F_MIMO

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Multiple-Input-Multiple-Output(MIMO) transmission over

fading channels

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ECSE610 MIMO - 3

Multiple-Antenna systems:

Antenna diversity

a) Receive diversity (SIMO); b) Transmit diversity (MISO);c) Transmit and Receive diversity (MIMO)

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ECSE610 MIMO - 4

Single-Input Multiple-Output (SIMO)

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ECSE610 MIMO - 5

Spatial diversity-on-receive SIMO channel

MT=1, MR=L, h=[h1,h2,…,hL]T

, y=hx+w, RW=W2

IL: white Gaussian noiseMaximum Likelihood (ML) decision:

Consider equally probable transmission of x from A={ak, k=1,2,…,K}

Receiver knows h, computes squared Euclidean distances: y-hak2

Select x=ak corresponding to the shortest Euclidean distance:From y-hx2=(y-hx)H(y-hx)=h2{y2/h2+x2-(xyHh+x*hHy)/h2}

= h2{x-xo2+yo2}, xo= hHy/h2: complex, yo2 ={y2/h2}-xo2

select ak A for min y-hak2 select ak A for min x-xo2

ML Rx: First, with channel knowledge, Rx uses matched filter (Maximum Ratio

Combiner, MRC): Rx beamforming → Rx diversity gain

Second, Rx selects akA nearest to xo.

2

22

received vector: [ ] [ ], { } , { }

[ ] [ ] '[ ], '[ ] [ ],

{[ ] } , { ' }

matched filter:

,

[ ]

]

/

[

H

s o L

o

s o M

H

RC s o

H

m m E x E E N

x m

x m

m w m w m m

E LE E w N SN x L

x

R E

m

N

y w ww Ih

h hhh h

y w

h

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ECSE610 MIMO - 6

SIMO channel: spatial diversity performance

In general, with ML detection, the conditional symbol error rate is

a: constant related to the number of nearest neighbors

d : constant related to the squared minimum distance (normalized to Es)

For Rayleigh fading channel, average symbol error rate

diversity order: indicated by the exponent L.

In general, this is an application of (L,1) repetitive coding. Diversity can be

obtained in time, frequency or space. Spatial diversity costs antennas (space)but save time and bandwidth (in time or frequency diversity).

2

2 2

2 2 -x /2

10 0

( ) exp , , Chernoff bound Q(x) e2

Ls s

e l

l

d E d E P aQ a h

N N

h hh h

1 2

L

se e

o

dE

P E P a N

h h

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ECSE610 MIMO - 7

Multiple-Input Single-Output (MISO) systems

Tx Diversity: With channel knowledge at Tx, Tx sends vector:

so that the received scalar signal is:

which maximizes the received SNR by in-phase addition of signals at thereceiver (Tx beamforming same as Rx beamforming).

*

( ) ,( )mm x h

hx

*

( ) ( ) ( ) ( ) (( ) )) (T T y m m w m w m w m x m x m hh h hh

x

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ECSE610 MIMO - 8

Diversity-on-transmit MISO channel:h known to Tx

MT=N, h=[h1,h2,…,hN], P/W2=Es /No Tx use a complex weight Nx1 vector p=[p1,p2,…,pN]

T to transmit x over Nantennas and keep p2=1 (for a total average Tx power of P or Es).

Rx signal:

Select p to maximize the received SNR:

→ Rx SNR: E{h2(Es /No)}=N(Es /No): N time better than SISO: Tx array gain of N

Conditional symbol error rate:

For Rayleigh fading channel, average symbol error rate:

diversity order of N

2 2

0 0

( ) exp2

s s

e

dE dE P aQ a

N N

h hh

12

N

se e

o

dE P E P a

N

h h

2

( ) ( ) ( ) ( ), Rx( ) sT T T

o

E x m y m m w m w m SN

N R pxh ph h

*2 2

maximumopti R mum , }x {s

o

SNR E E

N N

p h

h h

h

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ECSE610 MIMO - 9

Multiple-Input Multiple-Output (MIMO)

: Time delay-spread

t: time-variance

time-varying

MIMO channel:

Block frequency-flat fading MIMO channel:

where

Channel state information (CSI)

may be

unknown to both Tx and Rx

only known to Rx

known to both Rx and Tx

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ECSE610 MIMO - 10

zero-mean, circularly symmetric complex Gaussian

(ZMCSCG) Consider a continuous complex Gaussian vector Z=ZI+jZQ=[Z1,Z2,…,ZN]

T,Zk =ZkI+jZkQ with zero mean and NxN correlation matrix R Z=E{ZZ

H}

It can be represented by a real-valued vector

Z’ =[ZI,ZQ]=[Z1I,Z2I,…,ZNI,Z1Q,Z2Q,…,ZNQ]T

with zero mean and correlation 2Nx2Nmatrix R Z’ =E{Z’Z’

T} and pdf

R Z=E{ZZH}= E{(ZI+jZQ)(ZI-jZQ)

T}=(R II+R QQ)+j( R QI-R IQ)

If R II=R

QQand R

QI=-R

IQ R

Z=2(R

II-jR

IQ) and E{ZZT}=0 then Z is zero-mean,

circularly symmetric complex Gaussian (ZMCSCG):

ZI, ZQ: zero-mean, uncorrelated since R IQ=E{ ZIZQT}= E{ZI}[E{ZQ}]

T=0,

i.e., ZI,ZQ: orthogonal ZI,ZQ: statistically independent: R Z=2R II If Gaussian ZI,ZQ: statistically independent and identically distributed:

' , E

II IQ T

Z AB A B

QI QQ

R R R R Z Z

R R

T 1

'2

'

exp ' ' / 2( )

2 det Z

N p

Z'

Z

z R zz'

R

2-1H 1 exp / 2exp

( ) and ( ) ( ) ( )det 2 det

T

N N p p p p

I Q

I II IZ

Z Z Z I Z Q

ZII

z R zz R zz z z z

R R

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ECSE610 MIMO - 11

Eigen-decomposition andsingular-value decomposition (SVD): quick review

Eigen-decomposition of a MxM complex matrix A:

Eigenvalue problem: Ax= x, : scalar, x: Mx1 vector. AH= AT*, A is a Hermitian matrix if A= AH

In general, there are up to M distinct eigenvalues of A, n, n=1,2,…,M, whichare roots of the characteristic equation: det[ A-I]=0. Corresponding to each m,m=1,2,…,M, is the eigenvector xm.

Axm=m xm A[ x1, x2,…, xM]=[1 x1, 2 x2,…, M xM]or AU=U A=U UH (spectral decomposition theorem) where

U=[ x1, x2,…, xM]: unitary matrix,i.e., UUH=UHU=I, UH=U-1, and

=diag[1, 2,…,M]

Singular-value decomposition (SVD) of a complex-valued LxN matrix G:

xx x( )

( )x

1 1

1 2 min( , )

1 2 min( , )

for LN, or of if L

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ECSE610 MIMO - 12

Parallel Decomposition of a MIMO channel y=Gx+w MT=N, MR =L, LxN Channel Matrix: G=[gln] with E{gln2}=1 and (squared)

Frobenius norm

where dm is the mth singular values of G

Tx : X =[X1,X2,…,XN]T , Rx: Y =[Y 1,Y 2,…,Y L]

T noise: W=[W1,W2,…,WL]T

From y=Gx+w, using singular-value decomposition (SVD), we can obtain y’ =UH y=UHGV V H x+UHw = Dmin(L,N) x’+w’ where

y’ =UH y, w’ =UHw, x’ = V H x are column vectors of M=min(L,N) elements.

The SVD of G transform the MIMOchannel into M=min(L,N) parallelvirtual SISO channels:

Y’ m=dmX’ m+W’ m where

Y’ m, X’ m, W’ m are the mthcomponents of Y’ , X’ , W’ ,respectively, and

dm is the mth singular values of G.

X’ 1

X’ 2

X’ M

Y’ 1

Y’ 2

Y’ M

d2 W’ 2

d1 W’ 1

dM W’ M

min ,,

2 2 2

ln

1, 1

or , L N L N

H H

m m m

l n m

g tr tr d

G GG G G

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ECSE610 MIMO - 13

MIMO channel with Gaussian noise y=Gx+w:conditional mutual information for a given G

y=Gx+w

Additive Gaussian noise: W=[W1,W2,…,WL]T: ZMCSCG with LxL correlation matrix

R W=E{WWH}, R W=W2 IL for white Gaussian noise (W2=No /Ts)

Continuous source X =[X1,X2,…,XN]T, with correlation NxN matrix R X =E{ XX

H}, andTx power P=tr[R X ]: trace of R X : sum of N diagonal elements of R X

X , W: independent, continuous Rx Y =[Y 1,Y 2,…,Y L]T with LxL correlation matrix R Y

Conditional entropy H( Y X,G)= H(W)= [Llog(e)+log(det[R W])] conditional mutual information for a given G: I( X , Y G)=H( Y )–H( Y | X )

Capacity C= maxp(x) I( X , Y G)=maxp(x) [H( Y )–H( Y | X )] equivalent to maxp(x) H( Y )when X is ZMCSCG

X , W: independent, Y : Gaussian, zero-mean, H( Y )=E{-log[p Y ( y)]}=[Llog(e)+log(det[R Y ])] I( X , Y G)=H( Y )–H( Y | X )= log(det[R W+GR X G

H]/det[R W])

H H H H H H H H E E Y X WR (GX + W)(GX + W) GXX G + GXW WX G WW GR G R

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ECSE610 MIMO - 14

capacity of MIMO channel G with Gaussian noise:G: known at the transmitter C= log(det[R W+GR X G

H]/det[R W]) with

where P : total power, symbol energy: Es=P Ts R X : non-negative definite, to be designed

For R W

=W

2 IL: white Gaussian noise (

W

2=No /T

s), C= log(det[I

L+GR

X GH /

W

2])

From det[I+ AB]=det[I+BA] det[I+GR X GH /W2]=det[I+R X GHG /W2] Eigen-Decomposition of a Hermitian matrix: GHG=U UH

det[I+R X GHG /W2]=det[I+R X U UH /W2]=det[I+ UHR X U /W2] tr{UHR X U}= tr{U

HUR X }=tr{R X } and UHR X U: also non-negative definite

Recall: any non-negative definite NxN A satisfies the Hadamard inequality:

where ann: diagonal elements of A

det[I+ UHR X U /W2]

where 1 2…N: eigenvalues of GHG and Pnn: diagonal elements of UHR X U. Theequality holds only when UHR X U is a diagonal matrix.

max : constant Tx power tr Px

xR

R

1

det N

nn

n

a

A 2 2

1 1

1 / 1/ / N N

n nn W n n nn W

n n

P P

2

1 1

max log 1 / subject to the power constraintnn

N N

n nn W nnP

n n

C P P P

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ECSE610 MIMO - 15

capacity of MIMO channel G with Gaussian noise & Gknown at the transmitter: water-filling procedure As log(.) is monotonously increasing with its argument, to maximize C, we select

a set of {Pnn} to meet the total power constraint P , and maximize

Depending on P , some Pnn=0 selected to be corresponding to the smallest n.Let N+N such that Pnn>0 for nN+ and +=diag[1, 2,…,N+]. Then

The inequality follows the relation between the arithmetic and geometric means .The equality holds only if

water-filling procedure:

21

1/ / N

n n nn W

n

P

2 2

1 1 1 1

2 2

11 1 1

1/ / 1/ / 1/

1/ / 1/ / /

N N N N

n n nn W n n nn W n

n n n n N

N

N N N N

n n nn W n n nn W

nn n n

P P

P P N

1

2 2 2

1

1/ / 1/ / / / / N

n nn W n nn W W

n

P P N Tr P N

1

2 2Fill power (water) / where "water-fill level" = /nn W n W P Tr P N

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ECSE610 MIMO - 16

capacity of MIMO channel G with Gaussian noise & Gknown at the transmitter & using water-filling procedure

Transmit more signal power in the better channels (larger n). N+ can beimplicitly found from the above water-filling procedure: Start with N+=N, if

set N+= N+-1 and continue until .

The capacity of the channel G when G is known to the transmitter is

2 /W N 2 /W N

1

2( ) log det / /

N

T W C Tr P N

G Λ Λ

X’ 1

X’ 2

X’ M

Y’ 1

Y’ 2

Y’ M

d2 W’ 2

d1 W’ 1

dM W’ M

i.e., From y=Gx+w

y’ =UH y=UH {GVx’ +w}, x’ = V H x,the SVD of G transform the MIMOchannel into rank M≤min(L,N) virtual

SISO channels: Y’ m=[m]1/2X’ m+W’ m

where Y’ m, X’ m, W’ m are the mthcomponents of Y’ , X’ , W’= UH w,

optimum R X satisfiesUHR X U=diag{Pnn, n=1,2,…,N

+}

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ECSE610 MIMO - 17

Ergodic capacity of MIMO flat-fading Gaussian channel:G known to the receiver but unknown to the transmitter

G: is Zero Mean Spatially White (channel elements can be assumed to be i.i.d random variables), stationaryand ergodic (time averages of a sample function are equal to the corresponding ensemble average or

expectation at a particular time instant).

Without knowledge of G at Tx, Tx equally distributes power over Tx antennas, i.e., RX=X2I, X2=P/N, I(X,YG)=H(Y)–H(Y|X)= log(det[RW+GRXG

H]/det[RW]), AWGN: RW=W2 I

Ergodic capacity: C=EG{CUT(G)}, CUT(G)=log(det[RW+GRXGH]/det[RW]) subject to power constraint (i.e., total Txpower of P).

CUT(G)=log(det[IL +( P/NW2)GGH] for LN for GHG to be of full rank eigen-decomposition of a Hermitianmatrix: GGH=U UH for LN.

From det[I+ AB]=det[I+BA] CUT(G)=log(det[IL+(P/NW2)U UH] CUT(G)=log(det[IL+(P/NW2) UHU] for LN

MIMO ergodiccapacity =sum of capacities of min(L,N) virtual SISO channels defined by the spatialeigenmodesof GHG or GGH.

For N=L, as N, (Taylor’s series) and

i.e., C is linearly (rather than

logarithmically) increased with SNR.

min( , ) min( , )

2 2

11

( ) log 1 / max( , ) log 1 / max( , ) L N L N

UT m W m W

mm

C P L N P L N

G

2

2log 1 /

log ( )

ma m W

W e

PP N

N a

2 21 1

1log 1 / where

log ( )

N N avg

a m W avg m

m mW e

PC E P N E

a N

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ECSE610 MIMO - 18

Capacity comparison

FOR LARGE SNR: P/W2

FOR SMALL SNR: In CT(G), all Tx power allocated to the dominant eigenvector,corresponding to

1, of GGH and the channel is essentially reduced to a SISO

channel with gain 1.

CSI at Tx is more important for low SNR

min( , ) min( , )2

11

min( , ) 2

1

log 1/ / /

( )

( ) log 1 / max( , )

N L N L N

m m W

mm

T

L N UT

m W

m

P N

C

C P L N

G

G

min( , ) min( , )2 2

1 1

min( , ) min( , )min( , )2 2

1 1

log / log /( )

1( )

log / max( , ) log / max( , )

L N L N N N

m W m W

m mT

L N L N L N UT

m W m W m m

P N P N C

C P L N P L N

G

G

2 2

max max

min( , ) min( , )2 2

1 1

max

min( , )

1

( ) log 1 / / / log ( )

( ) log 1 / max( , ) / max( , ) log ( )

( ) ( )max( , ) for 1 or for 1

( ) ( )

T a W W e

L N L N

UT a m W m W e

m m

T T

L N

UT UT m

m

C P P a

C P L N P L N a

C C L N N L L N

C C

G

G

G G

G G

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ECSE610 MIMO - 19

SIMO, MISO: capacity

DIVERSITY-ON-RECEIVE SIMO CHANNEL:

MT=1, MR =L, h=[h1,h2,…,hL]T, y=hx+w, R W=W2 IL: white Gaussian noise

C=Eh{log[1+(P/W2)tr{hHh}]}=Eh{log[1+(Ph2 /W2)]}, P/W2=Es /No Additional Rx antennas provides only a log-increase in capacity. Channel knowledge at Tx for SIMO channels has no capacity benefit.

tr{hhH}= h12+h22+h32+…+hL2=h2: sum of eigen values of hHh The result confirms the maximal-ratio combining (MRC) principle.

DIVERSITY-ON-TRANSMIT MISO CHANNEL: MT=N, h=[h1,h2,…,hN]

T, P/W2=Es /No h unknown at Tx: equal power, P/N, over N Tx antennas

C= Eh{log[1+(Ph2 /NW2)]}: no improvement in capacity over a SISOchannel.

h known at Tx: C= Eh{log[1+(Ph2 /W2)]}, same as in SIMO

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ECSE610 MIMO - 20

Diversity in MIMO:

Diversity techniques rely on transmitting the signal over multiple (ideally)independently fading paths (in time/frequency/space).

Spatial (or antenna) diversity is preferred over time/frequency diversity as it

does not incur an expenditure in transmission time or bandwidth. Each of MR MT pairs of Tx-Rx antennas provides a signal path from transmitter

to receiver. By sending the SAME information through different paths, d independently-

faded replicas of the Tx signal can be obtained and combined at the receiver

such that the resultant signal exhibits considerably reduced amplitudevariability in comparison to a SISO link and we have a diversity order d ,where 1 ≤ d ≤ MR MT

Diversity gain (order) d implies that in the high SNR region, Pe ∼ 1/SNR d as

opposed to 1/SNR for a SISO system: The higher the diversity gain, the lower

the Pe Extracting spatial diversity gain in the absence of channel knowledge at the

transmitter is possible using suitably designed transmit signals, e.g., space–time coding techniques

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ECSE610 MIMO - 21

MIMO: Diversity &

Multiplexing gains

Spatial Diversity:

Sending the SAME information through different paths (selected from thetotal of MR MT pairs) to obtain a diversity of d to reduce Pe (∼ 1/SNR

d): Gofor Reliability, Counter fading

Spatial Multiplexing :

We can convert a general MIMO model to an equivalent diagonal model ofsome rank M ≤ min( MT,MR ) and send DIFFERENT information overdifferent links (selected from M) to increase the transmission rate, i.e., gofor Multiplexing (optimistic approach using rich scattering/fading toadvantage)

Diversity/Multiplexing Tradeoff in MIMO

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ECSE610 MIMO - 23

Space-Time Coding (STC)

Channel coding (FEC): with redundancy in time with coding rate rc 1: spatial multiplexing mode (max transmission rate)

Spectral efficiency BW=(f s Rrc rs)/B= Rrc rs in bits/s/Hz

1

2

MT

Channel

coding, r c

Symbol

mapping

Space-

Time

Coding

(STC)

.

.

Rbits/symbol

f s

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ECSE610 MIMO - 24

Alamouti 2x2 Orthogonal Space-Time Block Codefor 2 Tx antennas

Unitarity=Complex orthogonality

linearity:

S: transmission matrix:orthogonal in both the space

and time

*

1 2

*

2 1

s s

s s

S

(Siavash M. Alamouti, “A Simple Transmit Diversity Technique for Wireless Communications”, IEEE JSAC, Oct.1998, pp.1451-1458)

* * * *

2 2 2 211 2 1 2 1 2

1 2 2x2 1 2* *

2 1 2 1 2 1

/ H H s s s s s s

s s s ss s s s s s

S SS I S S

*

* *1 2

1 1 2 2*

2 1

1 0 0 0 0 0 0 1

0 0 0 1 1 0 0 0

s ss s s s

s s

S

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ECSE610 MIMO - 25

Decoding Alamouti OSTBC

For 1x2 complex channel G=[G1,G2], and Tx vector s=[s1,s2]T,

Rx signals in 2 consecutive time slots, Y 1, Y 2 are

Z1, Z2 can be independently decoded for s1,s2 , respectively

1 1 1 2 2 1 1 1 1 2 2 1 1 2 1 1

* * * * * * * *

2 1 2 2 1 2 1 2 2 1 2 2 1 2 2

1 1 2 1 1 2

* * * *

2 2 1

*

2

*

2 2 1

H

Y G s G s W Y G s G s W G G s W

Y G s G s W G s G s W G G s W

Z G G Y G G

Z G G Y G

Y

G

1 2 1 1 2 1

* * * * *

2 1 2 2 1 2

* *2 21 1 1 1 2 2

1 2 * *2 2 2 1 1 2

H H G G s G G W

G G s G G W

Z s G W G W G G

Z s G W G W

2 * *

1 2 1 1 2 2

* * * * * *

2 1 1 2 1 1 2 2 2 1 1 2

2* * * * * *

1 1 2 2 1 1 1 1 2 2 1 1 1

For iid zero-mean Gaussian noise components, W , W with variance , and

are Gaussian components with E{ } E{ } 0 and

E =E

W G W G W

G W G W G W G W G W G W

G W G W G W G W G W G W G

2 2* * * * 2

1 2 2 2 2 2 2 1 2

2 2 2* * 2

2 1 1 2 1 2E

W

W

W G W G W G W G G

G W G W G G

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ECSE610 MIMO - 26

Decoding Alamouti OSTBC: ML decision

Equally probable Tx of s1, s2 from A={ak , k=1,2,…,K}

Receiver knows G, computes Euclidean distances: Z j -(G12+G22)ak 2

ML decision rule: select ak for the shortest Euclidean distance.

Rx SNR: (G4Es /2)/(G2No)=(G2 /2)/(Es /No) E{(G2 /2)(Es /No)}=(Es /No) Absence of channel knowledge does not provide array gain

For Rayleigh fading channel, avg. symbol error rate diversity order of 2 and noarray gain

Full diversity, full-rate complex STBC, i.e., code rate=1.

For MT=2, MR =1, the Alamouti code is the only optimal STBC that satisfies the

capacity formula (3dB worse than MRC diversity-on-receive channel with MT=1,MR =2)

It is the only orthogonal STBC that achieves rate 1, i.e., can achieve its fulldiversity gain without needing to sacrifice its data rate.

2 2

0 0

( ) exp2 4

s s

e

dE dE P aQ a

N N

G GG

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ECSE610 MIMO - 27

ST coding in MIMO frequency flat fading channels

no channel knowledge at Tx over N Tx antennas, L Rx antennas, J symbol time intervals.

Symbol set A={ai, i=1,2,…, 2q}

Block of qk information bits encoded to qn bits.

Received LxJ matrix Y =[ y(1), y(2),…, y(J)]=GS+W, Transmitted NxJ matrix S=[ s(1), s(2),…, s(J)], E{s}=Es /N=PTs /N

LxJ noise matrix: W= [w(1), w(2),…, w(J)]

y(j)=Gs(j)+w(j), j=1, 2,…,J; w(j): zero-mean WGN with NoIL G: LxN channel matrix with E{g

ln}=0, E{g

ln2}=1

Rx ML decoding: Rx knows G 2 2

1

arg min arg min ( ) ( ) J

j

j j

S S

Y Y - GS y - Gs

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ECSE610 MIMO - 28

PEP: pairwise error probability:

for G, : Gaussian

2/ 42

2 2 2

1

Pr / 2 ,

where =tr

where ( ) and

Notes: Kronecker product of mxn matrix and m'xn' matrix mm'xnn' ma

o N

o

L H H H H

l l

l

T T

L

Q N e

vec

γΣ

S S' G G S - S'

G S - S' g S - S' S - S' g G S - S' γΣ Σ γ γΣ

γ G Σ I S - S'

A B

11 1

1 2

1

1

2

trix

= , , ,

( ) :stack the mxn matrix into a mnx1 vector columwise

Pr Pr det / 4

n

n

m mn

n

L H H

G NL G o

a a

a a

vec

E I E N

B B

C = A B A a a a

B B

a

aA A

a

S S' S S' G Σ γ γΣ

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ECSE610 MIMO - 29

ST code design criteria:

For iid spatial white channel G= GW, i.e., E{glngij*}=(l-i)(n-j)

determinant criterion: coding gain depends on ; therefore tomaximize the coding gain, we maximize the minimum of the determinant of

rank criterion: spatial diversity gain is indicated by LI(S,S’), with I(S,S’)N;therefore, design code for difference matrix of any pair (S,S’) to be full-rank,i.e., I(S,S’)=N

V. Tarokh, N. Seshadri, A. R. Calderbank, “Space–Time Codes for High Data Rate Wireless Communication: Performance

Criterion and Code Construction”, IEEE Transactions on Information Theory , Vol. 44, No. 2, March 1998, pp. 744-765

( ')

1

1th

Pr det / 4 1 ( ') / 4 /

where ( ') is the i non-zero eigenvalue of / ' ' for i=1,2,..., ( ')

for high / >>1 r (, P

H H H

G

L I L

H

NL o i s oi

H

i s

s o i

E

I N N E N

E N I

E N

S,S

Σ γ γΣ Σ Σ

S S' Σ Σ S, S

S, S S S S S S, S

S S' (

( ')

1

')

/' / 4) L

L

s o

I

i

I

E N N

S,S

SS,

S, S

( ')

1

( ') I

i

i

S,S

S, S

' ' H

S S S S

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ECSE610 MIMO - 30

Orthogonal STBC Designs Alamouti code: 2x2orthogonal STBC, spatial rate=1

(s – s’: codeword different matrix) effective channel is rendered orthogonal regardless of the channel realization Rx: simple

scalar ML detection

achieve full-diversitywithout wasting bandwidth achieve full-rate(two symbols over two time slots)

Extension to the case of more than 2 antennas?

For N=4: 4x4 orthogonal STBC, spatial rate=1 (full-rate)

1 2 3 4

2 1 4 3

3 4 1 2

4 3 2 1

For N=4, , : real-valued ' orthogonal,i

s s s s

s s s ss

s s s s

s s s s

S S S

V. Tarokh, H. Jafarkhani, A. R. Calderbank, “Space–Time Block Codes from Orthogonal Designs”, IEEE Trans on Info. The., July 1999, pp. 1456-1467

*

2 21 21 2 2x2*

2 1

' ' ' : orthogonal H s s s ss s

S S S S S I S S

1 2

1 2 2

( ') 4, / ' / for high / >>1,( '

Pr / ' / / / 4 ' / / 4 , 4

) s s o LN LN

LN

s s o o

i I N E N N E N

E N N E N N N N N

S, S S S

S S' S S S S

S,S

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ECSE610 MIMO - 31

orthogonal STBC with spatial rate2

does not exist for spatial rate=1,

exist for spatial rate 1/2

V. Tarokh, H. Jafarkhani, A. R. Calderbank, “Space–Time Block Codes from Orthogonal Designs”, IEEE Transactions onInformation Theory , Vol. 45, No. 5, July 1999, pp. 1456-1467

* * * *1 2 3 4 1 2 3 4

* * * *

2 1 4 3 2 1 4 3

* * * *

3 4 1 2 3 4 1 2

spatial rate=1/2, N=3

s s s s s s s s

s s s s s s s s

s s s s s s s s

S

* * *2 21 2 3 3

* *22 1 3

spatial rate=3/4, N=3

/ /

/

s s s s

s s sS * 23* * * *

2 23 3 1 1 2 2 1 1 2 2

/

/ / / 2 / 2

s

s s s s s s s s s s

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Orthogonal STBC Designs (cont.) For N = 8: 8x4 STBC, complex-value, spatial rate=1/2

It is not always possible to find full-rate, full-diversity STBC using (square)orthogonal designs.

For real constellation (PAM), full-rate, full-diversity designs exist for N = 2, 4, 8

For complex constellation (e.g. QAM, 8-PSK…), full-rate designs exist if and onlyif N = 2 (Alamouti scheme)

*

1

*

2

*

3

*

4

*

2

*

1

*

4

*

3

*

3

*

4

*

1

*

2

*

4

*

3

*

2

*

1

1234

2143

3412

4321

S

ssss

ssss

ssss

ssssssss

ssss

ssss

ssss

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ECSE610 MIMO - 33

Space-Time Trellis Codes (STTC)

Trellis codes applied to MIMO

A basic trellis structure determining the coded symbols to be transmittedfrom different antennas

Memorized from one block to another

Coding advantage vs. Decoding complexity

S ll C d

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Space-Time Trellis Codes(STTC) design: 2b/s/Hz, with4PSK and N=2:

INPUT SIGNAL: 0 1 2 3

OUTPUTSIGNALS:

s0as0b s1as1b s2as2b s3as3b

4-STATE

8-STATE

16-STATE

32-STATEV. Tarokh, N. Seshadri, A. R. Calderbank, “Space–Time Codes for High Data Rate Wireless Communication: Performance

Criterion and Code Construction”, IEEE Transactions on Information Theory, Vol. 44, No. 2, March 1998, pp. 744-765

b/ / h d f

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STTC 2b/s/Hz, with 4PSK and N=2: performance

Performance: simulation results

Frame of 130 transmissions out of each transmit antenna

V. Tarokh, N. Seshadri, A. R. Calderbank, “Space–Time Codes for High Data Rate Wireless Communication: PerformanceCriterion and Code Construction”, IEEE Transactions on Information Theory , Vol. 44, No. 2, March 1998, pp. 744-765

L=2L=1

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DELAY DIVERSITY CODING: L=1, N=2

00,01,02,03,04,05,06,07

10,11,12,13,14,15,16,17

20,21,22,23,24,25,26,27

30,31,32,33,34,35,36,37

40,41,42,43,44,45,46,47

50,51,52,53,54,55,56,57

60,61,62,63,64,65,66,67

70,71,72,73,74,75,76,77

1 2 3

1 2 3

... 0

0 ...

J

J

s s s s

s s s s

S

STTCrepresentation:e.g., 8PSK

Transmitting the symbol stream over 1st antenna

and its one-symbol delayed version over the 2nd

antenna.

[S-S’] has rank 2 diversity order of 2L=2

Rx: y=GS+w, G=[g1,g2], ML decoding: find S for min y-GS2:complex

y=[y1,y2,…], y1=g1s1+w1, yi+1=g2si+g1si+1+wi+1

Channel model:

ZF-DFE (Zero-Forcing Decision-Feedback Equalizer) using

Symbol ML decoding and iterative canceling:

select s1 for miny1-g1s12: SLICER (symbol MLD) using previously detected si’ to compute zi+1=yi+1-g2si (DFE)

and find si+1 for minzi+1-g1si2

(SLICER) If g1>g2: minimum-phase channel: good. Otherwise, error

propagation, use all-pass filter (g2+g1D-1)/ (g1+g2D

-1)

Case:g1>g2

Case:g1

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ST coding for spatial multiplexing

objective of spatial multiplexing: to maximize transmission rate.

MIMO channels offer a linear increase in capacity for no additional power orbandwidth expenditure, Rate (SNR) = r log SNR.

The gain, r, referred to as spatial multiplexing gain (spatial rate), is realized bytransmitting independent data signals from the individual antennas, rN. Under conducive channel conditions, such as rich scattering, the receiver can

separate the different streams, yielding a linear increase in capacity.

Rx performs ML decoding of each Rx signal (codeword)

J=1, Nx1 codeword difference matrix S-S’, [S-S’] [S-S’]H: rank 1 no coding gain, diversity order of L

1

2

only one ( ') : non-zero eigenvalue of / ' ' for i= ( ') 1

for high / >>1, Pr ( ') / / 4 ' / 4

H

s

L L L

s o s o o

E N I

E N E N N N

S, S S S S S S, S

S S' S,S S S

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Vertical Bell Labs Layered Space-Time (V-BLAST) Coding

Encoding with symbol set

A={ai, i=1,2,…, 2q} and block of qkinfo bits.

Transmitter: Split data into MT=N

streams

Rx: Y=GS+W

ML decoding: select S corresponding to

the min Y-GS 2: very complex

1:NDEMUX

TEMPORALENCODER INTER-LEAVER SYMBOLMAPPER

qk infobits

TEMPORALENCODER

INTER-LEAVER

SYMBOLMAPPER

1

N

qk/N info bits

qk/N info bits

n/N coded symbols

n/N coded symbols

N independentrow codewords

If G can be GS-decomposed,G=QB, then

Z=QHY=BS+QHW

Gram-Schmidt (GS) decomposition: An mxnmatrix A

with rank n can be factored uniquely as A =QB where Q

is an mxnmatrix whose columns are unit length andorthogonal and span the range of A, andQHQ=I.

Layered decoding:

1st row: z1=b11s1+n1 can be independently decoded for s1’

2nd row: removings1(using the previously decoded s

1’) from z

2 z2-b21s1=b22s2+n2 can be decoded for s2’ , … ith row: removings1,s2,…,si-1 (using their previously decoded values s1’,s2’,…,s i-1’) from zi zi-(bi1s1+ bi2s2+…+bi(i-1)si-1)=biis i+n i can be decoded for s i’ A small value of bii could yield unreliable decoded si’, which will create a chain effect in decoding the

subsequent rows.

1 2

11

21 22

1 2

( 1)1 ( 1)2 ( 1)( 1)

1 2

, , ,

0 0 0

0 0

, , ,0

n

n

n n n n

n n nn

b

b b

b b b

b b b

A a a a QB

q q q

G Foschini “Layered space time architecture for wireless communication in a

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ECSE610 MIMO - 39

diagonal D-BLAST In V-BLAST, there is no coding across these sub-channels: outage therefore occurs whenever one of these

sub-channels is in a deep fade and cannot support the rate of the stream using that sub-channel. D-BLAST: achievable diversity order of ML for optimum temporal coding and rotation with infinite block-size

Gaussian code books,

coding gain from temporal code; achievable array gain of L

1:NDEMUX

TEMPORALENCODER INTER-LEAVER SYMBOLMAPPER

qk infobits

TEMPORALENCODER

INTER-LEAVER

SYMBOLMAPPER

Out 1

Out N

qk/N info bits

qk/N info bits

s11

,s12

,…,s1J R

OT A TOR

sN1,sN2,…,sNJ

Out 1: s11 s21 s31 s41 s15 s25 s35 s45 0 0 0

Out 2: 0 s12 s22 s32 s42 s16 s26 s36 s46 0 0

Out 3: 0 0 s13 s23 s33 s43 s17 s27 s37 s47 0

Out 4: 0 0 0 s14 s24 s34 s44 s18 s28 s38 s48

The coded stream from Tx#nis

subdivided into J blocks of

several symbols snj.

The rotator fills the ST codeword

from N coded streams in a block-diagonal fashion as shown in the

following example with J=8, N=4.

ROTATOR OUTPUTS with N=4:

Rx: Diagonal-Layer decoding: 2 ways:

• column by column tentative decisions and cancellation, and then decoding the complete stream, or

• (more complex, better performance) detecting the diagonally layered stream, and applying cancellation of

previously detected streams

By coding across the sub-channels, D-BLAST can average over the randomness of the individual sub-channels

and get better outage performance. The D-BLAST scheme suffers from a rate loss because in the initialization

phase some of the antennas have to be kept silent.

G. Foschini, Layered space-time architecture for wireless communication in a

fading environment when using multi-element antennas”,Bell Labs Tech. J.,

pp.41-59, 1996

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ECSE610 MIMO - 40

horizontal encoding

achievable diversity order >L ,

coding gain from temporal code; achievable array gain of L

can reach optimality since each info bit can potentially be spread across allantennas, but requires complex joint decoding of the sub-streams

1:NDEMUX

TEMPORALENCODER

INTER-LEAVER

SYMBOLMAPPER

qk infobits

1

N

STC for frequency selective fading:

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ECSE610 MIMO - 41

STC for frequency-selective fading:

gln(t ): equivalent baseband impulse response of the channel from Tx antenna n to Rxantenna l: gln(t )=0 for t DTs

01

( ) : signal transmitted from Tx antenna ,

signal received at Rx antenna : ( ) ( ) ( ) ( ), 1,2,...,s

n

DT

n

ln l

N

l n

x t n

l y t g x t d w t l L

Discrete-time representation: 1 ( ) ( ) ( ) ( ), 1, 2,...,

l ln n

N

ln

y t i t i w t l L

g x

( ) ( ), ( 1 ), , ( ) T

n n n ncolumn vector t x t D x t D x t x

( ), ( 1), , (0)ln ln ln lng D g D g g : symbol-spaced samples of the equivalent baseband impulse

response of the channel from Tx antenna n to Rx antenna l: assumed to be zero-mean

complex circular Gaussian. At time t, for a block of T contiguous symbol intervals

11 1 1

1

( ), ( 1), , ( 1) , ,

( ) ( 1) ( 1)

: signal block from Tx antenna( 1) ( ) ( 2)

( ) ( 1) ( 1)

N

L LN N

n n n

n

n n n

n n n

t t t T

x t D x t D x t D T

x t x t x t T

x t x t x t T

g g S

Y y y y GS W G S

g g S

S

n

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ECSE610 MIMO - 43

Delay diversity codes for frequency-selective fading:standard delay diversity code: Transmitting the symbol stream over 1st antenna, its 1-symboldelayed version over the 2nd antenna, and its 2-symbol delayed version over the 3rd antenna, and soon.e.g., N=2, L=1, D=1, T=4

1 1 1 1 1 1 2 3 4

2 1 2 2 2 2 2 1 2 3 4

antenna 1: ( ), ( 1), ( 2), ( 3), ( 4), , , , ,0,

antenna 2: ( ) ( 1) ( ), ( 1), ( 2), ( 3), ( 4), 0, , , , ,

Note: addit

0 0

0 0

ional

x t x t x t x t x t s s s s

x t d x t d x t x t x t x t x t s s s s

1 2 3 4

1 2 3 41

2 1 2 3 4

1 2 3 4

at the end for guard of D=10 0

0: row 1= row 4, rank of 3 (low)

0 0

0

0

00

s s s s

s s s s

s s s s

s s s s

SS

S

generalized delay diversity code: Transmitting the symbol stream over 1st antenna, its (D+1)-

symbol delayed version over the 2nd antenna, and its 2(D+1)-symbol delayed version over the 3rd

antenna, and so on.e.g., N=2, L=1, D=1

1 1 1 1 1 1 1 2 3 4

2 2 2 2 2 1 1 2 3 4

antenna 1: ( ), ( 1), ( 2), ( 3), ( 4), ( 5), , , , , 0,0,

antenna 2: ( ), ( 1), ( 2), ( 3), ( 4), ( 5), 0,0, , , , ,

Note: additional 0 at the end for guard of 1

0 0

0 0

s t s t s t s t s t s t s s s s

s t s t s t s t s t s t s s s s

D

S

1 2 3 4

1 2 3 41

2 1 2 3 4

1 2 3 4

0 0 0

0 0 0: full rank of 4=( 1)

0 00

0 00

s s s s

s s s s D LN

s s s s

s s s s

S

S

MIMO-OFDM:SFBC

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ECSE610 MIMO - 44

MIMO OFDM:SFBCMIMO-OFDM: N Tx, L Rx, with F tones

For tone f, f=1,2,…,F

In frequency domain, ( ) ( ) ( ) ( ) f f f f y H s v (1) (1) (1)

(1) 0 0(2) (2) (2)

x1 , x1 x1 ZMCSCG x 0 0

0 0 ( )( ) ( ) ( )

LF NF LF LF NF

F F F F

y s wH

y s wy Hs w s w H

Hy s w

spatial diversity coding: e.g., Alamouti code with tone index, N=2, L=1

Tx:

* *11 2 2

* *22 1 1

(1, ) (1, 1)( ) , ( 1)

(2, ) (2, 1)

ss f s f s s s f f

ss f s f s s s

S S

then IFFT+CPI for 2 Tx

Rx: removes CP and FFT, and processes

1 2 1

* ** * * * *

2 1 2

* *21 2 1 1 2

* * *2 1 2 2

( ) ( )( ) ( ) ( ) ( ) ( )

( 1) ( 1)( 1) ( 1) ( 1) ( 1) ( 1)

( ) ( ) ( ) ( 1)( )

( 1) ( 1) (

f

H

f

H f H f s y f f f w f w f

H f H f s y f f f w f w f

H f H f s H w f H w f f

H f H f s H w

H sy

H s

y H*

1

if ( ) ( 1)) ( 1)

f f f H w f

H H

the noise components are iid ZMCSCG with variance of2( ) o f N H

Beamforming (BF)

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ECSE610 MIMO - 45

g ( ) BF techniques are used to create a certain required antenna directive pattern

to give the required performance under the given conditions, e.g., usingphased-array systems.

Switched BF requires the overall system to determine the direction of arrival ofthe incoming signal and then switch in the most appropriate beam:

Finite number of fixed predefined patterns

the fixed beam is unlikely to exactly match the required direction. Adaptive BF:

large number of patterns adjusted to the scenario in real time.

are able to direct the beam in the exact direction needed, and also move

the beam in real time - this is a particular advantage for moving systems -mobile telecommunications. The cost is the considerable extra complexity

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MIMO in WiMAX

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MIMO in LTE

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Downlink

- Single antenna transmission, no MIMO

- Transmit diversity

- Open-loop spatial multiplexing, no UE feedback required

- Closed-loop spatial multiplexing, UE feedback required

- Multi-user MIMO (more than one user is assigned to the same resource block)

- N Beamforming

Uplink - In order to keep the complexity low at the UE end, MU-MIMO is used in the

uplink. To do this, multiple UEs, each with only one Tx antenna, use the same

channel.

SU-MIMO vs. MU-MIMO

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ECSE610 MIMO - 50

SU MIMO vs. MU MIMO

Single - user MIMO

The data rate is to increased for a single user

In the 3GPP release 8 (LTE) standard, allows

us to achieve 300Mbps for DL and 75Mbps

for UL

Using the Spatial Multiplexing to combine the

bandwidth of each antennas.

Multi- user MIMO: the individual streams

are assigned to various users.

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4G Systems: SU-MIMO vs. MU-MIMO

3GPP LTE: General Structure for Downlink

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Physical Channels

Closed Loop MIMO

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p

Precoded Spatial Multiplexing: Rank 1

Closed Loop MIMO

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Precoded Spatial Multiplexing: Rank 2

Far-field Analysis of Linear Array

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ECSE610 MIMO - 55

N321t EEEEE ...

tE

00

3

30

2

2

1

1 ...321

EEEEE 0t N

jkr

N

jkr jkr jkr

r

ew

r

ew

r

ew

r

ew

N

In far field region, r i >> array dimension:

svariationamplitudefor...

svariation phasefor

cos)1(

cos2

cos

...

321

3

2

1

321

r r r r r

d N r r

d r r

d r r

r r

N

N

N

(AF)FactorArray

cos1cos2

3

cos

21

point)referenceatelement(single

... d N jk

N

d jk jkd jkr

ewewewwr

e

E

0t EE

Factor Array point)referenceatelement(singletotal EE

Reference: C.A. Balanis, Antenna Theory Analysis and Design, 3rd edition, John Wiley & Son, Inc, 2005

Two-Element Array

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ECSE610 MIMO - 56

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

1

1

2

1w

1

1

2

1w

jw

1

2

1

j

w1

2

12 :spacingElement d

3D Radiation Patterns of 2 Element Array

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ECSE610 MIMO - 57

11

21w 1

1

21w

jw

1

2

1

j

w1

2

1

2D Radiation Patterns of 2 Element ArrayXZ Cut, = 0 degree

XZ Cut, = 0 degree

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ECSE610 MIMO - 58

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

-20

-15

-10

-5

0

30

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

11

21w

11

21w

jw

1

2

1

j

w1

2

1

Four-Element Array, W0

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ECSE610 MIMO - 59

2 :spacingElement d

5.05.05.05.05.05.05.05.0

5.05.05.05.0

5.05.05.05.0

0w

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree


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ECSE610 MIMO - 60

5.0

5.0

5.0

5.0

00w

5.0

5.0

5.0

5.0

01w

5.0

5.0

5.0

5.0

02w

5.0

5.0

5.0

5.0

03w

2D Patterns of 4 Element Array, W0XZ Cut = 0 degree

XZ Cut, = 0 degree

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ECSE610 MIMO - 61

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

, g

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

-20

-15

-10

-5

0

30

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

5.0

5.0

5.0

5.0

00w

5.0

5.0

5.0

5.0

01w

5.0

5.0

5.0

5.0

02w

5.0

5.05.0

5.0

03w

Four Element Array W1

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ECSE610 MIMO - 62

Four-Element Array, W1

2 :spacingElement

d

5.05.05.05.0

5.05.05.05.0

5.05.05.05.05.05.05.05.0

1

j j

j j

j j j j

w

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree


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ECSE610 MIMO - 63

5.0

5.0

5.0

5.0

10

j

jw

5.0

5.0

5.0

5.0

11 j

j

w

5.0

5.0

5.0

5.0

12

j

jw

5.0

5.05.0

5.0

13 j

j

w

2D Patterns of 4 Element Array, W1XZ Cut, = 0 degree

XZ Cut, = 0 degree

8/20/2019 F_MIMO

64/64

ECSE610 MIMO - 64

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

-20

-15

-10

-5

0

30

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

-20

-15

-10

-5

0

30

150

60

120

90 90

120

60

150

30

180

0

XZ Cut, = 0 degree

-20

-15

-10

-5

030

150

60

120

90 90

120

60

150

30

180

0

5.0

5.0

5.05.0

10

j

jw

5.0

5.0

5.05.0

11 j

j

w

5.0

5.0

5.0

5.0

12

j

jw

5.0

5.0

5.0

5.0

13 j

j

w

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