Practical and Theoretical Advances in MIMO Systems Enoch Lu Ph.D. research talk Qualcomm San Diego, CA Mar. 1, 2012
MIMO? Hm… Mayo Ice cream Mash potatoes Oysters
Collaborators in my Ph.D. Coauthors Prof. I-Tai Lu Dr. Jialing Li
Special Thanks Prof. Dante Youla Prof. Peter Voltz
Dr. Yingxue (Kevin) Li Zihao You Tianxiang Ma Yiran Xu Dr. Onur Sahin Dr. Philip Pietraski
2
Goal •Introduce you to my Ph.D. research •Convey breadth, depth, and theme •Pass on some useful info to you
Enoch’s Zoo
Lake 3
Outline A. Breadth
Enoch’s Zoo
1. Unpacking the theme 2. Brief look at some works
B. Depth 1. Decentralized CBF 2. Channel estimation at transmitter
Lake
C. Conclusion
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A1. UNPACKING THE THEME: “PRACTICAL AND THEORETICAL ADVANCES IN MIMO SYSTEMS” 5
A lot of MIMO Applications Single-user MIMO and Multi-user MIMO Systems
MIMO Relaying Systems
Heterogeneous Networks
Coordinated Multipoint Transmission / Reception Systems 6
Practical Implementation • In LTE, WiMAX, IEEE 802.11n • Not easy – Want low feedback signaling designs – Want to obtain high performance
High throughput, etc. Low network load
7
CoMP NEWS
Not Mature Enough for LTE Rel. 10
Feedback: Main Remaining Open Issue for Standardization
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Perfect and LTE Rel. 8 feedback: High Performance Gap • M. Boldi et al. John Wiley & Sons, 2011. • A. Ziera. LTE World Summit, 2010. • Newspaper template from http://www.presentationmagazine.com/editable-powerpointnewspapers-407.htm
Need for Theoretical Work • How to know how good a practical design is? • Work on what is theoretical performance – Gives performance benchmarks for heuristics and practical implementations – Shows whether it is worth it to pursue practical implementations – Gives physical insights
9
Work was on 4 groups Single-user MIMO and Multi-user MIMO Systems
MIMO Relaying Systems
Heterogeneous Networks
Coordinated Multipoint Transmission / Reception Systems 10
For each group … 1. Propose transceiver and signaling design(s) Maximize performance while minimizing signaling PRACTICAL
2. Propose theoretical transceiver design(s) THEORETICAL
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2 journals 5 conferences 1 system-level test bench
Results 1 journal 4 conferences
SU MIMO and MU MIMO Systems
MIMO Relaying Systems
Heterogeneous Networks
CoMP Systems
1 submitted journal 1 system-level test bench
1 journal 5 conferences 1 submitted journal 12
A2. BRIEF LOOK AT SOME WORKS Enoch’s Zoo
Lake 13
MMSE based transceiver designs THEORETICAL
• Problem:
– For a given MIMO system, how to design the MIMO processing matrices if you have all the CSI at one location?
• Approach: – Iteratively design the matrices to minimize the system MSE subject to transmit power constraints
• Done for1: – Single-user, Multi-user, Relaying, CoMP – Perfect and imperfect CSI 1
Some done with IDCC
14
Coordinated Beamforming (CBF) Example CBF := coordination of the transceiver designs to mitigate the inter-site interference experienced by their users
cell site 1 Desired Interference
cell site 2 15
n1
user 1
G1
G2 user 2
cell site 1
H11 y1
F1
s1
F2
s2
H21 H12
y2 H22 n2
cell site 2
Iteratively design F1, F2, G1, G2 16
Transceiver and Signaling for Analog Network Coding PRACTICAL
• Problem: – How to get each node the info it needs to • design its MIMO processing matrix/matrices and • perform the network coding?
17
Analog Network Coding • 1st time slot: both end nodes transmit y
End Node E1
Relay Node
End Node E2
• 2nd time slot: relay transmits and end nodes cancel self-interference Ty
End Node E1
Relay Node
End Node E2
18
• Approach – TDD -> channel reciprocity – Make relay processing matrix symmetric
L is a complex N x N matrix. tr{LS}=0 for all symmetric N x N complex matrices iff L is skewsymmetric. 19
Interference Mitigation for Heterogeneous Networks1 • Problem:
THEORETICAL
– Closed subscriber group Femto eNBs can cause coverage holes for Macro UEs.
• Approach: – Message splitting2: • each transmitter does superposition coding • 1 part is for desired and interfered UEs to decode • Other part is only for desired UE 1
Done with IDCC. For confidentiality, many details omitted. 2 Used my group before my involvement.
20
m1+m2 f1+f2 Femto eNB Macro eNB Macro UE
Femto UE
• Macro UE: Decode f2 and cancel it. Then, decode m1 and m2. • Femto UE: Decode m2 and cancel it. Then, decode f1 and f2. 21
• Evaluated using system-level simulation – Scheduler1 – MIMO channel generation – Throughput calculation without actual data transmission1
1
For confidentiality, details omitted.
22
B. DEPTH
Enoch’s Zoo
Lake
23
B1. DECENTRALIZED COORDINATED BEAMFORMING PRACTICAL 24
Why are most CBF designs impractical? 1. 2. 3. 4.
Need too much information exchange Too complex Designs of different cells are not decoupled Only constrain average transmit power per cell site
cell site 1 Desired Interference
cell site 2 25
In this work: 1. We propose a framework for how to do CBF How a cell site designs its F How a user designs its G Information exchange used user 1
n1
G1 y 1 y2 G2 user 2
n2
H11
cell site 1
F1
H21 H12 F2
H22
s1 s2
cell site 2
26
2. Theoretical analysis Feasibility conditions Optimality
3. Numerical analysis
27
Phase 1: users’ channel sounding • Users perform channel soundings • Cell sites estimate their channels
H’11
user 1
H’12
cell site 1
H11, H21
cell site 2
H22, H12
H’21
user 2
H’22
28
Phase 2: design of precoders 1. Split each precoder: F1= F1,LF1,R 2. Pick F1,L: columns are orthonormal basis for nullspace of H21 H21F1,L= 0
user k
Gk
nk
H21F1s1= H21F1,LF1,Rs1= 0
cell site k
Hkk
Fk,L
Fk,R
sk 29
3. Pick F1,R – Optimize own link’s performance – Satisfy power constraint on F1
30
Phase 3: cell sites’ channel sounding • Cell sites perform channel soundings with precoders • Users estimate their equivalent channels
Equivalent channel
HkkFk
cell site k user k
Hkk
Fk
31
Phases 4-5 • Phase 4: Each user designs its decoder • Phase 5: Data transmission
32
Optimal right precoders Given a metric and power constraint on Fk, what is an optimum Fk,R? user k
nk
Gk user k
Gk
nk
cell site k
Hkk
Fk,L
Fk,R
sk
cell site k
HkkFk,L
Fk,R
sk 33
Overcomes 4 trouble issues 1. Need too much information exchange + (Equivalent) channels obtained only through sounding. + Only initial sounding needs to be orthogonal (e.g., via time). + Each node designs its precoder or decoder 2. Too complex + SVD to get left precoders. + Closed-form solutions for right precoders and decoders. 3. Designs of different cells are not decoupled + Pair doesn’t need to care for others MCS, # data streams, transmit powers, etc. 4. Only constrain average transmit power per cell site + Instantaneous antenna power can be constrained. 34
Centralized
30
Proposed
sum capacity
25 20 15
GIA-ATP PMMSE-ATP P-IAP PMBER-ATP PSVD-ATP
10 5 0 0
5
10
15 SNR (dB)
20
25
30 35
B2. CHANNEL ESTIMATION AT TRANSMITTER FOR LTE REL. 10 BASED SYSTEMS1 1
done with IDCC
36
Outline for B2 (CHEST) • • • •
Motivation Proposed Scheme Simulation Result Conclusion
37
MIMO in FDD systems • single transmitter and receiver’s equation for y k = H k Fk s k + n k a subcarrier, kth TTI: noise
nk
Decoder
Gk
Data vector
Channel
Hk yk Received vector
Fk
sk
Precoder 38
Double codebook based approach Suggested precoder
HkVkWk •Fed back from Rx •Characterizes wideband and/or long-term channel properties
•Fed back from Rx •Characterizes frequency-selective and/or short-term channel properties
39
Quantization and Performance • These approaches are fine but depend heavily on the size of codebooks • They don’t necessarily give the Tx the actual channel matrix Exploit the double codebook structure to estimate channel at Tx
40
• Knowing what a matrix does to a set of basis vectors -> know the matrix • Hk is r x t Basis: x1,…,xt • Know Hkx1 = a1 Hkxt = at -> know Hk −1
H k = [a1 … at ][ x1 … xt ]
41
• Q: How can we get such a set of equations? • A: Double Codebook V i Wi Fed back from Rx
• What about H1V1 ≈ z1W1* HkVk ≈ zkWk*
42
H1V1 ≈ z1W1* HkVk ≈ zkWk* Problems: 1. H1,…,Hk
Solutions: 1. Long enough coherence time -> H1≈… ≈ Hk
2. Too much feedback
2. Change V’s every TTI so that we cycle through the codebook in a predetermined way 43
HkV1 ≈ z1W1* HkVk ≈ zkWk* Problems: 1. H1,…,Hk
Solutions: 1. Long enough coherence time -> H1≈… ≈ Hk
2. Too much feedback
2. Change V’s every TTI so that we cycle through the codebook in a predetermined way 44
HkV1 ≈ z1W1* HkVk ≈ zkWk* Problems: 3. Still too much feedback
Solutions: 3. Get rid of phases of z’s. Estimate Hk*Hk instead of Hk
V1*H1*H1V1≈ |z1|2 W1W1* Vk*Hk*HkVk ≈ |zk|2 WkWk*
45
V1*H1*H1V1≈ |z1|2 W1W1* Vk*Hk*HkVk ≈ |zk|2 WkWk* Problems: 4. Still too much feedback and its not transparent to Rx
Solutions: 4. Proposed scheme
46
Rx does at ith TTI (i = 1,…,k) • Rx picks wi to maximize capacity • Rx calculates CQIi which is the SINR if Vi and wi are used • Rx feeds back wi and CQIi 2
( CQI iσ e
j hi Vi w i
*
, w i ) = arg min h i Vi − zw (l ) z ,w (l )
h i Vi ≈ CQI iσ 2 e jhi Vi wi w *i * i
* i
2
V h h i Vi ≈ CQI iσ w i w
* i
47
2 F
Tx does at kth TTI • Since coherence time is long enough, Tx knows * * i k
2
* i
V h h k Vi ≈ CQI iσ w i w ,
i = 1,..., k
• But it doesn’t know σ2, so it estimates hk*hk using * i
* k
* i
V h h k Vi ≈ CQI i w i w ,
i = 1, … , k 48
MU transmission mode. Single Antenna Rx Avg capacity (bits/sec/Hz)
25
20
Proposed (true beta) Proposed (approx beta) Rel. 8 based Perfect CSIT
15
10
5
0 0
10
20 transmit SNR (dB)
30
Rel. 8 based scheme couldn’t do ZF ~ 15% of the time
40
49
Conclusion for B2 (CHEST) • Proposed novel scheme to get CSI at Tx • Can be used for both single-user and multiuser MIMO • Greatly increases the amount of times ZF can be employed
50
C. CONCLUSION
51
Summary •Introduce you to my Ph.D. research •Convey breadth, depth, and theme •Pass on some useful info to you
Enoch’s Zoo
Lake 52
Takeaways • From theoretical work: – The transceiver design iterative approach is versatile • From practical work: –TDD is a very suitable enabler of advanced MIMO systems –The double codebook can be used to estimate CSI at the transmitter in FDD systems 53
Topics not covered • CoMP/HetNet scheduling • MIMO channel modeling • Channel estimation and data detection for WCDMAHSDPA MIMO • Relationship between interference alignment and optimum transceiver designs • Maximum Sum Mutual Info design • Minimum Symbol Error Probability design
[email protected] https://sites.google.com/site/enochluinfo/ 54
BACKUP FOR ANALOG NETWORK CODING WITH PRACTICAL CONSIDERATIONS 55
Outline (C. Analog Network Coding) • • • •
Motivation Analog Network Coding Formulation Proposed Protocol Conclusion
56
Motivation • Analog network coding may help save time slots
Figure copied from “Wireless Network Coding by Amplify-and-Forward for Bi-Directional Traffic Flows”
57
Motivation • How to do the design? • How to get each node the information it needs?
We propose a transceiver and signaling protocol
58
Outline (C. Analog Network Coding) • • • •
Motivation Analog Network Coding Formulation Proposed Protocol Conclusion
59
1st time slot Forward channel Data vector
s1
Precoder
F1
H1
y
x1
Transmitted End Node E1 vector
H2 x2
Relay Node R
F2
s2
End Node E2
y = H1F1s1 + H 2 F1s1 + w 60
2nd time slot Reverse channel
H1’
Transmitted vector
γTy
z1 End Node E1
Received vector
H2’ z2
Relay Node R
End Node E2
61
2nd time slot • Received vector at Ei
Noise from relay
z1 = H1' γ TH1x1 + H1' γ TH 2 x 2 + H1' γ Tw + a1 Self-interference
Desired signal
Received noise
• Self-interference is known to E1 and can be subtracted ˆz1 = H1' γ TH 2 x 2 + n1 Effective channel
Effective noise 62
Outline (C. Analog Network Coding) • • • •
Motivation Analog Network Coding Formulation Proposed Protocol Conclusion
63
Step 1 E1 and E2 to perform channel soundings so that the relay can estimate H1 and H2 H1
End Node E1
H2
Relay Node R
End Node E2
64
Step 2 • Relay chooses symmetric T – E.g.,
C1 ( T ) = I +
C2 ( T ) = I +
1
ρ
2
1
ρ
2
*
( H TH )( H TH ) ' 2
1
' 2
1
*
( H TH )( H TH ) ' 1
2
' 1
2
max ( C1 ( T ) + C2 ( T ) ) T = T′
subject to tr ( TT* ) = 1
• Relay chooses γ to satisfy relay’s power constraint 65
Step 3 • Relay performs 2 equivalent channel soundings – one precoded with γTH1 – one precoded with γTH2 γTH1 and γTH2 H1’
H2’
z1 End Node E1
z2 Relay Node R
E1 estimates H1’γTH1 and H1’γTH2 E2 estimates H2’γTH1 and H2’γTH2
66
Step 3 E1 estimates H1’γTH1 and H1’γTH2 E2 estimates H2’γTH1 and H2’γTH2 Noise from relay ' 1
' 1
' 1
z1 = H γ TH1x1 + H γ TH 2 x 2 + H γ Tw + a1 Self-interference
Desired signal
Received noise
67
Step 4 • Ei (i = 1,2) designs its precoder Fi and decoder Gi without any signaling with the other end node – E1 uses a reproducible algorithm to design F1 – E2 uses the same one so it also has F1 – E2 can thus design G2 to match F1 – Vice versa
68
Optional Step 5 • relay does an equivalent channel sounding with precoding matrix T or TΦ1/2 w Ei (i = 1,2)estimates covariance matrix of noise propagated from relay
69
Conclusion (C. Analog Network Coding) • Proposed novel scheme to do analog network coding • Tried to maximize performance while minimizing signaling
70
BACKUP FOR INTERFERENCE ALIGNMENT: A BUILDING BLOCK OF COORDINATED BEAMFORMING TRANSCEIVER DESIGNS
71
Outline (D. Interference Alignment) • • • •
Motivation Proof of IA-like behavior for MMSE design Numerical results Conclusion
72
Interference Alignment (IA) • In the theoretical category • Basic Idea: – Have all inter-user interference align so that at each user, they are all zero – Desired effective channels are full rank
• Problems: – Some approaches are seen to outperform it – Quite restrictive – Ignores noise 73
This work is on … • Show IA is a building block of multiple coordinated beamforming transceiver designs: – – – –
MMSE Max Min SINR Max Sum Capacity Min BER
• Each design seek to optimally balance A. B. C. D.
Preserving the desired data streams Aligning the inter-stream interferences Aligning the inter-user interferences Knocking out the noises 74
Outline (D. Interference Alignment) • • • •
Motivation Proof of IA-like behavior for MMSE design Numerical results Conclusion
75
System Model for CBF • • • •
K transmitters K receivers kth tx only has data for kth rx Each tx has a precoder and each rx has a decoder
76
n1
Rx 1
G1
GK Rx K
Tx 1
H11 y1
F1
s1
FK
sK
HK1 H1K
yK HKK nK
y k = H kk Fk s k + ∑ H kl Fl sl + n k , l ≠k
Tx K k = 1,..., K 77
IA for CBF • A set of precoders {Fk} and decoders {Gk} is said to achieve interference alignment if and only if rank(GkHkkFk) = mk GkHklFl = 0, ∀l≠k, ∀k.
78
MSE for CBF • the MSE for the kth pair ηk = G k H kk Fk - I m
k
2 F
2 12 nk F
+ GkΦ
K
+
∑ l =1, l ≠ k
G k H kl Fl
2 F
• MMSE problem: K
min
{Fk },{G k }
∑η
k
k =1
subject to some power constraint for each node 79
Proof of IA-like behavior for MMSE • All terms in ηk ≥ 0 ηk = G k H kk Fk - I m
k
Term 1
2 F
2 12 nk F
+ GkΦ
Term 2
K
+
∑
G k H kl Fl
l =1, l ≠ k
2 F
Term kl
• So, minimizing sum MSE means to balance minimizing each term in η1,…, and ηK
80
Proof of IA-like behavior for MMSE • All terms in ηk ≥ 0 ηk = G k H kk Fk - I Term 1
2 F
2 12 nk F
+ GkΦ
Term 2
K
+
∑
G k H kl Fl
l =1, l ≠ k
2 F
Term kl
• Term 1 = distance squared between GkHkkFk and I • Term 2 = E{(Gknk)*(Gknk)} • Term kl = distance squared between GkHklFl and 0
81
Outline (D. Interference Alignment) • • • •
Motivation Proof of IA-like behavior for MMSE design Numerical results Conclusion
82
numerical results setup • Feasible IA Case: – 3 pairs, 2 antennas at each node, 1 data stream for each pair
• Infeasible IA Case: – 4 pairs, 2 antennas at each node, 1 data stream for each pair
83
Sum Capacity results Per-Transmitter Power Constraint
Per-Antenna Power Constraint
Σ
Sum Capacity ( C )
20 K=3 GIA K=3 IA1 K=3 IA2 K=4 GIA K=4 IIA
15
10
5
0 -10
0
10
20 -10 SNR (dB)
0
10
20
84
IA-like behavior measure for IA feasible case σ
IO k ,1
= ∑ l =1,l ≠ k G k H kl Fl K
2 F
-1
after decoder (σIO ) k,1
Power of Inter-user Interference
10
-2
10
-3
10
-4
10
-5
0
5
10
SNR (dB)
15
20
85
IA-like behavior measure for IA infeasible case σ
IO k ,1
= ∑ l =1,l ≠ k G k H kl Fl K
2 F
0
IIA after decoder (σIO ) k,1
Power of Inter-user Interference
10
-1
10
GIA
-2
10
-3
10
-5
0
5
10
SNR (dB)
15
20
86
Conclusion (D. Interference Alignment) • Have rigorously and numerically showed the underlying physical mechanisms of MMSE designs (IA, etc.) – May be important to coming up with optimum, heuristic, or practical transceiver designs
• Have showed it is better to pursue MMSE instead of IA or joint MMSE and IA.
87
Backup for Decentralized CBF (B1)
88
• bk: # antennas at kth cell site • uk: # antennas at kth user • Φsk = E(sksk* ): source covariance matrix at kth cell site • Φnk = E(nknk* ): noise covariance matrix at kth user • Received signal vector at kth user: 2
y k = H kk Fk s k +
∑
H kl Fl sl + n k
l =1,l ≠ k 89
Power Constraints on Fk user k
Gk
nk
cell site k
Hkk
Fk,L
Fk,R
sk
Fk ith element is signal on ith antenna
= 90
Power Constraints on Fk • Average total power (ATP) constraint *
Pk = E{( Fk s k ) ( Fk s k )} = tr{Fk F }σ = tr{Fk ,R F }σ * k
2 sk
* k ,R
• Instantaneous antenna power (IAP) constraint * k
2 sk
* k ,R
Lk = λmax (Fk F )σ = λmax (Fk ,R F )σ
2 sk
91
2 sk
Power Constraints on Fk • How does it constrain the instantaneous antenna power? 2
max {Fk s k }i ≤ max s F F s i ,sk
sk
* k k
* k
* k k k
* k k
≤ λmax (F F ) max s s sk
92
Feasibility Conditions user k
Gk
nk
cell site k
Hkk
Fk,L
Fk,R
uk x bk
sk mk x 1
• How do you know if your equivalent channel after nulling will have enough rank? rank(HkkFk,L) ≥ mk iff bk − mk ≥ ∑ l =1,l ≠ k ul K
93
Background: interference alignment {Fk} and {Gk} is said to achieve interference alignment iff H13F3 rank(GkHkkFk) = mk, H F H11F1 11 1 GkHklFl = 0, ∀l≠k, ∀k. H F H H 21 31
F 2
12 2
H F 22 2
H F 23 3
H F 21 1 F 3
H F 31 1
H F 32 2 H F 33 3
Figure adapted from Omar El Ayach’s presentation at Milcom 09
94
Background: interference alignment rank(GkHkkFk) = mk and GkHklFl = 0, ∀l≠k, ∀k. H 11 H 21 H 31
F 1
H 13 F3
H 11F1
H F 12 2
F 2 H
F 3
F 31 1
H H
F 33 3
F 32 2 95
Optimality of Entire Design IF a) interference alignment is feasible and b) each pair’s # data streams = its user’s # antennas. THEN System performance with optimum right precoders and decoders will be at least as good as that of any interference alignment design under the same power constraint. Same is true for each user’s performance.
96
3 GIA-ATP PMMSE-ATP P-IAP PMBER-ATP PSVD-ATP
2.5
sum MSE
2 1.5 1 0.5 0 0
5
10
15 SNR (dB)
20
25
30 97
10
system BER
10
10
10
0
GIA-ATP PMMSE-ATP P-IAP PMBER-ATP PSVD-ATP
-2
-4
-6
0
5
10
15 SNR (dB)
20
25
30 98
BACKUP FOR CHEST AT TRANSMITTER (B2) 99
Tx does • Uses 1 of the following 2 approaches to estimate Hk*Hk • Takes the dominant eigenvector of Hk*Hk times the dominant eigenvalue as estimate of Hk • If Single-user – fk is chosen to be the estimate of Hk normalized to unit norm
100
Tx does • If Multi-user with 2 users ˆ H k ,1 Tk = ˆ H k ,2
– Form estimate composite channel
– Form ZF precoder Fk = f k ,1
f k ,2 ,
f k ,u = fk ,u / fk ,u *
( )
k F k = fk ,1 fk ,1 = T
F
Tk Tk *
( )
−1
101
Approach 1 to estimate Hk*Hk • Transform into standard set of linear equations – Take vec
Ai vec ( H*k H k ) ∼ bi , '
*
i = 1, … , n
(
2
*
A i V i ⊗ V i , bi vec β i w i w i
– Stack
)
A1 b1 vec H* H ∼ ( k k) A n b n
• Use inverse problem solver and take Hermitian approximation
102
Approach 2 to estimate Hk*Hk • Solve n
Q k = arg min ∑ Ci − V QV i * i
Q=Q*∈t ×t i =1
2
2 F
*
where Ci β i w i w i
• Use Qk as your estimate
103
Setup • • • • • • • • • •
1 eNB with 4 antennas 2 UE’s with 1 or 2 antennas each 50 RB’s 20 UE drops; 100 TTI’s per drop 3GPP WIM channel model: 3GPP case 1 Channel is normalized so that avg norm of channel per TTI, per RB is 1 v = 3 km/hr perfect CHEST at UE no feedback delay or error -For multi antenna UE, MRC decoder is used unless otherwise specified 104
SU transmission mode. Single Antenna Rx
Avg capacity (bits/sec/Hz)
14 12 10
Proposed (true beta) Proposed (approx beta) Rel. 8 based Perfect CSIT
8 6 4 2 0 0
10
20 transmit SNR (dB)
30
40 105
SU transmission mode. Multi Antenna Rx Avg capacity (bits/sec/Hz)
14 12 10
Proposed (true beta) Proposed (approx beta) Rel. 8 based Perfect CSIT
8 6 4 2 0 0
10
20 transmit SNR (dB)
30
40 106
MU transmission mode. Multi Antenna Rx Avg capacity (bits/sec/Hz)
25
20
Proposed (true beta) Proposed (approx beta) Rel. 8 based Perfect CSIT
15
10
5
0 0
10
20 transmit SNR (dB)
Rel. 8 based couldn’t do ZF ~ 23% of the time
30
40
107
Backup for Max Sum Mutual Info Design
Two types of work needed • Work on what is theoretical performance • Work on practical implementations – Considering all of these together: • • • •
Transceiver design Signaling Channel estimation Etc.
E1
R
E2 109
This work is on … • Theoretical transceiver design • Assume all CSI is known perfectly at an entity who designs all the MIMO processing matrices • Assume this entity can distribute the necessary info to every node • Propose an iterative approach – Design criteria: weighted sum mutual information – Power constraint: average total power 110
Outline • • • • • •
System Model Problem Formulation Important Observations Iterative Approach Convergence Conclusion
111
System Model • 2 end nodes • L relays • All have multiple antennas
112
System Model: 1st time slot Relay Node R1
y1 Data vector
s1
A11
A12
Precoder
F1
x1
z2
Transmitted End Node E1 vector
yL
AL2
F2
s2
End Node E2
AL1
Relay Node RL 113
System Model: 1st time slot Relay Node R1
y1 A11
s1
F1
A12
x1
z2 Received yL vector AL2
End Node E1
Forward channel
F2
s2
End Node E2
AL1
Relay Node RL 114
System Model: 1st time slot 2
y l = ∑ A lj F j s j + w l , l = 1,..., L j =1
Relay Node R1
y1 A11
s1
F1
A12
x1
z2
End Node E1
yL
AL2
s2
F2 End Node E2
AL1
Relay Node RL
115
System Model: 2nd time slot Relay Node R1
T1y1 B11
B21
T1
z1 End Node E1
z2 B1L
Transmitted vector
TLyL
B2L
TL
Processing Matrix
End Node E2
Relay Node RL 116
System Model: 2nd time slot Relay Node R1
T1y1 Received vector
B11
T1
B21
z1 End Node E1
z2 B1L Reverse channel
TLyL
B2L
End Node E2
TL Relay Node RL 117
System Model: 2nd time slot • Received vector at Ei
Noise from relay
L L L z1 = ∑ B1l Tl A l1F1s1 + ∑ B1l Tl Al 2 F2 s 2 + ∑ B1l Tl w l + a1 l =1 l =1 l =1 Self-interference
Desired signal
Received noise
• Self-interference is known to E1 and can be subtracted zˆ 1 = H12 F2 s 2 + n1 Effective channel
Effective noise 118
Problem Formulation • Mutual Info at Ei I Ei = log 2
G i (σ s2j H ij Fj F*j H*ij + Φni ) G*i G i Φ ni G
* i
, i ≠ j.
• If Gi is of the form G i = ΓF*j H*ij Φn−i1 where Γ is an invertible matrix and Fj has full column rank, I Ei = log 2
σ s2j H ij Fj F*j H*ij + Φni Φ ni
, i ≠ j. 119
Problem Formulation
max α I
1 E1
+ α2 IE2
{Fi },{Tk }
subject to p j =tr (σ Fj F ) , ql =E{T y T y l }, ∀j , l 2 sj
* j
* l
* l l
120
Important Observation # 1 • Given {Tl}, how to design Fj? • Recall, each direction looks like single-user MIMO transmission: zˆ 1 = H12 F2 s 2 + n1 Effective channel
Effective noise
• So, can just use available closed-form solution! • Drawback: such a solution may violate relays’ power constraints 121
Important Observation # 2 • Given {Fj}, T1,…, Tl-1, Tl+1,…, TL, how to design Tl? 1. Can introduce augmented cost function with Lagrange multiplier 2. get an expression for the Lagrange multiplier 3. get an expression for Tl 122
Iterative Approach • Due to the interdependence of the expressions, – just fix all but one matrix. – Update it using the expression. – Move on to next matrix.
123
Convergence • No proof but it always converged in our simulations, no matter what – Channel realization – Power – # of relays
124
Example Convergence Plot
125
Conclusion • Have proposed a theoretical transceiver design for the sum mutual information criterion • It can be used to benchmark future practical designs
126
