Table 3 Covariance matrices and ATP and IAP constraints of conventional and equivalent single-user MIMO systems

Source covariance

${\mathbf{\Phi }}_{\mathbf{s}}=E\left(\mathbf{s}{\mathbf{s}}^{*}\right)={\sigma }_{\mathbf{s}}^2{\mathbf{I}}_m$

${\mathbf{\Phi }}_{\mathbf{s}k}=E\left({\mathbf{s}}_k{\mathbf{s}}_k^{*}\right)={\sigma }_{\mathbf{s}k}^2{\mathbf{I}}_{m_k}$

${\underset{¯}{\mathbf{\Phi }}}_{\mathbf{s}k}=E\left({\underset{¯}{\mathbf{s}}}_k{\underset{¯}{\mathbf{s}}}_k^{*}\right)={\underset{¯}{\mathit{\sigma }}}_{\mathbf{s}k}^2{\mathbf{I}}_{{\underset{¯}{\mathit{m}}}_k}$

Noise covariance

Φ _n  = E(nn*) = β I _r

${\mathbf{\Phi }}_{\mathbf{n}k}=E\left({\mathbf{n}}_k{\mathbf{n}}_k^{*}\right)={\beta }_k{\mathbf{I}}_{u_k}$

$\begin{array}{l}E\left({\underset{¯}{\mathbf{G}}}_{k,\mathrm{R}}{\underset{¯}{\mathbf{n}}}_k{\underset{¯}{\mathbf{n}}}_k^{*}{\underset{¯}{\mathbf{G}}}_{k,\mathrm{R}}^{*}\right)={\underset{¯}{\mathit{\beta }}}_k{\mathbf{I}}_{{\underset{¯}{\mathit{d}}}_k}\\ \mathrm{where}\\ {\underset{¯}{\mathbf{\Phi }}}_{\mathbf{n}k}=E\left({\underset{¯}{\mathbf{n}}}_k{\underset{¯}{\mathbf{n}}}_k^{*}\right)={\underset{¯}{\mathit{\beta }}}_k{\mathbf{I}}_{{\underset{¯}{\mathit{b}}}_k}\left(14\right)\end{array}$

Transmit covariance

$E\left(\mathbf{Fs}{\mathbf{s}}^{*}{\mathbf{F}}^{*}\right)={\sigma }_{\mathbf{s}}^2\mathbf{F}{\mathbf{F}}^{*}$

$\begin{array}{l}E\left({\mathbf{F}}_{k,\mathrm{R}}{\mathbf{s}}_k{\mathbf{s}}_k^{*}{\mathbf{F}}_{k,\mathrm{R}}^{*}\right)={\sigma }_{\mathbf{s}k}^2{\mathbf{F}}_{k,\mathrm{R}}{\mathbf{F}}_{k,\mathrm{R}}^{*}\\ \mathrm{generally}\ne {\sigma }_{\mathbf{s}k}^2{\mathbf{F}}_k{\mathbf{F}}_k^{*}\end{array}$

$E\left({\underset{¯}{\mathbf{F}}}_k{\underset{¯}{\mathbf{s}}}_k{\underset{¯}{\mathbf{s}}}_k^{*}{\underset{¯}{\mathbf{F}}}_k^{*}\right)={\underset{¯}{\mathit{\sigma }}}_{\mathbf{s}k}^2{\underset{¯}{\mathbf{F}}}_k{\underset{¯}{\mathbf{F}}}_k^{*}$

ATP

$P=\mathrm{tr}\left\{\mathbf{F}{\mathbf{F}}^{*}\right\}{\sigma }_{\mathbf{s}}^2$

$\begin{array}{l}{P}_k=\mathrm{tr}\left\{{\mathbf{F}}_{k,\mathrm{R}}{\mathbf{F}}_{k,\mathrm{R}}^{*}\right\}{\sigma }_{\mathbf{s}k}^2\\ =\mathrm{tr}\left\{{\mathbf{F}}_k{\mathbf{F}}_k^{*}\right\}{\sigma }_{\mathbf{s}k}^2(9,10)\end{array}$

${\underset{¯}{\mathit{P}}}_k=\mathrm{tr}\left\{{\underset{¯}{\mathbf{F}}}_k{\underset{¯}{\mathbf{F}}}_k^{*}\right\}{\underset{¯}{\mathit{\sigma }}}_{\mathbf{s}k}^2$

IAP

$L={\lambda }_{max}\left\{\mathbf{F}{\mathbf{F}}^{*}\right\}{\sigma }_{\mathbf{s}}^2$

$\begin{array}{l}{L}_k={\lambda }_{max}\left\{{\mathbf{F}}_{k,\mathrm{R}}{\mathbf{F}}_{k,\mathrm{R}}^{*}\right\}{\sigma }_{\mathbf{s}k}^2\\ ={\lambda }_{max}\left\{{\mathbf{F}}_k{\mathbf{F}}_k^{*}\right\}{\sigma }_{\mathbf{s}k}^2(12,13)\end{array}$

${\underset{¯}{\mathit{L}}}_k={\lambda }_{max}\left\{{\underset{¯}{\mathbf{F}}}_k{\underset{¯}{\mathbf{F}}}_k^{*}\right\}{\underset{¯}{\mathit{\sigma }}}_{\mathbf{s}k}^2$