{\beta \left|\mathbf{G}{\mathbf{G}}^{*}\right|}$

ATP (9a)

IAP (11a)

PMBER

$\underset{\mathbf{G},\mathbf{F}}{max}{\lambda }_{min}\left\{{\sigma }_{\mathbf{s}}^2{\mathbf{F}}^{*}{\mathbf{H}}^{*}{\mathbf{G}}^{*}{\left(\beta \mathbf{G}{\mathbf{G}}^{*}\right)}^{-1}\mathbf{GHF}\right\}$

ATP (9a)

IAP (11a)

PSVD

$\begin{array}{l}\mathrm{SVD}\mathrm{on}\mathbf{H}:\mathbf{H}=\mathbf{W\Sigma }{\mathbf{T}}^{*}\mathrm{with}\mathbf{T}=\left[{\mathbf{t}}_1\dots {\mathbf{t}}_c\right]\\ \left(\mathrm{singular}\mathrm{values}\mathrm{are}\mathrm{in}\mathrm{decending}\mathrm{order}\right)\end{array}$

ATP (9a)

IAP (11a)