Transcript 2816_Chapter9

9: Diversity-Multiplexing Tradeoff
9. MIMO III: Diversity-Multiplexing
Tradeoff
Fundamentals of Wireless Communication, Tse&Viswanath.
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9: Diversity-Multiplexing Tradeoff
Slow Fading MIMO Channel
• So far we have emphasized the spatial multiplexing
aspect of MIMO channels.
• But we also learnt that in slow fading scenario, diversity
is an important thing.
• How do the two aspects interact?
• It turns out that you can get both in a slow fading
channel but there is a fundamental tradeoff.
• We characterize the optimal diversity-multiplexing
tradeoff and find schemes that approach the optimal
tradeoff.
Fundamentals of Wireless Communication, Tse&Viswanath.
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9: Diversity-Multiplexing Tradeoff
Diversity and Freedom (Review)
Two fundamental resources of a MIMO fading channel:
– diversity
– degrees of freedom
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9: Diversity-Multiplexing Tradeoff
Diversity
A channel with more diversity has smaller probability in deep fades.
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Diversity
• Additional independent fading channels increase diversity.
• Spatial diversity: receive, transmit, or both.
• For a nt by nr channel, maximum diversity is nt ¢ nr.
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9: Diversity-Multiplexing Tradeoff
Degrees of Freedom
• 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.
• In a nt by nr channel, there are min{nt,nr} degrees of freedom.
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9: Diversity-Multiplexing Tradeoff
Diversity and Freedom
In a MIMO channel with rich scattering:
maximum diversity = nt ¢ nr
degrees of freedom = min{nt,nr}
The name of the game in space-time coding is to design
schemes which exploit as much of both these resources
as possible.
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9: Diversity-Multiplexing Tradeoff
Space-Time Code Examples: 2 x 1Channel
Repetition Scheme:
Alamouti Scheme:
diversity: 2
diversity: 2
data rate: 1/2 sym/s/Hz
data rate: 1 sym/s/Hz
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9: Diversity-Multiplexing Tradeoff
Performance Summary: 2 x 1Channel
Diversity gain
Degrees of freedom utilized /s/Hz
Repetition
2
1/2
Alamouti
2
1
channel itself
2
1
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9: Diversity-Multiplexing Tradeoff
Space-Time Code Examples: 2 x 2 Channel
Repetition Scheme:
Alamouti Scheme:
diversity: 4
diversity: 4
data rate: 1/2 sym/s/Hz
data rate: 1 sym/s/Hz
But the 2x2 channel has 2 degrees of freedom !
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V-BLAST with Nulling
Send two independent uncoded streams over the two
transmit antennas.
Demodulate each stream by nulling out the other stream.
Data rate: 2 sym/s/Hz
Diversity: 1
Winters et al 93:
Nulling out k interferers using nr receive antennas yields
a diversity gain of nr -k.
Fundamentals of Wireless Communication, Tse&Viswanath.
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9: Diversity-Multiplexing Tradeoff
Performance Summary: 2 x 2 Channel
Diversity gain
d.o.f. utilized /s/Hz
Repetition
4
1/2
Alamouti
4
1
V-Blast with nulling
1
2
channel itself
4
2
Questions:
•
•
•
Alamouti is clearly better than repetition, but how can it be compared to VBlast?
How does one quantify the ``optimal'' performance achievable by any
scheme?
We need to make the notions of “diversity gain” and “d.o.f. utilized” precise
and enrich them.
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9: Diversity-Multiplexing Tradeoff
Classical Diversity Gain
Motivation: PAM
General Definition:
A space-time coding scheme achieves (classical) diversity gain dmax, if
for a fixed data rate.
i.e. error probability deceases by 1/2dmax for every 3 dB increase in SNR, by
1/4dmax for every 6 dB increase, etc.
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Example: PAM in 1 by 1Channel
Every 6 dB increase in SNR doubles the distance between
constellation points for a given rate.
Both PAM and QAM have the same (classical) diversity gain of 1.
(classical) diversity gain does not say anything about the d.o.f.
utilized by the scheme.
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Ask a Dual Question
Every 6 dB doubles the constellation size for a given reliability, for
PAM.
But for QAM, every 6 dB quadruples the constellation size.
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Degrees of Freedom Utilized
Definition:
A space-time coding scheme utilizes rmax degrees of freedom/s/Hz if
the data rate scales like
for a fixed error probability (reliability).
In a 1x1 channel, rmax = 1/2 for PAM, rmax = 1 for QAM.
Note: A space-time coding scheme is a family of modulations within
a certain structure, with varying symbol alphabet as a function of
SNR.
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Diversity-Multiplexing Tradeoff
Every 3 dB increase in SNR yields
either
a 1/2dmax decrease in error probability for a fixed rate;
or
rmax additional bits/s/Hz for a fixed reliability.
But these are two extremes of a rate-reliability tradeoff.
More generally, one wants to increase reliability and the
data rate at the same time.
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Diversity-Multiplexing Tradeoff of A Scheme
Definition
A space-time coding scheme achieves a diversity-multiplexing
tradeoff curve d(r) if for each multiplexing gain r, simultaneously
and
The largest multiplexing gain is rmax, the d.o.f. utilized by the
scheme.
The largest diversity gain is dmax = d(0), the classical diversity gain.
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Diversity-Multiplexing Tradeoff of the
Channel
Definition
The diversity-multiplexing tradeoff d*(r) of a MIMO channel is the
best possible diversity-multiplexing tradeoff achievable by any
scheme.
r*max is the largest multiplexing gain achievable in the channel.
d*max = d*(0) is the largest diversity gain achievable.
For a nt x nr MIMO channel, it is not difficult to show:
What is more interesting is how the entire curve looks like.
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Example: 1 x 1 Channel
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Example: 2 x 1 Channel
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Example: 2 x 2 Channel
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ML vs Nulling in V-Blast
Winters, Salz and Gitlins 93:
Nulling out k interferers using nr receive antennas provides a diversity gain
of nr-k.
Tse,Viswanath and Zheng 03:
Jointly detecting all users provides a diversity gain of nr to each.
There is free lunch. (?)
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Optimal D-M Tradeoff for General nt x nr Channel
As long as block length
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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D-M Tradeoff Optimal Code Design
How does one design space-time codes that are tradeoff
optimal?
Needs a code design criterion.
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DM-Tradeoff and Outage
• So far we have looked at tradeoff for finite block length S-T codes.
• For codes with arbitrary long block length, the optimal tradeoff can be
computed from the outage probability:
• This turns out to be exactly the same as the D-M tradeoff for block
length nt.
• This suggests that optimal tradeoff codes can be designed from an
outage point of view.
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Univeral Code Design Criterion
Simplifies in the high SNR regime.
• Parallel channel: product distance of codeword
difference (same as Rayleigh!)
• MISO channel: smallest singular value of codeword
matrix difference.
• MIMO channel: product of min(nt,nr) smallest singular
values.
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Outage Revisited
• Outage probability:
• Operational interpretation: there is a universal code that can reliably
communicate whenever H is not in outage.
• A pairwise criterion between two codewords can be derived by looking at
the worst-case error probability over all H not in outage.
• This is in contrast to the classic code design criterion, which is computed
by averaging over the channel statistics.
• A more robust approach.
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Achieving Optimal Diversity-Multiplexing
Tradeoff
• Hao and Wornell 03: MIMO rotation code (2 x 2 channel
only)
• Tavildar and Viswanath 04: D-Blast plus permutation
code.
• El Gamal, Caire and Damen 03: Lattice codes.
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Tavildar and Viswanath 04
• First use D-BLAST to convert the MIMO channel into a
parallel channel.
• Then design permutation codes to achieve the optimal
diversity-multiplexing tradeoff on the parallel channel.
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D-BLAST
Gains g1 and g2 are correlated and have a complicated distribution but
universal code design is oblivious to this.
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Permutation Codes for Parallel Channel
It is shown that there are universal permutation codes that can achieve
the optimal diversity-multiplexing tradeoff of the parallel channel.
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Multiple Antennas in Cellular Systems
• In a point-to-point link, multiple antennas provide
diversity and degrees of freedom gain.
• In a cellular systems, they can also perform SDMA and
out-of-cell interference suppression.
• The D-M tradeoff framework can be extended to
incorporate these gains as well.
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Example: MIMO + SDMA ?
Question: what does adding one
more antenna at each mobile buy
me?
No increase in total d.o.f. if the
number of users is greater than the
number of receive antennas.
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DM-Tradeoff Analysis
• D.o.f. unchanged, but maximum diversity gain increases from n to 2n.
• More generally, it improves the diversity gain d(r) for every r.
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Summary
Diversity-multiplexing tradeoff is a unified way to look at
performance of MIMO channels.
It puts diversity and multiplexing on an equal footing.
It provides a framework to compare existing schemes as
well as stimulates the design of new schemes.
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