diversity and multiplexing; a fundamental tradeoff in wireless systems

8/18/2019 Diversity and Multiplexing; A Fundamental Tradeoff in Wireless Systems

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Diversity and Freedom:

A Fundamental Tradeoff in Wireless Systems

David Tse

Department of EECS, U.C. Berkeley

September 23, 2003

University of Toronto


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Wireless Fading Channels

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

• Two important resources of a fading channel: diversity and

degrees of freedom.


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Wireless Fading Channels

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

• Two important resources of a fading channel: diversity and

degrees of freedom.


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Diversity

Channel Quality

t

A channel with more diversity has smaller probability in deep fades.


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Example: Spatial Diversity

Fading Channel: h1

• Additional independent fading channels increase diversity.

• Spatial diversity: receive, transmit or both.

• Repeat and Average: compensate against channel unreliability.


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1Fading Channel: h

2Fading Channel: h

• Additional independent fading channels increase diversity.• Spatial diversity: receive, transmit or both.



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1Fading Channel: h

2Fading Channel: h

• Additional independent fading channels increase diversity.• Spatial diversity : receive, transmit or both.



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1Fading Channel: h

2Fading Channel: h




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1

Fading Channel: h

Fading Channel: h

4

Fading Channel: h

Fading Channel: h

2

3




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1

Fading Channel: h

Fading Channel: h

4

Fading Channel: h

Fading Channel: h

2

3




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Degrees of Freedom

y2

y1

Signals arrive in multiple directions provide multiple degrees of freedom

for communication.

Same effect can be obtained via scattering even when antennas are

close together.


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Degrees of Freedom

y2

y

Signature 1

1


for communication.


close together.


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Degrees of Freedom

y2

y1

Signature 2

Signature 1


for communication.


close together.


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Degrees of Freedom

y2

y1

Signature 2Signature 1


for communication.


close together.


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Degrees of Freedom

y2

y1

Signature 2

Signature 1

Environment

Fading


for communication.


close together.


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Diversity vs. Freedom

Fading Channel: h 1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Spatial Channel

Spatial Channel

2

3

4

The two resources have been considered mainly in isolation: existing

schemes focus on maximizing either the diversity gain or the

multiplexing gain.

The right way of looking at the problem is a tradeoff between the two

types of gain.

The optimal tradeoff achievable by a coding scheme gives a

fundamental performance limit on communication over fading channels.


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Fading Channel: h 1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Spatial Channel

Spatial Channel

2

3

4



multiplexing gain.


types of gain.




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Fading Channel: h 1

Fading Channel: h

Fading Channel: h

Fading Channel: h

Spatial Channel

Spatial Channel

2

3

4



multiplexing gain.


types of gain.




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Talk Outline

• point-to-point MIMO channels (Zheng and Tse 02)

• multiple access MIMO channels (Tse, Viswanath, Zheng 03)

• cooperative relaying systems (Laneman,Tse, Wornell 02)


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Point-to-point MIMO Channel

n1

x

x

x

y

y

y

nm

w

w

wm

1

2

1

2

2

1

n

n

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.


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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.


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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.


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Diversity Gain

Motivation: Binary Detection

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

y1 = h1x+w1y2 = h2x+w2

P e ≈ P (h1, h2 are both small)∝ SNR−2

Definition

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

P e(SNR) ∼ SNR−d


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Diversity Gain

Motivation: Binary Detection

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

y1 = h1x+w1

y2 = h2x+w2

P e ≈ P (h1, h2 are both small)∝ SNR−2

General Definition

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

P e(SNR) ∼ SNR−d


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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 logSNR


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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


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Fundamental Tradeoff

A space-time coding scheme achieves

Spatial Multiplexing Gain r : R = r logSNR

and

Diversity Gain d : P e ≈ SNR−d

Fundamental tradeoff: for any r, the maximum diversity gain

achievable: d∗m,n(r).

r → d∗

m,n(r)

It is a tradeoff between data rate and error probability.


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andDiversity Gain d : P e ≈ SNR−d



r → d∗m,n(r)



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and

Diversity Gain d : P e ≈ SNR−d



r → d∗m,n(r)



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Main Result: Optimal Tradeoff

(Zheng and Tse 02)As long as block length l ≥ m + n− 1:

Spatial Multiplexing Gain: r=R/log SNR

D i v e r s i t y G a i n :

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.


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(Zheng and Tse 02)As long as block length l ≥ m + n− 1:



d * (

r )

(min{m,n},0)

(0,mn)

(1,(m−1)(n−1))




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(Zheng and Tse 02)

As long as block length l ≥ m + n− 1:



d * (

r )

(min{m,n},0)

(0,mn)

(2, (m−2)(n−2))

(1,(m−1)(n−1))




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(Zheng and Tse 02)




d * (

r )

(min{m,n},0)

(0,mn)

(r, (m−r)(n−r))

(2, (m−2)(n−2))

(1,(m−1)(n−1))




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(Zheng and Tse 02)




d * (

r )

(min{m,n},0)

(0,mn)

(r, (m−r)(n−r))

(2, (m−2)(n−2))

(1,(m−1)(n−1))




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(Zheng and Tse 02)




d * ( r )

(min{m,n},0)

(0,mn)

(r, (m−r)(n−r))

1

Multiple Antenna

m x n channel

Single Antenna

channel

1




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What do I get by adding one more antenna at thetransmitter and the receiver?


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Adding More Antennas


D i v e r s i t y

A d v a n t a g e :

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.


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D i v e r s i t y A

d v a n t a g e :

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



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D i v e r s i t y A

d v a n t a g e :

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



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Geometric Picture

ε

Scalar Channel

Bad H Good H


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Geometric Picture

ε

Scalar Channel

Bad H Good H

2 = SNR−1

P e ∼ SNR−1


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Geometric Picture

ε

Scalar Channel

Bad H Good H

2 = SNR−1

P e ∼ SNR−1

Good H

εBad H

h2

1h

x1 2 Channel

P e ∼ SNR−2


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Geometric Picture for General m× n Channels

Multiplexing gain = r (r integer)

Rank H =r

Good H

Full Rank

MIMO Channel

The co-dimension of the sub-manifold of rank r matrices within the set

of all m× n matrices is (m− r)(n− r).

P e ∼ SNR−(m−r)(n−r)


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Multiplexing gain = r (r integer)

Good H

Full Rank

Typical Bad H

MIMO Channel

Rank H =rε





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Multiplexing gain = r

(r

integer)

Good H

Full Rank

Typical Bad H

MIMO Channel

Rank H =rε





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Typical Error Events

• In 1× 1 and 1× n channels, error occurs when the channel gain

h2

is small.• In general m× n channel, error occurs when some or all of the

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

happen.

• High SNR analysis shows that errors occur typically when H is

close to a rank r matrix.


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Tradeoff Analysis of Specific Designs

Focus on two transmit antennas.

Y = HX+W

Repetition Scheme:

X =x 0

0 x

time

space

1

1

y1 = HF x1 + w1

Alamouti Scheme:

X =

time

space

x -x*

x x2

1 2

1*

[y1y2] = HF [x1x2] + [w1w2]


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Comparison: 2× 1 System

Repetition: y1 = HF x1 + w

Alamouti: [y1y2] = HF [x1x2] + [w1w2]



d * ( r )

(1/2,0)

(0,2)

Repetition


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d * ( r )

(1/2,0)

(0,2)

(1,0)

Alamouti

Repetition


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d * ( r )

(1/2,0)

(0,2)

(1,0)

Optimal Tradeoff

Alamouti

Repetition


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D i v

e r s i t y G a i n :

d * ( r )

(1/2,0)

(0,4)

Repetition


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D i v


d * ( r )

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

(0,4)

Alamouti

Repetition


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D i v


d * ( r )

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

(0,4)

(1,1)

(2,0)

Optimal Tradeoff

Alamouti


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Talk Outline

• point-to-point MIMO channels

• multiple access MIMO channels

• cooperative relaying systems


55/79

Multiple Access

User K

Tx

User 2

Tx

User 1

Tx

M Tx Antenna

M Tx Antenna

N Rx Antenna

Rx

In a point-to-point link, multiple antennas provide diversity and

multiplexing gain.

In a system with K users, multiple antennas can be used to discriminate

signals from different users too.

Continue assuming i.i.d. Rayleigh fading, n receive antennas, m

transmit antennas per user.


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Multiuser Diversity-Multiplexing Tradeoff

Suppose we want every user to achieve an error probability:

P e ∼ SNR−d

and a data rate

R = r log SNR bits/s/Hz.

What is the optimal tradeoff between the diversity gain d and the

multiplexing gain r?

Assume a coding block length l ≥ Km + n− 1.


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Optimal Multiuser D-M Tradeoff: m ≤ n/(K + 1)

(Tse, Viswanath and Zheng 02)



d * ( r )

(min{m,n},0)

(0,mn)

(r, (m−r)(n−r))

(2, (m−2)(n−2))

(1,(m−1)(n−1))

In this regime, the diversity-multiplexing tradeoff of each user is as

though it is the only user in the system, i.e. d∗m,n(r)


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Multiuser Tradeoff: m > n/(K + 1)

K+1n

Spatial Multiplexing Gain : r = R/log SNR

D i v e r s

i t y G a i n : d ( r ) Single User

Performance

(0,mn)

(2,(m−2)(n−2))

(r,(m−Kr)(n−r))

* d (r)*

m,n(1,(m−1)(n−1))

Single-user diversity-multiplexing tradeoff up to r∗ = n/(K + 1).

For r from N/(K + 1) to min{N/K, M }, tradeoff is as though the K users

are pooled together into a single user with KM antennas and rate Kr,

i.e. d∗KM,N (Kr) .


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Multiuser Tradeoff: m > n/(K + 1)

K+1n

d (r)*

m,n

*

Km,nd (Kr)


D i v e r s

i t y G a i n : d ( r ) Single User

Performance

(0,mn)

(2,(m−2)(n−2))

(r,(m−r)(n−r))

Antenna Pooling

(min(m,n/K),0)

*(1,(m−1)(n−1))

Single-user diversity-multiplexing tradeoff up to r∗ = m/(K + 1).

For r from n/(K + 1) to min{n/K, m}, tradeoff is as though the K users

are pooled together into a single user with Km antennas and rate Kr,

i.e. d∗Km,n(Kr) .


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Benefit of Dual Transmit Antennas

User K

Tx

User 2

Tx

User 1Tx

1 Tx Antenna

1 Tx Antenna

N Rx Antenna

Rx

Question: what does adding one more antenna at each mobile buy me?

Assume there are more users than receive antennas.


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Benefit of Dual Transmit Antennas

User K

Tx

User 2

Tx

User 1Tx

M Tx Antenna

M Tx Antenna

N Rx Antenna

Rx

Question: what does adding one more antenna at each mobile buy me?

Assume there are more users than receive antennas.


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Answer

K+1n


D i v e r s i t y G a i n : d ( r )

*

Optimal tradeoff

1 Tx antenna

n

Adding one more transmit antenna does not increase the number of degrees of freedom for each user.

However, it increases the maximum diversity gain from N to 2N .

More generally, it improves the diversity gain d(r) for every r.


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Answer

K+1n



* 2 Tx antenna

2n

Optimal tradeoff

n

1 Tx antenna

Adding one more transmit antenna does not increase the number of degrees of freedom for each user.

However, it increases the maximum diversity gain from n to 2n.

More generally, it improves the diversity gain d(r) for every r.


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Suboptimal Receiver: the Decorrelator/Nuller

User K

Tx

User 2

Tx

User 1

TxDecorrelator

User 1

Decorrelator

User 2

Decorrelator

User K

1 Tx Antenna

1 Tx Antenna

N Rx Antenna

Rx

Data for user 1

Data for user 2

Data for user K

Consider only the case of m = 1 transmit antenna for each user and

number of users K < n.


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Tradeoff for the Decorrelator



*

n−K+1

1

Decorrelator

Maximum diversity gain is n−K + 1: “costs K − 1 diversity gain to null

out K − 1 interferers.” (Winters, Salz and Gitlin 93)

Adding one receive antenna provides either more reliability per user oraccommodate 1 more user at the same reliability. Optimal tradeoff

curve is also a straight line but with a maximum diversity gain of N .

Adding one receive antenna provides more reliability per user and

accommodate 1 more user.


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*

n−K+1

1

Decorrelator



Adding one receive antenna provides either more reliability per user or

accommodate 1 more user at the same reliability.

Optimal tradeoff curve is also a straight line but with a maximum

diversity gain of n.




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*

n−K+1

1

n

Optimal tradeoff

Decorrelator



Adding one receive antenna provides either more reliability per user or

accommodate 1 more user at the same reliability.

Optimal tradeoff curve is also a straight line but with a maximum

diversity gain of n.




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Talk Outline

• point-to-point MIMO channels

• multiple access MIMO channels

• cooperative relaying systems


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Cooperative Relaying

Tx 1

Tx 2

Rx

Channel 2

Channel 1

Cooperative relaying protocols can be designed via a

diversity-multiplexing tradeoff analysis.

(Laneman, Tse, Wornell 02)


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Tx 1

Tx 2

Rx

Channel 2

Channel 1

Cooperation

Cooperative relaying protocols can be designed via adiversity-multiplexing tradeoff analysis.

(Laneman, Tse and Wornell 02)


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Tradeoff Curves of Relaying Strategies

Multiplexing

gain

D i v e r s i t y

g a i n

1½

1

2

direct

transmission


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Tx 1

Tx 2

Rx

Channel 2

Channel 1

Cooperation


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Multiplexing

gain

D i v e r s i t y

g a i n

1½

1

2

direct

transmission


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Multiplexing

gain

D i v e r s i t y

g a i n

1½

1

2

direct

transmission

amplify +

forward


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Multiplexing

gain

D i v e r s i t y

g a i n

1½

1

2

direct

transmission

amplify +

forward

?


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Tx 1

Tx 2

Rx

Channel 2

Channel 1

Cooperation


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Multiplexing

gain

D i v e r s i t y

g a i n

1½

1

2

direct

transmission

amplify +

forward

?


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Multiplexing

gain

D i v e r s i t y

g a i n

1½

1

2

direct

transmission

amplify +

forward

amplify + forward

+ ack


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Conclusion

Diversity-multiplexing tradeoff is a unified way to look at performance

over wireless channels.

Future work:

• Code and receiver design to achieve good tradeoffs.

• Application to other wireless scenarios.

diversity and multiplexing; a fundamental tradeoff in wireless systems

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