Modern Wireless Communication Chapter 6

Modern Wireless Communications


Modern Wireless Communication

Chapter 6 Diversity, Capacity and Space-

Simon Haykin, Michael Moher

Division Multiple Access

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Contents •

6.1 Introduction

•

6.2 “Space Diversity on Receive” Techniques

Content (Cont.) – 6.4.3 Outage Capacity – 6.4.4 Channel Known at the Transmitter

– 6.2.1 Selection Combining

•

6.5 Singular-Value Decomposition of the Channel Matrix

•

6.6 Space-Time Codes for MIMO Wireless Communications

– 6.2.2 Maximal-Ratio Combining

– 6.5.1 Eigendecomposition of the Log-det Capacity Formula

– 6.2.3 Equal-Gain Combining – 6.2.4 Square-Law Combining •

– 6.6.1 Preliminaries

6.3 Multiple-Input, Multiple-Output Antenna Systems

– 6.6.2 Alamouti Code

– 6.3.1 Coantenna Interference

– 6.6.3 Performance Comparison of Diversity-on-Receive and Diversity-onTransmit Schemes

– 6.3.2 Basic Baseband Channel Mode •

6.4 MIMO Capacity for Channel Known at the Receiver

– 6.6.4 Generalized Complex Orthogonal Space-Time Block Codes

– 6.4.1 Ergodic Capacity

– 6.6.5 Performance Comparisons of Different Space-Time Block Codes Using a Single Receiver

– 6.4.2 Two other special case of log-det formula: Capacities of Receiver and Transmit Diversity Links

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Content (Cont.) •

6.7 Differential Space-Time Block Codes – 6.7.1 Differential Space-Time Block Coding – 6.7.2 Transmitter and Receiver Structures

6.1 Introduction

– 6.7.3 Noise Performance •

6.8 Space-Division Multiple Access and Smart Antennas – 6.8.1 Antenna Arrays – 6.8.2 Multipath with Direction Antennas

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1


•


Diversity – Frequency diversity


– Time (signal-repetition) diversity – Space diversity • Receive diversity • Transmit diversity • Diversity on both transmit and receive

•

Multiple-input, multiple-output (MIMO) – User terminal of limited battery power – Channel of limited RF bandwidth

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6.2 “Space Diversity on Receive” Techniques 6.2.1 Selection Combining

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• Given Nr receiver outputs produced by common transmitted signal, logic circuit selects particular receiver output with largest signal-to-noise ratio as received signal.

•

Complex envelope of received signal of kth diversity branch is ~ ~ (t ) 0 ≤ t ≤ T and k = 1,2,K , N xk (t ) = α k e jθ k ~ s (t ) + w k r

(6.1)

•

With fading assumed to be slowly varying relative to symbol duration T, we simply the above Eq. to ~ ~ (t ) 0 ≤ t ≤ T and k = 1,2,K , N x (t ) ≈ α ~ s (t ) + w

(6.2)

•

The average signal-to-noise ratio at kth receiver output

• Assumption

k

– Frequency-flat: all frequency components constituting the transmitted signal are characterized by same random attenuation and phase shift. – Slow-fading: fading remains unchanged during transmission

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k

[ [

r

] ]

2 2  E αk ~ s (t )  E ~s (t ) = E α k2 2 2 ~   E wk (t )  E wk (t )

(SNR )k =  •

– Fading phenomenon is described by Rayleigh distribution.

k

[ ]

k = 1,2,K , N r

As mean-square value of w~k (t ) is the same for all k , we have (SNR )k = E E α k2 k = 1,2,K , N r N0

[ ]

(6.3)

(6.4) CH06-12

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•

Probability that all the diversity branches have a signal-to-noise ratio less than threshold γis N prob(γ k < γ for k = 1,2,K , N r ) = ∏ prob(γ k < γ ) r

•

Instantaneous signal-to-noise ratio γk =

•

k = 1,2,K , N r

1

γ av

 γ  exp − k   γ av 

γ k≥ 0 k = 1,2,K , N r

γ

−∞

f Γ f (γ k )dγ k

 γ   = 1 − exp −  γ av 

(6.7)

γ ≥0

  γ  = 1 − exp −   γ av  

(6.8)

Nr

γ ≥0

•

Cumulative distribution function of random variable ΓSC

•

Cumulative distribution function of selection combiner

γ sc = max{γ 1 , γ 1 ,K, γ 1 }

(6.6)

For some signal-to-noise ratio, the associated cumulative distributions of individual branches are prob (γ k ≤ γ ) = ∫

Nr   γ  = ∏ 1 − exp −  k =1   γ av 

(6.5)

Assume average signal-to-noise ratio over short-term fading is same, the probability density functions of random variables Γ k pertaining to individual branches as f Γk (γ k ) =

•

E 2 αk N0

k =1

  γ FΓ (γ sc ) = 1 − exp − sc  γ av 

  

(6.9)

Nr

γ sc ≥ 0

(6.10)

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•

Probability density function of

f Γ (γ sc ) = = •

d dγ sc Nr

γ av

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f Γ (γ sc )

FΓ (γ sc )

 γ   γ  exp − sc  1 − exp − sc    γ av    γ av  

(6.11)

N r −1

γ sc ≥ 0

For convenience of graphical presentation, we use the scaled probability density function f X ( x ) = γ av f Γsc (γ sc ) where the normalized variable x is x = γ sc γ av

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•


From figure 6.2 – As the number of branches increase, probability density function of normalized random variable moves progressively to right. – Probability density function becomes more symmetrical and Gaussian as Nr increase.

•

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Procedures of scanning version of selection-combining: – Selecting receiver with strongest output signal.

6.2.2 Maximal-Ratio Combining

– Maintain the procedure by using the output of this particular receiver as combiner’s output. – When the instantaneous signal-to-noise ratio of combiner falls below the threshold, select a new receiver with strongest output signal.

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•

Limitation of selection combiner can be mitigated by maximal-ratio combiner

•

Complex envelope of linear combiner output Nr

~y (t ) = a ~ ∑ k xk (t ) k =1

[

Nr

]

~ (t ) = ∑ ak α k e jθ k ~s (t ) + w k

(6.12)

k =1 Nr

Nr

k =1

k =1

~ (t ) = ~s (t )∑ ak α k e jθ k + ∑ ak w k

•

From Eq. (6.12), we notices that

Nr

jθ k ~ – Complex envelope of output signal equals s (t )∑ akα k e k =1

Nr

∑ a w~ (t )

– Complex envelope of output noise equals

k

k

k =1

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~ (t )are mutually independent for k = 1,2,K , N , the output signal - to - noise ratio of linear combiner is Assuming w k r 2 Nr   E  ~s (t )∑ akα k e jθ k   k =1   (SNR )c = 2 N  r  ~ (t )  E ∑ ak w k  k =1 

•

[ [

] ]

s (t ) E~ ⋅ ~ (t ) 2 Ew k 2

2

Nr

∑a α e θ j

k

 Nr  E  ∑ a k α k e jθ k   k =1  2  Nr  E ∑α k    k =1

Nr

and

k

k

∑a

k =1

2

=

Let γ c denote the instantaneous output signal-to-noise ratio of linear combiner, then using

(6.13)

2 k

k =1

as the instantaneous values of expectations in numerator and denominator of eq.(6.13), we can write 2

Nr

2  Nr  E  ∑ a k α k e jθ k   E   k =1  =   2 Nr N    0 E ∑α k   k =1 

∑ a k α k e jθ k

 E 

k =1

γ c =    N0 

(6.14)

Nr

∑a

2 k

k =1

where E/N 0 is the symbol energy - to - noise spectral density ratio

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•

According to Cauchy-Schwarz inequality for complex number, we 2 have Nr Nr Nr 2 2 (6.15) ∑ ak bk ≤ ∑ ak ∑ bk k =1

•


k =1

a ∑α e θ E ∑ 2

 

0



j

k

(

= cα k e •

∑a

(6.16)

k = 1,2,K , N r

(6.18)

Instantaneous output signal-to-noise ratio of maximal-ratio combiner

 E  Nr 

0

 k =1

(6.19)

According to Eq.(6.5), the maximal-ratio combiner produces an γmrc that is the sum of instantaneous signal-to-noise ratios of individual Nr branches γ mrc = γ k (6.20)

•

The term “maximal-ratio combiner” has been coined to describe the combiner of Fig. 6.4 that produces the produces optimum result.

k

∑

Canceling common terms in Eq.(6.16)

 E  Nr γ c ≤  ∑ α k2  N 0  k =1

∗

•

2

k =1

•

)

− jθ k

γ mrc =  ∑ α k2 N

k

k

k =1 Nr

Therefore, the equality in Eq.(6.17) holds for

ak = c α k e j θ k

k =1

Applying Cauchy-Schwarz inequality to instantaneous output signalto-noise ratio Nr Nr 2

γ c ≤   k =1 N

•

k =1

(6.17)

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•

Chi-Square with 2Nr degrees of freedom N r −1  γ  1 γ mrc f Γmrc (γ mrc ) = exp − mrc  (N r − r )! γ avN r  γ av 


(6.1)

6.2 “Space Diversity on Receive” Techniques 6.2.3 Equal-Gain Combining

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•

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Three issues for maximal-ratio combiner 1. Significant instrumentation is needed to adjust the complex weighting parameters of maximal-ratio combiner to their exact values.


2. Additional improvement in : 1. output signal-to-noise ration gained by mrc over sc is not large 2. Receiver performance is lost in inability to achieve the exact setting of mrc

6.2.4 Square-Law Combining

3. other details of the combiner may result in a minor improvement in overall receiver performance. •

Equal-gain combiner: all the complex weighting parameters have their phase angles set opposite to those of their respective multipath branches. CH06-27



•

No need phase estimation.

•

Applicable to orthogonal modulation.

•

With binary orthogonal signaling, receiver generates 1 T ~ ~∗ 1 T ~ ~∗ Q0 k = ∫0 xk (t )s0 (t )dt and Q1k = ∫0 xk (t )s1 (t )dt

•

In orthogonal modulation, two signaling waveforms approximately satisfy the condition T i= j E ~ (6.29) s (t )~ s ∗ (t )dt = b

N0

(6.27,28)

N0

∫

0

•

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i

 0

j

i≠ j

If binary symbol 0 is transmitted, the two decision variables are Eb α k e jθ w w Q0 k = + 0k and Q1k = 1k (6.30,31) k

N0

N0

N0

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•

Square-law receiver makes a decision between symbols 0 and 1 as

if •

2

Q0 k > Q1k

2

say 0 otherwise say 1

Nr

2

Nr

and Q1 = ∑ Q1k

k =1

2

•

• (6.33,34)

=

Eb Var (w0 k ) Var α k e jθ k + N0 N0

(

)

and

Var(Q1k ) =

Eb Var [w0 k ] E α k2 + N0 N0

( [ ])

1 Var (w1k ) N0

=1

= γ av + 1

(6.35,36)

•

q N r −1

1

(N r − 1)! (γ av + 1)

 q   exp −  γ av + 1 

1

(N r − 1)!

q N r −1 exp(− q )

(6.38)

As Q1 and Q0 are independent random variables, the probability of error is ∞ ∞ prob(Q0 < Q1 ) = ∫ f Q0 (q0 ) ∫ f Q1 (q1 )dq1 dq0 0  q0 

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(6.37)

For incorrect symbol, it is

f Q1 ( q) =

k =1

If s0(t) was transmitted, the variances are given by

Var (Q0 k ) =

Probability density function for correct symbol is

f Q0 (q ) =

(6.32)

With square-lave combining, we form the decision variable

Q0 = ∑ Q0 k •


(6.39)

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• MIMO wireless communications include space diversity • Three important points:

6.3 Multiple-Input, MultipleOutput Antenna Systems

– Fading phenomenon is an environmental source of possible enrichment. – Space diversity provides the basis for a significant increase in channel capacity or spectral efficiency. – Increasing channel capacity with MIMO is achieved by increasing computational complexity with maintaining primary communication resources fixed.

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6.3 Multiple-Input, MultipleOutput Antenna Systems 6.3.1 Co-antenna Interference

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•

Spatial parameter, defines as a new degree of freedom.

n = min{N t , N r }

6.3 Multiple-Input, MultipleOutput Antenna Systems

•

(6.40)

Nt-by-1 vector, denotes the complex signal vector transmitted by the Nt antennas at discrete time n. T s(n) = [~ s1 (n), ~ s2 (n),K, ~ st (n)]

6.3.2 Basic Baseband Channel Model •

Total transmit power is fixed at the value

P = N tσ s2

(6.42)

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•


•

~

For flat-fading, we can use hik (n) denote sampled complex gain of channel, thus express the Nr-by-Nt complex channel matrix as

~ ~ ~  h11 (n ) h21 (n ) L hN 1 (n )   ~  Nr ~ ~t h (n ) h22 (n ) L hN t 2 (n )   H (n) =  21  receive  M M M  ~  antennas ~ ~ hN r 1 (n ) hN r 2 (n ) L hN r N t (n )  14444442444444 3

(6.41)

The system of equations, defines the complex signal received at ith sk (n )radiated by kth antenna antenna due to transmitted symbol ~ Nt ~ ~ ~ (n ) xi (n ) = ∑ hik (n )~ sk ( n) + w i

• (6.43)

k =1

Complex received signal vector

i = 1,2,K, N r k = 1,2,K, N t

(6.44)

x(n ) = [~ x1 (n), ~ x1 (n),K, ~ x1 (n)]

(6.45)

•

Complex channel noise vector ~ (n), w ~ (n),K, w ~ ( n) T w (n ) = w 1 2 Nr

(6.46)

•

Therefore, the compact matrix form of the system equation which is the basic complex channel model for MIMO wireless communications

•

To simply the exposition, we get

N t transmit antennas

[

]

x(n) = H (n)s(n) + w(n)

x = Hs + w

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(6.47) (6.48)

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•

For mathematical tractability, we assume Gaussian model made up of three elements 1. Transmitter 2. Channel 3. Receiver

–

1. Correlation matrix of transmitted signal vector s is Rs = E[ss†] =σ2sINt where I N t is the Nt-by-Nt identity matrix

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•

2. The Nt x Nt elements of channel matrix H is

(

hik : N 0,1 •

•


)

(

2 + jN 0,1

2

)

i = 1,2,K , N r k = 1,2,K , N r

•

3. Correlation matrix of noise vector w is Rw = E[ww†]

(6.50)

=σ2wINr

On this basis, the amplitude component hik is rayleigh distributes, this is the reason why MIMO channel is a rich Rayleigh scattering environment.

where I Nt is the Nr-by-Nr identity matrix •

The average signal-to-noise ratio (SNR) at each receiver is

ρ=

Mean of squared amplitude component is a chi-square random variable

[ ]= 1

E hik

2

for all i and k

(6.51)

=

P

σ w2 N tσ s2

σ

(6.53)

2 w

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6.4 MIMO Capacity for Channel Known at the Receiver 6.4.1 Ergodic Capacity

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•

Information capacity of AWGN channel with fixed transmit power is defined as  P  C = B log 2 1 + 2 bit / s (6.54)



•

σw 

With the sampling theorem, we can rewrite to

 1 P  C = log 2 1 + 2 bit / s / Hz 2  σw  •

•


•

(

•

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)

  det R W + HR S H +  C = Elog 2  tr ( R S ) ≤ P bit / s / Hz which is subject to constraint max Rs det(R W )   

(6.55)

Capacity of complex, flat-fading channel is 2   h P  (6.56) C = E log 2 1 + 2 bits / s / Hz   σ  w    Assume the channel is stationary and ergodic, C is commonly referred to ergodic capacity of single-input, single-output (SISO) flat fading channel.

Ergodic capacity of MIMO channel

Substituting Eqs. (6.49) and (6.52) into Eq.(6.57)

    σ C = Elog 2 det  I N r + HH + bit / s / Hz σ     2 s 2 w

•

(6.58)

Invoking definition of average signal-to-noise ratio, we get

     ρ C = E log 2 det  I N r + HH + bit / s / Hz N t     •

(6.57)

(6.59)

This is the log-det capacity formula for a Gaussian MIMO channel. CH06-48

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•

Asymptotic formula

lim

N →∞

•


C ≥ constant N


(6.61)

The ergodic capacity of a MIMO flat-fading wireless link with an equal number N of transmit and receive antennas grows roughly proportionately with N

6.4.2 Two other special case of log-det formula: Capacities of Receiver and Transmit Diversity Links

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•


1. Diversity-on-receive channel

•

– Applying log-det capacity formula to this case, Eq. (6.60) reduces to Nr   C = E log 2 1 + ρ ∑ hi i =1  

2

 bit / s / Hz 

2. Diversity-on-transmit channel – The log-det capacity formula reduced to

  ρ C = Elog 2 1 + N  t 

(6.62)

Nt

∑h

k

k =1

2

 bits / s / Hz 

(6.63)

– Eq. (6.62) expresses the ergodic capacity due to linear combination of receive-antenna outputs. – It designed to maximize the information contained in Nr received signals about the transmitted signal. CH06-51

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•

Outage probability of MIMO link is defined as the probability for which the link is in a state of outage for data transmitted across the link at a certain rate, R.

•

To proceed on this probabilistic basis, we invoke a quasi-static model: – The burst is long enough to accommodate the transmission of large number of symbols.

6.4.3 Outage Capacity

– Yet the burst is short enough that can be treated as quasi static. – Channel matrix is permitted to change. – The different realizations of transmitted signal vector are drawn from a white gaussian codebook.

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•

In light of the log-det capacity formula, we may view the random variable     ρ +  Ck = E log 2 det I N r + H k H k  bit / s / Hz (6.64) Nt     as the expression for a sample of the wireless link. •


Outage probability at rate R is Poutage ( R ) = prob{Ck < R for some burst k }

6.4.4 Channel Known at the Transmitter

or equivalently,     ρ +  Poutage ( R ) = problog 2 det  I N r + H k H k  < R for some burst k  Nt     

•

(6.65)

Outage capacity of the MIMO link is the maximum bit rate that can be maintained across the link for all bursts of data transmissions for a prescribed outage probability. CH06-55

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• Log-det capacity formula is based on the premise that the transmitter has no knowledge of channel state.

6.5 Singular-Value Decompostion of the Channel Matrix

• Knowledge of channel state can be gathered by – first estimating the channel matrix at receiver – then sending to the transmitter via feedback channel. – Capacity is optimized.

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•


Diagonalize HH+ by invoking eigendecomposition of Hermitian Matrix

U + HH + U = A •

The matrixΛis a diagonal matrix whose Nr elements are eigenvalues of matrix product HH+

•

The matrix U is a unitary matrix whose Nr columns are the eigenvectors associated with eigenvalues of HH+.

•

Inverse of unitary matrix is equal to Hermitian transpose of matrix, as shown by

U −1 = U + or, equivalent ly,

UU + = U + U = I N r

•

(6.68,69)

Let Nt-by-Nt matrix V be another unitary matrix, that is

VV + = V + V = I N r

(6.67)

Inject the matrix product VV+ into the center of left-hand side of Eq. (6.67) (6.71) U + H(VV + )H + U = Α

•

Let the Nt-by-Nt matrix D denate a new diagonal matrix related to Nrby-Nr diagonal matrixΛwith Nr ≤ Nt by

Α = [D 0][D 0]

+

•

(6.72)

Examining Eqs.(6.71) and (6.72) and comparing terms, we deduce the new decomposition, which is singular-value decomposition (SDV)

U + HV = [D 0] CH06-59

(6.70)

•

(6.73) CH06-60

10


•


According to SVD theorem, we have – Elements of diagonal matrix

(

D = diag d1 , d 2 , K , d Nt

•

)

(6.74)

~ ~ xi = d i ~ si + w i

are the singular values of channel matrix H. – Columns of unitary matrix

[

U = u , u ,K , u N r

1 1 are the left singular vectors of matrix H

Using the definitions of Eqs. (6.74) through (6.76), the decomposed channel model can changed to scalar form

]

(6.75)

]

(6.76)

– Columns of second unitary matrix

[

V = v , v ,K, v

1 2 Nt are the right singular vectors of matrix H

i = 1,2,K , N r

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•

6.5 Singular-Value Decompostion of the Channel Matrix

Substituting Eq. (6.59)into Eq. (6.67) leads to spectral decomposition of HH+ in terms of Nr eigenmodes, we may write HH + = UΑU + Nr

= ∑ λi u iu i+

•

6.5.1 Eigendecomposition of the Logdet Capacity Formula

Substituting the first line of this decomposition into determinant part of Eq. (6.59) yields

    ρ ρ det I N r + HH +  = det I N r + UΛU +  Nt Nt     •

CH06-63

(6.84)

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And then using the defining Eq. (6.69), we can rewrite Eq. (6.83)

•

    ρ ρ + det I N r + HH +  = det I N r + U UΛ  Nt Nt      ρ  = det I N r + Λ N t  

(6.83)

Invoking the determinant identity.

det(I + AB ) = det (I + BA )

•

(6.82)

i =1

(6.85)

Nr  ρ  = ∏ 1 + λi  N t  i =1 

 Nr  ρ  (6.86) C = E ∑ log 2 1 + λi  bits / s / Hz  N t   i =1 • Ergodic capacity of a MIMO wireless communication system is the sum of capacity of Nr virtual single-input, single-output channels defined by the spatial eigenmodes of the matrix product HH+. •

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Finally, subsitute Eq. (6.85) into Eq. (6.59)

Using the log-det capacity formula of Eq.(6.60), for Nt ≤ Nr, we can show that  Nt  ρ  C = E ∑ log 2 1 + λi bits / s / Hz (6.87)  N r   i =1

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11



• Space-Time codes – Joint channel encoding of multiple transmit antennas


– Employ redundancy for purpose of providing protection against channel fading, noise, interference. – Maximize outage capacity – Classified into • Space-time trellis codes • Space-time block codes

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• Space-time trellis code

• Space-time block code

– Permits serial transmission of symbols.

– Transmission of signal takes place in blocks. Code is defined by transmission matrix with 3 parameters

– Designed for two to four transmit antennas

• Number of transmitted symbols l

– Well for slow-fading environment but not indoor data transmission.

• Number of transmit antennas, Nt, defines the size of transmission matrix.

– For decoding, multidimensional version of Viterbi algorithm is required – Decoding complexity of space-time trellis codes increases exponentially as the function of spectral efficiency.

• Number of time slots in data block m

– Ratio l/m defines of rate of code. – For efficient transmission, transmitted symbols are expressed in complex form.

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– Linear processing • Estimate transmitted symbols at receiver and simplify the receiver design.

– Different receiver design procedures: • Complex orthogonal design • Generalized complex orthogonal design

– Complex orthogonality of transmission matrix in temporal sense is a sufficient condition for linear processing at receiver. – Alamouti code: code with code rate of unity.

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6.6 Space-Time Codes for MIMO Wireless Communications 6.6.1 Preliminaries

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• Two functional units: – Mapper


• Take incoming binary data stream and generate a new sequence of blocks.

– Block encoder • Converts each block of complex symbols produced by mapper into an l-by- Nt transmission matrix S, where l and Nt are temporal dimension and spatial dimension

6.6.2 Alamouti Code

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•


•

Two-by-one orthogonal space-time block code.

~ s* S =  1~ *  − s2

−~ s2*   → Space ~ s1* 

~ s* ~ s *  ~ s* −~ s * SS + =  1~ * ~2*   ~1* ~ *2  s1  − s2 s1   s2 2 2  ~ s1 s1 s1 ~ s2 + ~ −~ =  ~ *~ * 2 *~ * ~ ~ s1 − s2 s1 + s1 s2

(6.88)

↓ Time •

Hermitian transpose of S

~ s* S + = ~1*  s2

−~ s2*   → Time ~ s1*  ↓ Space

(

s2 ~ s1  + ~ 2  + ~ s1 

(6.90)

)

0 2 2 1 s1 + ~ s2  = ~  0 1

(6.89)

•

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Multiply code matrix S by its Hermitian transpose, obtaining

This result also holds for alternative matrix product S+S, which is proof of orthogonality in the temporal sense CH06-78

13


•

The transmission matrix of Alamouti code satisfies the unique condition

(

)

2 2 SS + = S + S = ~ s1 + ~ s2 I

•


• Four Properties of Alamouti code • Property 1. Unitarity (Complex Orthogonality) (6.91)

– Alamouti code is an orthogonal space-time block code – Product of transmission matrix with its Hermitian transpose is equal to the two-by-two identity matrix scaled by the sum of squared amplitudes of transmitted symbols.

And Note that

S −1 =

1 ~ s1 + ~ s2 2

2

S+

• Property 2: Full-Rate Complex Code (6.92)

– Only complex space-time block code with a code rate unity.

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With s%1 and s% 2 transmitted simultaneously at time t , the complex received signal at time t' > t , is

• Property 3. Linearity

x%1 = α1e jθ1 s%1 + α 2e jθ 2 s%2 + w1

– Alamouti code is linear in transmitted symbols.

(6.95)

The complex signal received at time t' + T is

• Property 4. Optimality of Capacity

x%2 = −α1e jθ1 s%2* + α 2e jθ2 s%1* + w2 Reformulate the variable x%1 and complex

– For two transmit antennas and a single receive antenna, the Alamouti code is the only optimal space-time block code that satisfies the log-det capacity formula of Eq. (6.63).

(6.96)

conjugate of the second variable x%2 in matrix form  x%1   α1e jθ1  ∗ =  − jθ  x%2  α 2e 2

α1e jθ 2   s%1   w%1 

   +  * −α1e− jθ1   s%2   w% 2 

(6.97)

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•

•


Multiply Eq.(6.97) by hermitian transpose of two-by-two channel matrix− jθ ~ α1e 1 α 2e − jθ2   ~ x1  1 0  ~ s1  α1e − jθ1 α 2e − jθ2   w 1 2 2   − jθ 2   *  = (α1 + α 2 )  ~  +  − jθ − jθ   ~  x2  − α1e − jθ1   ~ 0 1  s2  α 2 e 2 − α1e 1   w2  α 2 e ~ ~∗  s  α e − jθ1 w ~ α 2 e − jθ 2 w 2 = (α12 + α 22 )~1  +  1 − jθ ~1 − jθ ~ ∗  (6.98)  s2  α 2e 2 w1 − α1e 1 w2  And let

x1   ~y1   α1e − jθ1 α 2e − jθ 2   ~  *  ~  =  − jθ 2 x2  − α1e − jθ1   ~  y2  α 2 e  α e − jθ1 ~ x + α 2 e − jθ 2 ~ x2∗  =  1 − jθ ~1 − jθ1 ~ ∗  2 α e x − α e x  2 1 1 2 

and

~  v~1  α1e − jθ1 w 1 v~  =  − jθ 2 ~  2  α 2 e w1

~∗  + α 2 e − jθ 2 w 2 − jθ1 ~ ∗  − α1e w2  (6.99)

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•

Recast Eq.(6.98) in matrix form of input-output relations describing the overall behavior of Alamouti code ~ y1  ~  v~1  2 2 s 1 2 2 ~ ~ ~ ~  = α1 + α 2  ~  + ~  in expanded form, yk = α1 + α 2 sk + vk k = 1,2  y2   s2  v2  (6.102)

(

)

(

)

•

Due to complex orthogonality of Alamouti code, the unwanted symbols are cancelled out in the equations. This cancellations are responsible for the simplification of receiver.

•

The detrimental effect of fading arises when diversity paths suffer from it. This means a wireless communication system based on the Alamouti code enjoys a two-level diversirty gain.

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•

Maximum-likelihood decoding rule: – Given that receiver has knowledge of • Channel fading coefficients α1andα2. • Set of all possible transmitted symbols in the mapper’s constellation denoted by S, the maximum-likelihood estimates of transmitted symbols defined by sˆ = arg min {d 2 (~ y , (α 2 + α 2 )ϕ )} and sˆ = arg min {d 2 (~ y , (α 2 + α 2 )ϕ )} (6.1) 1

ϕ∈S

1

1

2

2

ϕ∈S

2

1

2

where the φ denote the different hypotheses for the linear combiner output.

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6.6 Space-Time Codes for MIMO Wireless Communications 6.6.3 Performance Comparison of Diversity-on-Receive and Diversity-onTransmit Schemes

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• Generalized complex orthogonal designs of space–time block codes distinguish themselves from the Alamouti code in three respects:


– Nonsquare transmission matrix – Fractional code rate – Orthogonality of transmission matrix only in temporal sense.

6.6.4 Generalized Complex Orthogonal Space-Time Block Codes

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15


•


Let G be a an m-by- Nt matrix, Nt is the number of transmit antennas and m is the number of time slots, the entries of the matrix

•

Construction of space-time block codes using generalized complex orthogonal design is exemplified by rate-1/2 codes.

•

Case 1: For three transmit antennas (l=4, m=8) ~ ~ s1 s2 s3   ~ − ~ ~ ~ s s − s4  1  2 ~ ~ − ~ s3 s4 s1   ~  ~ −s −~ s s G 3 =  ~ *4 ~ *3 ~2*  → Space  s1 s2 s3   ~* ~*  s1 −~ s4*  − s2 ~ − ~ s3* ~ s4* s1*   ~*  ~* ~ s2*  − s4 − s3 ↓ Time

0,± s1 ,± s1* ,± s2 ,± s2* , K ,± sl ,± sl*

• G is said to be generalized complex orthogonalized design of size Nt and code rate is l/m if

 l 2  G + G =  ∑ s j  I  j =1 

(6.106)

(6.107)

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•


Case 2: For four transmit antennas (l=4,m=8)

s1  ~ − ~  s2 − ~ s3  ~ s − G 4 =  ~*4  s1  ~*  − s2 − ~ s*  ~3* −  s4

~ s2 ~ s1 ~ s

~ s3 s4 −~ ~ s

s3 −~ ~ s2* ~ s* 1

~ s2 ~ s3* s* −~

~ s4* s* −~

~ s1* ~ s*

4

3

1

4

2

•

~ s4 ~ s3 s −~

    2  ~ s1  → Space *  ~ s4  * ~ s3  −~ s2*   ~ s1* 

Compare with Alamouti code, space-time codes G3 and G4 are at a disadvantage in two respects: 1. The bandwidth efficiency is reduced by a factor of two. 2. The number of time slots across which channel is required to have a constant fading envelope is increased by a factor of four.

(6.108)

↓ Time CH06-93

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•

To improve the bandwidth efficiency, we may use rate-3/4 generalized complex linear processing orthogonal designs referred to as sporadic codes:

•

Case 1: For three transmit antennas (l=3, m=4)

 ~ s1  ~* − s2  H 3 = ~* s 2 ~3*  s3 2

~ s2 ~ s1* ~ s3* 2 −~ s3* 2

~ s3 ~ s3

2 2

(− ~s − ~s + ~s − ~s ) (~s + ~s + ~s − ~s ) 1

2

↓ Time

* 1 * 2

2

1

* 2 * 1

   → Space 2  2 

•

Case 2: For four transmit antennas (l=3,m=4)

 ~ s1  ~* − s2  H 3 = ~* s 2  ~3*  s3 2

~ s2 ~ s1* ~ s3* 2 −~ s3* 2

~ s3 2 ~ s3 2 −~ s1 − ~ s1* + ~ s2 − ~ s2* ~ s2 + ~ s2* + ~ s1 − ~ s1* ↓

(

(

)

Time

)

2 2

~ s3 2 −~ s3 2 −~ s2 − ~ s2* + ~ s1 − ~ s1* − ~ s +~ s* +~ s −~ s*

(

(

2

2

1

1

) )

   → Space 2  2  (6.110)

(6.109)

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16



6.6 Space-Time Codes for MIMO Wireless Communications 6.6.5 Performance Comparisons of Different Space-Time Block Codes Using a Single Receiver

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6.7 Differential Space-Time Block Codes

6.7 Differential Space-Time Block Codes 6.7.1 Differential Space-Time Block Coding

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17


•


Given the new pair of complex signals to be transmitted at time t+2, the row vector can be expressed as

[~s

3, t + 2

[

,~ s4 ,t + 2 ] = a1,t + 2 [~ s1,t , ~ s2 , t ] + a 2 , t + 2 − ~ s2* , ~ s1*,t

]

(6.111)

•

•

• a1,t+2 and a2,t+2 are coefficients of linear combination and defined as inner products of the row vector, therefore

[a

1,t + 2

[

[

+ , a2,t + 2 ] = [~ s3,t + 2 , ~ s4,t + 2 ][~ s1,t , ~ s2,t ] , − ~ s2*,t +1 , ~ s1*,t +1 ~ s* −~ s2,t +1  = [~ s3,t + 2 , ~ s4,t + 2 ]~1*,t  ~ s s 1,t +1    2 ,t = [~ s ,~ s ]S + 3,t + 2

4 ,t + 2

]]

Similarly, we can write

[− a

* 2 ,t + 3

] [

(6.113)

Coefficients matrix is a product of two orthogonal Alamouti (quaternionic) matrices.

 a1,t + 2 At + 2,t + 3 =  *  − a2 ,t + 3 = S t + 2,t +3S t+,t +1

+

]

, a1*,t +3 = − ~ s4*.t +3 , ~ s3*,t +3 S t+,t +1

a2 ,t + 2  a1*,t + 3 

(6.114)

(6.112)

t ,t +1

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•


Since St,t+1 is a unitary matrix by virtue of orthonormality of its tow constituent row vectors, it follow that

•

Similarly, the received signal matrix in response to the next transmitted signal matrix St+2,t+3 is

•

New two-by-two matrix

X t + 2,t +3 = S t ,t +1H

S t−,+t +1 = S t ,t +1 •

Basis for differential space-time block encoding at the transmitter

S t + 2,t +3 = A t + 2,t +3S t−,+t +1 •

Yt + 2,t +3 = X t + 2,t +3 X t+,t +1

(6.115)

= A t + 2,t +3S t ,t +1

(6.117)

= S t + 2,t + 3 H (S t ,t +1H )

+

In the absence of channel noise, the received signal matrix in response to transmitted signal matrix St,t+1 is given by

X t ,t +1 = S t ,t +1H

+

= S t + 2,t + 3 HH S

(6.118)

+ t ,t +1

(6.116)

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•

•

From the solution to problem6.12, we note that

 α e − jθ 1 H =  1 − jθ 2 α 2 e •


α 2 e − jθ   − α 1 e − jθ  2

+

H H = HH

and

+

1

(

= α

2 1

+α

2 2

)I

( = (α

) )A

Yt + 2 ,t +3 = α12 + α 22 S t + 2 ,t +3S t+,t +1 2 1

+ α 22

– A, is spanned by pair of complex coefficients (a1,a2) constituting matrix A

(6.119, 6.120)

Accordingly. Eq.(6.118) reduces to

(6.121)

t + 2,t + 3

With two sets of signal points involved in formulation of Eqs (6.115) and (6.121), we identify two signal spaces

~ ~ – S, is spanned by the complex signals s1 , s2 constituting matrix S

•

Properties of these signals:

•

1. With M-ary PSK as the method of modulation transmitting Alamouticode, – points representing signal space S are uniformly distributed on circle of unit radius

•

– Points representing signal space A constitute a quadrature amplitude modulation (QAM) constellation

This is the basis for differential space-time block decoding at the receiver.

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• 2. Minimum distance between the points in signal space S is equal to the minimum distance between the points in signal space A. • 3. To construct matrix A

6.7 Differential Space-Time Block Codes

– Bijective mapping of 2b bits onto the signal space A.

6.7.2 Transmitter and Receiver Structures

– The mapping bijective in the sense that it is one-to-one and onto.

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Differential space-time block encoder

Differential space-time block decoder

1. Mapper

1. Differential decoder

•

Generate the entries that make up matrix At+2,t+3

2. Differential encoder •

•

Delay unit feeds forward the matrix Xt,t+1

•

Multiplier multiplies the matrices Xt+2,t+3 and Xt,t+1 to produce new matrix Yt+2,t+3

2. Signal estimator

Transforms the matric At+2,t+3 into matrix St+2,t+3 •

Delay unit feeds back the matrix St,t+1 to input of differential encoder

•

Multiplier multiplies matrix inputs At+2,t+3 and St,t+1 to provide the transmitted signal matrix St,t+3 in accordance with Eq.(6.115)

•

ˆ Computes the matrix that A t + 2,t + 3 is closest to Yt + 2,t + 3

3. Inverse mapper •

ˆ

Operates on estimate At + 2,t + 3 to produce corresponding estimates of original pair of data bits transmitted at time t+2 and t+3.

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6.7 Differential Space-Time Block Codes 6.7.3 Noise Performance

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19



6.8 Space-Division Multiple Access and Smart Antennas

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• Advantages of 120o sector antennas at base station – Can be applied with FDMA, TDMA or CDMA – Allows multiple users to operate on same frequency and/or time slot in same cell – More users in same spectrum and improved capacity – Can be applied at base station without affecting mobile terminals

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• Advantages of smart antenna

• SDMA

– Greater range

– user terminals can spatially separated by virtue of their angular directions.

– Fewer base stations

• SDMA relies on smart antennas

– Better building penetration

• Examples of smart antennas

– Less sensitivity to power control errors – More responsive to traffic hot spots

– Sector antenna

• SDMA improves system capacity by

– Switched-beam antennas

– Minimization of effects of interference.

– Adaptive antenna

– Increasing signal strength

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6.8 Space-Division Multiple Access and Smart Antennas 6.8.1 Antenna Arrays

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•


Complex envelope of transmitted signal

~ s (t ) = m(t )e j 2πft •

Received symbols

Under the above assumptions, analysis depends solely on phase relationship of different elements.

•

If antenna element k is at distance lk from the transmitting antenna, then the carrier phase is φ (t ) = 2πf (t − lk c )

(6.123)

~ r (t , l ) = A(l )m(t − l c )e j 2πf (l c ) •

• (6.122)

= 2πf (t − l0 c ) − 2πf (lk − l0 c )

Key assumptions

≡ φ0 (t ) − 2πf∆lk / c

1. Incident field is a plane wave

≡ φ0 (t ) − ∆φk

2. The attenuation A(l ) ≈ A(l0 ) ≡ A0

•

3. . m(t − l c ) ≈ (t − l0 c ) ≡ m0 (t ) 4. There is no mutual coupling between the antenna elements.

The phase offset is

∆φk = (2πf c )(kd sin θ )  kd  = 2π   sin θ  λ 

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•

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  j 2π   sin θ  λ 

 kd  = 2π   sin θ  λ 

Complex rotation

ak (θ ) = e •

(6.125)

The received signal at element k is kd sk (t ) = s0 (t )e

•

(6.124)

s0 (t ) = A0 m0 (t )e jφ0 (t ) (6.126,127)

where  kd  j 2π   sinθ  λ 

(6.128)

A phased array computes a linear sum of the signals received at each element in Fig 6.29, yielding the received signal N 2

r (t ) =

∑ w s (t ) * k k

k =− N 2

(

)

(6.129)

[

= w a s0 (t ) where w = w− N 2 ,K , wN 2 +

]

T

[

]

and a (θ ) = a− N 2 (θ ), K, a N 2 (θ )

T

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6.8 Space-Division Multiple Access and Smart Antennas 6.8.2 Multipath with Direction Antennas

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