Diversity and Freedom: A Fundamental Tradeoﬀ in Multiple Antenna...

Diversity and Freedom:

A Fundamental Tradeoff in Multiple Antenna Channels

Lizhong Zheng and David Tse

Department of EECS, U.C. Berkeley

Feb 26, 2002

MSRI Information Theory Workshop

Wireless Fading Channels

Channel Quality

Time

Fundamental characteristic of wireless channels: multipath fading due

to constructive and destructive interference.

Channel varies over time as well as frequency.

Multiple Antennas

• Multi-antenna communication is a hot field in recent years.

• But the research community has a split personality.

• There are two very different views of how multiple antennas can

be used.

Diversity

Fading Channel: h 1

• Additional independent signal paths increase diversity.

• Diversity: receive, transmit or both.

• Compensate against channel unreliability.

Diversity

1Fading Channel: h

2Fading Channel: h




Diversity

1

Fading Channel: h

Fading Channel: h

4

Fading Channel: h

Fading Channel: h

2

3




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

• Fading is exploited as a source of randomness.

Freedom

Spatial Channel

Spatial Channel

Another way to view a 2× 2 system:

• Increases the degrees of freedom in the system

• Multiple antennas provide parallel spatial channels: spatial

multiplexing

• Fading is exploited as a source of randomness.

Diversity vs. Multiplexing

1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Fading Channel: h

4

Spatial Channel

Spatial Channel

2

3

Multiple antenna channel provides two types of gains:

Diversity Gain vs. Spatial Multiplexing Gain

Existing schemes focus on one type of gain

Diversity vs. Multiplexing

1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Fading Channel: h

4

Spatial Channel

Spatial Channel

2

3

Multiple antenna channel provides two types of gains:

Diversity Gain vs. Spatial Multiplexing Gain

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.

We propose a unified framework which encompasses both diversity and

freedom and study the optimal 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.

We propose a unified framework which encompasses both diversity and

multiplexing and study the optimal tradeoff.

Outline

• Problem formulation and main result on optimal tradeoff.

• Sketch of proof.

• Comparison of existing schemes.

Channel Model

h22

hN1

h21

h11

hNM

W1

W2

WN

X1

X2

XM

Y1

Y2

YN

y = Hx + w, wi ∼ CN (0, 1)

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

• Focus on codes of T symbols, where H remains constant (slow,

flat fading)

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

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

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 scheme achieves diversity gain d, if

Pe ∼ SNR−d

Actual error probability instead of pairwise error probability (eg. Tarokh

et al 98, Guey et al 99)

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 scheme achieves diversity gain d, if

Pe ∼ SNR−d

Acutal error probability instead of pairwise error probability. (eg.

Tarokh et al 98, Guey et al 99)

How to Define Spatial Multiplexing Gain

Motivation: (Telatar’95, Foschini’96)

Ergodic capacity:

C(SNR) ≈ min{M, N} log SNR (bps/Hz),

Equivalent to min{M, N} parallel spatial channels.

A scheme is a sequence of codes, one at each SNR level.

Definition A scheme achieves spatial multiplexing gain r, if

R = r log SNR (bps/Hz)

Increasing data rates instead of fixed data rate. (cf. Tarokh et al 98)

Fundamental Tradeoff

A 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)

Main Result: Optimal Tradeoff

As long as T ≥ M + N − 1:

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.




Div

ersi

ty G

ain:

d

* (r)

(min{M,N},0)

(0,MN)

(1,(M−1)(N−1))






Div

ersi

ty G

ain:

d

* (r)

(min{M,N},0)

(0,MN)

(2, (M−2)(N−2))

(1,(M−1)(N−1))






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))






Div

ersi

ty G

ain:

d

* (r)

Single Antenna SystemM x N System

(min{M,N},0)

(0,MN)

(r, (M−r)(N−r))

(2, (M−2)(N−2))

(1,(M−1)(N−1))

(0,1)

(0,1)



Revisit the 2× 2 Example

Fading Channel: h 1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Spatial Channel

Spatial Channel

2

3

4

Revisit the 2× 2 Example (ctd.)


Div

ersi

ty G

ain:

d

* (r)

(2,0)

(0,4)

(1,1)

• Tradeoff bridges the gap between the two types of approaches.

Adding More Antennas


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.



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




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


Increasing vs Fixed Code Rate

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

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 T 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∗] .

Outage Analysis

• Scalar 1× 1 channel:

For target rate R = r log SNR, r < 1,

P{log(1 + SNR‖h‖2) < r log SNR}∼ P

n‖h‖2 < SNR−(1−r)

o

∼ SNR−(1−r)

=⇒ dout(r) = 1− r

• Outage occurs when the channel gain ‖h‖2 is small.

• More generally, outage occurs for the multi-antenna channel when

some or all of the singular values of H are small.

• But unlike the scalar channel, there are many ways for this to

happen in a vector channel.

Typical Outage Behavior

v = vector of singular values of H.

Laplace Principle:

pout = minv∈Out

SNR−f(v)

Result:

At target rate R = r log SNR, outage typically occurs when H is near a

rank brc matrix, i.e. out of the min{M, N} non-zero squared singular

values:

• brc of them are order 1;

• min{M, N} − brc+ 1 of them are are order SNR−1;

• 1 of them is order SNR−(r−brc) ( just small enough to cause

outage)

When r is integer, exactly r squared singular values are order 1 and

min{M, N} are order SNR−1.

Geometric Picture (integer r)

0

Scalar Channel

pout ∼ SNR−(M−r)(N−r),

(M − r)(N − r) is the dimension of the normal space to the sub-manifold

of rank r matrices within the set of all M ×N matrices.


-1/2

Bad H

SNR

Scalar Channel

Good H





-1/2

Rank(H)=r

SNR

All MxN Matrices

Good HBad H

Scalar Channel Vector Channel





-1/2

SNR-1/2

Vector Channel

SNRRank(H)=r

Good HBad H

Good HFull Rank

Typical Bad HScalar Channel




Piecewise Linearity of Tradeoff Curve


Div

ersi

ty G

ain:

d

* (r)

Single Antenna SystemM x N System

(min{M,N},0)

(0,MN)

(r, (M−r)(N−r))

(2, (M−2)(N−2))

(1,(M−1)(N−1))

(0,1)

(0,1)

Scalar channel: qualitatively same outage behavior for all r.

Vector channel: qualitatively different outage behavior for different r.

Achievability: Random Codes

• Outage performance achievable as codeword length T →∞.

• But what about for finite T?

• Look at the performance of i.i.d Gaussian random codes.

• Can the outage behavior be achieved?

Analysis of Random Codes

Errors can occur due to three events:

• Channel H is aytically bad (outage)

• Additive Gaussian noise atypically large.

• Random codewords are atypically close together.

Outage analysis only needs to focus on the first event, but for finite T

all three effects come into play.

Multiplicative Fading vs Additive Noise

Look at two codewords at Euclidean distance x.

• Error Event A: H typical, AWGN large

P (A) ∼ exp(−x).

• Error Event B: H near singular, AWGN typical

P (B) ∼ x−α.

At high SNR, x →∞

=⇒ P (B) À P (B).

Multiplicative Fading vs Code Randomness

• Distance between random codewords may deviate from typical

distance x.

• Error Event C: codewords atypically close

P (C) ∼ x−β ,

Also polynomial in x, just like the effect due to channel fading.

• As long as T ≥ M + N − 1, the typical error event is due to bad

channel rather than bad codewords.

• For T < M + N − 1, random codes are not good enough. (ISIT 02)

To Fade or Not to Fade?

Line-of-Sight vs Fading Channel

1

LOS Non-Fading

Channel

Channel

1

d

r

Scalar Fading

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

• In a vector M ×N system, it depends.

Line-of-Sight vs Fading Channel

1

Vector FadingChannel

Channel

LOS Non-FadingChannel

1

d

MN

rmin{M,N}

Scalar Fading

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

• In a vector M ×N system, it depends.

Outline




Using the Optimal Tradeoff Curve

Provide a unified framework to compare different schemes.

For a given scheme, compute

r → d(r)

Compare with d∗(r)

Use the Optimal Tradeoff Curve

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 =

24 x1 0

0 x1

35 r = ‖H‖x1 + w

Alamouti Scheme

X =

24 x1 −x∗2

x2 x∗1

35 [r1r2] = ‖H‖[x1x2] + w

Two Diversity-Based Schemes

Focus on two transmit antennas.

Y = HX + W

Repetition Scheme:

X =

24 x1 0

0 x1

35 r = ‖H‖x1 + w

Alamouti Scheme

X =

24 x1 −x∗2

x2 x∗1

35 [r1r2] = ‖H‖[x1x2] + [w1w2]

Comparison: 2× 1 System

Repetition: r1 = ‖H‖x1 + w

Alamouti: [r1r2] = ‖H‖[x1x2] + [w1w2]


Div

ersi

ty G

ain:

d

* (r)

(1/2,0)

(0,2)





Div

ersi

ty G

ain:

d

* (r)

(1/2,0)

(0,2)

(0,1)





Div

ersi

ty G

ain:

d

* (r)

RepetitiveAlamoutiOptimal

(1/2,0)

(0,2)

(0,1)





Div

ersi

ty G

ain:

d

* (r)

(1/2,0)

(0,4)





Div

ersi

ty G

ain:

d

* (r)

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

(0,4)





Div

ersi

ty G

ain:

d

* (r)

RepetitiveAlamoutiOptimal

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

(0,4)

(1,1)

(2,0)

V-BLAST

Antenna 2:

Antenna 1:

TT

Antenna 1:

Antenna 2:

• Nulling and Canceling

• Independent data streams transmitted over antennas

V-BLAST

Antenna 2:

Receive

Null

Antenna 1:



V-BLAST

Antenna 2:

Null

Receive

Antenna 1:



V-BLAST

Antenna 2:

Antenna 1:



V-BLAST

Antenna 2:

Antenna 1:

Cancel



V-BLAST

Antenna 2:

Antenna 1:

TT

Antenna 1:

Antenna 2:



Tradeoff Performance of V-BLAST (N ×N)

Original V-BLAST

(N,0)

1

Low diversity due to lack of coding over space


V-BLAST with optimal rate allocation

(N,0)

1

(0,N)



Compare to the optimal tradeoff

(N,0)

1

(0,N)

(0,N2)


D-BLAST

Antenna 2:

Antenna 1:

T

Assume T = ∞, ignore the overhead.

D-BLAST

Antenna 2:

Antenna 1:

Receive


D-BLAST

Antenna 2:

Antenna 1:

Receive

Null


D-BLAST

Antenna 2:

Antenna 1:


D-BLAST

Antenna 2:

Antenna 1:

Receive

Cancel


D-BLAST

Antenna 2:

Null

Antenna 1:

Receive


D-BLAST

Antenna 2:

Antenna 1:


D-BLAST

Antenna 2:

Cancel

Antenna 1:

Receive


D-BLAST

Antenna 2:

Null

Antenna 1:

Receive

Ignore the overhead for now.

D-BLAST: Square System


Div

ersi

ty A

dvan

tage

: d

* (r)

(N,0)

(N−1,1)

(N−2,3)

(1,N(N+1)/2)

• Can achieve full multiplexing gain

• Maximum diversity gain d =N(N+1)

2.

D-BLAST: Square System


Div

ersi

ty A

dvan

tage

: d

* (r)

DBLASTOptimal

(N,0)

(N−1,1) (1,N(N+1)/2)

(0,N2)

• Can achieve full multiplexing gain

• Maximum diversity gain d =N(N+1)

2.

Replace Nulling by MMSE?

D-BLAST+MMSE


Div

ersi

ty A

dvan

tage

: d

* (r)

DBLAST+NullingDBLAST+MMSE

(N,0)

(N−1,1) (1,N(N+1)/2)

(0,N2)

• Achieve the optimal: successive cancellation + MMSE has the

optimal outage performance.

• Difference between MMSE and Nulling.

D-BLAST+MMSE


Div

ersi

ty A

dvan

tage

: d

* (r)

DBLAST+NullingDBLAST+MMSEOptimal

(N,0)

(N−1,1) (1,N(N+1)/2)

(0,N2)

• Achieve the optimal: successive cancellation + MMSE has the

optimal outage performance.

• Difference between MMSE and Nulling.

Penalty due to Overhead


Div

ersi

ty A

dvan

tage

: d

* (r)

DBLAST+NullingDBLAST+MMSEOptimal

Conclusion

• The diversity-multiplexing tradeoff is a fundamental way of looking

at fading channels.

• Same framework can be applied to other scenarios: multiuser,

non-coherent, more complex channel models, etc.

Diversity and Freedom: A Fundamental Tradeoﬀ in Multiple Antenna...

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