diversity and freedom: a fundamental tradeoff in multiple antenna...
TRANSCRIPT
![Page 1: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/1.jpg)
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
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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.
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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.
![Page 4: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/4.jpg)
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.
![Page 5: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/5.jpg)
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.
![Page 6: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/6.jpg)
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.
![Page 7: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/7.jpg)
Diversity
Fading Channel: h 1
• Additional independent signal paths increase diversity.
• Diversity: receive, transmit or both.
• Compensate against channel unreliability.
![Page 8: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/8.jpg)
Diversity
1Fading Channel: h
2Fading Channel: h
• Additional independent signal paths increase diversity.
• Diversity: receive, transmit or both.
• Compensate against channel unreliability.
![Page 9: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/9.jpg)
Diversity
1Fading Channel: h
2Fading Channel: h
• Additional independent signal paths increase diversity.
• Diversity: receive, transmit or both.
• Compensate against channel unreliability.
![Page 10: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/10.jpg)
Diversity
1Fading Channel: h
2Fading Channel: h
• Additional independent signal paths increase diversity.
• Diversity: receive, transmit or both.
• Compensate against channel unreliability.
![Page 11: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/11.jpg)
Diversity
1
Fading Channel: h
Fading Channel: h
4
Fading Channel: h
Fading Channel: h
2
3
• Additional independent signal paths increase diversity.
• Diversity: receive, transmit or both.
• Compensate against channel unreliability.
![Page 12: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/12.jpg)
Diversity
1
Fading Channel: h
Fading Channel: h
4
Fading Channel: h
Fading Channel: h
2
3
• Additional independent signal paths increase diversity.
• Diversity: receive, transmit or both.
• Compensate against channel unreliability.
![Page 13: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/13.jpg)
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.
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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.
![Page 15: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/15.jpg)
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.
![Page 16: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/16.jpg)
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.
![Page 17: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/17.jpg)
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
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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.
![Page 19: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/19.jpg)
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.
![Page 20: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/20.jpg)
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.
![Page 21: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/21.jpg)
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.
![Page 22: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/22.jpg)
Outline
• Problem formulation and main result on optimal tradeoff.
• Sketch of proof.
• Comparison of existing schemes.
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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.
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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)
![Page 25: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/25.jpg)
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)
![Page 26: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/26.jpg)
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)
![Page 27: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/27.jpg)
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)
![Page 28: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/28.jpg)
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)
![Page 29: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/29.jpg)
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)
![Page 30: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/30.jpg)
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)
![Page 31: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/31.jpg)
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)
![Page 32: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/32.jpg)
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)
![Page 33: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/33.jpg)
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.
![Page 34: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/34.jpg)
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)
(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.
![Page 35: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/35.jpg)
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)
(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.
![Page 36: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/36.jpg)
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)
(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.
![Page 37: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/37.jpg)
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)
(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.
![Page 38: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/38.jpg)
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)
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)
For integer r, it is as though r transmit and r receive antennas were
dedicated for multiplexing and the rest provide diversity.
![Page 39: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/39.jpg)
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
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Revisit the 2× 2 Example (ctd.)
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
(2,0)
(0,4)
(1,1)
• Tradeoff bridges the gap between the two types of approaches.
![Page 41: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/41.jpg)
Revisit the 2× 2 Example (ctd.)
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
(2,0)
(0,4)
(1,1)
• Tradeoff bridges the gap between the two types of approaches.
![Page 42: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/42.jpg)
Adding More Antennas
Spatial Multiplexing Gain: r=R/log SNR
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.
![Page 43: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/43.jpg)
Adding More Antennas
Spatial Multiplexing Gain: r=R/log SNR
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.
![Page 44: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/44.jpg)
Adding More Antennas
Spatial Multiplexing Gain: r=R/log SNR
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
gain +1 for any diversity requirement d.
![Page 45: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/45.jpg)
Increasing vs Fixed Code Rate
SNR
SNR increases
Channel
SNR 1
Channel
SNR 2
Channel
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Increasing vs Fixed Code Rate
SNR
code
SNR increases
Channel
SNR 1
Channel
SNR 2
Channel
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Increasing vs Fixed Code Rate
SNR
code code code
SNR increases
Channel
SNR 1
Channel
SNR 2
Channel
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Increasing vs Fixed Code Rate
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
![Page 49: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/49.jpg)
Increasing vs Fixed Code Rate
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
![Page 50: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/50.jpg)
Outline
• Problem formulation and main result on optimal tradeoff.
• Sketch of proof.
• Comparison of existing schemes.
![Page 51: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/51.jpg)
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∗] .
![Page 52: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/52.jpg)
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∗] .
![Page 53: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/53.jpg)
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∗] .
![Page 54: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/54.jpg)
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.
![Page 55: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/55.jpg)
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.
![Page 56: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/56.jpg)
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.
![Page 57: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/57.jpg)
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.
![Page 58: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/58.jpg)
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.
![Page 59: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/59.jpg)
Geometric Picture (integer r)
-1/2
Bad H
SNR
Scalar Channel
Good H
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.
![Page 60: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/60.jpg)
Geometric Picture (integer r)
-1/2
Rank(H)=r
SNR
All MxN Matrices
Good HBad H
Scalar Channel Vector 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.
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Geometric Picture (integer r)
-1/2
SNR-1/2
Vector Channel
SNRRank(H)=r
Good HBad H
Good HFull Rank
Typical Bad HScalar 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.
![Page 62: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/62.jpg)
Geometric Picture (integer r)
-1/2
SNR-1/2
Vector Channel
SNRRank(H)=r
Good HBad H
Good HFull Rank
Typical Bad HScalar 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.
![Page 63: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/63.jpg)
Piecewise Linearity of Tradeoff Curve
Spatial Multiplexing Gain: r=R/log SNR
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.
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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?
![Page 65: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/65.jpg)
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.
![Page 66: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/66.jpg)
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.
![Page 67: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/67.jpg)
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.
![Page 68: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/68.jpg)
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.
![Page 69: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/69.jpg)
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.
![Page 70: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/70.jpg)
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).
![Page 71: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/71.jpg)
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).
![Page 72: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/72.jpg)
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).
![Page 73: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/73.jpg)
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)
![Page 74: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/74.jpg)
To Fade or Not to Fade?
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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.
![Page 76: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/76.jpg)
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.
![Page 77: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/77.jpg)
Outline
• Problem formulation and main result on optimal tradeoff.
• Sketch of proof.
• Comparison of existing schemes.
![Page 78: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/78.jpg)
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)
![Page 79: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/79.jpg)
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)
![Page 80: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/80.jpg)
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
![Page 81: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/81.jpg)
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
![Page 82: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/82.jpg)
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
![Page 83: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/83.jpg)
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]
![Page 84: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/84.jpg)
Comparison: 2× 1 System
Repetition: r1 = ‖H‖x1 + w
Alamouti: [r1r2] = ‖H‖[x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
(1/2,0)
(0,2)
![Page 85: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/85.jpg)
Comparison: 2× 1 System
Repetition: r1 = ‖H‖x1 + w
Alamouti: [r1r2] = ‖H‖[x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
(1/2,0)
(0,2)
(0,1)
![Page 86: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/86.jpg)
Comparison: 2× 1 System
Repetition: r1 = ‖H‖x1 + w
Alamouti: [r1r2] = ‖H‖[x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
RepetitiveAlamoutiOptimal
(1/2,0)
(0,2)
(0,1)
![Page 87: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/87.jpg)
Comparison: 2× 2 System
Repetition: r1 = ‖H‖x1 + w
Alamouti: [r1r2] = ‖H‖[x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
(1/2,0)
(0,4)
![Page 88: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/88.jpg)
Comparison: 2× 2 System
Repetition: r1 = ‖H‖x1 + w
Alamouti: [r1r2] = ‖H‖[x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
(1/2,0) (1,0)
(0,4)
![Page 89: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/89.jpg)
Comparison: 2× 2 System
Repetition: r1 = ‖H‖x1 + w
Alamouti: [r1r2] = ‖H‖[x1x2] + [w1w2]
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty G
ain:
d
* (r)
RepetitiveAlamoutiOptimal
(1/2,0) (1,0)
(0,4)
(1,1)
(2,0)
![Page 90: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/90.jpg)
V-BLAST
Antenna 2:
Antenna 1:
TT
Antenna 1:
Antenna 2:
• Nulling and Canceling
• Independent data streams transmitted over antennas
![Page 91: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/91.jpg)
V-BLAST
Antenna 2:
Receive
Null
Antenna 1:
• Nulling and Canceling
• Independent data streams transmitted over antennas
![Page 92: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/92.jpg)
V-BLAST
Antenna 2:
Null
Receive
Antenna 1:
• Nulling and Canceling
• Independent data streams transmitted over antennas
![Page 93: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/93.jpg)
V-BLAST
Antenna 2:
Antenna 1:
• Nulling and Canceling
• Independent data streams transmitted over antennas
![Page 94: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/94.jpg)
V-BLAST
Antenna 2:
Antenna 1:
Cancel
• Nulling and Canceling
• Independent data streams transmitted over antennas
![Page 95: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/95.jpg)
V-BLAST
Antenna 2:
Antenna 1:
TT
Antenna 1:
Antenna 2:
• Nulling and Canceling
• Independent data streams transmitted over antennas
![Page 96: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/96.jpg)
Tradeoff Performance of V-BLAST (N ×N)
Original V-BLAST
(N,0)
1
Low diversity due to lack of coding over space
![Page 97: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/97.jpg)
Tradeoff Performance of V-BLAST (N ×N)
V-BLAST with optimal rate allocation
(N,0)
1
(0,N)
Low diversity due to lack of coding over space
![Page 98: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/98.jpg)
Tradeoff Performance of V-BLAST (N ×N)
Compare to the optimal tradeoff
(N,0)
1
(0,N)
(0,N2)
Low diversity due to lack of coding over space
![Page 99: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/99.jpg)
Tradeoff Performance of V-BLAST (N ×N)
Compare to the optimal tradeoff
(N,0)
1
(0,N)
(0,N2)
Low diversity due to lack of coding over space
![Page 100: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/100.jpg)
D-BLAST
Antenna 2:
Antenna 1:
T
Assume T = ∞, ignore the overhead.
![Page 101: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/101.jpg)
D-BLAST
Antenna 2:
Antenna 1:
Receive
Assume T = ∞, ignore the overhead.
![Page 102: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/102.jpg)
D-BLAST
Antenna 2:
Antenna 1:
Receive
Null
Assume T = ∞, ignore the overhead.
![Page 103: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/103.jpg)
D-BLAST
Antenna 2:
Antenna 1:
Assume T = ∞, ignore the overhead.
![Page 104: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/104.jpg)
D-BLAST
Antenna 2:
Antenna 1:
Receive
Cancel
Assume T = ∞, ignore the overhead.
![Page 105: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/105.jpg)
D-BLAST
Antenna 2:
Null
Antenna 1:
Receive
Assume T = ∞, ignore the overhead.
![Page 106: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/106.jpg)
D-BLAST
Antenna 2:
Antenna 1:
Assume T = ∞, ignore the overhead.
![Page 107: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/107.jpg)
D-BLAST
Antenna 2:
Cancel
Antenna 1:
Receive
Assume T = ∞, ignore the overhead.
![Page 108: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/108.jpg)
D-BLAST
Antenna 2:
Null
Antenna 1:
Receive
Ignore the overhead for now.
![Page 109: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/109.jpg)
D-BLAST: Square System
Spatial Multiplexing Gain: r=R/log SNR
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.
![Page 110: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/110.jpg)
D-BLAST: Square System
Spatial Multiplexing Gain: r=R/log SNR
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.
![Page 111: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/111.jpg)
Replace Nulling by MMSE?
![Page 112: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/112.jpg)
D-BLAST+MMSE
Spatial Multiplexing Gain: r=R/log SNR
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.
![Page 113: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/113.jpg)
D-BLAST+MMSE
Spatial Multiplexing Gain: r=R/log SNR
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.
![Page 114: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/114.jpg)
D-BLAST+MMSE
Spatial Multiplexing Gain: r=R/log SNR
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.
![Page 115: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/115.jpg)
Penalty due to Overhead
Spatial Multiplexing Gain: r=R/log SNR
Div
ersi
ty A
dvan
tage
: d
* (r)
DBLAST+NullingDBLAST+MMSEOptimal
![Page 116: Diversity and Freedom: A Fundamental Tradeoff in Multiple Antenna Channelsdntse/papers/msri02.pdf · 2002. 2. 28. · Diversity Fading Channel: h1 ... † Multiple antennas provide](https://reader035.vdocuments.net/reader035/viewer/2022071423/611df3e9304a2b55a642a180/html5/thumbnails/116.jpg)
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.