Linear Precoding in MIMO Wireless Systemss3.amazonaws.com/sdieee/143-MIMO-Feedback-FInal.pdfLinear Precoding in MIMO Wireless Systems Bhaskar Rao Center for Wireless Communications

Linear Precoding in MIMO Wireless Systems

Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego

Acknowledgement: Y. Isukapalli, L. Yu, J. Zheng, J. Roh

Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 1 / 48

Outline

1 Promise of MIMO Systems

2 Point to Point MIMO

3 Limited Feedback MIMO Systems

4 MIMO-OFDM

5 Multi-User MIMO


Outline




4 MIMO-OFDM

5 Multi-User MIMO


Multiple Input Multiple Output (MIMO) Systems

A system with multiple antennas at the transmitter and multipleantennas at the receiver.

Enables Spatio-Temporal processing and the goal is to exploitthe spatial dimension to increase system throughput

Multi-Input Multi-Output System


Textbooks

Introduction to Space-Time Wireless Communications, A.Paulraj, R. Nabar and D. Gore, Cambridge University Press

Fundamentals of Wireless Communications, D. Tse and P.Vishwanath

Space-Time Coding, H. Jafarkhani

MIMO Wireless Communications, Edited by Biglieri, Calderbank,et al


Benefits of MIMO Systems

Increased Network Capacity

Improved Signal Quality

Increased Coverage

Lower Power Consumption

Higher Data Rates

These requirements are often conflicting. Need balancing tomaximize system performance


Technical Rationale

Spatial Diversity to Combat Fading

Spatial Signature for Interference Management

Array Gain enables Lower Power Consumption

Capacity Improvements using Spatial Multiplexing


Outage Capacity of MIMO SystemsCapacity of MIMO systems


Outline




4 MIMO-OFDM

5 Multi-User MIMO


MIMO Channel Model

Input-Output relation for a discrete-time frequency-flat r × tMIMO channel

y =

√Es

tHs + n

y = [y1, y2, · · · , yr ]T · · · r × 1 receive signal vectors = [s1, s2, · · · , st ]T · · · t × 1 transmit signal vectorn = [n1, n2, · · · , nr ]T · · · r × 1 noise vector at the receiver

H is the r × t channel matrix

Es average energy over a symbol period

ni ∼ NC(0,No) with E [nnH ] = No Ir


MIMO Options

Channel assumed known at Receiver

Channel unknown at transmitter

Diversity Gain: Orthogonal space-time block codes, Space timetrellis codesSpatial Multiplexing: V-Blast, D-Blast

Channel known at the transmitter- Transmit precoding


Transmitter With Channel Knowledge

SVD of H can be expressed as

H = UΣVH

UHU = VHV = IrΣ = diag(σm)k

m=1, σm > 0

Further, HHH is Hermitian with eigendecomposition

HHH = UΛUH

Λ = diag(λm)km=1, σm ≥ σm+1 with λm = 0 for m > k and

λm = σ2m


Transmitter With Channel Knowledge Cont’d

Transmitted vector s = Vs

Input vector s is of dimension r × 1 with E [ssH ] = Γt , Γt

diagonal

Received signal transformed to y = UHy

y =

√Es

tΣs + n


Transmitter With Channel Knowledge Cont’d

H is decomposed into k parallel sub-channels satisfying

ym =

√Es

tσmsm + nm, m = 1, 2, · · · , k

The channels are of different quality with the gain on eachchannel determined by σm

Number of channels depends on the rank of H.


Transmitter with Channel KnowledgeTransmitter with Channel Knowledge

sV

sH HU

rTransmitte Channel Receivern

y y~

1λ1~s

1~n

1~y

kλks~kn~

ky~

2λ2~s

2~n

2~y


Capacity of a deterministic MIMO Channels

The channel capacity is given by

C = maxγm

k∑m=1

log2

[1 +

Esλm

Notγm

]γm = E [|sm|2] is the transmit energy in the mth sub-channel∑k

m=1 γm = t is the transmit energy constraint

Optimum power allocation across the sub-channels is obtainedas a solution to the lagrangian optimization problem


Optimal Power Allocation

Optimal power allocation satisfies

γoptm =

(µ− Not

Esλm

)+

, m = 1, 2, · · · , k

k∑m=1

γoptm = t

where µ is a constant and (x)+ implies

(x)+ =

{x if x ≥ 00 if x < 0

γoptm is found iteratively by waterpouring algorithm


Waterpouring Solution

Waterpouring Solution


High SNR

At high SNR, equal power allocation is optimal

C =k∑

m=1

log2

[1 +

Esλm

Not

]≈

k∑m=1

log2

[Esλm

Not

]= k log2

[Es

No

]+

k∑m=1

log2

[λm

t

]Capacity grows linearly with k , the rank of the channel. Significant

increase in Capacity.


Special Cases

SIMO: H = h. Rank one and all power allocated to one mode

CSIMO = log2(1 +Es‖h‖2

No)

MISO: H = hH . Rank one and all power allocated to one mode

CMISO = log2(1 +Es‖h‖2

No)

When Channel known at Tx

CSIMO = CMISO


Maximum Ratio Transmission (MRT)

Input-Output relation for a r × t MIMO channel

y =

√Es

tHs + n

When the channel is known at the transmitter, the informationcan be used to design an optimum precoder w

The new Input-Output relation becomes

y =

√Es

tHws + n


Maximum Ratio Transmission Cont’d

The receiver forms a weighted sum of the antenna outputs

y = gHy

The objective is to maximize the received SNR

η =‖gHHw‖2F

t‖g‖2Fρ

Optimal scheme is given by

w = v1, g = u1

Where, v1 and u1 are the left and right singular vectors of Hcorresponding to the maximum singular value

The scheme achieves full diversity


MRT Transmission: 2× 2 MIMO

MRT Transmission: 2x2 MIMO


Outline




4 MIMO-OFDM

5 Multi-User MIMO


2

Importance of CSI Feedback

A. Improved system performance, in terms of capacity, SNR, BER, etc.Example: An MISO system with M transmit antennas and single receive antenna

NO CSIT Perfect CSIT

B. Reduced implementation complexity

Example: An MIMO system with M transmit and receive antennas,

No CSIT, capacity can be achieved by some 2-D (space-time) code

Pre-coder with perfect CSIT, system isequivalent to M parallel SISO channels

3

Importance of CSI Feedback

D. Greatly increase the system capacity region as well as the sum capacity

C. Enables exploitation of multi-user diversity

With CSIT, effective selection of active users and route selection can be made.

E. Improve the robustness of the communication link (QoS requirements)

Power and rate control is possible when CSIT is available and the network throughput is increased.

Example: A multi-user MISO broadcasting channel with M transmit and single receive antenna

users are not allowed to cooperate, and hencecause serious multi-userinterference.

CSI FeedbackProper pre-coding is possible, such as Zero-forcing, MMSE, etc

Block Diagram

Sources of feedback imperfection

Channel estimation

Channel quantization

Feedback delay

() 6 / 34

4

Nature of CSI FeedbackChannel state information (CSI) is a complex vector or matrix of continuous values

For example: An MIMO system with M transmit antennas and N receive antennas, .

It is not reasonable to feedback total 2MN real numbers of continuous values.

Each index represents a particular mode of the channel, which corresponds to a particular transmission strategy

Channel Quantizer

Integer Index

Adaptive Transmitter

Practical Feedback Schemes:

5

Considerations in Feedback Systems

A. Design of Optimal Quantizers (at the receiver) & Optimization of the Codebook?

1) The quantizer (or the encoder) should be simple as well as effective.

2) The quantizer and the codebook should be designed to match both the channel distribution and the system performance metrics, such as capacity, SNR, BER, etc.

B. Performance Analysis of Finite Rate Feedback Multiple Antenna Systems

1) To understand the effects of the finite rate feedback on the system performance, to be specific, performance metric vs feedback rate.

2) Shed insights on the choice of the feedback schemes as well as the quantizer design.

6

MISO Channel Quantizer

If ideal CSIT available, the transmit beamforming scheme is chosen to be:

MISO Channel System Model:

(vector)(scalar)

If only finite rate feedback is available, the beamforming vector is quantized to ,

capacity

(codebook)

capacity

7

Codebook Design (Optimization)

1). The capacity loss can be approximated by the following form in high resolution regimes,

2). A New Design Criterion that can minimize the system capacity loss:

Simplifications:

(MSwIP)

High SNR(MSIP)

The capacity loss due to the finite rate quantization of the beamforming vectors is:

Motivation: Minimize the capacity loss by optimizing the codebook vectors

It is a difficult problem (non-convex optimization problem)!

8

Codebook design using the Lloyd Algorithm

partitioning the regions

Nearest Neighborhood Condition (NNC):

For given codebook vectors

the optimum partitions are given by:

Centroid Condition (CC):

For given partitions ,

the optimal code matrices are given by:

Shifting new centers

9

Codebook Design Examples

10

MISO Capacity With Quantized Feedback

11

Extension to MIMO Channel Quantizer

Precoding Matrix Equal Power Allocation

MIMO Channel System Model:

Channel Model With Quantized Feedback:

12

Sequential Vector QuantizerA simple approach to quantize the precoding matrix:

How? Consider a unitary matrix whose first column is and the remainder columns are arbitarily chosen to satisfy . Then, has the form of

where is a orthogonormal column matrix.

13

The Sequential Quantization Method

Practical applications: Under consideration by the Broadband Wireless Group (802.16e)

Vector Parameterization: An orthonormal column matrix can be uniquely represented by by a set of unit-norm vectors with different dimensions, .

Statistical Property: For random channel with entries,, for , and they are statistically independent.

Quantization: For , unit-norm vector is quantized using a codebook that is designed for random unit-norm vectors In with the MSIP criterion.

14

Joint Quantization for MIMO Systems

Joint Quantization: by quantizing the entire precoding matrix at one shot

The codebook is designed to minimize the system mutual information rate loss

With ideal CSI Feedback With Quantized CSI Feedback

Under the high resolution assumptions, it can be approximated as

The first n eigen-valuesGeneralized Weighted Matrix Inner Product between and .

15

Codebook design using the Lloyd Algorithm

partitioning the regions

Shifting new centers

Nearest Neighborhood Condition (NNC):

For given code matrices ,

the optimum partitions are given by:

Centroid Condition (CC):

For given partitions ,

the optimal code matrices are given by:

16

Multi-mode Spatial Multiplexing

Case I: Low SNR

water level

power allocated

Case II: High SNR

water level

power allocated

Multi-mode SP transmission strategy:

1) The number of data streams n is determined by the system SNR:

2) In each mode, the simple equal power allocation over n spatial channels is employed.

Intuitive Explanation:Inverse Water-Filling Power Allocation (Optimal)

17

Performance of Multi-mode S-M

Ideal CSI Feedback Quantized CSI Feedback

18

Performance Analysis

Some Interesting Questions:

Finite Rate Effects: What is the performance (capacity, SNR, BER) versus the feedback rate ?

Mismatched Analysis: What happens if a codebook designed for one system is used in another system?

Transform Codebooks: The codebook for a particular system is transformed from another system through a linear or non-linear operation. What is the performance? & How to design?

Feedback With Error: What happens if the feedback information also suffers from error (delay)?

Quantization of Imperfect CSI: What happens if CSI to be quantized suffers from estimation error?

19

Capacity Loss Analysis for MISO Channels

Assume MISO channel with entries

Instantaneous Capacity (mutual information rate) Loss:

Capacity Loss: For a given codebook

Analysis is quite involved

Publications

1 J. C. Roh and B. D. Rao, ”Transmit Beamforming inMultiple-Antenna Systems with Finite Rate Feedback: A VQ-BasedApproach,” IEEE Transactions Information Theory. vol. 52, no. 3,Pages: 1101-1112, Mar. 2006

2 J. C. Roh and B. D. Rao, ”Design and Analysis of MIMO SpatialMultiplexing Systems with Quantized Feedback,” IEEE Transactionson Signal Processing, Vol. 54, no. 8, Pages. 2874-2886, Aug. 2006

3 J. C. Roh and B. D. Rao, ”Efficient Feedback Methods for MIMOChannels Based on Parameterizations,” IEEE Transactions onWireless Communications, Pages: 282 - 292, Jan. 2007

4 J. Zheng, E. Duni, and B. D. Rao, ”Analysis of Multiple AntennaSystems with Finite-Rate Feedback Using High ResolutionQuantization Theory,” IEEE Trans. on Signal Processing, vol.55,Issue 4,Pages: 1461 1476, April 2007.


Outline




4 MIMO-OFDM

5 Multi-User MIMO


Frequency Selective Channels: MIMO-OFDM

Next generation wireless communication system uses MIMO- OFDM

MIMO-OFDM transfers a wideband frequency-selective channelinto a number of parallel narrowband flat fading MIMO channels

Benefits of OFDM

Achieves high spectral efficiency

Cyclic prefix is capable of mitigating multi-path fading

Allows for efficient FFT-based implementations and simplefrequency domain equalization

Exploits frequency diversity, in addition to time and spatialdiversity


MIMO-OFDM Block Diagram

MIMO-OFDM Transceiver

Binary Data

Modulation& Mapping

S/P Space-TimeProcessing

Space-TimeDecoder

& Equalizer

P/S

Binary Data

Demodulation& Demapping

IFFT Add CP P/S

IFFT Add CP P/S

FFTS/P RemoveCP

FFTS/P RemoveCP

OFDM Modulation OFDM Demodulation


MIMO-OFDM Signaling

The input-output relation of a broadband MIMO channel is

y [k] =

√Es

t

L∑l=0

H[l ]s[k − l ] + n[k]

k - discrete time index

L - number of channel taps

t - number of transmit antennas


MIMO-OFDM Signaling Cont’d

OFDM with FFT/IFFT and CP insertion/removal operationsdecuples the frequency selective MIMO channel to a set of parallelMIMO channels as

y [l ] =

√Es

tH[l ]s[l ] + n[l ], l = 0, 1, ..,N − 1.

N - Number of subcarriers

H[l ] - DFT Coefficient of the channel

s[l ] - data on carrier l


Spatial Diversity in MIMO-OFDM

Take Alamouti scheme as an example, there are two ways to realizespatial diversity

1 Coding in frequency domain, rather than in time domain

It requires that the channel remains constant over at least twoconsecutive tones

2 Coding on a per-tone basis across OFDM symbols in time

It requires that the channel remains constant during twoconsecutive OFDM symbols


Outline




4 MIMO-OFDM

5 Multi-User MIMO


Multi-User MIMO

Main Issue is the utilization of the spatial degree of freedom in amulti-user environment

Resource ManagementInterference Management

Capacity of Multi-User systems

Multi-user Diversity


Multi-User SIMO Systems

r(t) =P∑

l=1

hlsl(t) + n(t)

To receive user j , can use beamformer wj

yj(t) = wHj r(t) = wH

j hjsj(t) +P∑

l=1,l 6=j

wHj hlsl(t) + wH

j n(t)

The beamforming vector can be optimized for each user separately.


Multi-User MISO Systems

Transmitted signal

s(t) =P∑

l=1

wlsl(t)

Signal received by user j

rl(t) = hHj s(t) = hH

j wjsj(t) +P∑

l=1,l 6=j

hHj wlsl(t) + nj(t)

The transmit beamformers for the other users do interfere with thedesired user. Beamformers have to be jointly selected. A morechallenging problem.


Problem Statement

University of California, San Diego

Problem StatementConsider a multiuser MIMO beamforming network

Arbitrary Network configurations (cellular networks, multi-hop networks, etc.)Heterogeneous communication nodes with different power costs

Minimize the network power cost while satisfying the minimum SINR requirements of all links

SINR (signal to interference plus noise ratio)Joint optimization of beamforming weights and transmit powers


Problem Statement


JOP:

Solved for SIMO and MISO cases for MISO problem is solved by using the virtual uplink concept

Problem Statement

LlSINR

J

ll

T

≤≤≥

=

1 allfor subject to

)( min,,

γ

pwpUVp

TL

L

L

TL

ww

pp

],...,[

},...,{ },...,{

],...,[ where

1

1

1

1

=

===

w

uuUvvV

p (network power vector, L: no. of links)

(unit norm tx. beamforming vectors)

(unit norm rx. beamforming vectors)

(weight vector defining power costs)

T]1,...,1[== 1w


SINR Expression for MIMO Beamforming


SINR Expression for MIMO BeamformingSINR (signal to interference plus noise ratio)

Problem isolation for optimal Rx. beamforming vectors UMMSE/MVDR beamforming at the receivers

No straightforward problem isolation for V

linl

Hl

lsl

Hl

liliili

Hl

llllHl

lilili

lllll np

pnpG

pGSINRΓuΦuuΦu

vHuvHu

=+

=+

=≡∑∑≠≠

2

2

||||

liiliHlli

l

l

lili

l

l

rtG

llrt

lrLllt

to fromgain link effective: ||

link ofctor weight veantenna receive : link ofctor weight veantenna transmit :

to frommatrix gain channelcomplex : link ofReceiver :

)1( link ofr Transmitte :

2vHu

uvH

=

≤≤


SIMO problem : Cellular Uplink (Rashid-Farrokhi

et al. 98)


SIMO problem : Cellular Uplink(Rashid-Farrokhi et al. ’98)

Problem :

Joint Beamforming & Power Control Algorithm

Convergence to the global optima is established.Desirable features

MVDR beamforming : implemented using adaptive filters power control : using a simple power control loop

)(*)(

)(

)()1(

)()(

)(min)( where

)(

nl

lnl

l

lll

llj

jllj

ln

l

nn

pSINRG

npGI

l uu

up

pIp

u

γγ =+

=

=

∑≠

+

lΓ

p

ll

ll

∀≥

∑ subject to

min,

γ

Up


MISO Problem & Virtual Uplink

Concept(Rashid-Farrokhi et al. 98)


MISO Problem & Virtual Uplink Concept(Rashid-Farrokhi et al. ’98)

Dual relation between cellular downlink and uplinkVirtual uplink : uplink with reciprocal channels and noise vector 1. Optimal transmit beamforming vectors are identical to the optimal receive beamforming vectors in the virtual uplink

(a) Downlink (Primal) (b) Virtual Uplink (Dual)

11H

22H

33HHH11

HH22

HH33


Generalization


GeneralizationWe generalize this idea to arbitrary multiuser MIMO networks with generalized cost function (e.g., MIMO multihop networks, energy-aware networking environment, etc.)

We derive the dual relation using the well-established duality concept in optimization theory

We take advantage of the dual relation for solving the stated problem

We developed an improved Decentralized Algorithm


Construction of a Dual Network


Construction of a Dual NetworkFor any multi-user MIMO network with linear beamformers, one can construct a dual network using the following three rules:

Reverse the direction of all linksReplace any MIMO channel matrix H by HH

Use transmit beamforming vectors as receive beamforming vectors, and vice versa.

44H22H11H

33H 55H

HH44

HH22HH11

HH33

HH55


Duality


Applications to JOP


Applications to JOPTheorem 2 suggests an iterative algorithm (Algorithm E)

Primal Network : Update p and U for fixed V, so that wTp is minimizedDual Network : Update q and V for fixed U, so that nTq is minimized

Lemma 3. In the proposed algorithm, once the solution becomes feasible, i.e., all SINR values meet or exceed the minimum requirements, it generates a sequence of feasible solutions with monotonic decreasing cost.

)(noutΓ

)()(~ nout

nin ΓΓ =

)(~ noutΓ

)()1( ~ nout

nin ΓΓ =+


Cellular Network -Downlink


Cellular Network - DownlinkMultiple wrapped around cells (19 three-sectored cells)Same channel is reused in every cell but only in one sectorThree co-channel users per sectorPropagation exponent = 3.5, 8dB shadow fading


Performance Comparison


Algorithm A, B, E and F

The proposed algorithm presents significant improvement in the complexity-performance tradeoff, thereby greatly improving practical value.

Performance Comparison


Current Trends

Multi-user OFDM systems

Coordinated Multi-Point Transmission (CoMP)

Cooperative MIMO

MIMO Ad-Hoc Networks


Summary

MIMO Systems offer unique opportunities in wirelesscommunication

Provides an opportunity to use spatial dimension to providediversity and hence reliability.

Can be used to significantly increase capacity in a rich scatteringenvironment


Documents

Linear Precoding in MIMO Wireless Systemss3.amazonaws.com/sdieee/143-MIMO-Feedback-FInal.pdfLinear Precoding in MIMO Wireless Systems Bhaskar Rao Center for Wireless Communications