Linear Precoding in MIMO Wireless Systems
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego
Acknowledgement: Y. Isukapalli, L. Yu, J. Zheng, J. Roh
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 1 / 48
Outline
1 Promise of MIMO Systems
2 Point to Point MIMO
3 Limited Feedback MIMO Systems
4 MIMO-OFDM
5 Multi-User MIMO
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 2 / 48
Outline
1 Promise of MIMO Systems
2 Point to Point MIMO
3 Limited Feedback MIMO Systems
4 MIMO-OFDM
5 Multi-User MIMO
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 3 / 48
Multiple Input Multiple Output (MIMO) Systems
A system with multiple antennas at the transmitter and multipleantennas at the receiver.
Enables Spatio-Temporal processing and the goal is to exploitthe spatial dimension to increase system throughput
Multi-Input Multi-Output System
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Textbooks
Introduction to Space-Time Wireless Communications, A.Paulraj, R. Nabar and D. Gore, Cambridge University Press
Fundamentals of Wireless Communications, D. Tse and P.Vishwanath
Space-Time Coding, H. Jafarkhani
MIMO Wireless Communications, Edited by Biglieri, Calderbank,et al
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Benefits of MIMO Systems
Increased Network Capacity
Improved Signal Quality
Increased Coverage
Lower Power Consumption
Higher Data Rates
These requirements are often conflicting. Need balancing tomaximize system performance
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Technical Rationale
Spatial Diversity to Combat Fading
Spatial Signature for Interference Management
Array Gain enables Lower Power Consumption
Capacity Improvements using Spatial Multiplexing
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Outage Capacity of MIMO SystemsCapacity of MIMO systems
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Outline
1 Promise of MIMO Systems
2 Point to Point MIMO
3 Limited Feedback MIMO Systems
4 MIMO-OFDM
5 Multi-User MIMO
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 9 / 48
MIMO Channel Model
Input-Output relation for a discrete-time frequency-flat r × tMIMO channel
y =
√Es
tHs + n
y = [y1, y2, · · · , yr ]T · · · r × 1 receive signal vectors = [s1, s2, · · · , st ]T · · · t × 1 transmit signal vectorn = [n1, n2, · · · , nr ]T · · · r × 1 noise vector at the receiver
H is the r × t channel matrix
Es average energy over a symbol period
ni ∼ NC(0,No) with E [nnH ] = No Ir
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MIMO Options
Channel assumed known at Receiver
Channel unknown at transmitter
Diversity Gain: Orthogonal space-time block codes, Space timetrellis codesSpatial Multiplexing: V-Blast, D-Blast
Channel known at the transmitter- Transmit precoding
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Transmitter With Channel Knowledge
SVD of H can be expressed as
H = UΣVH
UHU = VHV = IrΣ = diag(σm)k
m=1, σm > 0
Further, HHH is Hermitian with eigendecomposition
HHH = UΛUH
Λ = diag(λm)km=1, σm ≥ σm+1 with λm = 0 for m > k and
λm = σ2m
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Transmitter With Channel Knowledge Cont’d
Transmitted vector s = Vs
Input vector s is of dimension r × 1 with E [ssH ] = Γt , Γt
diagonal
Received signal transformed to y = UHy
y =
√Es
tΣs + n
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Transmitter With Channel Knowledge Cont’d
H is decomposed into k parallel sub-channels satisfying
ym =
√Es
tσmsm + nm, m = 1, 2, · · · , k
The channels are of different quality with the gain on eachchannel determined by σm
Number of channels depends on the rank of H.
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Transmitter with Channel KnowledgeTransmitter with Channel Knowledge
sV
sH HU
rTransmitte Channel Receivern
y y~
1λ1~s
1~n
1~y
kλks~kn~
ky~
2λ2~s
2~n
2~y
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Capacity of a deterministic MIMO Channels
The channel capacity is given by
C = maxγm
k∑m=1
log2
[1 +
Esλm
Notγm
]γm = E [|sm|2] is the transmit energy in the mth sub-channel∑k
m=1 γm = t is the transmit energy constraint
Optimum power allocation across the sub-channels is obtainedas a solution to the lagrangian optimization problem
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Optimal Power Allocation
Optimal power allocation satisfies
γoptm =
(µ− Not
Esλm
)+
, m = 1, 2, · · · , k
k∑m=1
γoptm = t
where µ is a constant and (x)+ implies
(x)+ =
{x if x ≥ 00 if x < 0
γoptm is found iteratively by waterpouring algorithm
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Waterpouring Solution
Waterpouring Solution
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High SNR
At high SNR, equal power allocation is optimal
C =k∑
m=1
log2
[1 +
Esλm
Not
]≈
k∑m=1
log2
[Esλm
Not
]= k log2
[Es
No
]+
k∑m=1
log2
[λm
t
]Capacity grows linearly with k , the rank of the channel. Significant
increase in Capacity.
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Special Cases
SIMO: H = h. Rank one and all power allocated to one mode
CSIMO = log2(1 +Es‖h‖2
No)
MISO: H = hH . Rank one and all power allocated to one mode
CMISO = log2(1 +Es‖h‖2
No)
When Channel known at Tx
CSIMO = CMISO
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Maximum Ratio Transmission (MRT)
Input-Output relation for a r × t MIMO channel
y =
√Es
tHs + n
When the channel is known at the transmitter, the informationcan be used to design an optimum precoder w
The new Input-Output relation becomes
y =
√Es
tHws + n
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 21 / 48
Maximum Ratio Transmission Cont’d
The receiver forms a weighted sum of the antenna outputs
y = gHy
The objective is to maximize the received SNR
η =‖gHHw‖2F
t‖g‖2Fρ
Optimal scheme is given by
w = v1, g = u1
Where, v1 and u1 are the left and right singular vectors of Hcorresponding to the maximum singular value
The scheme achieves full diversity
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 22 / 48
MRT Transmission: 2× 2 MIMO
MRT Transmission: 2x2 MIMO
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Outline
1 Promise of MIMO Systems
2 Point to Point MIMO
3 Limited Feedback MIMO Systems
4 MIMO-OFDM
5 Multi-User MIMO
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 24 / 48
2
Importance of CSI Feedback
A. Improved system performance, in terms of capacity, SNR, BER, etc.Example: An MISO system with M transmit antennas and single receive antenna
NO CSIT Perfect CSIT
B. Reduced implementation complexity
Example: An MIMO system with M transmit and receive antennas,
No CSIT, capacity can be achieved by some 2-D (space-time) code
Pre-coder with perfect CSIT, system isequivalent to M parallel SISO channels
3
Importance of CSI Feedback
D. Greatly increase the system capacity region as well as the sum capacity
C. Enables exploitation of multi-user diversity
With CSIT, effective selection of active users and route selection can be made.
E. Improve the robustness of the communication link (QoS requirements)
Power and rate control is possible when CSIT is available and the network throughput is increased.
Example: A multi-user MISO broadcasting channel with M transmit and single receive antenna
users are not allowed to cooperate, and hencecause serious multi-userinterference.
CSI FeedbackProper pre-coding is possible, such as Zero-forcing, MMSE, etc
Block Diagram
Sources of feedback imperfection
Channel estimation
Channel quantization
Feedback delay
() 6 / 34
4
Nature of CSI FeedbackChannel state information (CSI) is a complex vector or matrix of continuous values
For example: An MIMO system with M transmit antennas and N receive antennas, .
It is not reasonable to feedback total 2MN real numbers of continuous values.
Each index represents a particular mode of the channel, which corresponds to a particular transmission strategy
Channel Quantizer
Integer Index
Adaptive Transmitter
Practical Feedback Schemes:
5
Considerations in Feedback Systems
A. Design of Optimal Quantizers (at the receiver) & Optimization of the Codebook?
1) The quantizer (or the encoder) should be simple as well as effective.
2) The quantizer and the codebook should be designed to match both the channel distribution and the system performance metrics, such as capacity, SNR, BER, etc.
B. Performance Analysis of Finite Rate Feedback Multiple Antenna Systems
1) To understand the effects of the finite rate feedback on the system performance, to be specific, performance metric vs feedback rate.
2) Shed insights on the choice of the feedback schemes as well as the quantizer design.
6
MISO Channel Quantizer
If ideal CSIT available, the transmit beamforming scheme is chosen to be:
MISO Channel System Model:
(vector)(scalar)
If only finite rate feedback is available, the beamforming vector is quantized to ,
capacity
(codebook)
capacity
7
Codebook Design (Optimization)
1). The capacity loss can be approximated by the following form in high resolution regimes,
2). A New Design Criterion that can minimize the system capacity loss:
Simplifications:
(MSwIP)
High SNR(MSIP)
The capacity loss due to the finite rate quantization of the beamforming vectors is:
Motivation: Minimize the capacity loss by optimizing the codebook vectors
It is a difficult problem (non-convex optimization problem)!
8
Codebook design using the Lloyd Algorithm
partitioning the regions
Nearest Neighborhood Condition (NNC):
For given codebook vectors
the optimum partitions are given by:
Centroid Condition (CC):
For given partitions ,
the optimal code matrices are given by:
Shifting new centers
9
Codebook Design Examples
10
MISO Capacity With Quantized Feedback
11
Extension to MIMO Channel Quantizer
Precoding Matrix Equal Power Allocation
MIMO Channel System Model:
Channel Model With Quantized Feedback:
12
Sequential Vector QuantizerA simple approach to quantize the precoding matrix:
How? Consider a unitary matrix whose first column is and the remainder columns are arbitarily chosen to satisfy . Then, has the form of
where is a orthogonormal column matrix.
13
The Sequential Quantization Method
Practical applications: Under consideration by the Broadband Wireless Group (802.16e)
Vector Parameterization: An orthonormal column matrix can be uniquely represented by by a set of unit-norm vectors with different dimensions, .
Statistical Property: For random channel with entries,, for , and they are statistically independent.
Quantization: For , unit-norm vector is quantized using a codebook that is designed for random unit-norm vectors In with the MSIP criterion.
14
Joint Quantization for MIMO Systems
Joint Quantization: by quantizing the entire precoding matrix at one shot
The codebook is designed to minimize the system mutual information rate loss
With ideal CSI Feedback With Quantized CSI Feedback
Under the high resolution assumptions, it can be approximated as
The first n eigen-valuesGeneralized Weighted Matrix Inner Product between and .
15
Codebook design using the Lloyd Algorithm
partitioning the regions
Shifting new centers
Nearest Neighborhood Condition (NNC):
For given code matrices ,
the optimum partitions are given by:
Centroid Condition (CC):
For given partitions ,
the optimal code matrices are given by:
16
Multi-mode Spatial Multiplexing
Case I: Low SNR
water level
power allocated
Case II: High SNR
water level
power allocated
Multi-mode SP transmission strategy:
1) The number of data streams n is determined by the system SNR:
2) In each mode, the simple equal power allocation over n spatial channels is employed.
Intuitive Explanation:Inverse Water-Filling Power Allocation (Optimal)
17
Performance of Multi-mode S-M
Ideal CSI Feedback Quantized CSI Feedback
18
Performance Analysis
Some Interesting Questions:
Finite Rate Effects: What is the performance (capacity, SNR, BER) versus the feedback rate ?
Mismatched Analysis: What happens if a codebook designed for one system is used in another system?
Transform Codebooks: The codebook for a particular system is transformed from another system through a linear or non-linear operation. What is the performance? & How to design?
Feedback With Error: What happens if the feedback information also suffers from error (delay)?
Quantization of Imperfect CSI: What happens if CSI to be quantized suffers from estimation error?
19
Capacity Loss Analysis for MISO Channels
Assume MISO channel with entries
Instantaneous Capacity (mutual information rate) Loss:
Capacity Loss: For a given codebook
Analysis is quite involved
Publications
1 J. C. Roh and B. D. Rao, ”Transmit Beamforming inMultiple-Antenna Systems with Finite Rate Feedback: A VQ-BasedApproach,” IEEE Transactions Information Theory. vol. 52, no. 3,Pages: 1101-1112, Mar. 2006
2 J. C. Roh and B. D. Rao, ”Design and Analysis of MIMO SpatialMultiplexing Systems with Quantized Feedback,” IEEE Transactionson Signal Processing, Vol. 54, no. 8, Pages. 2874-2886, Aug. 2006
3 J. C. Roh and B. D. Rao, ”Efficient Feedback Methods for MIMOChannels Based on Parameterizations,” IEEE Transactions onWireless Communications, Pages: 282 - 292, Jan. 2007
4 J. Zheng, E. Duni, and B. D. Rao, ”Analysis of Multiple AntennaSystems with Finite-Rate Feedback Using High ResolutionQuantization Theory,” IEEE Trans. on Signal Processing, vol.55,Issue 4,Pages: 1461 1476, April 2007.
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 25 / 48
Outline
1 Promise of MIMO Systems
2 Point to Point MIMO
3 Limited Feedback MIMO Systems
4 MIMO-OFDM
5 Multi-User MIMO
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 26 / 48
Frequency Selective Channels: MIMO-OFDM
Next generation wireless communication system uses MIMO- OFDM
MIMO-OFDM transfers a wideband frequency-selective channelinto a number of parallel narrowband flat fading MIMO channels
Benefits of OFDM
Achieves high spectral efficiency
Cyclic prefix is capable of mitigating multi-path fading
Allows for efficient FFT-based implementations and simplefrequency domain equalization
Exploits frequency diversity, in addition to time and spatialdiversity
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MIMO-OFDM Block Diagram
MIMO-OFDM Transceiver
Binary Data
Modulation& Mapping
S/P Space-TimeProcessing
Space-TimeDecoder
& Equalizer
P/S
Binary Data
Demodulation& Demapping
IFFT Add CP P/S
IFFT Add CP P/S
FFTS/P RemoveCP
FFTS/P RemoveCP
OFDM Modulation OFDM Demodulation
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MIMO-OFDM Signaling
The input-output relation of a broadband MIMO channel is
y [k] =
√Es
t
L∑l=0
H[l ]s[k − l ] + n[k]
k - discrete time index
L - number of channel taps
t - number of transmit antennas
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MIMO-OFDM Signaling Cont’d
OFDM with FFT/IFFT and CP insertion/removal operationsdecuples the frequency selective MIMO channel to a set of parallelMIMO channels as
y [l ] =
√Es
tH[l ]s[l ] + n[l ], l = 0, 1, ..,N − 1.
N - Number of subcarriers
H[l ] - DFT Coefficient of the channel
s[l ] - data on carrier l
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Spatial Diversity in MIMO-OFDM
Take Alamouti scheme as an example, there are two ways to realizespatial diversity
1 Coding in frequency domain, rather than in time domain
It requires that the channel remains constant over at least twoconsecutive tones
2 Coding on a per-tone basis across OFDM symbols in time
It requires that the channel remains constant during twoconsecutive OFDM symbols
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Outline
1 Promise of MIMO Systems
2 Point to Point MIMO
3 Limited Feedback MIMO Systems
4 MIMO-OFDM
5 Multi-User MIMO
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 32 / 48
Multi-User MIMO
Main Issue is the utilization of the spatial degree of freedom in amulti-user environment
Resource ManagementInterference Management
Capacity of Multi-User systems
Multi-user Diversity
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Multi-User SIMO Systems
r(t) =P∑
l=1
hlsl(t) + n(t)
To receive user j , can use beamformer wj
yj(t) = wHj r(t) = wH
j hjsj(t) +P∑
l=1,l 6=j
wHj hlsl(t) + wH
j n(t)
The beamforming vector can be optimized for each user separately.
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Multi-User MISO Systems
Transmitted signal
s(t) =P∑
l=1
wlsl(t)
Signal received by user j
rl(t) = hHj s(t) = hH
j wjsj(t) +P∑
l=1,l 6=j
hHj wlsl(t) + nj(t)
The transmit beamformers for the other users do interfere with thedesired user. Beamformers have to be jointly selected. A morechallenging problem.
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Problem Statement
University of California, San Diego
Problem StatementConsider a multiuser MIMO beamforming network
Arbitrary Network configurations (cellular networks, multi-hop networks, etc.)Heterogeneous communication nodes with different power costs
Minimize the network power cost while satisfying the minimum SINR requirements of all links
SINR (signal to interference plus noise ratio)Joint optimization of beamforming weights and transmit powers
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 36 / 48
Problem Statement
University of California, San Diego
JOP:
Solved for SIMO and MISO cases for MISO problem is solved by using the virtual uplink concept
Problem Statement
LlSINR
J
ll
T
≤≤≥
=
1 allfor subject to
)( min,,
γ
pwpUVp
TL
L
L
TL
ww
pp
],...,[
},...,{ },...,{
],...,[ where
1
1
1
1
=
===
w
uuUvvV
p (network power vector, L: no. of links)
(unit norm tx. beamforming vectors)
(unit norm rx. beamforming vectors)
(weight vector defining power costs)
T]1,...,1[== 1w
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SINR Expression for MIMO Beamforming
University of California, San Diego
SINR Expression for MIMO BeamformingSINR (signal to interference plus noise ratio)
Problem isolation for optimal Rx. beamforming vectors UMMSE/MVDR beamforming at the receivers
No straightforward problem isolation for V
linl
Hl
lsl
Hl
liliili
Hl
llllHl
lilili
lllll np
pnpG
pGSINRΓuΦuuΦu
vHuvHu
=+
=+
=≡∑∑≠≠
2
2
||||
liiliHlli
l
l
lili
l
l
rtG
llrt
lrLllt
to fromgain link effective: ||
link ofctor weight veantenna receive : link ofctor weight veantenna transmit :
to frommatrix gain channelcomplex : link ofReceiver :
)1( link ofr Transmitte :
2vHu
uvH
=
≤≤
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SIMO problem : Cellular Uplink (Rashid-Farrokhi
et al. 98)
University of California, San Diego
SIMO problem : Cellular Uplink(Rashid-Farrokhi et al. ’98)
Problem :
Joint Beamforming & Power Control Algorithm
Convergence to the global optima is established.Desirable features
MVDR beamforming : implemented using adaptive filters power control : using a simple power control loop
)(*)(
)(
)()1(
)()(
)(min)( where
)(
nl
lnl
l
lll
llj
jllj
ln
l
nn
pSINRG
npGI
l uu
up
pIp
u
γγ =+
=
=
∑≠
+
lΓ
p
ll
ll
∀≥
∑ subject to
min,
γ
Up
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MISO Problem & Virtual Uplink
Concept(Rashid-Farrokhi et al. 98)
University of California, San Diego
MISO Problem & Virtual Uplink Concept(Rashid-Farrokhi et al. ’98)
Dual relation between cellular downlink and uplinkVirtual uplink : uplink with reciprocal channels and noise vector 1. Optimal transmit beamforming vectors are identical to the optimal receive beamforming vectors in the virtual uplink
(a) Downlink (Primal) (b) Virtual Uplink (Dual)
11H
22H
33HHH11
HH22
HH33
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Generalization
University of California, San Diego
GeneralizationWe generalize this idea to arbitrary multiuser MIMO networks with generalized cost function (e.g., MIMO multihop networks, energy-aware networking environment, etc.)
We derive the dual relation using the well-established duality concept in optimization theory
We take advantage of the dual relation for solving the stated problem
We developed an improved Decentralized Algorithm
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 41 / 48
Construction of a Dual Network
University of California, San Diego
Construction of a Dual NetworkFor any multi-user MIMO network with linear beamformers, one can construct a dual network using the following three rules:
Reverse the direction of all linksReplace any MIMO channel matrix H by HH
Use transmit beamforming vectors as receive beamforming vectors, and vice versa.
44H22H11H
33H 55H
HH44
HH22HH11
HH33
HH55
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Duality
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 43 / 48
Applications to JOP
University of California, San Diego
Applications to JOPTheorem 2 suggests an iterative algorithm (Algorithm E)
Primal Network : Update p and U for fixed V, so that wTp is minimizedDual Network : Update q and V for fixed U, so that nTq is minimized
Lemma 3. In the proposed algorithm, once the solution becomes feasible, i.e., all SINR values meet or exceed the minimum requirements, it generates a sequence of feasible solutions with monotonic decreasing cost.
)(noutΓ
)()(~ nout
nin ΓΓ =
)(~ noutΓ
)()1( ~ nout
nin ΓΓ =+
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 44 / 48
Cellular Network -Downlink
University of California, San Diego
Cellular Network - DownlinkMultiple wrapped around cells (19 three-sectored cells)Same channel is reused in every cell but only in one sectorThree co-channel users per sectorPropagation exponent = 3.5, 8dB shadow fading
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 45 / 48
Performance Comparison
University of California, San Diego
Algorithm A, B, E and F
The proposed algorithm presents significant improvement in the complexity-performance tradeoff, thereby greatly improving practical value.
Performance Comparison
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 46 / 48
Current Trends
Multi-user OFDM systems
Coordinated Multi-Point Transmission (CoMP)
Cooperative MIMO
MIMO Ad-Hoc Networks
Bhaskar RaoCenter for Wireless CommunicationsUniversity of California, San Diego ()Linear Precoding in MIMO Wireless Systems 47 / 48
Summary
MIMO Systems offer unique opportunities in wirelesscommunication
Provides an opportunity to use spatial dimension to providediversity and hence reliability.
Can be used to significantly increase capacity in a rich scatteringenvironment
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