space time coding - faculty.coe.drexel.edu2/65 • b. vucetic and j. yuan, space-time coding, wiley,...

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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

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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!∈!!!×!

||!−!"||!!

= argmin ||!! − !!!||!!

!!!!

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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

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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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Example

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Decoder Structure of STTC

• Maximum Likelihood Decoding

•  Employ Viterbi Algorithm

•  Minimize the path metric

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