Linear Precoding in MIMO Wireless Systems

Linear Precoding in MIMO Wireless Systems Bhaskar Rao Center for Wireless Communications University of California, San Diego

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

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

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Outline

1

Promise of MIMO Systems

2

Point to Point MIMO

3

Limited Feedback MIMO Systems

4

MIMO-OFDM

5

Multi-User MIMO


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Outline

1


2

Point to Point MIMO

3


4

MIMO-OFDM

5

Multi-User MIMO


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Multiple Input Multiple Output (MIMO) Systems A system with multiple antennas at the transmitter and multiple antennas at the receiver. Enables Spatio-Temporal processing and the goal is to exploit the spatial dimension to increase system throughput

Multi-Input Multi-Output System


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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


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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 to maximize system performance


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Technical Rationale

Spatial Diversity to Combat Fading Spatial Signature for Interference Management Array Gain enables Lower Power Consumption Capacity Improvements using Spatial Multiplexing


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Outage Capacity of MIMO Systems

Capacity of MIMO systems


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Outline

1


2

Point to Point MIMO

3


4

MIMO-OFDM

5

Multi-User MIMO


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MIMO Channel Model Input-Output relation for a discrete-time frequency-flat r × t MIMO channel r Es y= Hs + n t y = [y1 , y2 , · · · , yr ]T · · · r × 1 receive signal vector s = [s1 , s2 , · · · , st ]T · · · t × 1 transmit signal vector n = [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 ∼ N C(0, No ) with E [nnH ] = No Ir


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MIMO Options

Channel assumed known at Receiver Channel unknown at transmitter Diversity Gain: Orthogonal space-time block codes, Space time trellis codes Spatial Multiplexing: V-Blast, D-Blast

Channel known at the transmitter- Transmit precoding


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Transmitter With Channel Knowledge SVD of H can be expressed as H = UΣVH UH U = VH V = Ir Σ = diag(σm )km=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 2 λm = σm


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Transmitter With Channel Knowledge Cont’d

Transmitted vector s = V˜s Input vector ˜s is of dimension r × 1 with E [˜s˜sH ] = Γt , Γt diagonal Received signal transformed to y˜ = UH y r Es ˜ y˜ = Σ˜s + n t


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Transmitter With Channel Knowledge Cont’d

H is decomposed into k parallel sub-channels satisfying r Es y˜m = σm ˜sm + n˜m , m = 1, 2, · · · , k t The channels are of different quality with the gain on each channel determined by σm Number of channels depends on the rank of H.


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Transmitter with Channel Knowledge Transmitter with Channel Knowledge Transmitter

Receiver

Channel n

V

sˆ

H

s

y

UH

~ y

n~1 ~ s1

~ y1

λ1 n~2

~ s2

~ y2

λ2

n~k

~ sk

λk

~ yk


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Capacity of a deterministic MIMO Channels

The channel capacity is given by C = max γm

k X m=1

log2

Es λm 1+ γm No t

γ sm |2 ] is the transmit energy in the mth sub-channel m = E [|˜ P k m=1 γm = t is the transmit energy constraint

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


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Optimal Power Allocation Optimal power allocation satisfies + No t opt γm = µ − , Es λm k X

m = 1, 2, · · · , k

opt =t γm

m=1

where µ is a constant and (x)+ implies ( x if x ≥ 0 (x)+ = 0 if x < 0 opt γm is found iteratively by waterpouring algorithm Bhaskar RaoCenter for Wireless CommunicationsUniversity Linear Precoding of in California, MIMO Wireless San Diego Systems ()

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Waterpouring Solution Waterpouring Solution


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High SNR

At high SNR, equal power allocation is optimal C=

k X m=1

log2

X X k k Es λ m Es λm Es λ m ≈ log2 = k log2 + log2 1+ No t No t No t m=1

m=1

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


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Special Cases SIMO: H = h. Rank one and all power allocated to one mode CSIMO = log2 (1 +

Es khk2 ) No

MISO: H = hH . Rank one and all power allocated to one mode CMISO = log2 (1 +

Es khk2 ) No

When Channel known at Tx CSIMO = CMISO


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Maximum Ratio Transmission (MRT) Input-Output relation for a r × t MIMO channel r Es y= Hs + n t When the channel is known at the transmitter, the information can be used to design an optimum precoder w The new Input-Output relation becomes r Es y= Hws + n t


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Maximum Ratio Transmission Cont’d The receiver forms a weighted sum of the antenna outputs y˜ = gH y The objective is to maximize the received SNR η=

kgH Hwk2F ρ tkgk2F

Optimal scheme is given by w = v1 ,

g = u1

Where, v1 and u1 are the left and right singular vectors of H corresponding to the maximum singular value The scheme achieves full diversity Bhaskar RaoCenter for Wireless CommunicationsUniversity Linear Precoding of in California, MIMO Wireless San Diego Systems ()

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MRT Transmission: 2 × 2 MIMO MRT Transmission: 2x2 MIMO


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Outline

1


2

Point to Point MIMO

3


4

MIMO-OFDM

5

Multi-User MIMO


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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 is equivalent to M parallel SISO channels

2

Importance of CSI Feedback C. Enables exploitation of multi-user diversity With CSIT, effective selection of active users and route selection can be made.

D. Greatly increase the system capacity region as well as the sum capacity Example: A multi-user MISO broadcasting channel with M transmit and single receive antenna users are not allowed to cooperate, and hence cause serious multi-user interference.

CSI Feedback

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

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.

3

Block Diagram

Sources of feedback imperfection Channel estimation Channel quantization Feedback delay ()

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Nature of CSI Feedback Channel 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.

Practical Feedback Schemes:

Integer Index

Channel Quantizer

Adaptive Transmitter

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

4

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.

5

MISO Channel Quantizer MISO Channel System Model: (scalar)

(vector)

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

If only finite rate feedback is available, the beamforming vector

is quantized to

,

capacity (codebook)

6

Codebook Design (Optimization) 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)! Simplifications: 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:

(MSwIP) High SNR

(MSIP)

7

Codebook design using the Lloyd Algorithm Nearest Neighborhood Condition (NNC): For given codebook vectors

partitioning the regions

the optimum partitions are given by:

Centroid Condition (CC): For given partitions

,

the optimal code matrices are given by:

Shifting new centers

8

Codebook Design Examples

9

MISO Capacity With Quantized Feedback

10

Extension to MIMO Channel Quantizer MIMO Channel System Model:

Precoding Matrix

Equal Power Allocation

Channel Model With Quantized Feedback:

11

Sequential Vector Quantizer A simple approach to quantize the precoding matrix:

How? Consider a unitary matrix whose first column is columns are arbitarily chosen to satisfy . Then,

where

is a

and the remainder has the form of

orthogonormal column matrix. 12

The Sequential Quantization Method 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.

Practical applications: Under consideration by the Broadband Wireless Group (802.16e) 13

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

Generalized Weighted Matrix Inner Product between and .

The first n eigen-values 14

Codebook design using the Lloyd Algorithm Nearest Neighborhood Condition (NNC): For given code matrices

,

partitioning the regions

the optimum partitions are given by:

Centroid Condition (CC): For given partitions

,

the optimal code matrices are given by:

Shifting new centers

15

Multi-mode Spatial Multiplexing 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)

water level

water level

power allocated

power allocated

Case I: Low SNR

Case II: High SNR

16

Performance of Multi-mode S-M

Ideal CSI Feedback

Quantized CSI Feedback

17

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?

18

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

19

Publications 1

J. C. Roh and B. D. Rao, ”Transmit Beamforming in Multiple-Antenna Systems with Finite Rate Feedback: A VQ-Based Approach,” 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 Spatial Multiplexing Systems with Quantized Feedback,” IEEE Transactions on Signal Processing, Vol. 54, no. 8, Pages. 2874-2886, Aug. 2006

3

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

4

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


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Outline

1


2

Point to Point MIMO

3


4

MIMO-OFDM

5

Multi-User MIMO


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Frequency Selective Channels: MIMO-OFDM Next generation wireless communication system uses MIMO- OFDM MIMO-OFDM transfers a wideband frequency-selective channel into 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 simple frequency domain equalization Exploits frequency diversity, in addition to time and spatial diversity


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MIMO-OFDM Block Diagram MIMO-OFDM Transceiver Binary Data

Binary Data

Modulation & Mapping

Demodulation & Demapping OFDM Modulation

IFFT

S/P

Add CP

OFDM Demodulation

P/S

S/P

Remove CP

FFT

Space-Time Decoder & Equalizer

Space-Time Processing

IFFT

Add CP

P/S

S/P

Remove CP

P/S

FFT


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MIMO-OFDM Signaling

The input-output relation of a broadband MIMO channel is r L Es X y [k] = H[l]s[k − l] + n[k] t l=0 k - discrete time index L - number of channel taps t - number of transmit antennas


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MIMO-OFDM Signaling Cont’d OFDM with FFT/IFFT and CP insertion/removal operations decuples the frequency selective MIMO channel to a set of parallel MIMO channels as r Es ˜ y˜ [l] = H[l]˜s [l] + n˜[l], l = 0, 1, .., N − 1. t N - Number of subcarriers ˜ - DFT Coefficient of the channel H[l] ˜s [l] - data on carrier l


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Spatial Diversity in MIMO-OFDM

Take Alamouti scheme as an example, there are two ways to realize spatial diversity 1 Coding in frequency domain, rather than in time domain It requires that the channel remains constant over at least two consecutive tones 2

Coding on a per-tone basis across OFDM symbols in time It requires that the channel remains constant during two consecutive OFDM symbols


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Outline

1


2

Point to Point MIMO

3


4

MIMO-OFDM

5

Multi-User MIMO


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Multi-User MIMO

Main Issue is the utilization of the spatial degree of freedom in a multi-user environment Resource Management Interference Management

Capacity of Multi-User systems Multi-user Diversity


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Multi-User SIMO Systems

r(t) =

P X

hl sl (t) + n(t)

l=1

To receive user j, can use beamformer wj yj (t) = wjH r(t) = wjH hj sj (t) +

P X

wjH hl sl (t) + wjH n(t)

l=1,l6=j

The beamforming vector can be optimized for each user separately.


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Multi-User MISO Systems Transmitted signal s(t) =

P X

wl sl (t)

l=1

Signal received by user j rl (t) =

hHj s(t)

=

hHj wj sj (t)

+

P X

hHj wl sl (t) + nj (t)

l=1,l6=j

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


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Problem Statement

Problem Statement

Consider 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

University of California, San Diego


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Problem Statement

Problem Statement JOP:

min

J (p) = w T p

subject to

SINRl ≥ γ l for all 1 ≤ l ≤ L

p ,V ,U

where p = [ p1 ,..., pL ]T (network power vector, L: no. of links) V = {v1 ,..., v L } (unit norm tx. beamforming vectors) U = {u1 ,..., u L } (unit norm rx. beamforming vectors) w = [ w1 ,..., wL ]T (weight vector defining power costs)

Solved for SIMO and MISO cases for w = 1 = [1,...,1] MISO problem is solved by using the virtual uplink concept

T



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SINR Expression for MIMO Beamforming SINR Expression for MIMO Beamforming

SINR (signal to interference plus noise ratio) Γ l ≡ SINRl =

Gll pl = ∑ Gli pi + nl i ≠l

| u lH H ll v l |2 pl u H Φ su = Hl inl l H 2 ∑ | ul H li v i | pi + nl ul Φl ul i ≠l

tl : Transmitter of link l (1 ≤ l ≤ L) rl : Receiver of link l H li : complex channel gain matrix from ti to rl v l : transmit antenna weight vector of link l u l : receive antenna weight vector of link l Gli =| u lH H li v i |2 : effective link gain from ti to rl

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

No straightforward problem isolation for V



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

Problem :

∑p

min

l

p ,U

l

Γl ≥ γ l

subject to

∀l

Joint Beamforming & Power Control Algorithm p ( n +1) = I (p ( n ) ) where I l (p ( n ) ) = min γ l ul

∑ G (u ) p j ≠l

lj

l

j

Gll (u l )

+ nl =

γl SINRl( n ) (u*l )

pl( n )

Convergence to the global optima is established. Desirable features

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



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

Dual relation between cellular downlink and uplink

Virtual 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

H 11

H 33

H11H

H 33H H H 22

H 22

(a) Downlink (Primal)

(b) Virtual Uplink (Dual)



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Generalization

Generalization

We 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 University of California, San Diego


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Construction of a Dual Network

Construction of a Dual Network

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

Reverse the direction of all links Replace any MIMO channel matrix H by HH Use transmit beamforming vectors as receive beamforming vectors, and vice versa.

H 33 H 11

H 22

H 55

H 44

H 33H H11H

H 22H

H 55H

H 44H



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Duality


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Applications to JOP

Applications to JOP

Theorem 2 suggests an iterative algorithm (Algorithm E)

Primal Network : Update p and U for fixed V, so that wTp is minimized Dual Network : Update q and V for fixed U, so that nTq is minimized ) Γ (n out

~ (n) Γ in( n ) = Γ out

~ n) Γ (out

~ n) Γ in( n +1) = Γ (out

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. University of California, San Diego


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Cellular Network -Downlink

Cellular Network - Downlink

Multiple wrapped around cells (19 three-sectored cells) Same channel is reused in every cell but only in one sector Three co-channel users per sector Propagation exponent = 3.5, 8dB shadow fading



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Performance Comparison Performance Comparison

Algorithm A, B, E and F

The proposed algorithm presents significant improvement in the complexity-performance tradeoff, thereby greatly improving practical value. University of California, San Diego


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Current Trends

Multi-user OFDM systems Coordinated Multi-Point Transmission (CoMP) Cooperative MIMO MIMO Ad-Hoc Networks


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Summary

MIMO Systems offer unique opportunities in wireless communication Provides an opportunity to use spatial dimension to provide diversity and hence reliability. Can be used to significantly increase capacity in a rich scattering environment


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Linear Precoding in MIMO Wireless Systems

Linear Precoding in MIMO Wireless Systems

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