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Overview of Gaussian

MIMO (Vector) BC

Gwanmo Ku

Adaptive Signal Processing and Information Theory Research Group

Nov. 30, 2012

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

/

Gaussian MIMO (Vector) BC

3/11

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

Capacity Region of Gaussian MIMO BC

3/11

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

Case 1 : Gaussian Product BC

4/11

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)

Case 1 : Gaussian Product BC

4/11

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

Case 1 : Gaussian Product BC

4/11

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)

Case 1 : Gaussian Product BC

4/11

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

Case 1 : Gaussian Product BC

4/11

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 )

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)

Case 2 : Private Messages

4/11

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|

Vector Writing on Dirty Paper (1)

4/11

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|

Vector Writing on Dirty Paper (2)

4/11

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|

Vector Writing on Dirty Paper (3)

3/11

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|

Vector Writing on Dirty Paper

3/11

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|

Capacity Region of Gaussian MIMO BC

4/11

BC-MAC Duality

4/11

⊕

⊕

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

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

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|

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

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