limited feedback in wireless communication...

Adaptive Signal Processing Information Theory Group

Gwanmo Ku

May 14, 17, and 21, 2013

Limited Feedback in

Wireless Communication Systems

- Summary of “An Overview of Limited Feedback

in Wireless Communication Systems”


OutlineTransmitter Receiver

… …

Ant. 1

Ant. 2

Ant. 𝑴𝒕

Ant. 1

Ant. 2

Ant. 𝑴𝒓

Feedback Design

𝐇

Limited Feedback 𝐇

𝐅

Codebook Design

Channel

2

Narrowband (NB) Broadband (BB)

Single User (SU) / Multiple User (MU)

Single Antenna (SA) Multiple Antenna (MA)


𝐒 𝐗

𝐍

𝐘



Ant. 1 Ant. 1

Feedback Design

𝒉[𝒌]


𝒇[𝒌]

Codebook Design

3

Narrowband (NB) Broadband

Single User (SU) / Multiple User

Single Antenna (SA) Multiple Antenna

Narrowband Broadband

𝑠[𝑘] 𝑥[𝑘]

𝑛[𝑘]

𝑦[𝑘]

𝒚 𝒌 = 𝒉 𝒌 𝒙 𝒌 + 𝒏[𝒌]

Slow fading


SU - SA - NB with Perfect CSI12/22

Received Signal

Adaptive Power Allocation by Waterfilling

𝜌(ℎ) = arg max𝐄ℎ 𝑘 ,𝑥[𝑘] 𝑥 𝑘 2 ≤𝜌

log2(1 + 𝜌 ℎ[𝑘] ℎ[𝑘] 2)

How to measure Channel State ℎ[𝑘] at the receiver?

By Training Sequence from Tx.

𝑦 𝑘 = ℎ 𝑘 𝑥 𝑘 + 𝑛[𝑘]

𝑥 𝑘 = 𝜌(ℎ[𝑘]) 𝑠[𝑘]

4


Water-filling12/22

𝑌𝑗 = 𝑋𝑗 + 𝑍𝑗

𝐼 𝑋1, … , 𝑋𝑘; 𝑌1, … , 𝑌𝑘 ≤ 1

2log(1 +

𝑃𝑖

𝑁𝑖)

𝑘

𝑖=1

𝑗 = 0, … , 𝑘, 𝑍𝑗 ∼ 𝑁(0, 𝑁𝑗), 𝐄 𝑋𝑗2𝑘

𝑗=1 ≤ 𝑃

𝐿 𝑃1, … , 𝑃𝑘 = 1

2log(1 +

𝑃𝑖

𝑁𝑖)

𝑘

𝑖=1

+ 𝜆( 𝑃𝑖

𝑘

𝑖=1

− 𝑃)

𝑃𝑖 = 𝜈 − 𝑁𝑖+ 𝜈 − 𝑁𝑖

+

𝑘

𝑖=1

= 𝑃

1

2

1

𝑃𝑖 + 𝑁𝑖+ 𝜆 = 0

𝜈

Power

Channel

𝑃1

𝑁1 𝑁2

𝑃2

𝑃3 𝑃5

𝑃4 = 0

𝑁3

𝑁4

𝑁5

Channel 1 Channel 2 Channel 3 Channel 4 Channel 5

5

Ref : Elements of Information Theory by Thomas Cover


Limited Feedback : SU-SA-NB12/22

1. Quantization of ℎ[𝑘], 𝛾 = ℎ 𝑘 2

2. Rate Quantization by Lloyd algorithm

Find Quantization Level minimizing Distortion Measure

Optimal Rate Partitions associated with Limited feedback

3. On/Off Rate Adaptation

1 bit : On/Off Transmission subject to the channel condition

4. ARQ ACK/NACK Signalling

1 bit : Successful Transmission or Re-transmission Required

6

𝐼 𝛾 = 𝑖, 𝛾 ∈ [𝛾𝑖𝑏 , 𝛾𝑖+1

𝑏 ), 𝑖 = 0, … , 𝑄 − 1



Ant. 1

Ant. 1

Feedback Design

𝒉[𝒌, 𝒍]


𝒇[𝒌]

Codebook Design

Channel

7

Narrowband Broadband (BB)


Single Antenna (SA) Multiple Antenna


𝑠[𝑘] 𝑥[𝑘] 𝑛[𝑘]

𝑦[𝑘]

𝒚 𝒌 = 𝒉 𝒌, 𝒍 𝒙[𝒌 − 𝒍]

𝑳

𝒍=𝟎

+ 𝒏[𝒌]

𝑳 + 𝟏

Paths


SU-SA-BB with Perfect CSI14/22

Received Signal, OFDM Signalling

Post-processing Signal in Frequency Domain

Subcarrier Power Allocation

𝑦 𝑘 = ℎ 𝑘, 𝑙 𝑥[𝑘 − 𝑙]

𝐿

𝑙=0

+ 𝑛[𝑘]

𝐲 𝑘 = diag 𝐡 [𝑘 ] 𝐱 𝑘 + 𝐧 𝑘

8

𝑥𝑣[𝑘 ] = 𝑃𝑣 𝑠𝑣[𝑘 ] 𝐱 𝑘 =

𝑥1[𝒌 ]…

𝑥𝑣[𝒌 ]…

𝑥𝑁[𝒌 ]


Limited Feedback : SU-SA-BB12/22

1. Subcarrier On/Off Signaling

# subcarrier bits : Active or inactive subcarrier

2. Subcarrier Grouping : Subchannelization

# subchannel bits

3. Order of Pilot Channel Gain

Order index within 𝑵𝒑𝒊𝒍𝒐𝒕! sets, 𝒉𝒎𝒊𝒏,𝒑𝒊𝒍𝒐𝒕𝒔 & 𝒉𝒎𝒂𝒙,𝒑𝒊𝒍𝒐𝒕𝒔

4. Adaptive Modulation and Coding

Level Index : Modulation and Coding Scheme ~ SNR

9



… …

Ant. 1

Ant. 2

Ant. 𝑴𝒕

Ant. 1

Ant. 2

Ant. 𝑴𝒓

Feedback Design

𝐇[𝐤]


𝐅

Codebook Design

Channel

10



Single Antenna Multiple Antenna (MA)

Narrowband (NB) Broadband

𝐬[𝐤] 𝐱[𝐤]

𝐧[𝐤]

𝐲[𝐤]

𝐲 𝑘 = 𝐇 𝑘 𝐱 𝑘 + 𝐧[𝑘]


SU-MA-NB with Perfect CSI16/22

Received Signal

𝑀𝑡 × 𝑀𝑟 MIMO System

𝐲 𝑘 : 𝑀𝑟 × 1 Complex Received Vector

𝐇 𝑘 : 𝑀𝑟 × 𝑀𝑡 , each entry has flat fading property

𝐱 𝑘 : Transmitted Symbol with Power Constraint 𝐄𝐇,𝐱 | 𝐱 𝑘 |𝟐𝟐

≤ 𝜌

Average Power Constraint : 𝐄𝐱 𝐱 𝑘𝟐

𝟐|𝐇 𝑘 = 𝐇(𝑡) ≤ 𝜌𝑡

under 𝐄𝐇 𝜌𝑡 ≤ 𝜌

𝐧 𝑘 : 𝑀𝑟 × 1 Complex Gaussian Noise according to 𝐂𝐍(0,1)

𝐲 𝑘 = 𝐇 𝑘 𝐱 𝑘 + 𝐧[𝑘]

11


SU-MA-NB : Rate Maximizing17/22

Adaptive Power Allocation

• 𝐐 𝑘 : Covariance of the Tx. Sig. for each 𝐇[𝑘]

𝐐 𝑘 = arg max𝐐:tr 𝐐 ≤1,𝐐∗=𝐐,𝐐≥0

log2 det(𝐈 + 𝜌𝐇 𝑘 𝐐 𝐇∗ 𝑘 )

• Ergodic Capacity : 𝑅 = 𝐄𝐇[ max𝐐:tr 𝐐 ≤1 ,𝐐∗=𝐐,𝐐≥0

log2 det 𝐈 + 𝜌𝐇𝐐𝐇∗ ]

• s 𝑘 : Channel independent Codeword with 𝐄𝑠 𝑠 𝑘 𝟐 ≤ 1

𝐱 𝑘 = 𝜌 𝐐 𝑘12𝐬[𝑘]

12


Covariance Quantization17/22

Codebook of Possible Cov. Matrices

ℚ = {𝐐𝟏, … , 𝐐𝟐𝑩}

• Rate Maximizing Covariance Selecting

𝑛𝑜𝑝𝑡 𝑘 = arg max1≤𝑛≤2𝐵

log2 det(𝐈 + 𝜌𝐇 𝑘 𝐐𝑛𝐇∗ 𝑘 )

• Maximum Achievable Rate

𝑅ℚ = 𝐄𝐇[max𝐐∈ℚ

log2 det(𝐈 + 𝜌𝐇𝐐𝐇∗)]

• Codebook Generation ℚ based on VC using Lloyd Algorithm

13


Vector Quantization using Lloyd Algorithm17/22

Step 1. Determine { 𝐐𝟏, 𝑹𝟏 , … , 𝐐𝟐𝑩 , 𝑹𝟐𝑩 } for an

initial partition {ℋ𝟏, … , ℋ𝟐𝑩}

Define a distortion measure

14

𝓠∗, 𝓡∗ = arg max𝑄1,𝑅1 ,…,{𝑄

2𝐵 ,𝑅2𝐵}

𝐄𝐇 𝑑 𝐇, 𝑖 𝐇 ∈ ℋ𝑖 Pr[𝐇 ∈ ℋ𝑖]

2𝐵

𝑖=1

= arg max𝑄1,𝑅1 ,…,{𝑄

2𝐵 ,𝑅2𝐵}

𝑅𝑗 ⋅ Pr[𝑅𝑗 < log2 det 𝐈 + 𝐇𝐐𝑗𝐇∗ |𝐇 ∈ ℋ𝑖]

2𝐵

𝑖=1

2𝐵

𝑗=1

⋅ Pr 𝐇 ∈ ℋ𝑖 ⋅ 𝑃𝑖𝑗𝐶𝑆𝐼𝑇

𝓠∗ = {𝐐1∗ , … , 𝐐

2𝐵∗ } 𝓡∗ = {𝑅1

∗, … , 𝑅2𝐵∗ }

𝑑 𝐇, 𝑖 = 𝑅𝑗 ⋅ 1(𝑅𝑗 < log2 det 𝐈 + 𝐇𝐐𝑗𝐇∗ ) ⋅ 𝑃𝑖𝑗

𝐶𝑆𝐼𝑇

2𝐵

𝑖=1


Vector Quantization using Lloyd Algorithm17/22

Step 2. Determine partitions {ℋ𝟏, … , ℋ𝟐𝑩} for a

given 𝓠, 𝓡

15

ℋ𝑖∗ = {𝐇 ∈ ℂ𝑀𝑟×𝑀𝑡: 𝑑 𝐇, 𝑖 ≥ 𝑑 𝐇, 𝑘 , ∀𝑖, 𝑘 ∈ 1, … , 2𝐵 , 𝑖 ≠ 𝑘}

ℋ∗ = {ℋ𝟏∗, … , ℋ

𝟐𝑩∗ }

= {𝐇 ∈ ℂ𝑀𝑟×𝑀𝑡: 𝑅𝑗 ⋅ 1[𝑅𝑗 < log2 det 𝐈 + 𝐇𝐐𝑗𝐇∗ ] ⋅ 𝑃𝑖𝑗

𝐶𝑆𝐼𝑇

2𝐵

𝑗=1

≥ 𝑅𝑗 ⋅ 1[𝑅𝑗 < log2 det 𝐈 + 𝐇𝐐𝑗𝐇∗ ] ⋅ 𝑃𝑘𝑗

𝐶𝑆𝐼𝑇

2𝐵

𝑗=1

∀𝑖, 𝑘 ∈ 1, … , 2𝐵 , 𝑖 ≠ 𝑘}

Repeat Step 2 & 3 Until Convergence


Beamforming in MISO : Rank One 𝐐18/22

Beamforming Vector

• 𝐟[𝒌] : Channel Dependent Beamforming Vector,

𝐟[𝑘] 2 = 1

• 𝑀𝑡 × 1 MISO Case

• 𝐡 𝑘 : Perfect Channel Column Vector

𝐱 𝑘 = 𝜌𝐟[𝑘]𝑠[𝑘]

𝐟 𝑘 = arg max𝐟: 𝐟 =1

log2(1 + 𝜌 𝐡𝑇𝐟 2)

16


Limited Feedback for Beamforming20/22

1. Antenna Selection

2. Channel Vector Quantization ℋ = {𝐡𝟏, … , 𝐡𝟐𝑩}

17

𝑚𝑜𝑝𝑡 𝑘 = arg max1≤𝑚≤𝑀𝑡

ℎ𝑚 𝑘 2

𝑛𝑜𝑝𝑡 𝑘 = arg max1≤𝑛≤2𝐵

𝐡𝑛∗ 𝐡 𝑘 2

𝐟 𝑘 = arg max𝐟: 𝐟 =1

log2(1 + 𝜌 𝐡𝑛𝑜𝑝𝑡[𝑘]𝑇 𝐟

2)

=𝐡𝑛𝑜𝑝𝑡[𝑘]

𝑇∗

𝐡𝑛𝑜𝑝𝑡 𝑘2


Limited Feedback for Beamforming20/22

3. K-Phase Quantization for 𝟐 × 𝟏 MISO

4. Codebook Index within 𝓕 = {𝐟𝟏, … , 𝐟𝟐𝑩}

Grassmannian Line Packing maximizing min. 𝑑(ℱ)

18

𝑛𝑜𝑝𝑡[𝑘] = arg max1≤𝑖≤𝐾

𝐡𝑇 𝑘 𝐟𝒊2

𝐟𝑖 =1

2

1

𝑒𝑗2𝜋𝑖𝐾

𝑑 𝓕 = 1 − max1≤𝑖<𝑗≤2𝐵

𝐟𝑖∗𝐟𝑗

2= min

1≤𝑖<𝑗≤2𝐵sin 𝜃𝑖𝑗

𝓕 ∈ ℂ𝑀𝑡 𝐟𝑖 2 = 1


Limited Feedback for Spatial Multiplexing20/22

Linear Precoding for Spatial Multiplexing

• 𝐅 𝑘 : Precoding Matrix, 𝑀𝑡 × 𝑀, 𝐅 𝑘 𝐹2 ≤ 𝑀

• 𝐬[𝑘] : Signal Vector, 𝐄𝐬 𝐬 𝑘 𝐬∗ 𝑘 =1

𝑀𝐈

𝐲 𝑘 = 𝜌𝐇 𝑘 𝐅 𝑘 𝐬 𝑘 + 𝐧[𝑘]

19


Limited Feedback for Spatial Multiplexing20/22

1. Antenna Subset Selection

Choose 𝑀 antenna ports for Power Control or Rate Maximization

2. Codebook 𝓕 = {𝐅𝟏, … , 𝐅𝟐𝑩}

Grassmannian 𝑴-Dim. Line Packing

Householder Reflection Matrix

20

𝐅 𝑘 = 𝑐ℎ𝑜𝑜𝑠𝑒 𝑀 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 [𝐼𝑀𝑡×𝑀𝑡]

log2𝑀𝑡

𝑀 bits


Limited Feedback : Space-Time Coding20/22

Received Signal

Limited Feedback

1. Codebook 𝓕 = {𝐅𝟏, … , 𝐅𝟐𝑩}

Grassmannian Subspace Packing

2. Rate Adaptation : Adaptive Constellation

Feedback of Constellation Size ~ SNR

3. Quantized Phase

𝐘𝑀𝑟×𝑀𝑆𝑇𝑘 = 𝜌𝐇 𝑘 𝐅 𝑘 𝐒𝑀×𝑀𝑆𝑇

𝑘 + 𝐍𝑀𝑟×𝑀𝑆𝑇[𝑘]

21



… …

Ant. 1

Ant. 2

Ant. 𝑴𝒕

Ant. 1

Ant. 2

Ant. 𝑴𝒓

Feedback Design

𝐇


𝐅

Codebook Design

Channel

22





𝑺 𝐗

𝐍

𝐘

𝐲 𝑣 𝑘 = 𝐇 𝑣 𝑘 𝐱 𝑣 𝑘 +𝐧 𝑣 𝑘


SU-MA-BB in Frequency Domain 21/22

Received Signal in Frequency Domain

𝒗 : Subcarrier Index

Subcarrier Power Allocation

𝜌𝑣 : SNR on subcarrier 𝑣

𝐲 𝑣 𝑘 = 𝐇 𝑣 𝑘 𝐱 𝑣 𝑘 +𝐧 𝑣 𝑘

𝐱 𝑣 𝑘 = 𝜌𝑣 𝐅 𝑣 𝑘 𝐬 𝑣[𝑘 ]

23


Limited Feedback : SU-MA-BB22/22

Limited Feedback using Interpolation

𝐟𝑗

𝐟𝑖

𝐟𝑘

Interpolated Subcarriers

𝑖 th subcarrier Feedback

(Pilot)

Reported Pilot Feedback

Vectors

Interpolated vectors

24

𝐟 𝑘 =𝑏𝑖𝐟𝑖 + 𝑏𝑗𝐟𝑗

𝑏𝑖𝐟𝑖 + 𝑏𝑗𝐟𝑗 2

𝑏𝑖 , 𝑏𝑗 ≥ 0

𝑏𝑖 + 𝑏𝑗 = 1

𝑓𝑖 2 = 𝑓𝑗 2= 1

𝐰𝑙 𝐰𝑙+1

𝐰 𝑙𝐾 + 𝑘; 𝜃𝑙 =1 − 𝑐𝑘 𝐰𝑙 + 𝑐𝑘𝑒𝑗𝜃𝑙𝐰𝑙+1

1 − 𝑐𝑘 𝐰𝑙 + 𝑐𝑘𝑒𝑗𝜃𝑙𝐰𝑙+1

𝑐𝑘 =𝑘 − 1

𝐾 0 ≤ 𝑘 ≤ 𝐾

𝜃𝑙 :Phase Rotation

𝒘


OutlineTransmitter

User 1

…

Ant. 1

Feedback Design

𝐅

Codebook Design

25





𝐬𝟏

𝐬𝑈

… User 2

User U


Milti-user & Single Transmit Antenna21/22

Resource Scheduling for Multiusers

Ensure larger rate & better reliability

Maximum Throughput ~ Largest Received SNR

Needs Each User Receiver SNR

SNR Limited Feedback

One bit according to predefined threshold SNR

Quantized SNR of Each User

Quantized SNR of Subchannels in FDMA

26


OutlineTransmitter

…

Ant. 1

Ant. 2

Ant. 𝑴𝒕

𝐅

Codebook Design

27





𝐬𝟏

𝐬𝑈

… Ant. 𝑴𝒓

Receiver 1 …

Ant. 𝟏

Receiver U …

Ant. 𝟏 …

Ant. 𝑴𝒓


Milti-user MIMO21/22

Resource Scheduling for Multiusers

Maximizing Sum Rate

Spatial Resource Scheduling

Precoding (𝐅) for Spatial Interference Cancellation

Limited Feedback in MISO

Quantization of 𝐡𝑖[𝑘]

1 Bit Effective SNR or Quantized Effective SNR

28

𝑦𝑖 𝑘 = 𝐡𝑖𝑇 𝑘 𝐱 𝑘 + 𝑛𝑖[𝑘]

𝐱 𝑘 = 𝜌𝐅 𝑘 𝐬[𝑘]


Milti-user MIMO21/22

Limited Feedback in MIMO

Quantized Codebook Index based on VQ

Block Diagonalization Information

Quantized Elements of Channel Matrix

Antenna Selection Information

Limited Feedback associated with Relay

1 bit for Relay Selection

Codebook based feedback : Grassmannian or Lloyd Algorithm

29


21/22

30

Codebook Based Feedback in Standards

• 3GPP : WCDMA / LTE

• IEEE : WiMAX / WiFi

• 3GPP2 : CDMA


Limited Feedback in WCDMA21/22

Adaptive Technique

Open/Closed-loop Transmit Diversity (Tx.D)

𝟐 × 𝟐 MIMO

Limited Feedback for Closed-loop Tx.D

1 bit Phase Adjustment : Equal Gain Combining

0 or 𝜋 Phase Adjustment

4 bits Quantized Index : Amplitude & Phase

31


4 bit Quantization in WCDMA21/22

3 bits Phase & 1 bit Amplitude Quantization

32

Feedback Bits Phase

000 𝜋

001 −3𝜋

4

010 −2𝜋

4

011 −𝜋

4

100 0

101 𝜋

4

110 2𝜋

4

111 3𝜋

4

Feedback Bit 𝑷𝟏 / 𝑷𝟐

0 0.2 / 0.8

1 0.8 / 0.2


Limited Feedback in LTE21/22

Adaptive Technique

𝟒 × 𝟒 MIMO (𝟖 × 𝟖 for LTE Advanced)

Transmit Diversity & Spatial Multiplexing

Limited Feedback

Quantized 4 bit CQI Index

2 or 3 bit Differential CQI Feedback in Multiple CQI Reporting

Predefined Precoding Matrix Index (PMI)

PMI based on Householder Reflection Matrix

33


PMIs in LTE21/22

For 2 Antenna Ports

For 4 Antenna Ports (16 Possibilities)

A Set of Column Vectors from Householder Reflection Matrix

34

𝐖𝑛 = 𝐈 − 𝐮𝑛𝐮𝑛

𝐻

𝐮𝑛𝐻𝐮𝑛


Limited Feedback in IEEE 802-11n21/22

Adaptive Technique

𝟒 × 𝟒 MIMO


Limited Feedback per Subcarrier

Quantized Elements of Channel Matrix

A 3 bit Maximum Value

Scaling of Each Real and Imaginary Parts

35

3 + 2 ⋅ 𝑁𝑏 ⋅ 𝑀𝑅 ⋅ 𝑀𝑇 bits

𝑁𝑏 ∈ {4,5,6,8}


Limited Feedback in WiMAX21/22

Adaptive Technique

𝟒 × 𝟒 MIMO


Limited Feedback

Precoding Matrix

Householder Reflection Matrix

3 bits or 6 bits Indeces

36


PMIs in WiMAX21/22

37


Limited Feedback in 3GPP221/22

Adaptive Technique

𝟒 × 𝟒 MIMO

Limited Feedback

Precoding Matrix

Knockdown Codebook

- Identity or Q-level Fourier Matrix

Readymade Precoding Matrix : 64 entries

38


PMIs in 3GPP221/22

Q – Level Fourier Matrix Generation

39

𝒬 = {𝐄𝑀0

, 𝐄𝑀1

, … , 𝐄𝑀𝑄−1

}

𝐄𝑀(𝑞)

= 𝑓𝑛𝑚𝑞

= 𝑒𝑗2𝜋𝑛𝑀 (𝑚+

𝑞𝑄)

𝐄4(0)

=1

2

1 11 𝑗

1 1−1 −𝑗

1 −11 −𝑗

1 −1−1 𝑗

𝐄4(1)

=1

2

1 11 + 𝑗

2

−1 + 𝑗

2

1 1−1 − 𝑗

2

1 − 𝑗

2𝑗 −𝑗

−1 + 𝑗

2

1 + 𝑗

2

𝑗 −𝑗1 − 𝑗

2

−1 − 𝑗

2


Summary : Limited Feedback21/22

Scalar Quantization

User SNR Qunatization

Subchannelization

On/Off Signaling of User/Antenna/Subchannel Selection

ACK/NACK Signaling

Use AMC Table

40



Vector Quantization

By Lloyd Algorithm

By Grassmannian Line Packing

Beamformaing Vector Quantization

Phase Quantization

Interpollation Vector

41



Quantization in Multiuser MIMO

Element Quantization of Channel Vector

Quantization of Effective User SNR

Limited Feedback per Subcarrier in Broadband System

Limited Feedback in Standards

Codebook based Quantization Index

- Householder Reflection Matrix

- Grassmannian Line Packing

- Fourier Matrix

42

limited feedback in wireless communication...

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