认知无线电MIMO广播信道，基于MMSE的块对角化
K. -J. Lee and I. Lee, “MMSE Based Block Diagonalization for Cognitive Radio MIMO Broadcast Channels,” in IEEE Transactions on Wireless Communications, vol. 10, no. 10, pp. 3139-3144, October 2011, doi: 10.1109/TWC.2011.081611.101853.
这里写自定义目录标题
- 白化
- 迫零块对角化
- MMSE块对角化
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- The Karush-Kuhn-Tucker (KKT) conditions
Fig. 1. System description of a CR MU-MIMO downlink system with K K K SUs coexisting with a PU MIMO link.
H ‾ = [ H 1 T H 2 T ⋯ H K T ] T P ‾ = [ P 1 P 2 ⋯ P K ] s ‾ = [ s 1 T s 2 T ⋯ s K T ] T G ‾ = [ G 1 T G 2 T ⋯ G K T ] T n ‾ = [ n 1 T n 2 T ⋯ n K T ] T y ‾ = [ y 1 T y 2 T ⋯ y K T ] T y = H ‾ P ‾ s ‾ + G ‾ P 0 s 0 + n ‾ \underline {\bf{H}} = {[{\bf{H}}_1^T{\bf{H}}_2^T \cdots {\bf{H}}_K^T]^T}\\ \underline {\bf{P}} = [{{\bf{P}}_1}{\mkern 1mu} {\mkern 1mu} {{\bf{P}}_2}{\mkern 1mu} {\mkern 1mu} \cdots {\mkern 1mu} {\mkern 1mu} {{\bf{P}}_K}]\\ \underline {\bf{s}} = {[{\bf{s}}_1^T{\mkern 1mu} {\mkern 1mu} {\bf{s}}_2^T{\mkern 1mu} {\mkern 1mu} \cdots {\mkern 1mu} {\mkern 1mu} {\bf{s}}_K^T]^T}\\ \underline {\bf{G}} = {[{\bf{G}}_1^T{\bf{G}}_2^T \cdots {\bf{G}}_K^T]^T}\\ \underline {\bf{n}} = {[{\bf{n}}_1^T{\mkern 1mu} {\mkern 1mu} {\bf{n}}_2^T{\mkern 1mu} {\mkern 1mu} \cdots {\mkern 1mu} {\mkern 1mu} {\bf{n}}_K^T]^T}\\ \underline {\bf{y}} = {[{\bf{y}}_1^T{\bf{y}}_2^T \cdots {\bf{y}}_K^T]^T}\\ {\bf{y}} = \underline {\bf{H}} \underline {\bf{P}} \underline {\bf{s}} +\underline {\bf{G}} {{\bf{P}}_0}{{\bf{s}}_0} +\underline {\bf{n}} H=[H1TH2T⋯HKT]TP=[P1P2⋯PK]s=[s1Ts2T⋯sKT]TG=[G1TG2T⋯GKT]Tn=[n1Tn2T⋯nKT]Ty=[y1Ty2T⋯yKT]Ty=HPs+GP0s0+n
y 0 = G 0 P 0 s 0 + H 0 P ‾ s ‾ + n 0 {{\bf{y}}_0} = {{\bf{G}}_0}{{\bf{P}}_0}{{\bf{s}}_0} +{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} +{{\bf{n}}_0} y0=G0P0s0+H0Ps+n0
白化
G k Φ 0 G k † + σ n 2 I N R , k = L k L k † Φ 0 = P 0 E [ s 0 s 0 † ] P 0 † T r ( Φ 0 ) ≤ P 0 M ~ ‾ = { L 1 − 1 , L 2 − 1 , ⋯ , L K − 1 } H ~ ‾ = [ H ~ 1 T H ~ 2 T … H ~ K T ] T = M ~ ‾ H ‾ w ‾ = [ w 1 T w 2 T ⋯ w K T ] T = M ~ ‾ ( G ‾ P 0 s 0 + n ‾ ) M ~ ‾ y = M ~ ‾ H ‾ P ‾ s ‾ + M ~ ‾ ( G ‾ P 0 s 0 + n ‾ ) y ~ ‾ = H ~ ‾ P ‾ s ‾ + w ‾ y ~ k = H ~ k P k s k + H ~ k ∑ l = 1 , l ≠ k K P l s l + w k {{\bf{G}}_k}{\Phi _0}{\bf{G}}_k^\dag +\sigma _n^2{{\bf{I}}_{{N_{R,k}}}} = {{\bf{L}}_k}{\bf{L}}_k^\dag \\ {\Phi _0} = {{\bf{P}}_0}E[{{\bf{s}}_0}{\bf{s}}_0^\dag ]{\bf{P}}_0^\dag \\ {\rm{Tr}}({\Phi _0}) \le {P_0}\\ \underline {\widetilde {\bf{M}}} = \{ {\bf{L}}_1^{ - 1},{\bf{L}}_2^{ - 1}, \cdots ,{\bf{L}}_K^{ - 1}\} \\ \underline {\widetilde {\bf{H}}} = {[\widetilde {\bf{H}}_1^T\widetilde {\bf{H}}_2^T \ldots \widetilde {\bf{H}}_K^T]^T} = \underline {\widetilde {\bf{M}}} \underline {\bf{H}} \\ \underline {\bf{w}} = {[{\bf{w}}_1^T{\bf{w}}_2^T \cdots {\bf{w}}_K^T]^T} = \underline {\widetilde {\bf{M}}} (\underline {\bf{G}} {{\bf{P}}_0}{{\bf{s}}_0} +\underline {\bf{n}} )\\ \underline {\widetilde {\bf{M}}} {\bf{y}} = \underline {\widetilde {\bf{M}}} \underline {\bf{H}} \underline {\bf{P}} \underline {\bf{s}} +\underline {\widetilde {\bf{M}}} (\underline {\bf{G}} {{\bf{P}}_0}{{\bf{s}}_0} +\underline {\bf{n}} )\\ \underline {\widetilde {\bf{y}}} = \underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} +\underline {\bf{w}} \\ {\widetilde {\bf{y}}_k} = {\widetilde {\bf{H}}_k}{{\bf{P}}_k}{{\bf{s}}_k} +{\widetilde {\bf{H}}_k}\sum\limits_{l = 1,l \ne k}^K {{{\bf{P}}_l}} {{\bf{s}}_l} +{{\bf{w}}_k} GkΦ0Gk†+σn2INR,k=LkLk†Φ0=P0E[s0s0†]P0†Tr(Φ0)≤P0M ={L1−1,L2−1,⋯,LK−1}H =[H 1TH 2T…H KT]T=M Hw=[w1Tw2T⋯wKT]T=M (GP0s0+n)M y=M HPs+M (GP0s0+n)y =H Ps+wy k=H kPksk+H kl=1,l=k∑KPlsl+wk
迫零块对角化
H 0 P k = 0 N R , 0 × N R , k f o r k = 1 , 2 , ⋯ , K H ~ l P k = 0 N R , l × N R , k f o r a l l k ≠ l a n d 1 ≤ k , l ≤ K , Π k = [ H 0 T H ~ 1 T ⋯ H ~ k − 1 T H ~ k + 1 T ⋯ H ~ K T ] T H ~ k Π k ⊥ = U ‾ k Λ ˉ k V ‾ k † P k Z B = Π k ⊥ V ‾ k Φ k 1 2 M k Z B = U ‾ k † L k − 1 y ~ k = H ~ k P k Z B s k + H ~ k ∑ l = 1 , l ≠ k K P l Z B s l + w k = H ~ k Π k ⊥ V ‾ k Φ k 1 2 s k + w k = U ‾ k Λ ˉ k V ‾ k † V ‾ k Φ k 1 2 s k + w k = U ‾ k Λ ˉ k Φ k 1 2 s k + w k x k = Λ ˉ k Φ k 1 2 s k + U ‾ k † w k {{\bf{H}}_0}{{\bf{P}}_k} = {0_{{N_{R,0}} \times {N_{R,k}}}}{\mkern 1mu} {\mkern 1mu} {\rm{for}}{\mkern 1mu} {\mkern 1mu} k = 1,2, \cdots ,K\\ {\widetilde {\bf{H}}_l}{{\bf{P}}_k} = {0_{{N_{R,l}} \times {N_{R,k}}}}{\mkern 1mu} {\mkern 1mu} {\rm{for}}{\mkern 1mu} {\mkern 1mu} {\rm{all}}{\mkern 1mu} {\mkern 1mu} k \ne l{\mkern 1mu} {\mkern 1mu} {\rm{and}}{\mkern 1mu} {\mkern 1mu} 1 \le k,l \le K,\\ {\Pi _k} = {[{\bf{H}}_0^T\widetilde {\bf{H}}_1^T \cdots \widetilde {\bf{H}}_{k - 1}^T\widetilde {\bf{H}}_{k +1}^T \cdots \widetilde {\bf{H}}_K^T]^T}\\ {\widetilde {\bf{H}}_k}\Pi _k^ \bot = {\overline {\bf{U}} _k}{{\bar \Lambda }_k}\overline {\bf{V}} _k^\dag \\ {\bf{P}}_k^{{\bf{ZB}}} = \Pi _k^ \bot {\overline {\bf{V}} _k}\Phi _k^{\frac{1}{2}}\\ {\bf{M}}_k^{{\bf{ZB}}} = \overline {\bf{U}} _k^\dag {\bf{L}}_k^{ - 1}\\ {\widetilde {\bf{y}}_k} = {\widetilde {\bf{H}}_k}{\bf{P}}_k^{{\bf{ZB}}}{{\bf{s}}_k} +{\widetilde {\bf{H}}_k}\sum\limits_{l = 1,l \ne k}^K {{\bf{P}}_l^{{\bf{ZB}}}{{\bf{s}}_l}} +{{\bf{w}}_k}\\ = {\widetilde {\bf{H}}_k}\Pi _k^ \bot {\overline {\bf{V}} _k}\Phi _k^{\frac{1}{2}}{{\bf{s}}_k} +{{\bf{w}}_k}\\ = {\overline {\bf{U}} _k}{{\bar \Lambda }_k}\overline {\bf{V}} _k^\dag {\overline {\bf{V}} _k}\Phi _k^{\frac{1}{2}}{{\bf{s}}_k} +{{\bf{w}}_k}\\ = {\overline {\bf{U}} _k}{{\bar \Lambda }_k}\Phi _k^{\frac{1}{2}}{{\bf{s}}_k} +{{\bf{w}}_k}\\ {{\bf{x}}_k} = {{\bar \Lambda }_k}\Phi _k^{\frac{1}{2}}{{\bf{s}}_k} +\overline {\bf{U}} _k^\dag {{\bf{w}}_k} H0Pk=0NR,0×NR,kfork=1,2,⋯,KH lPk=0NR,l×NR,kforallk=land1≤k,l≤K,Πk=[H0TH 1T⋯H k−1TH k+1T⋯H KT]TH kΠk⊥=UkΛˉkVk†PkZB=Πk⊥VkΦk21MkZB=Uk†Lk−1y k=H kPkZBsk+H kl=1,l=k∑KPlZBsl+wk=H kΠk⊥VkΦk21sk+wk=UkΛˉkVk†VkΦk21sk+wk=UkΛˉkΦk21sk+wkxk=ΛˉkΦk21sk+Uk†wk
MMSE块对角化
min γ , P ‾ E [ ∣ ∣ s ‾ − γ − 1 y ~ ‾ ∣ ∣ 2 ] s . t . E [ ∣ ∣ H 0 P ‾ s ‾ ∣ ∣ 2 ] ≤ I t h E [ ∣ ∣ P ‾ s ‾ ∣ ∣ 2 ] ≤ P T {\min _{\gamma ,\underline {\bf{P}} }}{\rm{E}}\left[ {||\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{y}}} |{|^2}} \right]\\ {\rm{s}}.{\rm{t}}.{\rm{E}}[||{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} |{|^2}] \le {I_{th}}\\ {\rm{E}}[||\underline {\bf{P}} \underline {\bf{s}} |{|^2}] \le {P_T} minγ,PE[∣∣s−γ−1y ∣∣2]s.t.E[∣∣H0Ps∣∣2]≤IthE[∣∣Ps∣∣2]≤PT
f ( P ‾ ) = E [ ∣ ∣ s ‾ − γ − 1 y ~ ‾ ∣ ∣ 2 ] = T r { I N R − γ − 1 H ~ ‾ P ‾ − γ − 1 P ‾ † H ~ ‾ † + γ − 2 H ~ ‾ P ‾ P ‾ † H ~ ‾ † + γ − 2 I N R } g 1 ( P ‾ ) = I t h − E [ ∣ ∣ H 0 P ‾ s ‾ ∣ ∣ 2 ] ≥ 0 g 2 ( P ‾ ) = P T − E [ ∣ ∣ P ‾ s ‾ ∣ ∣ 2 ] ≥ 0 f(\underline {\bf{P}})={\rm{E}}\left[ {||\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{y}}} |{|^2}} \right] ={\rm{Tr}}\left\{ {{{\bf{I}}_{{N_R}}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} - {\gamma ^{ - 1}}{{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } +{\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } +{\gamma ^{ - 2}}{{\bf{I}}_{{N_R}}}} \right\}\\ {g_1}(\underline {\bf{P}}) = {I_{th}} - {\rm{E}}[||{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} |{|^2}] \ge 0\\ {g_2}(\underline {\bf{P}}) = {P_T} - {\rm{E}}[||\underline {\bf{P}} \underline {\bf{s}} |{|^2}] \ge 0 f(P)=E[∣∣s−γ−1y ∣∣2]=Tr{INR−γ−1H P−γ−1P†H †+γ−2H PP†H †+γ−2INR}g1(P)=Ith−E[∣∣H0Ps∣∣2]≥0g2(P)=PT−E[∣∣Ps∣∣2]≥0
把 y ~ ‾ = H ~ ‾ P ‾ s ‾ + w ‾ \underline {\widetilde {\bf{y}}} = \underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} +\underline {\bf{w}} y =H Ps+w代入上述问题求得Lagrange
L ( P ‾ ) = f ( P ‾ ) − λ 1 g 1 ( P ‾ ) − λ 2 g 2 ( P ‾ ) = T r { I N R − γ − 1 H ~ ‾ P ‾ − γ − 1 P ‾ † H ~ ‾ † + γ − 2 H ~ ‾ P ‾ P ‾ † H ~ ‾ † + γ − 2 I N R } + λ 1 { T r ( H 0 P ‾ P ‾ † H 0 † ) − I t h } + λ 2 { T r ( P ‾ P ‾ † ) − P T } L(\underline {\bf{P}} ) ={f(\underline {\bf{P}} ) - {\lambda _1}{g_1}(\underline {\bf{P}} ) - {\lambda _2}{g_2}(\underline {\bf{P}} )}\\ = {\rm{Tr}}\left\{ {{{\bf{I}}_{{N_R}}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} - {\gamma ^{ - 1}}{{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } +{\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } +{\gamma ^{ - 2}}{{\bf{I}}_{{N_R}}}} \right\}\\ +{\lambda _1}\left\{ {{\rm{Tr}}\left( {{{\bf{H}}_0}\underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{\bf{H}}_0^\dag } \right) - {I_{th}}} \right\} +{\lambda _2}\left\{ {{\rm{Tr}}\left( {\underline {\bf{P}} {{\underline {\bf{P}} }^\dag }} \right) - {P_T}} \right\} L(P)=f(P)−λ1g1(P)−λ2g2(P)=Tr{INR−γ−1H P−γ−1P†H †+γ−2H PP†H †+γ−2INR}+λ1{Tr(H0PP†H0†)−Ith}+λ2{Tr(PP†)−PT}
T r { w ‾ w ‾ H } = T r { s ‾ s ‾ H } = I N R {\rm{Tr}}\left\{ {\underline {\bf{w}} {{\underline {\bf{w}} }^H}} \right\} = {\rm{Tr}}\left\{ {\underline {\bf{s}} {{\underline {\bf{s}} }^H}} \right\} = {{\bf{I}}_{{N_R}}} Tr{wwH}=Tr{ssH}=INR
∣ ∣ s ‾ − γ − 1 y ~ ‾ ∣ ∣ 2 = ( s ‾ − γ − 1 H ~ ‾ P ‾ s ‾ − γ − 1 w ‾ ) H ( s ‾ − γ − 1 H ~ ‾ P ‾ s ‾ − γ − 1 w ‾ ) = T r { ( s ‾ − γ − 1 H ~ ‾ P ‾ s ‾ − γ − 1 w ‾ ) ( s ‾ − γ − 1 H ~ ‾ P ‾ s ‾ − γ − 1 w ‾ ) H } = T r { ( s ‾ − γ − 1 H ~ ‾ P ‾ s ‾ − γ − 1 w ‾ ) ( s ‾ H − γ − 1 s ‾ H P ‾ H H ~ ‾ H − γ − 1 w ‾ H ) } = T r { s ‾ s ‾ H − γ − 1 s ‾ s ‾ H P ‾ H H ~ ‾ H − γ − 1 s ‾ w ‾ H − γ − 1 H ~ ‾ P ‾ s ‾ s ‾ H + γ − 2 H ~ ‾ P ‾ s ‾ s ‾ H P ‾ † H ~ ‾ † + γ − 2 H ~ ‾ P ‾ s ‾ w ‾ H − γ − 1 w ‾ s ‾ H + γ − 2 w ‾ s ‾ H P ‾ H H ~ ‾ H + γ − 2 w ‾ w ‾ H } = T r { I N R − γ − 1 H ~ ‾ P ‾ − γ − 1 P ‾ H H ~ ‾ H + γ − 2 H ~ ‾ P ‾ P ‾ † H ~ ‾ † + γ − 2 I N R − γ − 1 s ‾ w ‾ H + γ − 2 H ~ ‾ P ‾ s ‾ w ‾ H − γ − 1 w ‾ s ‾ H + γ − 2 w ‾ s ‾ H P ‾ H H ~ ‾ H } ? = T r { I N R − γ − 1 H ~ ‾ P ‾ − γ − 1 P ‾ † H ~ ‾ † + γ − 2 H ~ ‾ P ‾ P ‾ † H ~ ‾ † + γ − 2 I N R } ||\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{y}}} |{|^2} = {\left( {\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\bf{w}} } \right)^H}\left( {\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} -{\gamma ^{ - 1}}\underline {\bf{w}} } \right)\\ = {\rm{Tr}}\left\{ {\left( {\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\bf{w}} } \right){{\left( {\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\bf{w}} } \right)}^H}} \right\}\\ = {\rm{Tr}}\left\{ {\left( {\underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} - {\gamma ^{ - 1}}\underline {\bf{w}} } \right)\left( {{{\underline {\bf{s}} }^H} - {\gamma ^{ - 1}}{{\underline {\bf{s}} }^H}{{\underline {\bf{P}} }^H}{{\underline {\widetilde {\bf{H}}} }^H} - {\gamma ^{ - 1}}{{\underline {\bf{w}} }^H}} \right)} \right\}\\ = {\rm{Tr}}\left\{ \begin{array}{l} \underline {\bf{s}} {\underline {\bf{s}} ^H} - {\gamma ^{ - 1}}\underline {\bf{s}} {\underline {\bf{s}} ^H}{\underline {\bf{P}} ^H}{\underline {\widetilde {\bf{H}}} ^H} - {\gamma ^{ - 1}}\underline {\bf{s}} {\underline {\bf{w}} ^H} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} {\underline {\bf{s}} ^H}\\ +{\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} {\underline {\bf{s}} ^H}{\underline {\bf{P}} ^\dag }{\underline {\widetilde {\bf{H}}} ^\dag } +{\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} {\underline {\bf{w}} ^H} - {\gamma ^{ - 1}}\underline {\bf{w}} {\underline {\bf{s}} ^H}\\ +{\gamma ^{ - 2}}\underline {\bf{w}} {\underline {\bf{s}} ^H}{\underline {\bf{P}} ^H}{\underline {\widetilde {\bf{H}}} ^H} +{\gamma ^{ - 2}}\underline {\bf{w}} {\underline {\bf{w}} ^H} \end{array} \right\}\\ = {\rm{Tr}}\left\{ \begin{array}{l} {{\bf{I}}_{{N_R}}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} - {\gamma ^{ - 1}}{\underline {\bf{P}} ^H}{\underline {\widetilde {\bf{H}}} ^H} +{\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} {\underline {\bf{P}} ^\dag }{\underline {\widetilde {\bf{H}}} ^\dag } +{\gamma ^{ - 2}}{{\bf{I}}_{{N_R}}}\\ -{\gamma ^{ - 1}}\underline {\bf{s}} {\underline {\bf{w}} ^H} +{\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \underline {\bf{s}} {\underline {\bf{w}} ^H} - {\gamma ^{ - 1}}\underline {\bf{w}} {\underline {\bf{s}} ^H} +{\gamma ^{ - 2}}\underline {\bf{w}} {\underline {\bf{s}} ^H}{\underline {\bf{P}} ^H}{\underline {\widetilde {\bf{H}}} ^H} \end{array} \right\}\\ ? = {\rm{Tr}}\left\{ {{{\bf{I}}_{{N_R}}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} - {\gamma ^{ - 1}}{{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } +{\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } +{\gamma ^{ - 2}}{{\bf{I}}_{{N_R}}}} \right\} ∣∣s−γ−1y ∣∣2=(s−γ−1H Ps−γ−1w)H(s−γ−1H Ps−γ−1w)=Tr{(s−γ−1H Ps−γ−1w)(s−γ−1H Ps−γ−1w)H}=Tr{(s−γ−1H Ps−γ−1w)(sH−γ−1sHPHH H−γ−1wH)}=Tr⎩ ⎨ ⎧ssH−γ−1ssHPHH H−γ−1swH−γ−1H PssH+γ−2H PssHP†H †+γ−2H PswH−γ−1wsH+γ−2wsHPHH H+γ−2wwH⎭ ⎬ ⎫=Tr{INR−γ−1H P−γ−1PHH H+γ−2H PP†H †+γ−2INR−γ−1swH+γ−2H PswH−γ−1wsH+γ−2wsHPHH H}?=Tr{INR−γ−1H P−γ−1P†H †+γ−2H PP†H †+γ−2INR}
∣ ∣ H 0 P ‾ s ‾ ∣ ∣ 2 = ( H 0 P ‾ s ‾ ) H ( H 0 P ‾ s ‾ ) = T r { ( H 0 P ‾ s ‾ ) ( H 0 P ‾ s ‾ ) H } = T r ( H 0 P ‾ P ‾ † H 0 † ) ||{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} |{|^2} = {\left( {{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} } \right)^H}\left( {{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} } \right)\\ = {\rm{Tr}}\left\{ {\left( {{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} } \right){{\left( {{{\bf{H}}_0}\underline {\bf{P}} \underline {\bf{s}} } \right)}^H}} \right\}\\ = {\rm{Tr}}\left( {{{\bf{H}}_0}\underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{\bf{H}}_0^\dag } \right) ∣∣H0Ps∣∣2=(H0Ps)H(H0Ps)=Tr{(H0Ps)(H0Ps)H}=Tr(H0PP†H0†)
∣ ∣ P ‾ s ‾ ∣ ∣ 2 = ( P ‾ s ‾ ) H ( P ‾ s ‾ ) = T r { ( P ‾ s ‾ ) ( P ‾ s ‾ ) H } = T r ( P ‾ P ‾ † ) ||\underline {\bf{P}} \underline {\bf{s}} |{|^2} = {\left( {\underline {\bf{P}} \underline {\bf{s}} } \right)^H}\left( {\underline {\bf{P}} \underline {\bf{s}} } \right)\\ = {\rm{Tr}}\left\{ {\left( {\underline {\bf{P}} \underline {\bf{s}} } \right){{\left( {\underline {\bf{P}} \underline {\bf{s}} } \right)}^H}} \right\}\\ = {\rm{Tr}}\left( {\underline {\bf{P}} {{\underline {\bf{P}} }^\dag }} \right) ∣∣Ps∣∣2=(Ps)H(Ps)=Tr{(Ps)(Ps)H}=Tr(PP†)
The Karush-Kuhn-Tucker (KKT) conditions
∂ ∂ P ‾ T r { I N R − γ − 1 H ~ ‾ P ‾ − γ − 1 P ‾ † H ~ ‾ † + γ − 2 H ~ ‾ P ‾ P ‾ † H ~ ‾ † + γ − 2 I N R } = − γ − 1 ∂ ∂ P ‾ T r { H ~ ‾ P ‾ } − γ − 1 ∂ ∂ P ‾ T r { P ‾ † H ~ ‾ † } + γ − 2 ∂ ∂ P ‾ T r { H ~ ‾ P ‾ P ‾ † H ~ ‾ † } = − γ − 1 H ~ ‾ † − γ − 1 H ~ ‾ † + 2 γ − 2 H ~ ‾ † H ~ ‾ P ‾ = − 2 γ − 1 H ~ ‾ † + 2 γ − 2 H ~ ‾ † H ~ ‾ P ‾ ∂ ∂ P ‾ T r ( H 0 P ‾ P ‾ † H 0 † ) = 2 H 0 † H 0 P ‾ ∂ ∂ P ‾ T r ( P ‾ P ‾ † ) = 2 P ‾ \frac{\partial }{{\partial \underline {\bf{P}} }}{\rm{Tr}}\left\{ {{{\bf{I}}_{{N_R}}} - {\gamma ^{ - 1}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} - {\gamma ^{ - 1}}{{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } + {\gamma ^{ - 2}}\underline {\widetilde {\bf{H}}} \underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag } + {\gamma ^{ - 2}}{{\bf{I}}_{{N_R}}}} \right\}\\ = - {\gamma ^{ - 1}}\frac{\partial }{{\partial \underline {\bf{P}} }}{\rm{Tr}}\left\{ {\underline {\widetilde {\bf{H}}} \underline {\bf{P}} } \right\} - {\gamma ^{ - 1}}\frac{\partial }{{\partial \underline {\bf{P}} }}{\rm{Tr}}\left\{ {{{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag }} \right\} + {\gamma ^{ - 2}}\frac{\partial }{{\partial \underline {\bf{P}} }}{\rm{Tr}}\left\{ {\underline {\widetilde {\bf{H}}} \underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{{\underline {\widetilde {\bf{H}}} }^\dag }} \right\}\\ = - {\gamma ^{ - 1}}{\underline {\widetilde {\bf{H}}} ^\dag } - {\gamma ^{ - 1}}{\underline {\widetilde {\bf{H}}} ^\dag } + 2{\gamma ^{ - 2}}{\underline {\widetilde {\bf{H}}} ^\dag }\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \\ = - 2{\gamma ^{ - 1}}{\underline {\widetilde {\bf{H}}} ^\dag } + 2{\gamma ^{ - 2}}{\underline {\widetilde {\bf{H}}} ^\dag }\underline {\widetilde {\bf{H}}} \underline {\bf{P}} \\ \frac{\partial }{{\partial \underline {\bf{P}} }}{\rm{Tr}}\left( {{{\bf{H}}_0}\underline {\bf{P}} {{\underline {\bf{P}} }^\dag }{\bf{H}}_0^\dag } \right) = 2{\bf{H}}_0^\dag {{\bf{H}}_0}\underline {\bf{P}} \\ \frac{\partial }{{\partial \underline {\bf{P}} }}{\rm{Tr}}\left( {\underline {\bf{P}} {{\underline {\bf{P}} }^\dag }} \right) = 2\underline {\bf{P}} ∂P∂Tr{INR−γ−1H P−γ−1P†H †+γ−2H PP†H †+γ−2INR}=−γ−1∂P∂Tr{H P}−γ−1∂P∂Tr{P†H †}+γ−2∂P∂Tr{H PP†H †}=−γ−1H †−γ−1H †+2γ−2H †H P=−2γ−1H †+2γ−2H †H P∂P∂Tr(H0PP†H0†)=2H0†H0P∂P∂Tr(PP†)=2P
∂ ∂ P ‾ L ( P ‾ ) = ∂ ∂ P ‾ ( f ( P ‾ ) − λ 1 g 1 ( P ‾ ) − λ 2 g 2 ( P ‾ ) ) = 2 ( − γ − 1 H ~ ‾ † + γ − 2 H ~ ‾ † H ~ ‾ P ‾ + λ 1 H 0 † H 0 P ‾ + λ 2 P ‾ ) \frac{\partial }{{\partial \underline {\bf{P}} }}L(\underline {\bf{P}} )=\frac{\partial }{{\partial \underline {\bf{P}} }}\left( {f(\underline {\bf{P}} ) - {\lambda _1}{g_1}(\underline {\bf{P}} ) - {\lambda _2}{g_2}(\underline {\bf{P}} )} \right)\\ = 2\left( { - {\gamma ^{ - 1}}{{\underline {\widetilde {\bf{H}}} }^\dag } + {\gamma ^{ - 2}}{{\underline {\widetilde {\bf{H}}} }^\dag }\underline {\widetilde {\bf{H}}} \underline {\bf{P}} + {\lambda _1}{\bf{H}}_0^\dag {{\bf{H}}_0}\underline {\bf{P}} + {\lambda _2}\underline {\bf{P}} } \right) ∂P∂L(P)=∂P∂(f(P)−λ1g1(P)−λ2g2(P))=2(−γ−1H †+γ−2H †H P+λ1H0†H0P+λ2P)
(11) and (12) are derived from the zero gradient conditions by using some rules of differentiation [13].
？？？？？？？？？？？？？？？？？？？12哪来的
− γ − 1 H ~ ‾ † + γ − 2 H ~ ‾ † H ~ ‾ P ‾ + λ 1 H 0 † H 0 P ‾ + λ 2 P ‾ = 0 , (11) γ − 2 T r ( H ~ ‾ P ‾ + P ‾ † H ~ ‾ † ) − 2 γ − 3 T r ( H ~ ‾ P ‾ P ‾ † H ~ ‾ † + I N R ) = 0 , (12) λ 1 ≥ 0 , T r ( H 0 P ‾ P ‾ † H 0 † ) − I t h ≤ 0 , (13) λ 1 { T r ( H 0 P ‾ P ‾ † H 0 † ) − I t h } = 0 , (14) λ 2 ≥ 0 , T r ( P ‾ P ‾ † ) − P T ≤ 0 , (15) λ 2 { T r ( P ‾ P ‾ † ) − P T } = 0 , (16) -\gamma^{-1}\underline{\tilde{{\bf H}}}^{\dagger}+\gamma^{-2}\underline{\tilde{{\bf H}}}^{\dagger}\underline{\tilde{{\bf H}}} \underline{{\bf P}}+\lambda_{1}{\bf H}_{0}^{\dagger}{\bf H}_{0} \underline{{\bf P}}+\lambda_{2}\underline{{\bf P}}=0, {\hbox{(11)}}\\ \gamma^{-2}{\rm Tr}\left(\underline{\tilde{{\bf H}}}\underline{{\bf P}} +\underline{{\bf P}}^{\dagger}\underline{\tilde{{\bf H}}}^{\dagger}\right) {-2}\gamma^{-3}{\rm Tr}\left(\underline{\tilde{{\bf H}}}\underline{{\bf P}} \underline{{\bf P}}^{\dagger}\underline{\tilde{{\bf H}}}^{\dagger}+{\bf I}_{N_{R}}\right)=0, {\hbox{(12)}}\\ \lambda_{1}\geq 0, {\rm Tr} ({\bf H}_{0}\underline{{\bf P}}\underline{{\bf P}}^{\dagger}{\bf H}_{0}^{\dagger})-I_{th}\leq 0, {\hbox{(13)}}\\ \lambda_{1}\left\{{\rm Tr}\left({\bf H}_{0}\underline{{\bf P}}\underline{{\bf P}}^{\dagger}{\bf H}_{0}^{\dagger}\right)-I_{th}\right\}=0,{\hbox{(14)}}\\ \lambda_{2}\geq 0,{\rm Tr} \left(\underline{{\bf P}}\underline{{\bf P}}^{\dagger}\right) -P_{T}\leq 0,{\hbox{(15)}}\\ \lambda_{2}\left\{{\rm Tr}\left(\underline{{\bf P}}\underline{{\bf P}}^{\dagger}\right) -P_{T}\right\}=0,{\hbox{(16)}} −γ−1H~†+γ−2H~†H~P+λ1H0†H0P+λ2P=0,(11)γ−2Tr(H~P+P†H~†)−2γ−3Tr(H~PP†H~†+INR)=0,(12)λ1≥0,Tr(H0PP†H0†)−Ith≤0,(13)λ1{Tr(H0PP†H0†)−Ith}=0,(14)λ2≥0,Tr(PP†)−PT≤0,(15)λ2{Tr(PP†)−PT}=0,(16)
− γ − 1 H ~ ‾ † + γ − 2 H ~ ‾ † H ~ ‾ P ‾ + λ 1 H 0 † H 0 P ‾ + λ 2 P ‾ = 0 ( γ − 2 H ~ ‾ † H ~ ‾ + λ 1 H 0 † H 0 + λ 2 I N T ) P ‾ = γ − 1 H ~ ‾ † ( H ~ ‾ † H ~ ‾ + λ 1 γ 2 H 0 † H 0 + λ 2 γ 2 I N T ) P ‾ = γ H ~ ‾ † P ‾ = γ ( H ~ ‾ † H ~ ‾ + λ 1 γ 2 H 0 † H 0 + λ 2 γ 2 I N T ) − 1 H ~ ‾ † = γ ( H ~ ‾ † H ~ ‾ + μ 1 H 0 † H 0 + μ 2 I N T ) − 1 H ~ ‾ † - {\gamma ^{ - 1}}{\underline {\widetilde {\bf{H}}} ^\dag } + {\gamma ^{ - 2}}{\underline {\widetilde {\bf{H}}} ^\dag }\underline {\widetilde {\bf{H}}} \underline {\bf{P}} + {\lambda _1}{\bf{H}}_0^\dag {{\bf{H}}_0}\underline {\bf{P}} + {\lambda _2}\underline {\bf{P}} = 0\\ \left( {{\gamma ^{ - 2}}{{\underline {\widetilde {\bf{H}}} }^\dag }\underline {\widetilde {\bf{H}}} + {\lambda _1}{\bf{H}}_0^\dag {{\bf{H}}_0} + {\lambda _2}{{\bf{I}}_{{N_T}}}} \right)\underline {\bf{P}} = {\gamma ^{ - 1}}{\underline {\widetilde {\bf{H}}} ^\dag }\\ \left( {{{\underline {\widetilde {\bf{H}}} }^\dag }\underline {\widetilde {\bf{H}}} + {\lambda _1}{\gamma ^2}{\bf{H}}_0^\dag {{\bf{H}}_0} + {\lambda _2}{\gamma ^2}{{\bf{I}}_{{N_T}}}} \right)\underline {\bf{P}} = \gamma {\underline {\widetilde {\bf{H}}} ^\dag }\\ \underline {\bf{P}} = \gamma {\left( {{{\underline {\widetilde {\bf{H}}} }^\dag }\underline {\widetilde {\bf{H}}} + {\lambda _1}{\gamma ^2}{\bf{H}}_0^\dag {{\bf{H}}_0} + {\lambda _2}{\gamma ^2}{{\bf{I}}_{{N_T}}}} \right)^{ - 1}}{\underline {\widetilde {\bf{H}}} ^\dag }\\ = \gamma {\left( {{{\underline {\widetilde {\bf{H}}} }^\dag }\underline {\widetilde {\bf{H}}} + {\mu _1}{\bf{H}}_0^\dag {{\bf{H}}_0} + {\mu _2}{{\bf{I}}_{{N_T}}}} \right)^{ - 1}}{\underline {\widetilde {\bf{H}}} ^\dag } −γ−1H †+γ−2H †H P+λ1H0†H0P+λ2P=0(γ−2H †H +λ1H0†H0+λ2INT)P=γ−1H †(H †H +λ1γ2H0†H0+λ2γ2INT)P=γH †P=γ(H †H +λ1γ2H0†H0+λ2γ2INT)−1H †=γ(H †H +μ1H0†H0+μ2INT)−1H †
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