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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Example: Spatial Diversity
1Fading Channel: h
2Fading Channel: h
• Additional independent fading channels increase diversity.• Spatial diversity: receive, transmit or both.
• Repeat and Average: compensate against channel unreliability.
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Example: Spatial Diversity
1Fading Channel: h
2Fading Channel: h
• Additional independent fading channels increase diversity.• Spatial diversity : receive, transmit or both.
• Repeat and Average: compensate against channel unreliability.
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Example: Spatial Diversity
1Fading Channel: h
2Fading Channel: h
• Additional independent fading channels increase diversity.• Spatial diversity: receive, transmit or both.
• Repeat and Average: compensate against channel unreliability.
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Example: Spatial Diversity
1
Fading Channel: h
Fading Channel: h
4
Fading Channel: h
Fading Channel: h
2
3
• Additional independent fading channels increase diversity.• Spatial diversity: receive, transmit or both.
• Repeat and Average: compensate against channel unreliability.
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Example: Spatial Diversity
1
Fading Channel: h
Fading Channel: h
4
Fading Channel: h
Fading Channel: h
2
3
• Additional independent fading channels increase diversity.• Spatial diversity: receive, transmit or both.
• Repeat and Average: compensate against channel unreliability.
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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
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
y1
Signature 2
Signature 1
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
y1
Signature 2Signature 1
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
y1
Signature 2
Signature 1
Environment
Fading
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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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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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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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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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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Fundamental Tradeoff
A space-time coding scheme achieves
Spatial Multiplexing Gain r : R = r logSNR
andDiversity 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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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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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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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)
(1,(m−1)(n−1))
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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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)
(2, (m−2)(n−2))
(1,(m−1)(n−1))
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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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)
(r, (m−r)(n−r))
(2, (m−2)(n−2))
(1,(m−1)(n−1))
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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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)
(r, (m−r)(n−r))
(2, (m−2)(n−2))
(1,(m−1)(n−1))
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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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)
(r, (m−r)(n−r))
1
Multiple Antenna
m x n channel
Single Antenna
channel
1
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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What do I get by adding one more antenna at thetransmitter and the receiver?
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Adding More Antennas
Spatial Multiplexing Gain: r=R/log SNR
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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Adding More Antennas
Spatial Multiplexing Gain: r=R/log SNR
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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Adding More Antennas
Spatial Multiplexing Gain: r=R/log SNR
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
gain +1 for any diversity requirement d.
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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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Geometric Picture for General m× n Channels
Multiplexing gain = r (r integer)
Good H
Full Rank
Typical Bad H
MIMO Channel
Rank H =rε
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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Geometric Picture for General m× n Channels
Multiplexing gain = r
(r
integer)
Good H
Full Rank
Typical Bad H
MIMO Channel
Rank H =rε
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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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]
Spatial Multiplexing Gain: r=R/log SNR
D i v e r s i t y G a i n :
d * ( r )
(1/2,0)
(0,2)
Repetition
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Comparison: 2× 1 System
Repetition: y1 = HF x1 + w
Alamouti: [y1y2] = HF [x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
D i v e r s i t y G a i n :
d * ( r )
(1/2,0)
(0,2)
(1,0)
Alamouti
Repetition
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Comparison: 2× 1 System
Repetition: y1 = HF x1 + w
Alamouti: [y1y2] = HF [x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
D i v e r s i t y G a i n :
d * ( r )
(1/2,0)
(0,2)
(1,0)
Optimal Tradeoff
Alamouti
Repetition
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Comparison: 2× 2 System
Repetition: y1 = HF x1 + w
Alamouti: [y1y2] = HF [x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
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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Comparison: 2× 2 System
Repetition: y1 = HF x1 + w
Alamouti: [y1y2] = HF [x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
D i v
e r s i t y G a i n :
d * ( r )
(1/2,0)(1,0)
(0,4)
Alamouti
Repetition
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Comparison: 2× 2 System
Repetition: y1 = HF x1 + w
Alamouti: [y1y2] = HF [x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
D i v
e r s i t y G a i n :
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
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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)
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)
(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)
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−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
Spatial Multiplexing Gain : r = R/log SNR
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
Spatial Multiplexing Gain : r = R/log SNR
D i v e r s i t y G a i n : d ( r )
* 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
Spatial Multiplexing Gain : r = R/log SNR
D i v e r s i t y G a i n : d ( r )
*
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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Tradeoff for the Decorrelator
Spatial Multiplexing Gain : r = R/log SNR
D i v e r s i t y G a i n : d ( r )
*
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 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.
Adding one receive antenna provides more reliability per user and
accommodate 1 more user.
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Tradeoff for the Decorrelator
Spatial Multiplexing Gain : r = R/log SNR
D i v e r s i t y G a i n : d ( r )
*
n−K+1
1
n
Optimal tradeoff
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 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.
Adding one receive antenna provides more reliability per user and
accommodate 1 more user.
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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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Cooperative Relaying
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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Cooperative Relaying
Tx 1
Tx 2
Rx
Channel 2
Channel 1
Cooperation
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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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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
amplify +
forward
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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
amplify +
forward
?
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Cooperative Relaying
Tx 1
Tx 2
Rx
Channel 2
Channel 1
Cooperation
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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
amplify +
forward
?
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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
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.
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