MIMO Wireless Communications · MIMO Wireless Communications • Single Input Single Output (SISO)...

MIMO Wireless Communications:

An IntroductionAn Introduction

Dr. Rakhesh Singh Kshetrimayum

MIMO Wireless Communications

• Single Input Single Output (SISO)

Tx Rx

• What will happen if Tx and Rx employs multiple antennas?

Ant 1 Ant 1

Fig. 1 SISO system


• Multiple Input Multiple Output (MIMO)?Ant 1

Ant 1

Ant 2

. .

Tx Rx

Ant 2

Ant NT

Ant NR

.

.

.

.

.

.

Fig. 2 NT × NR MIMO system

channel


• What are advantages of MIMO?

• Capacity (C) for Single Input Single Output (SISO) system

( )SNRBWC += 1log2

• Data rate increase when

– Bandwidth (BW) and

– Signal to noise (SNR) power increase

Inherent problems:

• BW is precious, almost and always fixed for different applications

( )SNRBWC += 1log2


• Signal power increase

– battery lifetime decrease

– creates higher interference

– needs expensive RF amplifier – needs expensive RF amplifier

• In MIMO, spectral efficiency increase

– Without increasing BW and

– Signal power


• But, how?

• Basically two gains for MIMO systems:

– (i) MUX/rate gain

– Capacity behavior ( )SNRrR log≈– Capacity behavior

( )( )SNR

SNRR

SNRr

2log

lim

∞→=

( )SNRrR 2log≈

MIMO Wireless Communications• For instance

• How much is the achievable rate gain?Ant 1

Ant 1

Ant 2

Tx RxAnt 2

Ant 3

Ant 3

Fig. 3 Achievable rate gain with 3 × 3 MIMO system

P/SS/P

r=3

( )SNRR 2log3≈


(ii) Diversity gain

• Error probability behavior (minimize it)

• Slope of symbol error rate/ bit error rate (SER/BER curve

increases)

( ) d

e SNRSNRP−≈

increases)

• Behavior of error probability w.r.t. average transmit

power in log-log scale

• for asymptotically high power

( ){ }( )

2

2

loglim

log

eP SNR

dSNR SNR

= −→ ∞

MIMO Wireless Communications• How much is diversity gain?

Ant 1Ant 1

Ant 2

( ) 9−≈ SNRSNRPe

• For SISO (Rayleigh fading case), d=1,

Tx Rx

Ant 2

Ant 3Ant 3

Fig. 4 Diversity gain of 3 × 3 MIMO system

d=9



• How much is rate and diversity gain?Ant 1

Ant 1

Ant 2

r=1

Tx Rx

Ant 2

Ant 3

Ant 3

Fig. 5 Rate and diversity gain of 3 ×

3 MIMO system (Case I)

d=4

( )SNRR 2log≈



• How much is rate and diversity gain?Ant 1

Ant 1

Ant 2r=2

Tx Rx

Ant 2

Ant 3

Ant 3

Fig. 6 Rate and diversity gain of

3 × 3 MIMO system (Case II)

d=1

( )SNRR 2log2≈



• Need for a proper design for MIMO systems

• As r decrease, d increase

• As r increase, d decrease• As r increase, d decrease

• Diversity-multiplexing trade-off [1]

( )( ) ( )RTRTopt NNrrNrNd ,min0, ≤≤−−=


• Narrowband MIMO System ModelAnt 1

Ant 1

h12

h11

x1

y1

Tx Rx

Ant 2

Ant 2

Fig. 7 Narrowband MIMO system model for 2 × 2

MIMO system

h21

h22x2 y2


12121111 nxhxhy ++=At receiving antenna 1 (receives mixture of signals 1 &2)

At receiving antenna 2 (receives mixture of signals 1 &2,

a major problem in MIMO detection)a major problem in MIMO detection)

22221212 nxhxhy ++=

In matrix form, y=Hx+n

+

=

2

1

2

1

2221

1211

2

1

n

n

x

x

hh

hh

y

y


• For NT × NR MIMO system

+

=

T

N

N

n

n

x

x

hhh

hhh

y

y

L

L

2

1

2

1

22221

11211

2

1

+

=

RTTRRR

T

R NNNNNN

N

N n

n

x

x

hhh

hhh

y

y

MM

L

MOOM

L

M

22

21

222212

Channel matrixReceived signal

vector

Transmitted

signal vector

Noise

signal

vector


• In order to have performance analysis of

MIMO channel

– we need Analytical MIMO channel models

• Analytical MIMO channel model• Analytical MIMO channel model

– i.i.d. MIMO channel model

– Separately correlated MIMO channel model

– Uncorrelated keyhole MIMO channel model


• i.i.d. MIMO channel model (each element of

the channel matrix H is complex random

variable)

==+= ,2,1;,2,1; TR

imag

ij

real

ijij NjNijhhh LL

( ) ( )

×

−

×

=⇒

2

12

exp

2

12

1

2

1,0~

2/

/

/

imagreal

ijimagreal

ij

imagreal

ij

TRijijij

hhp

Nh

π


( )

Complex Gaussian distribution

�Joint distribution of real and imaginary part

� They are assumed independent

�PDF can be multiplied

( )

( ) ( )( ) ( )( )

( ) ( ) ( )( )( )

( ) ( )2

22

22

exp1

exp1

exp1

exp1

1,0~

ijij

imag

ij

real

ijij

imag

ij

real

ijij

Cij

hhp

hhhp

hhhp

Nh

−=⇒

+−=⇒

−−=⇒

π

π

ππ


( ) { }, ,

2 2

, ,

, 1, 1

1 1exp exp

R T R T

R T

N N N N

i j i jN Ni ji j

p h hπ π ==

= − = − ∑∏H H

( ) ( )( )1 H

For i.i.d. case,

( ) ( )( )1exp

R T

H

N Np Trace

π= −H H HH

“etr” is the abbreviation for “exponential trace”.

( ) { }HNNetrp TR HHHH −= −π


( )*

11 12 1 11 21 1

21 22 2 12 22 2

1 2 1 2

T R

T R

H

N N

N N

N N N N N N N N

trace

h h h h h h

h h h h h htrace

h h h h h h

=

HH

L L

L L

M O O M M O O M

L L1 2 1 2

22 2

11 12 1

22 2

21 22 2

2

1

R R R T T T R T

T

T

R

N N N N N N N N

N

N

N N

h h h h h h

h h h

h h htrace

h h

+ + +

+ + +=

+

L L

L L L L

L L L L

M O O M

L L L2 2

2

,2

,

, 1

R R T

R T

N N

N N

i j

i j

h

h

=

+ +

= ∑

L


• MIMO channel parallel decomposition

• To see how much is the capacity increase in MIMO systems

yx~ xy~

Fig. 8 Transmit precoding and receiver shaping (needs CSIR and CSIT)

nHxy +=

y

xVx ~=

x~ x

yUyH=~

y


• From SVD of the channel matrix H, we have,

( ) ( )nxVΣUUnHxUyUy +=+== HHHH~

HVUH ∑=

( ) ( )nxVΣUUnHxUyUy +=+==

( )nxVVΣUUy +=⇒ ~~ HH

nxΣy ~~~ +=⇒


• Component-wise

n

n

x

x

y

y~

~

~

~

00000

00000~

~

2

1

2

1

2

1

2

1

σ

σ

+

=

R

H

T

HH

R

H

N

R

N

RR

N

R

n

n

n

x

x

x

y

y

y

~

~

~

~

000000

00000

00000

00000

00000

~

~

2222

M

M

M

M

O

O

M

M

σ

σ


• Parallel RH Gaussian channels

nxy

nxy~~~

~~~

2222

1111

+=

+=

MOM

σ

σ

• Capacity?

• increases RH fold

RR

HHHH

NN

RRRR

ny

nxy

~~

~~~

=

+=

MOM

MOM

σ


• Rate Gain √

• Diversity gain ×

Space-time codes

Implemented at the transmitter side, needs CSIR Implemented at the transmitter side, needs CSIR

and block fading

• Why space-time codes?

( ) dGc

eSG

cP ≈


• where S is the SNR

• c is a scaling constant specific to the

– modulation employed and

– the nature of the channel– the nature of the channel

• Gc≥ 1 denotes the coding gain and

• Gd is the diversity order of the system


• Diversity gain/order determines the

– negative slope of an error rate curve plotted vs

SNR on a log-log scale

( )

• Space-time coded scheme with diversity order

Gd has

– an error probability at high SNR behaving as

( )2 2 2 2log log log loge d c d

P c G G G S≈ − −

( ) dG

eP S−

≈


If there is some coding gain, then

– average probability of error will be of the form

• If there were no array or power gain then

– the probability of error expression will be of the

form

( )

1d

e G

c

PG S

≈

( )

1d

e G

c

PG S

≈


• The coding gain determines the

– horizontal shift of uncoded system error rate

curve to the

• space time coded error rate curve • space time coded error rate curve

• plotted on a log-log scale obtained for the same

diversity order

Fig. 9 Illustration of diversity and coding gains

• BER curves are usually waterfall type but

– we have shown straight lines for illustration purpose only


• Alamouti Space-Time Codes

Fig. 10 A block diagram of Alamouti space-

time encoder

( )1 2 1 2, 0H

= =s s s s

r

r%

sFig. 11 Alamouti’s space-time decoding


• Received signal vector

• Received signals in two time intervals

= +r Hs n

• the output of the combiner

1 1 2 1 1

* *

2 2 1 2 2

r s s h n

r s s h n

= + −

−

+=

−=

*211

*2

*221

*1

*2

1

1*2

2*1

2

1

~

~

rhrh

rhrh

r

r

hh

hh

r

r

H= +r H Hs n% %


• For 2×1 MIMO system, two signals are picked up by

– the receiving antenna at the receiver

( ) ( ) 22

2

2

2

1211

2

2

2

11~~;~~ nshhrnshhr ++=++=

– the receiving antenna at the receiver

• Two signals are completely decoupled after the combining operation

– for Alamouti Space Time Codes

• Simplifies greatly the detection strategy in comparison

– to conventional MIMO detection


• Applying MLD

• For 0≤t≤T, we have

( )2 2args r h h s= − +% { }M

• For T≤t≤2T, we have,

( )2 2

1 1 1 2

argˆ

min{ }m

s r h h sm

= − +%

( )2 2

2 2 1 2

argˆ

min{ }ms r h h s

m= − +%

{ }M

kkm ss1=∈


MIMO detection

• Detect signals jointly

– since many signals are transmitted from the transmitter to the receiver

• Maximum likelihood (ML) detection outputs the • Maximum likelihood (ML) detection outputs the vector which

• minimizes the Euclidean distance between

– the received vector and

– all possible combinations of the transmitted symbol vectors 2min

argˆ Hsrx

s −=


• Consider 2 × 2 MIMO system

=

=

=

=

2

1

2

1

2221

1211

2

1;;;

n

n

s

s

hh

hh

r

rnsHr

2222212

+

=

⇒

+=

2

1

2

1

2221

1211

2

1

n

n

s

s

hh

hh

r

r

nHsr


• At the detector,

– we want to detect s1 and s2 at time t,

– but there exist interference between these two

signalssignals

• Assume that sk are modulated in M-ary

constellation i.e., sk є {s1,s2,…,sM}

2222121212121111 ; nshshrnshshr ++=++=


• We need to find the minimum metric of the

Euclidean distance

{ }( ) ( )

+−++−

∈

2

22212

2

12111,,2,1,

min jijiMji

shshrshshrL

• For instance, 16-QAM, (s1,s2) are (1 of 16

symbols, 1 of 16 symbols) implies 16×16 pairs

{ }

{ }( ) ( ) ( ) ( )

+−++++−++

∈

∈

2

22212222121

2

12111212111,,2,1,

,,2,1,

min jijiMji

Mji

shshnshshshshnshshL

L


• Metric calculations of 256 are required

• For 3×3 MIMO system,

– NT=NR=3, metric calculations of 163=4096 are

requiredrequired

• For 5×5 MIMO system,

– NT=NR=5, metric calculations of 165 =10,48,576

are required

• which is obviously impractical


• ML detectors are optimal but impractical

• Low complexity suboptimal detectors like

– Zero forcing (ZF) and

– Minimum mean square error (MMSE) – Minimum mean square error (MMSE)

• are preferable

• In linear detector,

• a linear preprocessor (W) is first applied

– to the received signal vector

rWsH=ˆ


• For instance in ZF

• H+ is the Moore Penrose pseudo-inverse of H

nHsrHrWs++ +=== H

ZFˆ

• H is the Moore Penrose pseudo-inverse of H

• Performance is not so good

• Consider a 2×2 MIMO system with s є S={-3,-

1,1,3}

( ) HHHZF HHHHW

1−+ ==


• The channel matrix is given by H=[2 0.5; 1 2;]

• Suppose the received signal vector is r=[1 0.9]T

• The ZF detector’s output is given by

• Thus the hard decision of for s becomes [1 1]T

– which is different from the ML decision of [1 -1]T

• ZF less complex than MLD, but poor performance

[ ]T2.05.0ˆ == +rHs


Antenna selection [2]

• Reduce hardware complexity

• A subset of total available antennas selected

– Based on capacity maximization – Based on capacity maximization

– Based on maximum received SNR

– Done at transmitter, at receiver or both

MIMO Wireless Communications• Transmit Antenna Selection/Maximal Ratio

Combining (TAS/MRC) [3]Ant 1

Ant 1

Ant 2Select

antenna 2,

then use

Tx Rx

Ant 2

Ant 3Ant 3

Fig. 12 3 × 3 TAS/MRC MIMO system

then use

only this 3

path


• How to select antenna?

• By maximizing the received SNR

• For Rayleigh fading channel,

– C are i.i.d. Chi square distributed with the probability

2

,

1

arg max{ | | }rN

i i j

j

I C h=

= =∑

– Ci are i.i.d. Chi square distributed with the probability

density function (PDF) and cumulative distribution

function (CDF) as

1

1

0

1( ) , 0

( 1) !

( ) 1 , 0!

r

r

N x

r

N ix

i

p x x e xN

xP x e x

i

− −

−−

=

= ≥−

= − ≥∑


• Order statistics

• PDF of maximum received SNR C(Nt) such that

C(1)≤ C(2) ≤…≤ C(Nt) can be given as [4]

1N −= 1

( )

11 1

0

( ) [ ( )] ( )

(1 ) ·( 1)! !

t

t

r

t r

N

N t

N iN Nx xt

it

p x N P x p x

N xe x e

N i

−

−− −− −

=

=

= −−

∑


• Assume binary phase shift keying (BPSK) for

TAS/MRC MIMO system

• Instantaneous SNR at the MRC receiver

2

• The average BER

2

0

( 2 ) ( )bb b b

P Q p dγγ γ γ∞

= ∫

Conditional error probability (CEP) for BPSK

( )t

r

t N

N

j

jNb Ch γγγ == ∑=

2

1

,


• Thus, closed form expression for BER Rayleigh fading is

( )

( )

( )( )( )

( )11

2

0 0

11

, 1 ! 11 ! 12 1

rrt

r

k tN t

k NN

tt r rN

k tr

N

kNP a N k N t

N kk

γ

γ

+−−

= =

− − = × + − × − − + ++

∑ ∑

• where, is the coefficient of in the expansion of

•

1( )

0

12 1

1

r

jN t

rj

j

N t j

j k

γ

γ

+ −−

=

+ − + × × + + + ∑

( , )t r

a N k 2tz

( )

2

1

0

2 1

!

r

ki

N

i

z

k

i

−

=

+

∑


• η − µ distribution proposed by M. Yacoub in 2000 (can analyze non LOS propagation)

• Can model many distributions

• Some of the special cases of η − µ distribution are

– Rayleigh distribution for η → 1, µ = 0.5, – Rayleigh distribution for η → 1, µ = 0.5,

– one sided Gaussian distribution for η → 1, µ = 0.25,

– Nakagami-m distribution for η → 1, µ = m/2 and

– Hoyt distribution for η → q2, µ = 0.5

• BER analysis for TAS/MRC over η − µ fading channel can be carried out in the similar way


Fig. 13: BER performance of (2, 1; 1) TAS/MRC system over η

− µ fading channels for η = 1 with BPSK modulation


Spatial modulation

• Two information bearing units

– Transmit antenna index : estimated at receiver

– A symbol from signal constellation : transmitted from antenna corresponding to transmit antenna indexantenna corresponding to transmit antenna index

–Advantages

• Higher capacity

• Reduced hardware complexity

• Avoidance of transmit antenna synchronization

• How?


Fig. 14 SM system model [5]


• k bit information blocks

• Sub blocks of m bits and n bits

• n bits spatially modulated

• m bits modulated using digital modulation • m bits modulated using digital modulation

schemes

• m bits are transmitted physically, effectively similar to

transmitting k=m+n bits

• Restriction on the number of transmit antennas

• Integer exponent of 2


SM Receiver

• MRC diversity scheme

• Two step detection of transmitted symbol

• Antenna index estimation ((h )Hy is noise except for • Antenna index estimation ((hj)Hy is noise except for

actual transmit antenna)

• ML symbol detection, only one transmit antenna [6]

ˆ arg maxj

j =

H

j

2

j

h y

h

{ }*ˆ

2

ˆˆ Re2min

arg q

H

jqjq xxj

yhhx −=


• Assume two estimation processes are independent

– Transmit antenna index estimate

– Estimation of the transmit symbol– Estimation of the transmit symbol

• Notation

– Pa is probability that the antenna index estimation is incorrect

– Ps is probability is that the transmitted symbol estimation is incorrect


• Probability of correct estimation

• Probability of error

( )( )sac PPP −−= 11

• Assume QAM

• CEP

( )( ) sasasace PPPPPPPP −+=−−−=−= 1111

( ) ( )2( )eP aQ b cQ bγ γ γ= −


• Table 2 MODULATION PARAMETERS FOR

VARIOUS MODULATION SCHEMES [8]


• Q-function as sum of exponentials [7]

• CEP becomes

( )22 2

321 1

12 4

xx

Q x e e−−

≈ +

• CEP becomes

• where a=2, b=1, and c=1 for 4-QAM

2 4 7

3 3 62( )12 4 144 16 24

,b b bb

b

e

a a c c cP e e e e e

γ γ γγγγ

− − −−−≅ + − − −


• the integral of fits the definition of MGF [9] of

the received SNR, MGF(γ).

( ) ( ) ( )dxxpxMGF X∫∞

−= exp γγ

• Probability of error in symbol detection

( )1 1 2 1 1 1 4 1 7

6 2 2 3 4 36 4 3 6 6s

MGF MGF MGFP MGF MGFγ γ γ γ

γ

+ − + +

≈

∫0


• Probability of error in transmit antenna index

estimation

• where

( )

+

≈=

34

1

412

1 γγγ MGFMGFQP effa

• where

• MGF for Rice fading case

2

ˆ2

eff

γγ = −j j

h h

( ) 1

1 1exp

Rice

K KMGF

K K

γγ

γ γ

+= −

+ + + +


• Fig. 15 SER vs SNR (dB) for 2×2 SM MIMO system considering Rician fading


• SM with antenna selection [10]

• Some of the selected antennas for SM may be

completely down

• SM systems combined with antenna selection• SM systems combined with antenna selection

Antennas selected at transmitter

SM applied over selected antennas with

better links

• CSI assumed to be available at transmitter


• SM with antenna selectionAnt 1

Ant 1

Ant 2Ant 2

Apply SM

on

selected

antennas

Tx Rx

Ant 4

Ant 4

Fig. 16 4 × 4 TAS SM MIMO system

Ant 3 Ant 3

antennas

which

give the

best links


• Order statistics

• Antenna Selection Parameter

2rN

∑

( )( )

( )

2

,

1

1

1

0

, 0

1 , 0!

r

r

j

r

Aj

N

j i j

i

N x

A

r

N ix

i

A h

x ef x x

N

xF x e x

i

=

− −

−−

=

=

= ≥Γ

= − ≥

∑

∑


• Ajs are arranged in ascending order

• S antennas corresponding to highest Ajs are

selected

• PDF of A(r) such that A ≤ A ≤…≤ A can be • PDF of A(r) such that A(1)≤ A(2) ≤…≤ A(Nt) can be

given as

• where r = Nt-S+1

( )( )

( )( ){ } ( ){ } ( )

111

, 1

t

j j jr

r N r

A A A A

t

f x F x F x f xB r N r

− −

= −− +


• The PDF of received SNR can be given as

• where and C (j,N ) is

( )( ) ( ) ( )

( ) ( ) ( )1

11

0 0

11 11 ,

1 , 1

t

tr

N i Mj x N i jN tr

t r

i r j tt r t

if x C j N x e

jN r N B i N iγ

−− − + ++ −

= = =

− = −

− + Γ − + ∑∑∑

• where and Ct(j,Nr) is

coefficient of xt in

( ) ( )1r tM N N i j= − − +

0

1

!

tr

N i jN i

i

x

i

− +

=

− ∑


• Outage probability

( ) ( )

( )

r

2 1, P

1 1t

R

out r

N

P R Aγγ

γ

−= <

= ∑

• where

( )( ) ( ) ( )

( ) ( )( )( )

( )( )

1

0 0

1 1,

1 , 1

, 111 ,

1

t

r

N

out

i rt r t

i Mj r th t

t r N tj t

t

P RN r N B i N i

N t N i jiC j N

j N i j

γ

γ γ

=

−

+= =

=− + Γ − +

+ − + +− × −

− + +

∑

∑ ∑

2 1R

thγ

γ

−=


Fig. 17: Outage Probability Vs. SNR curve for TAS SM MIMO

systems with antenna selection (2 bits/s/Hz)


• References:

1. L. Zheng and D. N. Tse, “Diversity and multiplexing: A fundamental

trade-off in multiple antenna channels,” IEEE Trans. Information

Theory, pp. 1073-96, May 2003.

2. S. Sanayei and A. Nosratinia, “Antenna selection in MIMO 2. S. Sanayei and A. Nosratinia, “Antenna selection in MIMO

systems,” IEEE Commun. Mag., vol. 42, no. 10, pp. 68–73, 2004.

3. Z. Chen, Z. Chi, Y. Li, and B. Vucetic, “Error performance of

maximal-ratio combining with transmit antenna selection in flat

Nakagami-m fading channels,” IEEE Trans. Wireless Commun., vol.

8, pp. 424–431, Jan 2009.

4. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. New York:

Wiley Interscience, 2003.


5. R. Mesleh, S. Engelken, S. Sinanovic, and H. Haas, “Analytical ser calculation of spatial modulation,” in Spread Spectrum Techniques and Applications, 2008. ISSSTA ’08. IEEE 10th International Symposium on, pp. 272 –276, Aug. 2008.

6. J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Spatial modulation: optimal detection and performance analysis,” Communications Letters, IEEE, vol. 12, pp. 545 –547, Aug. 2008.Letters, IEEE, vol. 12, pp. 545 –547, Aug. 2008.

7. M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 4, pp. 840–845, 2003.

8. Y. Chen and C. Tellambura, “Distribution functions of selection combiner output in equally correlated Rayleigh, Rician, and Nakagami-m fading channels,” IEEE Transactions on Communications, vol. 52, no. 11, pp. 1948–1956, 2004.


9. A. Papoulis and S. U. Pillai, Probability, Random Variables and

Stochastic Processes, Tata McGraw Hill, 2002.

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