space time codes

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SPACE TIME CODING
Jie Ren
ASPITRG Drexel

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•  B. Vucetic and J. Yuan, Space-Time Coding, Wiley, 2003
•  Erik G. Larsson and Petre Stoica Space-Time Block
Coding for Wireless Communications, Cambridge, 2005

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Outline
• MIMO Wireless Communication Systems
• Space-Time Coding Performance Analysis
• Space-Time Block Codes
• Space-Time Trellis Codes

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Outline
o MIMO System Model
o MIMO System Capacity Derivation
o MIMO Capacity Examples

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Outline

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MIMO System Model
•  Notations
nT transmit antennas
nR receive antennas
x transmitted signals, N(0,µ) i.i.d.
n noise
r received signals
Rxx, Rnn, Rrr covariance matrix of x, n and r

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MIMO System Model
•  Covariance matrix of the transmitted signal
•  Transmitted power constraint
•  Channel is unknown at the transmitter

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MIMO System Model
•  Noise n
•  independent complex zero-mean Gaussian
•  No correlation between components of n

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MIMO System Model
•  MIMO Channel H
•  nR by nT complex matrix
•  perfectly known at the receiver
•  not known at the transmitter
•  normalization:

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MIMO System Model
•  Average SNR at each receive antenna
•  Received vector
! =
!!
!! =
!
!!!

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Outline

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MIMO System Capacity Derivation
•  Theorem: Singular value decomposition
•  Suppose M is an m×n matrix whose entries come from the field K.
(either the field of real numbers or the field of complex numbers)
Then,
•  where U is an m×m unitary matrix over K, V* is the conjugate
transpose of the n×n unitary matrix V over K, Σ is an m×n diagonal
matrix with non-negative real numbers on the diagonal.
! = !!!!!
!

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•  Singular value decomposition
! = !"!!!
! = !"!!! + !!
!!! = !!!"!!! + !!! = ! !!! + !!!!
!
!

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•  Equivalent channel

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•  Equivalent channel

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•  Covariance Matrix
•  Power constraint

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•  Capacity

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•  Capacity: Relates to the channel matrix H
! =
!!!, !! < !!
!!!, !! ≥ !!
!
! − !! = det!(!!! − !)
!
!!!
!
!"#!$%$"$&!! = −
!!!!
!
!
! = ! log! det!(!! +
!
!!!! !)!

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Outline

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Examples 1
•  SISO channel
•  1 receive antennas and 1 transmit antennas

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Examples 2
•  MIMO channel with unity H
•  Coherent combining
•  Reduces to a single effective channel

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Example 3
•  Receive Diversity
•  n receive antennas and 1 transmit antennas

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Example 4
•  Transmit Diversity
•  n transmit antennas and 1 receive antennas

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Outline

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Outline
o Diversity-Multiplexing Tradeoff
o ML Detection
o Error Analysis
o Space-Time Code Design Criteria

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Diversity-Multiplexing Tradeoff
•  Why MIMO?
•  Utilize multiple antennas to improve wireless system performance
•  Higher capacity
•  Lower error probability

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Definitions
•  Diversity Gain d
•  Change in slope of the error probability
•  Multiplexing Gain r
•  Change in slope of the rate

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Beamforming
•  Antennas transmit the same signal
•  Pre-coding and shaping matrices (vectors): u, v
•  Corresponding SNR

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Diversity-Multiplexing Trade-offs
•  Obtain full multiplexing gain
•  Decompose the MIMO into parallel SISO
•  multiplexing different data streams
•  each SISO quality depends on the singular values of HHH
•  may have poor performance
•  Obtain full diversity gain
•  Apply beamforming

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•  Fundamental design question:
•  Should the antennas be used for diversity gain, multiplexing gain or
both?
•  Assume block fading channels with receiver CSI only
•  Maximum d for fixed r:

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Outline
o ML Detection
o Error Analysis

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Space-Time Coded Systems
•  Information symbols
•  Input vector
•  Received vector

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ML Detection
! = argmin
!∈!!!×!
||! − !"||!
!
= arg min ||!! − !!!||!
!
!!!
!

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Space-Time Coded Systems
•  Decision Metrics
•  Selects a code word with the minimum decision metric

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Outline
o ML Detection
o Error Analysis

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Error Analysis
•  AWGN fading channel
•  General error probability
!! = ! ∙ !( ! ∙
!!
!!
)!
!!
= !!
! = |!|!
!
!
ℎ~!!(!, !)!

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Error Analysis
•  Theorem: The error probability, averaged over h, is
bounded by:

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Error Analysis
•  Diversity gain: Gd
•  Coding gain: Gc

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Outline
o Diversity-Mutiplexing Tradeoff
o ML Detection
o Error Analysis

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Space-Time Code Design Criteria
•  Pair-wise error probability for STC
•  Rank criterion: the difference matrix must be full rank to obtain the
maximum diversity gain MrMt
•  Determinant criterion: maximize the minimum of the Det(Δ) to
obtain a high coding gain

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Outline
•  Alamouti’s Space-Time Code
•  STBC

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Alamouti Space-Time Code

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•  Orthogonal Property
•  Received Signal

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•  Define
•  where e is white noise

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•  ML detection

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•  Decision Statistics
•  Decision Rules

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•  Achieve a full diversity gain

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Outline
•  Alamouti’s Space-Time Code
•  STBC

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Space-Time Block Codes
•  Code Matrix
•  Orthogonal Property

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Decoding of STBC
•  Decision Statistics
•  Decision Rules

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Outline
o Delay Diversity Code
o General STTC

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Delay Diversity
•  STTC: a steam of data is encoded via Nt convolutional
encoders
•  Delay Diversity for Nt=2
•  First convolutional encoder: absent
•  Second convolutional encoder: replace by time delay

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Delay Diversity
•  covariance matrix of he full rank

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Outline
o Delay Diversity Code
o General STTC

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Encoder Structure of STTC

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Encoder Structure of STTC
•  Generator Description
•  Generator Polynomial Description

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Example
•  4-state space-time trellis coded QPSK scheme with 2
transmit antennas
•  Generator sequences:

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Decoder Structure of STTC
•  Maximum Likelihood Decoding
•  Employ Viterbi Algorithm
•  Minimize the path metric

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