Diversity-Multiplexing Tradeoff in MIMO Channels
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1 Diversity-Multiplexing Tradeoff in MIMO Channels David Tse Department of EECS, U.C. Berkeley February 26, 2004 Intel Smart Antenna Workshop
2 Two objectives of the talk: Present a new performance metric for evaluating MIMO coding schemes. Give some examples of new coding schemes designed to optimize the metric.
3 Diversity and Freedom Two fundamental resources of a MIMO fading channel: diversity degrees of freedom
4 Diversity Channel Quality A channel with more diversity has smaller probability in deep fades. t
5 Diversity Fading Channel: h 1 Additional independent channel paths increase diversity. Spatial diversity: receive, transmit or both. For a m by n channel, maximum diversity is mn.
6 Diversity Fading Channel: h 1 Fading Channel: h 2 Additional independent fading channels increase diversity. Spatial diversity: receive, transmit or both. For a m by n channel, maximum diversity is mn.
7 Diversity Fading Channel: h 1 Fading Channel: h 2 Additional independent fading channels increase diversity. Spatial diversity : receive, transmit or both. For a m by n channel, maximum diversity is mn.
8 Diversity Fading Channel: h 1 Fading Channel: h 2 Additional independent fading channels increase diversity. Spatial diversity: receive, transmit or both. For a m by n channel, maximum diversity is mn.
9 Diversity Fading Channel: h 1 Fading Channel: h 2 Fading Channel: h 3 Fading Channel: h 4 Additional independent fading channels increase diversity. Spatial diversity: receive, transmit or both. For a m by n channel, maximum diversity is mn.
10 Diversity Fading Channel: h 1 Fading Channel: h 2 Fading Channel: h 3 Fading Channel: h 4 Additional independent fading channels increase diversity. Spatial diversity: receive, transmit or both. For a m by n channel, diversity is mn.
11 Degrees of Freedom y 2 y 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. In a m by n channel with rich scattering, there are min{m, n} degrees of freedom.
12 Degrees of Freedom y 2 Signature 1 y 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. In a m by n channel with rich scattering, there are min{m, n} degrees of freedom.
13 Degrees of Freedom y 2 Signature 1 y 1 Signature 2 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 m by n channel with rich scattering, there are min{m, n} degrees of freedom.
14 Degrees of Freedom y 2 Signature 1 Signature 2 y 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. In a m by n channel with rich scattering, there are min{m, n} degrees of freedom.
15 Degrees of Freedom y 2 Fading Environment Signature 1 y 1 Signature 2 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 m by n channel with rich scattering, there are min{m, n} degrees of freedom.
16 Degrees of Freedom y 2 Fading Environment Signature 1 y 1 Signature 2 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 m by n channel with rich scattering, there are min{m, n} degrees of freedom.
17 Diversity and Freedom In a MIMO channel with rich scattering: maximum diversity = mn degrees of freedom = min{m, n} The name of the game in space-time coding is to design schemes which exploit as much of both these resources as possible.
18 Space-Time Code Examples: 2 1 Channel Repetition Scheme: Alamouti Scheme: X = x x 1 time X = x -x * x x * 1 time space space diversity: 2 data rate: 1/2 sym/s/hz diversity: 2 data rate: 1 sym/s/hz
19 Performance Summary: 2 1 Channel Diversity gain Degrees of freedom utilized /s/hz Repetition 2 1/2 Alamouti 2 1 channel itself 2 1
20 Space-Time Code Examples: 2 2 Channel Repetition Scheme: Alamouti Scheme: X = x x 1 time X = x -x * x x * 1 time space space diversity gain : 4 data rate: 1/2 sym/s/hz But the 2 2 channel has 2 degrees of freedom! diversity gain : 4 data rate: 1 sym/s/hz
21 Space-Time Code Examples: 2 2 Channel Repetition Scheme: Alamouti Scheme: X = x x 1 time X = x -x * x x * 1 time space space diversity: 4 data rate: 1/2 sym/s/hz But the 2 2 channel has 2 degrees of freedom! diversity: 4 data rate: 1 sym/s/hz
22 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, Salz and Gitlins 93: Nulling out k interferers using n receive antennas yields a diversity gain of n k.
23 Performance Summary: 2 2 Channel Questions: 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 Alaomuti is clearly better than repetition, but how can it be compared to V-Blast? How does one quantify the optimal performance achievable by any scheme? We need to make the notions of fiversity gain and d.o.f. utilized precise and enrich them.
24 Performance Summary: 2 2 Channel Questions: 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 Alaomuti is clearly better than repetition, but how can it be compared to V-Blast? How does one quantify the optimal performance achievable by any scheme? We need to make the notions of fiversity gain and d.o.f. utilized precise and enrich them.
25 Performance Summary: 2 2 Channel Questions: 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 Alaomuti is clearly better than repetition, but how can it be compared to V-Blast? How does one quantify the optimal performance achievable by any scheme? We need to make the notions of fiversity gain and d.o.f. utilized precise and enrich them.
26 Classical Diversity Gain Motivation: PAM y = hx + w P e P ( h is small ) SNR 1 y 1 = h 1 x + w 1 y 2 = h 2 x + w 2 P e P ( h 1, h 2 are both small) SNR 2 Definition A space-time coding scheme achieves (classical) diversity gain d, if P e (SNR) SNR d for a fixed data rate. i.e. error probability deceases by 2 d for every 3 db increase in SNR, by 1/4 d for every 6dB increase, etc.
27 Classical Diversity Gain Motivation: PAM y = hx + w P e P ( h is small ) SNR 1 y 1 = h 1 x + w 1 P e P ( h 1, h 2 are both small) y 2 = h 2 x + w 2 SNR 2 General Definition A space-time coding scheme achieves (classical) diversity gain d max, if P e (SNR) SNR d max for a fixed data rate. i.e. error probability deceases by 2 d max by 4 d max for every 6dB increase, etc. for every 3 db increase in SNR,
28 Example: PAM vs QAM in 1 by 1 Channel Every 6 db increase in SNR doubles the distance between constellation points for a given rate. PAM P e 1 4 -a +a -2a +2a 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.
29 Example: PAM vs QAM in 1 by 1 Channel Every 6 db increase in SNR doubles the distance between constellation points for a given rate. PAM P e 1 4 -a +a -2a +2a QAM P e 1 4 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.
30 Example: PAM vs QAM in 1 by 1 Channel Every 6 db increase in SNR doubles the distance between constellation points for a given rate. PAM P e 1 4 -a +a -2a +2a QAM P e 1 4 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.
31 Ask a Dual Question Every 6 db doubles the constellation size for a given reliability, for PAM. PAM +1 bit -a +a -3a -a +a +3a But for QAM, every 6 db quadruples the constellation size.
32 Ask a Dual Question Every 6 db doubles the constellation size for a given reliability, for PAM PAM +1 bit -a +a -3a -a +a +3a QAM +2 bits But for QAM, every 6 db quadruples the constellation size.
33 Degrees of Freedom Utilized Definition: A space-time coding scheme utilizes r max degrees of freedom/s/hz if the data rate scales like R(SNR) r max log 2 SNR bits/s/hz for a fixed error probability (reliability) In a 1 1 channel, r max = 1/2 for PAM, r max = 1 for QAM. Note: A space-time coding scheme is a family of codes within a certain structure, with varying symbol alphabet as a function of SNR.
34 Degrees of Freedom Utilized Definition: A space-time coding scheme utilizes r max degrees of freedom/s/hz if the data rate scales like R(SNR) r max log 2 SNR bits/s/hz for a fixed error probability (reliability) In a 1 1 channel, r max = 1/2 for PAM, r max = 1 for QAM. Note: A space-time coding scheme is a family of codes within a certain structure, with varying symbol alphabet as a function of SNR.
35 Diversity-Multiplexing Tradeoff Every 3 db increase in SNR yields either a 2 d max decrease in error probability for a fixed rate; or r max 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.
36 Diversity-Multiplexing Tradeoff Every 3 db increase in SNR yields either a 2 d max decrease in error probability for a fixed rate; or r max 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.
37 Diversity-Multiplexing Tradeoff Every 3 db increase in SNR yields either a 2 d max decrease in error probability for a fixed rate; or r max additional bits/s/hz for a fixed reliability. But these are two extremes of a rate-reliability tradeoff. More generally, one can increase reliability and the data rate at the same time.
38 Diversity-Multiplexing Tradeoff of A Scheme (Zheng and Tse 03) Definition A space-time coding scheme achieves a diversity-multiplexing tradeoff curve d(r) if for each multiplexing gain r, simultaneously R(SNR) r log 2 SNR bits/s/hz and P e (SNR) SNR d(r). The largest multiplexing gain is r max, the d.o.f. utilized by the scheme. The largest diversity gain is d max = d(0), the classical diversity gain.
39 Diversity-Multiplexing Tradeoff of A Scheme (Zheng and Tse 03) Definition A space-time coding scheme achieves a diversity-multiplexing tradeoff curve d(r) if for each multiplexing gain r, simultaneously R(SNR) r log 2 SNR bits/s/hz and P e (SNR) SNR d(r). The largest multiplexing gain is r max, the d.o.f. utilized by the scheme. The largest diversity gain is d max = d(0), the classical diversity gain.
40 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 m n MIMO channel, it is not difficult to show: r max = min{m, n} d max = mn What is more interesting is how the entire curve looks like.
41 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 m n MIMO channel, it is not difficult to show: r max = min{m, n} d max = mn What is more interesting is how the entire curve looks like.
42 Example: 1 1 Channel Fixed Rate (0,1) Diversity Gain: d * (r) PAM QAM Fixed Reliability (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
43 Example: 2 1 Channel Diversity Gain: d * (r) (0,2) Repetition (1/2,0) Spatial Multiplexing Gain: r=r/log SNR
44 Example: 2 1 Channel Diversity Gain: d * (r) (0,2) Alamouti Repetition (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
45 Example: 2 1 Channel Optimal Tradeoff Diversity Gain: d * (r) (0,2) Alamouti Repetition (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
46 Example: 2 2 Channel (0,4) Diversity Gain: d * (r) Repetition (1/2,0) Spatial Multiplexing Gain: r=r/log SNR
47 Example: 2 2 Channel (0,4) Diversity Gain: d * (r) Alamouti Repetition (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
48 Example: 2 2 Channel (0,4) Repetition Alamouti Diversity Gain: d * (r) (0,1) V BLAST(Nulling) (2,0) (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
49 Example: 2 2 Channel Diversity Gain: d * (r) (0,1) (0,4) Repetition Alamouti Optimal Tradeoff (1,1) V BLAST(Nulling) (2,0) (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
50 Example: 2 2 Channel Diversity Gain: d * (r) (0,2) (0,1) (0,4) Repetition Alamouti Optimal Tradeoff V BLAST(ML) (1,1) V BLAST(Nulling) (2,0) (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
51 ML vs Nulling in V-Blast Diversity Gain: d * (r) (0,2) (0,1) V BLAST(ML) V BLAST(Nulling) (2,0) Spatial Multiplexing Gain: r=r/log SNR Winters, Salz and Gitlins 93: Nulling out k interferers using n receive antennas provides a diversity gain of n k. Tse,Viswanath and Zheng 03: Jointly detecting all users provides a diversity gain of n to each. There is free lunch. (?)
52 ML vs Nulling in V-Blast Diversity Gain: d * (r) (0,2) (0,1) V BLAST(ML) V BLAST(Nulling) (2,0) Spatial Multiplexing Gain: r=r/log SNR Winters, Salz and Gitlins 93: Nulling out k interferers using n receive antennas provides a diversity gain of n k. Tse,Viswanath and Zheng 03: Jointly detecting all users provides a diversity gain of n to each. There is free lunch. (?)
53 ML vs Nulling in V-Blast Diversity Gain: d * (r) (0,2) (0,1) V BLAST(ML) V BLAST(Nulling) (2,0) Spatial Multiplexing Gain: r=r/log SNR Winters, Salz and Gitlins 93: Nulling out k interferers using n receive antennas provides a diversity gain of n k. Tse,Viswanath and Zheng 03: Jointly detecting all users provides a diversity gain of n to each. There is free lunch. (?)
54 Optimal D-M Tradeoff for General m n Channel (Zheng and Tse 03) As long as block length l m + n 1: (0,mn) Diversity Gain: d * (r) (min{m,n},0) Spatial Multiplexing Gain: r=r/log SNR For integer r, it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity.
55 Optimal D-M Tradeoff for General m n Channel (Zheng and Tse 03) As long as block length l m + n 1: (0,mn) Diversity Gain: d * (r) (1,(m 1)(n 1)) (min{m,n},0) Spatial Multiplexing Gain: r=r/log SNR For integer r, it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity.
56 Optimal D-M Tradeoff for General m n Channel (Zheng and Tse 03) As long as block length l m + n 1: (0,mn) Diversity Gain: d * (r) (1,(m 1)(n 1)) (2, (m 2)(n 2)) (min{m,n},0) Spatial Multiplexing Gain: r=r/log SNR For integer r, it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity.
57 Optimal D-M Tradeoff for General m n Channel (Zheng and Tse 03) As long as block length l m + n 1: (0,mn) Diversity Gain: d * (r) (1,(m 1)(n 1)) (2, (m 2)(n 2)) (r, (m r)(n r)) (min{m,n},0) Spatial Multiplexing Gain: r=r/log SNR For integer r, it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity.
58 Optimal D-M Tradeoff for General m n Channel (Zheng and Tse 03) As long as block length l m + n 1: (0,mn) Diversity Gain: d * (r) (1,(m 1)(n 1)) (2, (m 2)(n 2)) (r, (m r)(n r)) (min{m,n},0) Spatial Multiplexing Gain: r=r/log SNR For integer r, it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity.
59 Achieving Optimal Diversity-Multiplexing Tradeoff Hao and Wornell 03: MIMO rotation code (2 2 channel only). Tavildar and Viswanath 04: D-Blast plus permutation code. El Gamal, Caire and Damen 03: Lattice codes.
60 Hao and Wornell 03 Alamouti scheme: x 1 x 2 x 2 x 1 Hao and Wornell s scheme: x 1 x 2 x 3 x 4 where x 1 x 4 = Rotate(θ 1) u 1 u 4 x 2 x 3 = Rotate(θ 2) u 2 u 3 and u 1, u 2, u 3, u 4 are independent QAM symbols.
61 Tavildar and Viswanth 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.
62 D-BLAST Antenna 1: Antenna 2: Receive
63 D-BLAST Receive Antenna 1: Antenna 2: Null
64 D-BLAST Antenna 1: Antenna 2:
65 D-BLAST Cancel Antenna 1: Antenna 2: Receive
66 Original D-Blast is sub-optimal. D-Blast with MMSE suppression is information lossless h 12 h 21 h 11 D BLAST g 1 h 22 g 2
67 Permutation Coding for Parallel Channel The channel is parallel but the fading at the different sub-channels are correlated. Nevertheless it is shown that the permutation codes can achieve the optimal diversity-multiplexing tradeoff of the parallel channel.
68 Conclusion Diversity-multiplexing tradeoff is a unified way to look at space-time code design for 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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