Diversity-Multiplexing Tradeoff in MIMO ChannelsDavid Tse Department of EECS, U.C. Berkeley February 26, 2004 Intel Smart Antenna Workshop 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. Diversity and Freedom Two fundamental resources of a MIMO fading channel: diversity degrees of freedom .Diversity Channel Quality t A channel with more diversity has smaller probability in deep fades. • For a m by n channel. transmit or both.Diversity Fading Channel: h 1 • Additional independent channel paths increase diversity. • Spatial diversity: receive. . maximum diversity is mn. maximum diversity is mn. • For a m by n channel. .Diversity Fading Channel: h 1 Fading Channel: h 2 • Additional independent fading channels increase diversity. transmit or both. • Spatial diversity: receive. Diversity Fading Channel: h 1 Fading Channel: h 2 • Additional independent fading channels increase diversity. • Spatial diversity : receive. . • For a m by n channel. maximum diversity is mn. transmit or both. • Spatial diversity: receive. transmit or both.Diversity Fading Channel: h 1 Fading Channel: h 2 • Additional independent fading channels increase diversity. maximum diversity is mn. . • For a m by n channel. .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. maximum diversity is mn. • For a m by n channel. transmit or both. . diversity is mn. transmit or both.Diversity Fading Channel: h 1 Fading Channel: h 2 Fading Channel: h 3 Fading Channel: h 4 • Additional independent fading channels increase diversity. • For a m by n channel. • Spatial diversity: receive. Degrees of Freedom y2 y1 Signals arrive in multiple directions provide multiple degrees of freedom for communication. there are min{m. Same effect can be obtained via scattering even when antennas are close together. In a m by n channel with rich scattering. n} degrees of freedom. . there are min{m. In a m by n channel with rich scattering.Degrees of Freedom y2 Signature 1 y1 Signals arrive in multiple directions provide multiple degrees of freedom for communication. n} degrees of freedom. . 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. . Same effect can be obtained via scattering even when antennas are close together.Degrees of Freedom y2 Signature 1 y1 Signature 2 Signals arrive in multiple directions provide multiple degrees of freedom for communication. n} degrees of freedom. Same effect can be obtained via scattering even when antennas are close together. n} degrees of freedom. In a m by n channel with rich scattering.Degrees of Freedom y2 Signature 1 Signature 2 y1 Signals arrive in multiple directions provide multiple degrees of freedom for communication. . there are min{m. n} degrees of freedom. In a m by n channel with rich scattering. there are min{m. . Same effect can be obtained via scattering even when antennas are close together.Degrees of Freedom y2 Fading Environment Signature 1 y1 Signature 2 Signals arrive in multiple directions provide multiple degrees of freedom for communication. there are min{m. .Degrees of Freedom y2 Fading Environment Signature 1 y1 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. n} degrees of freedom. In a m by n channel with rich scattering. 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. . Space-Time Code Examples: 2 × 1 Channel Repetition Scheme: Alamouti Scheme: time X= x1 0 0 time X= x1 space diversity: 2 data rate: 1/2 sym/s/Hz x 1 -x *2 x x* 2 1 space diversity: 2 data rate: 1 sym/s/Hz . Performance Summary: 2 × 1 Channel Diversity gain Degrees of freedom utilized /s/Hz Repetition 2 1/2 Alamouti 2 1 channel itself 2 1 . Space-Time Code Examples: 2 × 2 Channel Repetition Scheme: Alamouti Scheme: time X= x1 0 0 time X= x1 space diversity gain : 4 data rate: 1/2 sym/s/Hz x 1 -x *2 x x* 2 1 space diversity gain : 4 data rate: 1 sym/s/Hz But the 2 × 2 channel has 2 degrees of freedom! . Space-Time Code Examples: 2 × 2 Channel Repetition Scheme: Alamouti Scheme: time X= x1 0 0 time X= x1 2 space diversity: 4 data rate: 1/2 sym/s/Hz x 1 -x *2 x x* 1 space diversity: 4 data rate: 1 sym/s/Hz But the 2 × 2 channel has 2 degrees of freedom! . Salz and Gitlins 93: Nulling out k interferers using n receive antennas yields a diversity gain of n − k.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. f.o.Performance Summary: 2 × 2 Channel Diversity gain d. utilized /s/Hz Repetition 4 1/2 Alamouti 4 1 V-Blast with nulling 1 2 channel itself 4 2 Questions: • Alaomuti is clearly better than repetition.o.f. 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. utilized” precise and enrich them. . utilized” precise and enrich them. utilized /s/Hz Repetition 4 1/2 Alamouti 4 1 V-Blast with nulling 1 2 channel itself 4 2 Questions: • Alaomuti is clearly better than repetition.Performance Summary: 2 × 2 Channel Diversity gain d.f.f.o. .o. 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. 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.Performance Summary: 2 × 2 Channel Diversity gain d. utilized” precise and enrich them.f.o.f. utilized /s/Hz Repetition 4 1/2 Alamouti 4 1 V-Blast with nulling 1 2 channel itself 4 2 Questions: • Alaomuti is clearly better than repetition. .e. i. etc. kh2 k are both small) ∝ SNR−2 Definition A space-time coding scheme achieves (classical) diversity gain d.Classical Diversity Gain Motivation: PAM Pe ≈ P (khk is small ) ∝ SNR−1 y = hx + w y1 = h1 x + w1 y2 = h2 x + w2 Pe ≈ P (kh1 k. by 1/4d for every 6dB increase. error probability deceases by 2− d for every 3 dB increase in SNR. if Pe (SNR) ∼ SNR−d for a fixed data rate . . error probability deceases by 2−dmax for every 3 dB increase in SNR. i. by 4−dmax for every 6dB increase.e.Classical Diversity Gain Motivation: PAM Pe ≈ P (khk is small ) ∝ SNR−1 y = hx + w y1 = h1 x + w1 y2 = h2 x + w2 Pe ≈ P (kh1 k. etc. if Pe (SNR) ∼ SNR−dmax for a fixed data rate. kh2 k are both small) ∝ SNR−2 General Definition A space-time coding scheme achieves (classical) diversity gain dmax . f. (classical) diversity gain does not say anything about the d. Pe ↓ PAM -a +a 1 4 -2a +2a Both PAM and QAM have the same (classical) diversity gain of 1.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. utilized by the scheme. .o. .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. Pe ↓ PAM -a 1 4 +a -2a Pe ↓ +2a 1 4 QAM Both PAM and QAM have the same (classical) diversity gain of 1. utilized by the scheme.o.f. (classical) diversity gain does not say anything about the d. f. . utilized by the scheme. (classical) diversity gain does not say anything about the d. Pe ↓ PAM -a 1 4 +a -2a Pe ↓ +2a 1 4 QAM Both PAM and QAM have the same (classical) diversity gain of 1.o.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. Ask a Dual Question Every 6 dB doubles the constellation size for a given reliability. for PAM. every 6 dB quadruples the constellation size. . +1 bit PAM -a +a -3a -a +a +3a But for QAM. every 6 dB quadruples the constellation size. .Ask a Dual Question Every 6 dB doubles the constellation size for a given reliability. for PAM +1 bit PAM -a +a ∼-3a ∼-a ∼+a ∼+3a +2 bits QAM But for QAM. rmax = 1 for QAM. rmax = 1/2 for PAM. with varying symbol alphabet as a function of SNR.Degrees of Freedom Utilized Definition: A space-time coding scheme utilizes rmax degrees of freedom/s/Hz if the data rate scales like R(SNR) ∼ rmax log2 SNR bits/s/Hz for a fixed error probability (reliability) In a 1 × 1 channel. Note: A space-time coding scheme is a family of codes within a certain structure. . with varying symbol alphabet as a function of SNR. . rmax = 1 for QAM. rmax = 1/2 for PAM.Degrees of Freedom Utilized Definition: A space-time coding scheme utilizes rmax degrees of freedom/s/Hz if the data rate scales like R(SNR) ∼ rmax log2 SNR bits/s/Hz for a fixed error probability (reliability) In a 1 × 1 channel. Note: A space-time coding scheme is a family of codes within a certain structure. one wants to increase reliability and the data rate at the same time. or rmax additional bits/s/Hz for a fixed reliability. But these are two extremes of a rate-reliability tradeoff. More generally. .Diversity-Multiplexing Tradeoff Every 3 dB increase in SNR yields either a 2−dmax decrease in error probability for a fixed rate. 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.Diversity-Multiplexing Tradeoff Every 3 dB increase in SNR yields either a 2−dmax decrease in error probability for a fixed rate. or rmax additional bits/s/Hz for a fixed reliability. . Diversity-Multiplexing Tradeoff Every 3 dB increase in SNR yields either a 2−dmax 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 can increase reliability and the data rate at the same time. The largest multiplexing gain is rmax .o. The largest diversity gain is dmax = d(0). . simultaneously R(SNR) ∼ r log2 SNR bits/s/Hz and Pe (SNR) ∼ SNR−d(r) . the d. utilized by the scheme.f. the classical diversity gain.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. the d.f.o. simultaneously R(SNR) ∼ r log2 SNR bits/s/Hz and Pe (SNR) ∼ SNR−d(r) . The largest multiplexing gain is rmax .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. . the classical diversity gain. The largest diversity gain is dmax = d(0). utilized by the scheme. d∗max = d∗ (0) is the largest diversity gain achievable. ∗ rmax is the largest multiplexing gain achievable in the channel. n} d∗max = mn What is more interesting is how the entire curve looks like. it is not difficult to show: ∗ rmax = min{m.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. . For a m × n MIMO channel. it is not difficult to show: ∗ rmax = min{m. For a m × n MIMO channel. d∗max = d∗ (0) is the largest diversity gain achievable. .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. ∗ rmax is the largest multiplexing gain achievable in the channel. n} d∗max = mn What is more interesting is how the entire curve looks like. 0) Spatial Multiplexing Gain: r=R/log SNR .Example: 1 × 1 Channel Fixed Rate Diversity Gain: d*(r) (0.1) QAM PAM Fixed Reliability (1/2.0) (1. 2) Repetition (1/2.d*(r) Example: 2 × 1 Channel Diversity Gain: (0.0) Spatial Multiplexing Gain: r=R/log SNR . 0) (1.2) Diversity Gain: Alamouti Repetition (1/2.d*(r) Example: 2 × 1 Channel (0.0) Spatial Multiplexing Gain: r=R/log SNR . Example: 2 × 1 Channel d*(r) Optimal Tradeoff (0.0) (1.0) Spatial Multiplexing Gain: r=R/log SNR .2) Diversity Gain: Alamouti Repetition (1/2. 4) Repetition (1/2.0) Spatial Multiplexing Gain: r=R/log SNR .Example: 2 × 2 Channel Diversity Gain: d*(r) (0. 0) (1.4) Diversity Gain: Alamouti Repetition (1/2.0) Spatial Multiplexing Gain: r=R/log SNR .Example: 2 × 2 Channel d*(r) (0. 0) (1/2.Example: 2 × 2 Channel (0.0) (1.4) Repetition Diversity Gain: d*(r) Alamouti V−BLAST(Nulling) (0.1) (2.0) Spatial Multiplexing Gain: r=R/log SNR . Example: 2 × 2 Channel (0.0) Spatial Multiplexing Gain: r=R/log SNR .4) Repetition Alamouti Diversity Gain: d*(r) Optimal Tradeoff (1.1) V−BLAST(Nulling) (0.0) (1.1) (2.0) (1/2. 0) (1/2.1) V−BLAST(Nulling) (0.2) (1.4) Repetition d*(r) Optimal Tradeoff Diversity Gain: Alamouti V−BLAST(ML) (0.0) (1.1) (2.Example: 2 × 2 Channel (0.0) Spatial Multiplexing Gain: r=R/log SNR . (?) .Diversity Gain: * d (r) ML vs Nulling in V-Blast V−BLAST(ML) (0. Salz and Gitlins 93: Nulling out k interferers using n receive antennas provides a diversity gain of n − k. There is free lunch.2) V−BLAST(Nulling) (0. Tse.0) Spatial Multiplexing Gain: r=R/log SNR Winters.Viswanath and Zheng 03: Jointly detecting all users provides a diversity gain of n to each.1) (2. 1) (2.Diversity Gain: * d (r) ML vs Nulling in V-Blast V−BLAST(ML) (0.Viswanath and Zheng 03: Jointly detecting all users provides a diversity gain of n to each. There is free lunch. (?) . Salz and Gitlins 93: Nulling out k interferers using n receive antennas provides a diversity gain of n − k.0) Spatial Multiplexing Gain: r=R/log SNR Winters.2) V−BLAST(Nulling) (0. Tse. 1) (2. Salz and Gitlins 93: Nulling out k interferers using n receive antennas provides a diversity gain of n − k. There is free lunch.Viswanath and Zheng 03: Jointly detecting all users provides a diversity gain of n to each.0) Spatial Multiplexing Gain: r=R/log SNR Winters. Tse. (?) .Diversity Gain: * d (r) ML vs Nulling in V-Blast V−BLAST(ML) (0.2) V−BLAST(Nulling) (0. Optimal D-M Tradeoff for General m × n Channel (Zheng and Tse 03) As long as block length l ≥ m + n − 1: Diversity Gain: d*(r) (0,mn) (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. Optimal D-M Tradeoff for General m × n Channel (Zheng and Tse 03) As long as block length l ≥ m + n − 1: (1,(m−1)(n−1)) Diversity Gain: d*(r) (0,mn) (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. Optimal D-M Tradeoff for General m × n Channel (Zheng and Tse 03) As long as block length l ≥ m + n − 1: Diversity Gain: d*(r) (0,mn) (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. (m−1)(n−1)) (2.n}. . (m−r)(n−r)) (min{m.Optimal D-M Tradeoff for General m × n Channel (Zheng and Tse 03) As long as block length l ≥ m + n − 1: Diversity Gain: d*(r) (0.mn) (1.0) Spatial Multiplexing Gain: r=R/log SNR For integer r. (m−2)(n−2)) (r. it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity. 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. (m−r)(n−r)) (min{m.(m−1)(n−1)) (2. .n}.Optimal D-M Tradeoff for General m × n Channel (Zheng and Tse 03) As long as block length l ≥ m + n − 1: Diversity Gain: d*(r) (0.mn) (1. (m−2)(n−2)) (r. . • El Gamal. Caire and Damen 03: Lattice codes.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. u2 u3 . u4 are independent QAM symbols. u2 .Hao and Wornell 03 Alamouti scheme: x1 −x∗2 x2 x∗1 Hao and Wornell’s scheme: x1 x2 x3 x4 where x1 x4 = Rotate(θ1∗ ) u1 u4 x2 x3 = Rotate(θ2∗ ) and u1 . u3 . • Then design permutation codes to achieve the optimal diversity-multiplexing tradeoff on the parallel channel. .Tavildar and Viswanth 04 • First use D-Blast to convert the MIMO channel into a parallel channel. D-BLAST Antenna 1: Antenna 2: Receive . D-BLAST Receive Antenna 1: Antenna 2: Null . D-BLAST Antenna 1: Antenna 2: . D-BLAST Cancel Antenna 1: Antenna 2: Receive . Original D-Blast is sub-optimal. D-Blast with MMSE suppression is information lossless g1 h11 D−BLAST h12 h21 h22 g2 . Nevertheless it is shown that the permutation codes can achieve the optimal diversity-multiplexing tradeoff of the parallel channel. ⊕ ⊗ .Permutation Coding for Parallel Channel The channel is parallel but the fading at the different sub-channels are correlated. ♣ ♠ ⊕ ♣ ♠ ¶ ⊗ ¶ . . . It puts diversity and multiplexing on an equal footing.Conclusion Diversity-multiplexing tradeoff is a unified way to look at space-time code design for MIMO channels. It provides a framework to compare existing schemes as well as stimulates the design of new schemes.
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