overview of gaussian mimo (vector) bc · overview of gaussian mimo (vector) bc ... → writing on...
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Overview of Gaussian
MIMO (Vector) BC
Gwanmo Ku
Adaptive Signal Processing and Information Theory Research Group
Nov. 30, 2012
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Outline
Capacity Region of Gaussian MIMO BC
System Structure
Know Capacity Regions
- Aligned and Inconsistently Degraded MIMO BC → Superposition
- Aligned MIMO BC without Common Message
→ Writing on Dirty Paper
- Degraded Message Sets (A Common & One Private Message)
Duality of Gaussian MIMO BC & MAC
Gaussian MIMO MAC
Gaussian MIMO BC & MAC
/
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Gaussian MIMO (Vector) BC
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System Structure
Encoder
Decoder 1
Decoder 2
⊕
⊕
𝑀0, 𝑀1, 𝑀2 𝐗𝑛
𝐘1𝑛
𝐘2𝑛
𝐙2𝑛
𝐙1𝑛
𝑀 01, 𝑀 1
𝑀 02, 𝑀 2
𝐺1
𝐺2 𝑀0 : A Common Message
𝑀1 : A Private Message to Rx. 1
𝑀2 : A Private Message to Rx. 2
𝑡 : # Tx. Ant. 𝑟 : # Rx. Ant.
channel
𝐘1 = 𝐺1𝐗 + 𝐙1 𝐘2 = 𝐺2𝐗 + 𝐙2
Power Constraint
1
𝑛 𝐱𝑇 𝑚0, 𝑚1, 𝑚2, 𝑖 𝐱(𝑚0,𝑚1, 𝑚2, 𝑖 )
𝑛
𝑖=1
≤ 𝑃
𝑚0,𝑚1, 𝑚2 ∈ 1: 2𝑛𝑅0 × 1: 2𝑛𝑅1 × [1: 2𝑛𝑅2]
𝐙1 ∼ 𝓝(0, 𝐼𝑟)
𝐙2 ∼ 𝓝(0, 𝐼𝑟)
dim 𝐲1 = 𝑟 × 1 dim 𝐲2 = 𝑟 × 1
dim 𝐳1 = 𝑟 × 1
dim 𝐳2 = 𝑟 × 1
𝑟 : # Rx. Ant.
dim 𝐺1 = 𝑟 × 𝑡 dim 𝐺2 = 𝑟 × 𝑡 dim 𝐱1 = 𝑡 × 1
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Capacity Region of Gaussian MIMO BC
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Special Cases Known Capacity Region
Aligned and Inconsistently Degraded MIMO BC
𝒕 = 𝒓, diagonal 𝑮𝟏, 𝑮𝟐 (𝐺1𝑇𝐺1 and 𝐺2
𝑇𝐺2 have the same set of Eigenvalue)
: A Product of Gaussian BC → Superposition Coding
Aligned MIMO BC (𝑀0 = 0)
Only Private Messages without a Common Message
→ Vector Writing on Dirty Paper
Degraded a Private Message and a Common Message
Either 𝑀0 = 0 or 𝑀0 = 0
→ Degraded Message Set → Superposition Coding
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Case 1 : Gaussian Product BC
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Parallel Gaussian BCs
Not Degraded, but Inconsistently Degraded BC
𝑌1𝑘 = 𝑋𝑘 + 𝑍1𝑘
𝑌2𝑘 = 𝑋𝑘 + 𝑍2𝑘 𝑘 ∈ [1: 𝑟] 𝑍𝑗𝑘 ∼ 𝓝(0,𝑁𝑗𝑘) 𝑗 = 1,2 𝑀. 𝐼.
𝑵𝟏𝒌 ≤ 𝑵𝟐𝒌
𝑵𝟐𝒌 > 𝑵𝟏𝒌
𝑘 ∈ [1: 𝑙]
𝑘 ∈ [𝑙 + 1: 𝑟]
+ 𝑌2𝑙
+
𝑍1𝑙 ∼ 𝒩(0,𝑁1)
𝑌1𝑙 𝑋1
𝑙
𝑍 2𝑙 ∼ 𝒩(0,𝑁2 − 𝑁1)
+ 𝑌1,𝑙+1𝑟
+
𝑍2,𝑙+1𝑟 ∼ 𝒩(0,𝑁2)
𝑌2,𝑙+1𝑟 𝑋𝑙+1
𝑟
𝑍 1,𝑙+1𝑟 ∼ 𝒩(0,𝑁1 − 𝑁2)
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Case 1 : Gaussian Product BC
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Capacity Region
𝑅0 + 𝑅1 ≤ 𝐶𝛽𝑘𝑃
𝑁1𝑘
𝑙
𝑘=1
+ 𝐶(𝛼𝑘𝛽𝑘𝑃
1 − 𝛼𝑘 𝛽𝑘𝑃 + 𝑁1𝑘)
𝑟
𝑘=𝑙+1
𝑅0 + 𝑅2 ≤ 𝐶𝛼𝑘𝛽𝑘𝑃
1 − 𝛼𝑘 𝛽𝑘𝑃 + 𝑁2𝑘
𝑙
𝑘=1
+ 𝐶𝛽𝑘𝑃
𝑁2𝑘
𝑟
𝑘=𝑙+1
𝑅0 + 𝑅1 + 𝑅2 ≤ 𝐶𝛽𝑘𝑃
𝑁1𝑘
𝑙
𝑘=1
+ [𝐶𝛼𝑘𝛽𝑘𝑃
1 − 𝛼𝑘 𝛽𝑘𝑃 + 𝑁1𝑘+ 𝐶(1 − 𝛼𝑘 𝛽𝑘𝑃
𝑁2𝑘)
𝑟
𝑘=𝑙+1
]
𝑅0 + 𝑅1 + 𝑅2 ≤ [𝐶𝛼𝑘𝛽𝑘𝑃
1 − 𝛼𝑘 𝛽𝑘𝑃 + 𝑁2𝑘+ 𝐶𝛽𝑘𝑃
𝑁1𝑘]
𝑙
𝑘=1
+ 𝐶(𝛽𝑘𝑃
𝑁2𝑘)
𝑟
𝑘=𝑙+1
For some 𝛼𝑘, 𝛽𝑘 ∈ [0,1], 𝑘 ∈ [1: 𝑟], with 𝛽𝑘𝑟𝑘=1 = 1
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Case 1 : Gaussian Product BC
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Rate Region
Achievability & Converse Proof
→ Superposition Coding (Degraded Gaussian BC)
𝑅0 + 𝑅1 ≤ 𝐼 𝑋1; 𝑌11 + 𝐼(𝑈2; 𝑌12)
𝑅0 + 𝑅2 ≤ 𝐼 𝑋2; 𝑌22 + 𝐼(𝑈1; 𝑌21)
𝑅0 + 𝑅1 + 𝑅2 ≤ 𝐼 𝑋1; 𝑌11 + 𝐼 𝑈2; 𝑌12 + 𝐼 𝑋2; 𝑌22 𝑈2)
𝑅0 + 𝑅1 + 𝑅2 ≤ 𝐼 𝑋2; 𝑌22 + 𝐼 𝑈1; 𝑌21 + 𝐼 𝑋1; 𝑌11 𝑈1)
For some pmf 𝑝 𝑢1, 𝑥1 𝑝( 𝑢2, 𝑥2)
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Case 1 : Gaussian Product BC
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Achievability Proof
Rate Splitting
𝑝(𝑦11|𝑥1)
𝑝(𝑦22|𝑥2)
𝑝(𝑦21|𝑦11)
𝑝(𝑦12|𝑦22)
𝑋1
𝑋2
𝑌11
𝑌22
𝑌21
𝑌12
(𝓧1, 𝑝 𝑦11 𝑥1 𝑝 𝑦21 𝑦11 , 𝓨11 ×𝓨21)
(𝓧2, 𝑝 𝑦22 𝑥2 𝑝 𝑦12 𝑦22 , 𝓨12 ×𝓨22)
Divide 𝑀𝑗, 𝑗 = 1,2 into two indep. Messages :
𝑀𝑗0 at rate 𝑅𝑗0, 𝑀𝑗𝑗 at rate 𝑅𝑗𝑗
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Case 1 : Gaussian Product BC
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Codebook Generation
Fix a pmf 𝑝 𝑢1, 𝑥1 𝑝(𝑢2, 𝑥2).
Randomly and indep. Generate 2𝑛(𝑅0+𝑅10+𝑅20) sequence pairs
𝑢1𝑛, 𝑢2𝑛 𝑚0, 𝑚10, 𝑚20
𝑚0, 𝑚10, 𝑚20 ∈ 1: 2𝑛𝑅0 × 1: 2𝑛𝑅10 × [1: 2𝑛𝑅20]
according to 𝑝𝑈1 𝑢1𝑖 𝑝𝑈2(𝑢2𝑖)𝑛𝑖=1
For 𝑚0, 𝑚10, 𝑚20 , randomly and conditionally indep. Generate 2𝑛𝑅𝑗𝑗 sequences
𝑥𝑗𝑛(𝑚0, 𝑚10, 𝑚20, 𝑚𝑗𝑗)
𝑚𝑗𝑗 ∈ [1: 2𝑛𝑅𝑗𝑗], 𝑗 = 1,2
according to 𝑝𝑋𝑗|𝑈𝑗(𝑥𝑗𝑖|𝑢𝑗𝑖(𝑚0, 𝑚10, 𝑚20)𝑛𝑖=1
Encoding
To send the message triple 𝑚0, 𝑚1, 𝑚2 = (𝑚0, 𝑚10, 𝑚11 , 𝑚20, 𝑚22 )
Transmit (𝑥1𝑛 𝑚0, 𝑚10, 𝑚20, 𝑚11 , 𝑥2
𝑛 𝑚0, 𝑚10, 𝑚20, 𝑚22 )
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Case 1 : Gaussian Product BC
4/11
Decoding and analysis of the probability of error
Decoder 1 : find unique triple (𝑚 01, 𝑚 10, 𝑚 11)
such that ((𝑢1𝑛, 𝑢2𝑛)(𝑚 01, 𝑚 10, 𝑚 11),𝑥1
𝑛 𝑚 01, 𝑚 10, 𝑚20, 𝑚 11), 𝑦1𝑛, 𝑦2𝑛 ∈ 𝑇𝜖
(𝑛)
For some 𝑚10.
Probability error for decoder 1
𝑅0 + 𝑅1 + 𝑅20 < 𝐼 𝑈1, 𝑈2, 𝑋1; 𝑌11, 𝑌12 − 𝛿(𝜖)
= 𝐼 𝑋1; 𝑌11 + 𝐼 𝑈2; 𝑌12 − 𝛿(𝜖)
𝑅11 < 𝐼 𝑋1; 𝑌11|𝑈1 − 𝛿(𝜖)
Probability error for decoder 2
𝑅0 + 𝑅10 + 𝑅2 < 𝐼(𝑋2; 𝑌22) + 𝐼(𝑈1; 𝑌21) − 𝛿(𝜖)
𝑅22 < 𝐼 𝑋2; 𝑌22|𝑈2 − 𝛿(𝜖)
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Case 2 : Private Messages
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Capacity Region
𝐑𝟏 : DPC with Non-causal State 𝐗𝟐𝒏
𝐑𝟐 : DPC with Non-causal State 𝐗𝟏𝒏
𝐂 = 𝐑𝑾𝑫𝑷 = 𝒄𝒐(𝐑𝟏 ∪ 𝐑𝟐)
𝑅2 <1
2log|𝐺2𝐾2𝐺2
𝑇 + 𝐺2𝐾1𝐺2𝑇 + 𝐼𝑟|
|𝐺2𝐾1𝐺2𝑇 + 𝐼𝑟|
𝑅1 <1
2log|𝐺1𝐾1𝐺1
𝑇 + 𝐺1𝐾2𝐺1𝑇 + 𝐼𝑟|
|𝐺1𝐾2𝐺1𝑇 + 𝐼𝑟|
𝑅1 <1
2log |𝐺1𝐾1𝐺1
𝑇 + 𝐼𝑟|
𝑅2 <1
2log |𝐺2𝐾2𝐺2
𝑇 + 𝐼𝑟|
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Vector Writing on Dirty Paper (1)
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Vector Writing on Dirty Paper
⊕ ⊕ Encoder Decoder
𝐒𝐧 𝐙 ∼ 𝓝(𝟎, 𝑰𝒓)
𝑀 𝒀 𝑊
𝐘 = 𝐺𝐗 + 𝐒 + 𝐙
Second noise channel (AWGN)
𝐗𝒏 Average power constraint
𝑷
𝐒 ∼ 𝓝(𝟎,𝑲𝑺)
𝐂 = max𝑡𝑟 𝐾𝑋 ≤𝑷
𝟏
𝟐𝐥𝐨𝐠 |𝑮 𝑲𝑿𝑮
𝑻 + 𝑰𝒓|
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Vector Writing on Dirty Paper (2)
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Proof of Capacity
𝐂 = max𝑡𝑟 𝐾𝑋 ≤𝑷
𝟏
𝟐𝐥𝐨𝐠 |𝑮 𝑲𝑿𝑮
𝑻 + 𝑰𝒓|
𝐶 = sup𝑝 𝐮 𝐬 ,𝐱 𝐮 𝐬 :E 𝐗𝑇𝐗 ≤𝑃
[ 𝐼 𝐔; 𝐘 − 𝐼 𝐔; 𝐒 ]
Let 𝐔 = 𝐗 + 𝐴𝐒, where 𝑋 ∼ 𝓝(0, 𝐾𝑋) is independent of 𝐒
𝐴 = 𝐾𝑋𝐺𝑇 𝐺 𝐾𝑋𝐺
𝑇 + 𝐼𝑟−1
𝐼 𝐔; 𝐘 − 𝐼 𝐔; 𝐒 = ℎ 𝐔 𝐒 − ℎ(𝐔|𝐘)
= ℎ 𝐗 + 𝐴𝐒 𝐒 − ℎ(𝐗 + 𝐴𝐒|𝐘)
= ℎ(𝐗) − ℎ(𝐗|𝐺𝐗 + 𝐙)
ℎ 𝐗 + 𝐴𝐒 𝐘 = ℎ(𝐗 + 𝐴𝐒 − 𝐴𝐘|𝐘)
= ℎ(𝐗 + 𝐴(𝐒 − 𝐘)|𝐘)
= ℎ(𝐗 + 𝐴(𝐺𝑿 + 𝒁)|𝐘)
= ℎ(𝐗 + 𝐴(𝐺𝑿 + 𝒁))
= ℎ(𝐗 + 𝐴(𝐺𝑿 + 𝒁)|𝐺𝐗 + 𝐙)
= ℎ(𝐗|𝐺𝐗 + 𝐙)
= 𝐼(𝐗; 𝐺𝐗 + 𝐙)
=𝟏
𝟐𝐥𝐨𝐠 |𝑰𝒓 + 𝑮 𝑲𝑿𝑮
𝑻|
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Vector Writing on Dirty Paper (3)
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𝐑𝟏
𝑴𝟏- Encoder 𝑀1 ⊕
⊕
𝐗𝑛
𝐘1𝑛
𝐘2𝑛
𝐙2𝑛
𝐙1𝑛
𝐺1
𝐺2 𝑴𝟐-Encoder
𝑀2 ⊕
𝐗𝟏𝒏
𝐗𝟐𝒏
𝐘1 = 𝐺1𝐗1 + 𝐺1𝐗2 + 𝐙1
𝐘2 = 𝐺2𝐗2 + 𝐺2𝐗1 + 𝐙2
𝑅1 < 𝐼 𝐗1; 𝐺1𝐗1 + 𝐙1 =1
2log |𝐺1𝐾1𝐺1
𝑇 + 𝐼𝑟|
𝑅2 < 𝐼 𝐗2; 𝐺2𝐗1 + 𝐺2𝐗2 + 𝐙2 =1
2log|𝐺2𝐾1𝐺2
𝑇 + 𝐺2𝐾2𝐺2𝑇 + 𝐼𝑟|
|𝐺2𝐾1𝐺2𝑇 + 𝐼𝑟|
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Vector Writing on Dirty Paper
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𝐑𝟐
𝑴𝟐- Encoder
𝑀1 ⊕
⊕
𝐗𝑛
𝐘1𝑛
𝐘2𝑛
𝐙2𝑛
𝐙1𝑛
𝐺1
𝐺2
𝑴𝟏-Encoder
𝑀2 ⊕
𝐗𝟏𝒏
𝐗𝟐𝒏
𝐘1 = 𝐺1𝐗1 + 𝐺1𝐗2 + 𝐙1
𝐘2 = 𝐺2𝐗2 + 𝐺2𝐗1 + 𝐙2
𝑅2 < 𝐼 𝐗2; 𝐺2𝐗2 + 𝐙2 =1
2log |𝐺2𝐾2𝐺2
𝑇 + 𝐼𝑟|
𝑅1 < 𝐼 𝐗1; 𝐺1𝐗1 + 𝐺1𝐗2 + 𝐙1 =1
2log|𝐺1𝐾1𝐺1
𝑇 + 𝐺1𝐾2𝐺1𝑇 + 𝐼𝑟|
|𝐺1𝐾2𝐺1𝑇 + 𝐼𝑟|
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Capacity Region of Gaussian MIMO BC
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BC-MAC Duality
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⊕
⊕
𝐗
𝐘1
𝐘2
𝐙2 ∼ 𝒩(0, 𝐼𝑟)
𝐙1 ∼ 𝒩(0, 𝐼𝑟)
𝐺1
𝐺2
⊕
𝐙 ∼ 𝒩(0, 𝐼𝑡) 𝐺1𝑇
𝐺2𝑇
𝐘
𝐗1
𝐗2
𝐶𝐵𝐶𝐷𝑃 𝑃; 𝐺1, 𝐺2 = 𝐶𝑀𝐴𝐶(𝑃1, 𝑃2; 𝐺1
𝑇 , 𝐺2𝑇)
𝑡𝑟 𝑃𝑖 ≤𝑃2𝑖=1
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MIMO Multiple Access Channel
3/11
System Structure
channel
𝐘 = 𝐺1𝐗1 + 𝐺2𝐗2 + 𝐙
Power Constraint
1
𝑛 𝐱𝑗
𝑇 𝑚𝑗, 𝑖 𝐱𝑗(𝑚𝑗, 𝑖 )
𝑛
𝑖=1
≤ 𝑃
𝑚𝑗 ∈ 1: 2𝑛𝑅𝑗 , 𝑗 = 1,2
𝐙 ∼ 𝓝(0, 𝐼𝑟)
𝑀1
𝑀2
⊕
𝐙 ∼ 𝒩(0, 𝐼𝑟) 𝐺1
𝐺2
𝐘𝑛
𝐗1𝑛
𝐗2𝑛
Decoder
Encoder 1
Encoder 2
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MIMO MAC
4/11
Capacity Region
Boundary Point 𝑅∗
𝑅1 ≤1
2log |𝐺1𝐾1𝐺1
𝑇 + 𝐼𝑟|
𝑅2 ≤1
2log |𝐺2𝐾2𝐺2
𝑇 + 𝐼𝑟|
𝑅1 + 𝑅2 ≤1
2log |𝐺1𝐾1𝐺1
𝑇 + 𝐺2𝐾2𝐺2𝑇 + 𝐼𝑟|
𝑅1∗ =1
2log |𝐺1𝐾1
∗𝐺1𝑇 + 𝐼𝑟|
𝑅2∗ =1
2log |𝐺1𝐾1𝐺1
𝑇 + 𝐺2𝐾2𝐺2𝑇 + 𝐼𝑟| −
1
2log |𝐺1𝐾1𝐺1
𝑇 + 𝐼𝑟|
=1
2log|𝐺1𝐾1
∗𝐺1𝑇 + 𝐺2𝐾2
∗𝐺2𝑇 + 𝐼𝑟|
|𝐺1𝐾1∗𝐺1𝑇 + 𝐼𝑟|
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Achievability Proof : DPC Capacity Region
/
Using Dual MAC
𝐑𝑊𝐷𝑃 = 𝐂𝐷𝑀𝐴𝐶 = 𝐑(𝐾1, 𝐾2)
𝐾1,𝐾2≽0:𝑡𝑟 𝐾1 +𝑡𝑟 𝐾2 ≤𝑃
𝑅1∗, 𝑅2∗ of 𝐂𝐷𝑀𝐴𝐶 lies on the boundary of (𝐾1, 𝐾2)
max𝛼∈ 0,1 , 𝑅1,𝑅2 ∈𝐂𝐷𝑀𝐴𝐶
[𝛼𝑅1 + 𝛼 𝑅2]
max𝛼∈ 0,1 ,𝑡𝑟 𝐾1 +𝑡𝑟 𝐾2 ≤𝑃,𝐾1,𝐾2≽0}
[𝛼
2log 𝐺1𝐾1𝐺1
𝑇 + 𝐺2𝐾2𝐺2𝑇 + 𝐼𝑟 +
𝛼 − 𝛼
2log |𝐺2𝐾2𝐺2
𝑇 + 𝐼𝑟|]
Introducing Dual Variables
𝑡𝑟 𝐾1 + 𝑡𝑟 𝐾2 ≤ 𝑃
𝐾1, 𝐾2 ≽ 0
𝜆 ≥ 0
𝛾1, 𝛾2 ≽ 0
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Achievability Proof : DPC Capacity Region
4/11
𝜆∗𝐺1𝑆1𝐺1𝑇 + 𝛾1
∗ − 𝜆∗𝐼𝑟 = 0
𝐿 𝐾1, 𝐾2, 𝛾1, 𝛾2, 𝜆 =𝛼
2log 𝐺1𝐾1𝐺1
𝑇 + 𝐺2𝐾2𝐺2𝑇 + 𝐼𝑟 +
𝛼 − 𝛼
2log |𝐺2𝐾2𝐺2
𝑇 + 𝐼𝑟|
Applying KKT
+𝑡𝑟 𝛾1𝐾1 + 𝑡𝑟 𝛾2𝐾2 − 𝜆[𝑡𝑟 𝐾1 + 𝑡𝑟 𝐾2 − 𝑃)
𝜆∗𝐺2𝑆2𝐺2𝑇 + 𝛾2
∗ − 𝜆∗𝐼𝑟 = 0
𝜆∗ 𝑡𝑟 𝐾1∗ + 𝑡𝑟 𝐾2
∗ − 𝑃 = 0
𝑡𝑟 𝛾1𝐾1 = 𝑡𝑟(𝛾2𝐾2) = 0
𝑆1 =𝛼
2𝜆∗𝐺1𝑇𝐾1∗𝐺1 + 𝐺2
𝑇𝐾2∗𝐺2 + 𝐼𝑟
−1
𝑆2 =𝛼
2𝜆∗𝐺1𝑇𝐾1∗𝐺1 + 𝐺2
𝑇𝐾2∗𝐺2 + 𝐼𝑟
−1 +𝛼 − 𝛼
2𝜆∗𝐺2𝑇𝐾2∗𝐺2 + 𝐼𝑟
−1
𝐾1∗∗ =𝛼
2𝜆∗𝐺2𝑇𝐾2∗𝐺2 + 𝐼𝑟
−1 − 𝑆1
𝐾2∗∗ =𝛼
2𝜆∗𝐼𝑟 − 𝐾1
∗∗ − 𝑆2