Diversity and Freedom: A Fundamental Tradeoﬀ in Multiple ...dntse/papers/epfl02.pdf · Diversity and Freedom: A Fundamental Tradeoﬀ in Multiple Antenna Channels Lizhong Zheng

Diversity and Freedom:

A Fundamental Tradeoff in Multiple Antenna Channels

Lizhong Zheng and David Tse

Department of EECS, U.C. Berkeley

July, 2002

EPFL

Wireless Fading Channel

• Fundamental characteristic of wireless channels: multi-path fading.

• Two different views of fading have emerged.

Wireless Fading Channel

• Fundamental characteristic of wireless channels: multi-path fading.

• Two different views of fading have emerged.

View #1: Fading is Bad

Channel Quality

t

Fading Channels have poor reliability

Diversity

Fading Channel: h 1

• Additional independent fading channels increase diversity.

• Spatial diversity: receive, transmit or both.

• Repeat and Average: compensate against channel unreliability.

Diversity

1Fading Channel: h

2Fading Channel: h




Diversity

1Fading Channel: h

2Fading Channel: h


• Spatial diversity : receive, transmit or both.


Diversity

1Fading Channel: h

2Fading Channel: h




Diversity

1

Fading Channel: h

Fading Channel: h

4

Fading Channel: h

Fading Channel: h

2

3




Diversity

1

Fading Channel: h

Fading Channel: h

4

Fading Channel: h

Fading Channel: h

2

3




View #2: Fading is Good

y2

y1

Multiple spatial channels created in fading environment


y2

y

Signature 1

1



y2

y1

Signature 2

Signature 1



y2

y1

Signature 2Signature 1



y2

y1

Signature 2

Signature 1

EnvironmentFading

Multiple spatial channels created even when antennas are close together.

Freedom

Another way to view a 2× 2 system:

• Increases the degrees of freedom in the system.

• Multiple antennas provide parallel spatial channels: spatial

multiplexing

Freedom

Spatial Channel

Spatial Channel




multiplexing

Freedom

Spatial Channel

Spatial Channel




multiplexing

Diversity vs. Multiplexing

Fading Channel: h 1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Spatial Channel

Spatial Channel

2

3

4

Multiple antenna channel provides two types of gains:

Diversity Gain vs. Spatial Multiplexing Gain

The research community has a split personality: existing schemes focus

on one type of gain

Diversity vs. Multiplexing

Fading Channel: h 1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Spatial Channel

Spatial Channel

2

3

4

Multiple antenna channel provides two types of gains:

Diversity Gain vs. Spatial Multiplexing Gain

The research community has a split personality: existing schemes focus

on one type of gain.

A Different Point of View

Both types of gains can be achieved simultaneously in a given multiple

antenna channel, but there is a fundamental tradeoff.

A Different Point of View

Both types of gains can be achieved simultaneously in a given multiple

antenna channel, but there is a fundamental tradeoff.

Outline

• Precise problem formulation.

• A simple characterization of the optimal tradeoff.

• Sketch of proof.

• Tradeoff as a unified framework to compare different schemes.

Channel Model

n1

x

x

x

y

y

y

nm

w

w

wm

1

2

1

22

1

nn

h

h

h

h

11

22

yt = Htxt + wt, wt ∼ CN (0, 1)

• Rayleigh flat fading i.i.d. across antenna pairs (hij ∼ CN (0, 1)).

• SNR is the average signal-to-noise ratio at each receive antenna.

Coherent Block Fading Model

• Focus on codes over l symbols, where H remains constant.

• H is known to the receiver but not the transmitter.

• Assumption valid as long as

l ¿ coherence time × coherence bandwidth.

Space-Time Block Code

Y = HX + W

Y

l

H WX

m x space

time

Focus on coding over a single block of length l.

How to Define Diversity Gain

Motivation: Binary Detection

y = hx + w Pe ≈ P (‖h‖ is small ) ∝ SNR−1

y1 = h1x + w1

y2 = h2x + w2

9=;

Pe ≈ P (‖h1‖, ‖h2‖ are both small)

∝ SNR−2

Definition

A space-time coding scheme achieves diversity gain d, if

Pe(SNR) ∼ SNR−d

How to Define Diversity Gain

Motivation: Binary Detection

y = hx + w Pe ≈ P (‖h‖ is small ) ∝ SNR−1

y1 = h1x + w1

y2 = h2x + w2

9=;

Pe ≈ P (‖h1‖, ‖h2‖ are both small)

∝ SNR−2

General Definition

A space-time coding scheme achieves diversity gain d, if

Pe(SNR) ∼ SNR−d

How to Define Spatial Multiplexing Gain

Motivation: Channel capacity (Telatar ’95, Foschini’96)

C(SNR) ≈ min{m, n} log SNR(bps/Hz)

min{m, n} degrees of freedom to communicate.

Definition A space-time coding scheme achieves spatial multiplexing

gain r, if

R(SNR) = r log SNR(bps/Hz)

How to Define Spatial Multiplexing Gain

Motivation: Channel capacity (Telatar’ 95, Foschini’96)

C(SNR) ≈ min{m, n} log SNR(bps/Hz)

min{m, n} degrees of freedom to communicate.

Definition A space-time coding scheme achieves spatial multiplexing

gain r, if

R(SNR) = r log SNR(bps/Hz)

Fundamental Tradeoff

A space-time coding scheme achieves

Spatial Multiplexing Gain r : R = r log SNR (bps/Hz)

and

Diversity Gain d : Pe ≈ SNR−d

Fundamental tradeoff: for any r, the maximum diversity gain

achievable: d∗(r).

r → d∗(r)

A tradeoff between data rate and error probability.




and




r → d∗(r)





and




r → d∗(r)


Outline





Main Result: Optimal Tradeoff

m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain

Pe ≈ SNR−d

r: multiplexing gain

R = r log SNR

Spatial Multiplexing Gain: r=R/log SNR

Div

ersi

ty G

ain:

d

* (r)

(min{m,n},0)

(0,mn)

For integer r, it is as though r transmit and r receive antennas were

dedicated for multiplexing and the rest provide diversity.

Recall: 2× 2 system

Fading Channel: h 1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Spatial Channel

Spatial Channel

2

3

4


m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain

Pe ≈ SNR−d


R = r log SNR


Div

ersi

ty G

ain:

d

* (r)

(min{m,n},0)

(0,mn)




m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain

Pe ≈ SNR−d


R = r log SNR


Div

ersi

ty G

ain:

d

* (r)

(min{m,n},0)

(0,mn)

(1,(m−1)(n−1))




m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain

Pe ≈ SNR−d


R = r log SNR


Div

ersi

ty G

ain:

d

* (r)

(min{m,n},0)

(0,mn)

(2, (m−2)(n−2))

(1,(m−1)(n−1))




m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain

Pe ≈ SNR−d


R = r log SNR


Div

ersi

ty G

ain:

d

* (r)

(min{m,n},0)

(0,mn)

(r, (m−r)(n−r))

(2, (m−2)(n−2))

(1,(m−1)(n−1))




m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain

Pe ≈ SNR−d


R = r log SNR


Div

ersi

ty G

ain:

d

* (r)

(min{m,n},0)

(0,mn)

(r, (m−r)(n−r))

(2, (m−2)(n−2))

(1,(m−1)(n−1))




m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain

Pe ≈ SNR−d


R = r log SNR


Div

ersi

ty G

ain:

d

* (r)

(min{m,n},0)

(0,mn)

(r, (m−r)(n−r))

1

Multiple Antenna m x n channel

Single Antenna channel

1




Theorem: In a channel with m transmit and n receive antennas, for any

space-time coding scheme with block length l ≥ m + n− 1 that achieves

spatial multiplexing gain of r, the error probability is lower bounded by:

limSNR→∞

log Pe(SNR)

log SNR≥ −d∗(r)

Furthermore, there exists a coding scheme that achieves this lower

bound with equality.

A Conceptual Picture

SNR

SNR increases

Channel

SNR 1

Channel

SNR 2

Channel


SNR

code

SNR increases

Channel

SNR 1

Channel

SNR 2

Channel


SNR

code code code

SNR increases

Channel

SNR 1

Channel

SNR 2

Channel


code

-d

code code

SNR increases

Channel

SNR 1

Channel

SNR 2

Channel

SNR

log(SNR)

log(P )e

P = SNR

Slope= Diversity Gain

e


1 2

code 2 code

Slope= Spatial Multiplexing Gain

1

SNR increases

Channel

SNR

Channel

SNR

Channel

SNR

log(SNR)

log(P )e

Slope= Diversity Gain

code

What do I get by adding one more antenna?

Adding More Antennas

m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain



Div

ersi

ty A

dvan

tage

: d

* (r)

• Capacity result: increasing min{M, N} by 1 adds 1 more degree of

freedom.

• Tradeoff curve: increasing both M and N by 1 yields multiplexing

gain +1 for any diversity requirement d.


m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain



Div

ersi

ty A

dvan

tage

: d

* (r)

• Capacity result : increasing min{m, n} by 1 adds 1 more degree of

freedom.

• Tradeoff curve : increasing both M and N by 1 yields multiplexing



m: # of Tx. Ant.

n: # of Rx. Ant.

l: block length

l ≥ m + n− 1

d: diversity gain



Div

ersi

ty A

dvan

tage

: d

* (r)

d

• Capacity result: increasing min{m, n} by 1 adds 1 more degree of

freedom.

• Tradeoff curve: increasing both m and n by 1 yields multiplexing


To Fade or Not to Fade?

Fading

LOS Non-Fading AWGN Channel

y = x + w

Error occurs when noise is large,

with probability

Pe ≈ exp(−SNR)

Infinite diversity.

Fading Channel

y = hx + w

Error occurs when the channel is in

deep fade, with probability

Pe ≈ SNR−1

Needs to increase diversity

Deep fades are major cause of error

Fading


y = x + w


with probability

Pe ≈ exp(−SNR)

pdf

|| w ||t2

Infinite diversity.

Fading Channel

y = hx + w



Pe ≈ SNR−1



Fading


y = x + w


with probability

Pe ≈ exp(−SNR)

pdf

|| w ||t2

Infinite diversity.

Fading Channel

y = hx + w



Pe ≈ SNR−1

pdf

t2|| h ||



Fading


y = x + w


with probability

Pe ≈ exp(−SNR)

pdf

|| w ||t2

Infinite diversity.

Fading Channel

y = hx + w



Pe ≈ SNR−1

pdf

t2|| h ||



Fading


y = x + w


with probability

Pe ≈ exp(−SNR)

pdf

|| w ||t2

Infinite diversity.

Fading Channel

y = hx + w



Pe ≈ SNR−1

pdf

t2|| h ||

Need to increase diversity


Line-of-Sight vs Fading Channel

m: # of Tx. Ant.

n: # of Rx. Ant.

d: Diversity gain

r: Multiplexing Gain

oo

1

1

LOS Non-FadingChannel

d

r

Scalar FadingChannel

• In a scalar 1× 1 system, line-of-sight AWGN is better.

• In a vector M ×N system, fading is exploited as a source of

randomness to provide multiple degrees of freedom.

• Whether fading is good or bad depends on where you operate.


m: # of Tx. Ant.

n: # of Rx. Ant.

d: Diversity gain


oo

1

mn

Vector FadingChannel


d

rmin{m,n}


1


• In a vector m× n system, fading is exploited as a source of




m: # of Tx. Ant.

n: # of Rx. Ant.

d: Diversity gain


oo

1

mn

Vector FadingChannel


d

rmin{m,n}


1


• In a vector m× n system, fading is exploited as a source of



Outline





Converse: Outage Bound

• Outage formulation for quasi-static scenarios. (Ozarow et al 94,

Telatar 95)

• Look at the mutual information per symbol I(Q,H) as a function

of the input distribution and channel realization.

• Error probability for finite block length l is asymptotically lower

bounded by the outage probability:

infQ

PH [I(Q,H) < R] .

• At high SNR, i.i.d. Gaussian input Q∗ is asymptotically optimal,

and

I(Q∗,H) = log det [I + SNRHH∗] .



Telatar 95)





infQ

PH [I(Q,H) < R] .


and




Telatar 95)





infQ

PH [I(Q,H) < R] .


and


Outage Analysis

P (Outage) = P{I(H) = log det[I + SNRHH†] < R}

• In scalar 1× 1 channel, outage occurs when the channel gain ‖h‖2is small.

• In general m× n channel, outage occurs when some or all of the

singular values of H are small. There are many ways for this to

happen.

• Let v = vector of singular values of H:

Laplace Principle:

P (Outage) ≈ minv∈Out

SNR−f(v)

Outage Analysis





happen.


Laplace Principle:


SNR−f(v)

Outage Analysis





happen.


Laplace Principle:


SNR−f(v)

Geometric Picture (integer r)

0

Scalar Channel

Result: At rate R = r log SNR, for r integer, outage occurs typically

when H is in or close to the set {H : rank(H) ≤ r}, with ε2 = SNR−1.

The dimension of the normal space to the sub-manifold of rank r

matrices within the set of all M ×N matrices is (M − r)(N − r).

P (Outage) ≈ SNR−(M−r)(N−r)


Scalar Channel

ε

Good HBad H


when H is close to the set {H : rank(H) ≤ r}, with ε2 = SNR−1.





ε

All n x m Matrices

Good HBad H

Scalar Channel Vector Channel

Rank(H)=r







ε

ε

Good HBad H

Good HFull Rank

Typical Bad HScalar Channel Vector Channel

Rank(H)=r







ε

ε

Good HBad H

Good HFull Rank


Rank(H)=r







ε

ε

Good HBad H

Good HFull Rank


Rank(H)=r


when H is close to the set {H : rank(H) ≤ r} , with ε2 = SNR−1.


matrices within the set of all m× n matrices is (m− r)(n− r).

P (Outage) ≈ SNR−(m−r)(n−r)

Achievability: Random Codes

• Outage performance achievable as codeword length l →∞.

• For random codes of finite length, errors can occur in three ways:

• Channel H is aytically bad (outage)

• Additive Gaussian noise atypically large.

• Random codewords are atypically close together.

• We show that as long as l ≥ m + n− 1, the first event is the

dominating one.

Achievability: Random Codes

• Outage performance achievable as codeword length l →∞.

• For random codes of finite length, errors can occur in three ways:

• Channel H is aytically bad (outage)

• Additive Gaussian noise atypically large.

• Random codewords are atypically close together.

• We show that as long as l ≥ m + n− 1, the first event is the

dominating one.

Outline





Tradeoff Performance of Specific Designs

Provide a unified framework to compare different schemes.

For a given scheme, compute

r → d(r)

Compare with d∗(r)

Tradeoff Performance of Specific Designs

Provide a unified framework to compare different schemes.

For a given scheme, compute

r → d(r)

Compare with d∗(r)

Two Diversity-Based Schemes

Focus on two transmit antennas.

Y = HX + W

Repetition Scheme:

X = x 0

0 x

time

space

1

1

y1 = ‖H‖x1 + w1

Two Diversity-Based Schemes

Focus on two transmit antennas.

Y = HX + W

Repetition Scheme:

X = x 0

0 x

time

space

1

1

y1 = ‖H‖x1 + w1

Alamouti Scheme:

X =

time

space

x -x *

x x2

1 2

1*

[y1y2] = ‖H‖[x1x2] + [w1w2]

Comparison: 2× 1 System

Repetition: y1 = ‖H‖x1 + w

Alamouti: [y1y2] = ‖H‖[x1x2] + [w1w2]


Div

ersi

ty G

ain:

d

* (r)

(1/2,0)

(0,2)

Repetition





Div

ersi

ty G

ain:

d

* (r)

(1/2,0)

(0,2)

(1,0)

Alamouti

Repetition





Div

ersi

ty G

ain:

d

* (r)

(1/2,0)

(0,2)

(1,0)

Optimal Tradeoff

Alamouti

Repetition





Div

ersi

ty G

ain:

d

* (r)

(1/2,0)

(0,4)

Repetition





Div

ersi

ty G

ain:

d

* (r)

(1/2,0) (1,0)

(0,4)

Alamouti

Repetition





Div

ersi

ty G

ain:

d

* (r)

(1/2,0) (1,0)

(0,4)

(1,1)

(2,0)

Optimal Tradeoff

Alamouti

Multiplexing-Based Schemes: BLAST (n× n)

(n,0)

1

(0,n2) Optimal Tradeoff

D−BLAST

V−BLAST

• Make comparison between diversity-based and multiplexing-based

schemes

• Pinpoint the problem and improve tradeoff performance.


(n,0)

1

(0,n2) Optimal Tradeoff

D−BLAST

V−BLAST

1

Alamouti


schemes



(n,0)

1

(0,n2)

1

Optimal Tradeoff

D−BLAST

V−BLAST


schemes


Documents

Diversity and Freedom: A Fundamental Tradeoﬀ in Multiple ...dntse/papers/epfl02.pdf · Diversity and Freedom: A Fundamental Tradeoﬀ in Multiple Antenna Channels Lizhong Zheng