Diversity and Multiplexing: A Fundamental Tradeoff in Wireless Systems
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1 Diversity and Multiplexing: A Fundamental Tradeoff in Wireless Systems David Tse Department of EECS, U.C. Berkeley June 6, 2003 UCSB
2 Wireless Fading Channels Fundamental characteristic of wireless channels: multi-path fading. Two important resources of a fading channel: diversity and degrees of freedom.
3 Wireless Fading Channels Fundamental characteristic of wireless channels: multi-path fading. Two important resources of a fading channel: diversity and degrees of freedom.
4 Diversity Channel Quality A channel with more diversity has smaller probability in deep fades. t
5 Example: Spatial Diversity Fading Channel: h 1 Additional independent fading channels increase diversity. Spatial diversity: receive, transmit or both. Repeat and Average: compensate against channel unreliability.
6 Example: Spatial Diversity Fading Channel: h 1 Fading Channel: h 2 Additional independent fading channels increase diversity. Spatial diversity: receive, transmit or both. Repeat and Average: compensate against channel unreliability.
7 Example: Spatial Diversity Fading Channel: h 1 Fading Channel: h 2 Additional independent fading channels increase diversity. Spatial diversity : receive, transmit or both. Repeat and Average: compensate against channel unreliability.
8 Example: Spatial Diversity Fading Channel: h 1 Fading Channel: h 2 Additional independent fading channels increase diversity. Spatial diversity: receive, transmit or both. Repeat and Average: compensate against channel unreliability.
9 Example: Spatial 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. Repeat and Average: compensate against channel unreliability.
10 Example: Spatial 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. Repeat and Average: compensate against channel unreliability.
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.
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.
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.
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.
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.
16 Diversity vs. Multiplexing Fading Channel: h 1 Fading Channel: h Fading Channel: h 2 3 Spatial Channel Spatial Channel Fading Channel: h 4 The two resources have been considered mainly in isolation: existing schemes focus on maximizing either the diversity gain or the multiplexing gain. The right way of looking at the problem is a tradeoff between the two types of gain. The optimal tradeoff achievable by a coding scheme gives a fundamental performance limit on communication over fading channels.
17 Diversity vs. Multiplexing Fading Channel: h 1 Fading Channel: h Fading Channel: h 2 3 Spatial Channel Spatial Channel Fading Channel: h 4 The two resources have been considered mainly in isolation: existing schemes focus on maximizing either the diversity gain or the multiplexing gain. The right way of looking at the problem is a tradeoff between the two types of gain. The optimal tradeoff achievable by a coding scheme gives a fundamental performance limit on communication over fading channels.
18 Diversity vs. Multiplexing Fading Channel: h 1 Fading Channel: h Fading Channel: h 2 3 Spatial Channel Spatial Channel Fading Channel: h 4 The two resources have been considered mainly in isolation: existing schemes focus on maximizing either the diversity gain or the multiplexing gain. The right way of looking at the problem is a tradeoff between the two types of gain. The optimal tradeoff achievable by a coding scheme gives a fundamental performance limit on communication over fading channels.
19 Talk Outline point-to-point MIMO channels (Zheng and Tse 02) multiple access MIMO channels (Tse, Viswanath, Zheng 03) cooperative relaying systems (Laneman,Tse, Wornell 02)
20 Point-to-point MIMO Channel w 1 y 1 x x 1 2 h h w 2 y 2 x m h nm h n1 w n y n y t = H t x t + w t, w t CN (0, 1) Rayleigh flat fading i.i.d. across antenna pairs (h ij CN (0, 1)). SNR is the average signal-to-noise ratio at each receive antenna.
21 Coherent Block Fading Model Focus on codes over l symbols, where H remains constant. H is known to the receiver but not the transmitter. Assumption valid as long as l coherence time coherence bandwidth.
22 Space-Time Block Code Y = HX + W time Y H X W space m x l Focus on coding over a single block of length l.
23 Diversity Gain Motivation: Binary Detection 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 diversity gain d, if P e (SNR) SNR d
24 Diversity Gain Motivation: Binary Detection 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 General Definition A space-time coding scheme achieves diversity gain d, if P e (SNR) SNR d
25 Spatial Multiplexing Gain Motivation: Channel capacity (Telatar 95, Foschini 96) C(SNR) min{m, n} log SNR(bps/Hz) min{m, n} degrees of freedom to communicate. Definition A space-time coding scheme achieves gain r, if spatial multiplexing R(SNR) = r log SNR(bps/Hz)
26 Spatial Multiplexing Gain Motivation: Channel capacity (Telatar 95, Foschini 96) C(SNR) min{m, n} log SNR(bps/Hz) min{m, n} degrees of freedom to communicate. Definition A space-time coding scheme achieves spatial multiplexing gain r, if R(SNR) = r log SNR(bps/Hz)
27 Fundamental Tradeoff A space-time coding scheme achieves Spatial Multiplexing Gain r : R = r log SNR (bps/hz) and Diversity Gain d : P e SNR d Fundamental tradeoff: for any r, the maximum diversity gain achievable: d m,n (r). r d m,n(r) A tradeoff between data rate and error probability.
28 Fundamental Tradeoff A space-time coding scheme achieves Spatial Multiplexing Gain r : R = r log SNR (bps/hz) and Diversity Gain d : P e SNR d Fundamental tradeoff: for any r, the maximum diversity gain achievable: d m,n (r). r d m,n(r) A tradeoff between data rate and error probability.
29 Fundamental Tradeoff A space-time coding scheme achieves Spatial Multiplexing Gain r : R = r log SNR (bps/hz) and Diversity Gain d : P e SNR d Fundamental tradeoff: for any r, the maximum diversity gain achievable: d m,n (r). r d m,n(r) A tradeoff between data rate and error probability.
30 Main Result: Optimal Tradeoff (Zheng and Tse 02) m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain P e SNR d Diversity Gain: d * (r) (0,mn) r: multiplexing gain R = r log SNR (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.
31 Main Result: Optimal Tradeoff (Zheng and Tse 02) m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain P e SNR d Diversity Gain: d * (r) (0,mn) (1,(m 1)(n 1)) r: multiplexing gain R = r log SNR (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.
32 Main Result: Optimal Tradeoff (Zheng and Tse 02) m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain P e SNR d Diversity Gain: d * (r) (0,mn) (1,(m 1)(n 1)) (2, (m 2)(n 2)) r: multiplexing gain R = r log SNR (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.
33 Main Result: Optimal Tradeoff (Zheng and Tse 02) m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain P e SNR d Diversity Gain: d * (r) (0,mn) (1,(m 1)(n 1)) (2, (m 2)(n 2)) (r, (m r)(n r)) r: multiplexing gain R = r log SNR (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.
34 Main Result: Optimal Tradeoff (Zheng and Tse 02) m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain P e SNR d Diversity Gain: d * (r) (0,mn) (1,(m 1)(n 1)) (2, (m 2)(n 2)) (r, (m r)(n r)) r: multiplexing gain R = r log SNR (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.
35 Main Result: Optimal Tradeoff (Zheng and Tse 02) m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain P e SNR d Diversity Gain: d * (r) (0,mn) Multiple Antenna m x n channel (r, (m r)(n r)) r: multiplexing gain R = r log SNR 1 Single Antenna channel 1 Spatial Multiplexing Gain: r=r/log SNR (min{m,n},0) For integer r, it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity.
36 What do I get by adding one more antenna at the transmitter and the receiver?
37 Adding More Antennas m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain r: multiplexing gain Diversity Advantage: d * (r) Spatial Multiplexing Gain: r=r/log SNR 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.
38 Adding More Antennas m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain r: multiplexing gain Diversity Advantage: d * (r) Spatial Multiplexing Gain: r=r/log SNR 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.
39 Adding More Antennas m: # of Tx. Ant. n: # of Rx. Ant. l: block length l m + n 1 d: diversity gain r: multiplexing gain Diversity Advantage: d * (r) d Spatial Multiplexing Gain: r=r/log SNR 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.
40 Sketch of Proof Lemma: For block length l m + n 1, the error probability of the best code satisfies at high SNR: where P e (SNR) P (Outage) = P (I(H) < R) I(H) = log det [I + SNRHH ] is the mutual information achieved by the i.i.d. Gaussian input.
41 Outage Analysis P (Outage) = P {log det[i + SNRHH ] < R} In scalar 1 1 channel, outage occurs when the channel gain h 2 is small. In general m n channel, outage occurs when some or all of the singular values of H are small. There are many ways for this to happen. Let v = vector of singular values of H: Laplace Principle: P (Outage) min v Out SNR f(v)
42 Outage Analysis P (Outage) = P {log det[i + SNRHH ] < R} In scalar 1 1 channel, outage occurs when the channel gain h 2 is small. In general m n channel, outage occurs when some or all of the singular values of H are small. There are many ways for this to happen. Let v = vector of singular values of H: Laplace Principle: P (Outage) min v Out SNR f(v)
43 Outage Analysis P (Outage) = P {log det[i + SNRHH ] < R} In scalar 1 1 channel, outage occurs when the channel gain h 2 is small. In general m n channel, outage occurs when some or all of the singular values of H are small. There are many ways for this to happen. Let v = vector of singular values of H: Laplace Principle: P (Outage) min v Out SNR f(v)
44 Geometric Picture (integer r) Scalar Channel 0 Result: At rate R = r log SNR, for r integer, outage occurs typically when H is in or close to the set {H : rank(h) r}, with ɛ 2 = SNR 1. The dimension of the normal space to the sub-manifold of rank r matrices within the set of all M N matrices is (M r)(n r). P (Outage) SNR (M r)(n r)
45 Geometric Picture (integer r) Scalar Channel ε Bad H Good H Result: At rate R = r log SNR, for r integer, outage occurs typically when H is close to the set {H : rank(h) r}, with ɛ 2 = SNR 1. The dimension of the normal space to the sub-manifold of rank r matrices within the set of all M N matrices is (M r)(n r). P (Outage) SNR (M r)(n r)
46 Geometric Picture (integer r) Scalar Channel Vector Channel ε All n x m Matrices Bad H Good H Rank(H)=r Result: At rate R = r log SNR, for r integer, outage occurs typically when H is close to the set {H : rank(h) r}, with ɛ 2 = SNR 1. The co-dimension of the manifold of rank r matrices within the set of all m n matrices is (m r)(n r). P (Outage) SNR (M r)(n r)
47 Geometric Picture (integer r) Scalar Channel Vector Channel Typical Bad H ε Bad H Good H Good H Full Rank ε Rank(H)=r Result: At rate R = r log SNR, for r integer, outage occurs typically when H is close to the set {H : rank(h) r}, with ɛ 2 = SNR 1. The co-dimension of the manifold of rank r matrices within the set of all m n matrices is (m r)(n r). P (Outage) SNR (M r)(n r)
48 Geometric Picture (integer r) Scalar Channel Vector Channel Typical Bad H ε Bad H Good H Good H Full Rank ε Rank(H)=r Result: At rate R = r log SNR, for r integer, outage occurs typically when H is close to the set {H : rank(h) r}, with ɛ 2 = SNR 1. The co-dimension of the manifold of rank r matrices within the set of all M N matrices is (M r)(n r). P (Outage) SNR (M r)(n r)
49 Geometric Picture (integer r) Scalar Channel Vector Channel Typical Bad H ε Bad H Good H Good H Full Rank ε Rank(H)=r Result: At rate R = r log SNR, for r integer, outage occurs typically when H is close to the set {H : rank(h) r}, with ɛ 2 = SNR 1. The co-dimension of the manifold of rank r matrices within the set of all m n matrices is (m r)(n r). P (Outage) SNR (m r)(n r)
50 Piecewise Linearity of Tradeoff Curve (0,mn) Diversity Gain: d * (r) Multiple Antenna m x n channel (r, (m r)(n r)) 1 Single Antenna channel (min{m,n},0) 1 Spatial Multiplexing Gain: r=r/log SNR For non-integer r, qualitatively same outage behavior as r but with larger ɛ. Scalar channel: qualitatively same outage behavior for all r. Vector channel: qualitatively different outage behavior in different segments of the tradeoff curve.
51 Tradeoff Analysis of Specific Designs Focus on two transmit antennas. Y = HX + W Repetition Scheme: Alamouti Scheme: X = x x 1 time X = x -x * x x * 1 time space space y 1 = H x 1 + w 1 [y 1 y 2 ] = H [x 1 x 2 ] + [w 1 w 2 ]
52 Comparison: 2 1 System Repetition: y 1 = H x 1 + w Alamouti: [y 1 y 2 ] = H [x 1 x 2 ] + [w 1 w 2 ] Diversity Gain: d * (r) (0,2) Repetition (1/2,0) Spatial Multiplexing Gain: r=r/log SNR
53 Comparison: 2 1 System Repetition: y 1 = H x 1 + w Alamouti: [y 1 y 2 ] = H [x 1 x 2 ] + [w 1 w 2 ] Diversity Gain: d * (r) (0,2) Alamouti Repetition (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
54 Comparison: 2 1 System Repetition: y 1 = H x 1 + w Alamouti: [y 1 y 2 ] = H [x 1 x 2 ] + [w 1 w 2 ] Optimal Tradeoff Diversity Gain: d * (r) (0,2) Alamouti Repetition (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
55 Comparison: 2 2 System Repetition: y 1 = H x 1 + w Alamouti: [y 1 y 2 ] = H [x 1 x 2 ] + [w 1 w 2 ] (0,4) Diversity Gain: d * (r) Repetition (1/2,0) Spatial Multiplexing Gain: r=r/log SNR
56 Comparison: 2 2 System Repetition: y 1 = H x 1 + w Alamouti: [y 1 y 2 ] = H [x 1 x 2 ] + [w 1 w 2 ] (0,4) Diversity Gain: d * (r) Alamouti Repetition (1/2,0) (1,0) Spatial Multiplexing Gain: r=r/log SNR
57 Comparison: 2 2 System Repetition: y 1 = H x 1 + w Alamouti: [y 1 y 2 ] = H [x 1 x 2 ] + [w 1 w 2 ] (0,4) Optimal Tradeoff Diversity Gain: d * (r) Alamouti (1,1) (1/2,0) (1,0) (2,0) Spatial Multiplexing Gain: r=r/log SNR
58 Talk Outline point-to-point MIMO channels multiple access MIMO channels cooperative relaying systems
59 Multiple Access User 1 Tx M Tx Antenna User 2 Tx Rx N Rx Antenna User K Tx M Tx Antenna In a point-to-point link, multiple antennas provide diversity and multiplexing gain. In a system with K users, multiple antennas can be used to discriminate signals from different users too. Continue assuming i.i.d. Rayleigh fading, n receive antennas, m transmit antennas per user.
60 Multiuser Diversity-Multiplexing Tradeoff Suppose we want every user to achieve an error probability: and a data rate P e SNR d R = r log SNR bits/s/hz. What is the optimal tradeoff between the diversity gain d and the multiplexing gain r? Assume a coding block length l Km + n 1.
61 Optimal Multiuser D-M Tradeoff: m n/(k + 1) (Tse, Viswanath and Zheng 02) (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 In this regime, diversity-multiplexing tradeoff of each user is as though it is the only user in the system, i.e. d m,n(r)
62 Multiuser Tradeoff: m > n/(k + 1) (0,mn) Diversity Gain : d (r) * (1,(m 1)(n 1)) (2,(m 2)(n 2)) Single User Performance (r,(m Kr)(n r)) * m,n d (r) n K+1 Spatial Multiplexing Gain : r = R/log SNR Single-user diversity-multiplexing tradeoff up to r = n/(k + 1). For r from N/(K + 1) to min{n/k, M}, tradeoff is as though the K users are pooled together into a single user with KM antennas and rate Kr, i.e. d KM,N (Kr).
63 Multiuser Tradeoff: m > n/(k + 1) (0,mn) Diversity Gain : d (r) * (1,(m 1)(n 1)) (2,(m 2)(n 2)) Single User Performance (r,(m r)(n r)) n K+1 * m,n d (r) * Km,n d (Kr) Antenna Pooling (min(m,n/k),0) Spatial Multiplexing Gain : r = R/log SNR Single-user diversity-multiplexing tradeoff up to r = m/(k + 1). For r from n/(k + 1) to min{n/k, m}, tradeoff is as though the K users are pooled together into a single user with Km antennas and rate Kr, i.e. d Km,n (Kr).
64 Benefit of Dual Transmit Antennas User 1 Tx 1 Tx Antenna User 2 Tx Rx N Rx Antenna User K Tx 1 Tx Antenna Question: what does adding one more antenna at each mobile buy me? Assume there are more users than receive antennas.
65 Benefit of Dual Transmit Antennas User 1 Tx M Tx Antenna User 2 Tx Rx N Rx Antenna User K Tx M Tx Antenna Question: what does adding one more antenna at each mobile buy me? Assume there are more users than receive antennas.
66 Answer Diversity Gain : d (r) * n Optimal tradeoff 1 Tx antenna n K+1 Spatial Multiplexing Gain : r = R/log SNR Adding one more transmit antenna does not increase the number of degrees of freedom for each user. However, it increases the maximum diversity gain from N to 2N. More generally, it improves the diversity gain d(r) for every r.
67 Answer 2n Diversity Gain : d (r) * 2 Tx antenna n 1 Tx antenna Optimal tradeoff n K+1 Spatial Multiplexing Gain : r = R/log SNR Adding one more transmit antenna does not increase the number of degrees of freedom for each user. However, it increases the maximum diversity gain from n to 2n. More generally, it improves the diversity gain d(r) for every r.
68 Suboptimal Receiver: the Decorrelator/Nuller User 1 Tx 1 Tx Antenna Decorrelator User 1 Data for user 1 User 2 Tx Decorrelator User 2 Data for user 2 Rx N Rx Antenna Decorrelator User K Data for user K User K Tx 1 Tx Antenna Consider only the case of m = 1 transmit antenna for each user and number of users K < n.
69 Tradeoff for the Decorrelator Diversity Gain : d (r) * n K+1 Decorrelator 1 Spatial Multiplexing Gain : r = R/log SNR Maximum diversity gain is n K + 1: costs K 1 diversity gain to null out K 1 interferers. (Winters, Salz and Gitlin 93) Adding one receive antenna provides either more reliability per user or accommodate 1 more user at the same reliability. Optimal tradeoff curve is also a straight line but with a maximum diversity gain of N. Adding one receive antenna provides more reliability per user and accommodate 1 more user.
70 Tradeoff for the Decorrelator Diversity Gain : d (r) * n K+1 Decorrelator 1 Spatial Multiplexing Gain : r = R/log SNR Maximum diversity gain is n K + 1: costs K 1 diversity gain to null out K 1 interferers. (Winters, Salz and Gitlin 93) Adding one receive antenna provides either more reliability per user or accommodate 1 more user at the same reliability. Optimal tradeoff curve is also a straight line but with a maximum diversity gain of n. Adding one receive antenna provides more reliability per user and accommodate 1 more user.
71 Tradeoff for the Decorrelator n Diversity Gain : d (r) * n K+1 Optimal tradeoff Decorrelator 1 Spatial Multiplexing Gain : r = R/log SNR Maximum diversity gain is n K + 1: costs K 1 diversity gain to null out K 1 interferers. (Winters, Salz and Gitlin 93) Adding one receive antenna provides either more reliability per user or accommodate 1 more user at the same reliability. Optimal tradeoff curve is also a straight line but with a maximum diversity gain of n. Adding one receive antenna provides more reliability per user and accommodate 1 more user.
72 Talk Outline point-to-point MIMO channels multiple access MIMO channels cooperative relaying systems
73 Cooperative Relaying Channel 1 Rx Tx 1 Channel 2 Tx 2 Cooperative relaying protocols can be designed via a diversity-multiplexing tradeoff analysis. (Laneman, Tse, Wornell 01)
74 Cooperative Relaying Channel 1 Rx Tx 1 Cooperation Channel 2 Tx 2 Cooperative relaying protocols can be designed via a diversity-multiplexing tradeoff analysis. (Laneman, Tse and Wornell 01)
75 Tradeoff Curves of Relaying Strategies 2 Diversity gain 1 direct transmission ½ 1 Multiplexing gain
76 Cooperative Relaying Channel 1 Rx Tx 1 Cooperation Channel 2 Tx 2
77 Tradeoff Curves of Relaying Strategies 2 Diversity gain 1 direct transmission ½ 1 Multiplexing gain
78 Tradeoff Curves of Relaying Strategies 2 Diversity gain 1 amplify + forward direct transmission ½ 1 Multiplexing gain
79 Tradeoff Curves of Relaying Strategies 2 Diversity gain 1 amplify + forward? direct transmission ½ 1 Multiplexing gain
80 Cooperative Relaying Channel 1 Rx Tx 1 Cooperation Channel 2 Tx 2
81 Tradeoff Curves of Relaying Strategies 2 Diversity gain 1 amplify + forward? direct transmission ½ 1 Multiplexing gain
82 Tradeoff Curves of Relaying Strategies 2 Diversity gain 1 amplify + forward amplify + forward + ack direct transmission ½ 1 Multiplexing gain
83 Conclusion Diversity-multiplexing tradeoff is a unified way to look at performance over wireless channels. Future work: Code design. Application to other wireless scenarios. Extension to channel-uncertainty-limited rather than noise-limited regime.
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