This document outlines and describes space-time coding techniques for MIMO wireless systems. It introduces MIMO system models and derives MIMO capacity. It then discusses space-time coding performance analysis, including diversity-multiplexing tradeoffs and error analysis. Finally, it describes specific space-time coding schemes, including Alamouti codes, space-time block codes, and space-time trellis 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
• MIMO Wireless Communication Systems
o MIMO System Model
o MIMO System Capacity Derivation
o MIMO Capacity Examples
• Space-Time Coding Performance Analysis
• Space-Time Block Codes
• Space-Time Trellis Codes

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Outline
• MIMO Wireless Communication Systems
o MIMO System Model
o MIMO System Capacity Derivation
o MIMO Capacity Examples
• Space-Time Coding Performance Analysis
• Space-Time Block Codes
• Space-Time Trellis Codes

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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

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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
• MIMO Wireless Communication Systems
o MIMO System Model
o MIMO System Capacity Derivation
o MIMO Capacity Examples
• Space-Time Coding Performance Analysis
• Space-Time Block Codes
• Space-Time Trellis Codes

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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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MIMO System Capacity Derivation
• Singular value decomposition
! = !"!!!
! = !"!!! + !!
!!! = !!!"!!! + !!! = ! !!! + !!!!
!
!

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MIMO System Capacity Derivation
• Singular value decomposition
• Equivalent channel

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MIMO System Capacity Derivation
• Singular value decomposition
• Equivalent channel

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MIMO System Capacity Derivation
• Covariance Matrix
• Power constraint

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MIMO System Capacity Derivation
• Capacity

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

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Outline
• MIMO Wireless Communication Systems
o MIMO System Model
o MIMO System Capacity Derivation
o MIMO Capacity Examples
• Space-Time Coding Performance Analysis
• Space-Time Block Codes
• Space-Time Trellis Codes

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

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Outline
• MIMO Wireless Communication Systems
• Space-Time Coding Performance Analysis
o Diversity-Multiplexing Tradeoff
o ML Detection
o Error Analysis
o Space-Time Code Design Criteria
• Space-Time Block Codes
• Space-Time Trellis Codes

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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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Diversity-Multiplexing Trade-offs
• 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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Diversity-Multiplexing Trade-offs

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Outline
• MIMO Wireless Communication Systems
• Space-Time Coding Performance Analysis
o Diversity-Multiplexing Tradeoff
o ML Detection
o Error Analysis
o Space-Time Code Design Criteria
• Space-Time Block Codes
• Space-Time Trellis Codes

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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
• MIMO Wireless Communication Systems
• Space-Time Coding Performance Analysis
o Diversity-Multiplexing Tradeoff
o ML Detection
o Error Analysis
o Space-Time Code Design Criteria
• Space-Time Block Codes
• Space-Time Trellis Codes

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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
• MIMO Wireless Communication Systems
• Space-Time Coding Performance Analysis
o Diversity-Mutiplexing Tradeoff
o ML Detection
o Error Analysis
o Space-Time Code Design Criteria
• Space-Time Block Codes
• Space-Time Trellis Codes

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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
• MIMO Wireless Communication Systems
• Space-Time Coding Performance Analysis
• Space-Time Block Codes
• Alamouti’s Space-Time Code
• STBC
• Space-Time Trellis Codes

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Alamouti Space-Time Code

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Alamouti Space-Time Code
• Orthogonal Property
• Received Signal

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Alamouti Space-Time Code
• Define
• where e is white noise

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Alamouti Space-Time Code
• ML detection

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Alamouti Space-Time Code
• Decision Statistics
• Decision Rules

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Alamouti Space-Time Code
• Achieve a full diversity gain

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Alamouti Space-Time Code

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Outline
• MIMO Wireless Communication Systems
• Space-Time Coding Performance Analysis
• Space-Time Block Codes
• Alamouti’s Space-Time Code
• STBC
• Space-Time Trellis Codes

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Space-Time Block Codes

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

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