MIMO Mobile Radio Systems · 10/12/07 Weber: MIMO Mobile Radio Systems 22 Optimization Task Question: How large is the total channel capacity C for limited total transmitted power

Institut fürNachrichtentechnik

MIMO Mobile Radio SystemsProf. Tobias WeberUniversity of RostockEmail: [email protected]


Weber: MIMO Mobile Radio Systems 210/12/07

Topics

1st lesson: introduction system modelling channel capacity

2nd lesson: channel models

3rd lesson: canonical system implementation signal processing with non cooperative inputs (BLAST) signal processing with non cooperative outputs diversity



1. Introduction



Application of MIMO: 802.11n

characteristics:• 20 MHz to 40 MHz bandwidth• OFDM• 2 to 4 antennas per station• spatial multiplexing• up to 540 Mbit/s



Terms

number of inputs

N = 1 N > 1

number of outputs

M = 1 SISO MISO

M > 1 SIMO (N, M) MIMO

spatial signal processingCzech:

mimo = except



Cable Binder

insufficient shielding cross couplings

R parallel wires (R, R) MIMO systemcapacity proportional to R (for fixed transmitted power per input)



Multiple Antenna System

increasing the capacity by increasing the bandwidth is expensive (UMTS in Germany: approx. 50 000 000 000 € for 120 MHz)

alternative: use spectrum more efficiently

RxTx

1

N

1

M



2. System Modelling



Quadrature Modulator

0j2

R 0 I 0

2Re e

2 cos 2 2 sin 2

f ta t u t

u t f t u t f t

Ru t

Iu t

02 cos 2 f t

02 sin 2 f t

a t

: bandpass signal

: equivalent lowpass signal

a t

u t

in-phase component quadrature component



Quadrature Demodulator

high frequency

0j2R 0 I 0

R 0 I 0

I 0 R 0

R

I

0 0

cos 4 sin 4

j co

2 2 cos 2 2 sin 2

2 cos 2 j 2 sin 2

s 4 j sin 4j

f ta t e u t f t u t f t

f t f t

u t f t u t f t

u

u t

t f t u t f tu t

a t

Ru t

Iu t

02 cos 2 f t

02 sin 2 f t



Sampling in Lowpass Domain

bandwidth B sampling interval T = 1/B:

time limited signal L samples:

signal vector:

the signal lies in a L-dimensional vector space spanned by the basis functions

sincl

l

tu t u lT lT

u

1

0sinc

L

ll

tu t u lT

T0 1Lu u u

sinc , 0 1ltb t l l LT

0b t

/t T-1 1

1



Time Shift

0

0 0

0 0

j2

-j2 j2

lowpass equivalent of -

-j2 j2

2Re e

2Re e e

2Re e e

f t

f f t

a t

f f t

a t u t

u t

u t

Small time shifts correspond to phase rotations byin equivalent lowpass domain.

0j2e f



Linear Time Invariant Systems

s t h t

e t

1

01

0with

W

w

W

w wl l ww

e lT T h wT s l w T

e h s h T h wT

matrix vector model:

00

010

11 0

11

12

0

0

W

WL

WL W

e he h

ss

h hh

s

e h

s

He

the channel convolution matrix H has Toeplitz structure



MIMO System

Tx1,Kh

1

KTx

1

KRx

1,1h

Rx Tx,K Kh

Rx ,1Kh

SISO subsystem:

MIMO system:

Rx Rx Tx Tx,k k k k e H s

Tx

Rx Rx Rx Tx Tx

1 1,1 1, 1

,1 ,

K

K K K K K

e H H s

e H H se H s

received vector channel matrix transmitted vector



MIMO System, Single Tap Channel

Tx1,Kh1

KTx

1

KRx

1,1h

Rx Tx,K Kh

Rx ,1KhSISO subsystem:

MIMO system:

Rx TxRx Tx,k kk ke h s

Tx

Rx Rx TxRx Tx

1,1 1,1 1

,1 ,

K

K K KK K

e h h s

e h h s

He s

channel matrix H is aKRx KTx matrix



White Multivariate Gaussian Noise

2

2

notation: 0,

0,

(circular symmetric complex normal)independent identically distributed, i.i.d.

mn

n E

2 20 0

2 2

2 21 1

2 2

*T2

0 0n

1 1

Re Im

2 2

Re Im

2 2

1

n 2

p p Re p Im

p Re p Im

1 1e e

1 1e e

1p e

M M

M M

n n

n n

M

n n

n n

n n

n

n



Multi Carrier Transmission, OFDM

h(t)

H(f)



3. Channel Capacity



SISO Channel Capacity

Shannon (1948): A Mathematical Theory of Communication

transmitted signal,power S

channel coefficient

receivedsignal

per channel use (Nyquist rate):

equivalent lowpass channel of bandwidth B

2

2bitld 1 ,

s Hzh S

C C

hGaussian noise,

power 2



Uncoupled (R, R) MIMO Channel

rece

ived

vec

tor

12h1

1S

R2hR

RStrans

mitt

ed v

ecto

r

total transmitted power:

1

R

rr

S S

total channel capacity:

2

21 1

ld 1R R

r rr

r r r

h SC C

power

power

power

power



Total Channel Capacity without Transmitter Side Channel State Information

All R parallel channels get the same transmitted power:

2 2

2 211ld 1 ld 1

r

RRr r

rr r r

SSR

h hS SCR R



Optimization Task

Question: How large is the total channel capacity C for limited total transmitted power S and how can it be achieved?

Idea: Allocate the total transmitted power S in a smart way to the R parallel channels!

2

21ld 1

Rr r

r r

h SC

subject to the constraint

1

R

rr

S S

maximize

method of Lagrangian multipliers



Optimization (1)

definition:

1

R

S

SS

maximize 2

21

f ld 1R

r r

r r

h S

S

subject to

1

g 0R

rr

S SS

R = 2; identical channelsf(S) = const., g(S) = const.

1 1

Lagrange: grad f grad g

f g

grad f , grad gf g

R R

S S

S S

S S 0

S S

S SS S

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1S

2S



Optimization (2)

2

2 22

22

f 1 1 1 1ln2 ln2

1

g1

r

rr rr rr

rr

r

hS h S S

h

S

S

S

with

21

121

2

2

1

11grad f grad g

ln21 1

RR

R

Sh

Sh

S S 0



Optimization (3)

2

W2

2

W 2

1 constln2

rr

r

rr

r

S Sh

S Sh

attention:2

W 2

W

1

0

max 0,

where is chosen such that

r

rr

r

R

rr

S

S Sh

S

S S



Waterfilling

22

h2

2

2S

32

h3

2

3 0S

42

h4

2

4S

12

h1

2

1S

R2

hR

2

RS

power

r

WS

2

W 2max 0, rr

r

S Sh

Holsinger (1964): Digital communication over fixed time-continous channels with memory - with special application to telephone channels

1

R

rr

S S



Total Channel Capacity with Waterfilling

2 22W

W 22 21 1ld 1 max 0, max 0, ld

R Rr rr

r rr rr

r

h h SC S

h

S

special case: all R channels used2 2 2

W W2 2 21 1

2 2w w

2 21 1

1

ld ld

R Rr r r

r Wr rr r r

RRr r

r rr r

SS S S S SR Rh h h

h S h SC

requires transmitter side channel state information



Challenge

t V s H e U*T r

Σt r

s should havesame power as t:

find unitary matrices U, V and a diagonal matrix such that

diagonal matrix

2m, m R E

m should be white:*T2

m*T (unitary)

R U UU U E

N M MN

*T *T Σ U H V H U Σ V

2n, (white)n R E

*T *T *T

*T (unitary)

t V Vt t tV V E



Singular Value Decomposition (1)

Singular Value Decomposition Theorem(Eckart & Young: 1939):For every M N matrix H there are two unitary matrices U and V, such that

is a M N diagonal matrix with nonnegative real diagonal elements.

*TΣ U H V



Singular Value Decomposition (2)


U: unitary M M matrix, columns are named left singular vectors and correspond to eigenvectors of *T *T*THH UΣΣ UV: unitary N N matrix, columns are named right singular vectors and correspond to eigenvectors of *T *T*TH H VΣ ΣV

: M N diagonal matrix, diagonal elements are named singular values and correspond to the square roots of the eigenvalues of or q HH*T H*TH

*T :HH Grammian of the row vectors*T :H H Grammian of the column vectors



Matrix Structures (1)*T *T Σ U H V H U Σ V

1

example: M = N

H

q

Q

VU*T

0

0

rank min ,R Q N M Hrank of the channel:

=

1 2 1 0R R Q



Matrix Structures (2)


example: M > N

H

VU*T

1

q

Q

0

0=



Matrix Structures (3)


example: M < N

H

VU*T

=

1

q

Q

0

0



MIMO Channel Capacity

without transmitter side channel state information(Foschini, 1996):all inputs with same power

2 211

*T2

*T *T *T *T *T2 2

*T *T2 2

ld 1 ld 1

ld det

ld det ld det

ld det det det ld det

RRr r

rr

S SCN NS

N

S SN N

S SN N

E

E V H UU HV V E H H V

V E H H V E H

*T

*T2ld det S

N

H

E HH



MIMO Channel Capacity

W2

1max 0, ld

Rr

r

SC

with transmitter side channel stateinformation (Telatar, 1995, 1999):

SW such that2

W1 1

max 0,R R

rr r r

S S S



Capacity of Stochastic Channels

instantaneous channel capacity:

complementary cumulative distribution function:

ergodic channel capacity:

outage channel capacity, outage probability :

W2

1inst

21

max 0, ld with TxCSI

ld 1 without TxCSI

Rr

r

Rr

r

S

CSN

erg instEC C

inst out outPr 1C C P

inst inst instPr p dC

C C C C

outP


0 1 2 3 4 50

0.2

0.4

0.6

0.8

1


Graphical Evaluation

parameters:• N = M = 1• S/2 = 4• E{|h|2} = 1• Rayleigh

Pr{

Cin

st>

C}

CCerg = 1,9415

1 - Pout = 0,9

Cou

t=

0,50

93



4. Channel Models



Deterministic SISO Channel Model

time domain not band limited:

band limited:

in general time dispersive, i.e., impulse response spread in time

frequency domain not band limited:

band limited:

in general frequency selective, i.e., frequency dependent transfer function

1

P

p pp

h t a t

1

sincP

p pp

h t a B B t

j2

1e p

Pf

pp

H f a

j2

1e rectp

Pf

pp

fH f aB

s t e tLOS

NLOS LOS: line of sight,NLOS: non line of sight



Single Tap Channel

time domain single tap channel:

impulse response:

frequency domain flat fading channel:

transfer function:'

1 für alle , 'p p p pB

constH f

sinch t h B B t - j2e rectf fH f hB

1

P

pp

h a



Stochastic Model for Non Frequency Selective NLOS Channels

neglect access delay

if the ap are independent it follows for P→ (central limit theorem):

1

P

pp

h a

2h0,h

2

2h

2h

is Rayleigh distributed:

20p h

0 else

h

h

he h

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

h

p h2h 1



Capacity of Rayleigh Channels

average SNR:

complementary cumulative distribution function:

outage channel capacity:

ergodic channel capacity:

2 1

inste 0Pr

1 else

C

CC C

2h

2S

out outld 1 ln 1C P

1

erg 11E

ln2eC

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

bits Hz

C

ergodic

out 0,01; 0,05; 0,1; 0,25P



Geometrical Non Frequency SelectiveMIMO Channel Models

1

N

Tx RP RP Rx

1

M

here: micro architectures, antenna arrays 1RPh

RPph

RPPh

Tx

p Rx

p

direction of departure, DOD: direction of arrival, DOA: directional channel coefficient:

RPph

Tx

p Rx

p



Steering Factors and Steering Vector

,Rx Rx Rx Rx

, ,Rx Rx

cos

2

m p m p m

m p m p

l l

l

, ,Rx Rx

T1, ,Rx Rx Rx

exp jm p m p

p p M p

a

a a

a

wave front

RP RL

Rx

p

Rx

m Rx

mlm ,

Rxm pl

due to reciprocity dual results hold for transmitter side



Weighting Network 1 *Rxw

*RxMw

2*Rx 1w

T* 1 * *Rx Rx Rx

Mw ww

1 *Txw

*TxNw

transmitter side weighting vector:

2*Tx 1w

T* 1 * *Tx Tx Tx

Nw ww

receiver sideweighting vector:



Antenna Gain

Rx RP

*Rx

1

*Rx Rx RP

1*TRxRP Rx

scalar

m m

Mm m

mM

m m

m

e a e

e w e

w a e

e e

w a

due to reciprocity dual results hold for transmitter antennas

2*TRx RxRx

*T *TRx RxRx Rx

g

w a

w a a w

antenna gain:



-20

0

45

225

90

270

135

315

180 0

Antenna Diagram

consider antenna gain as a function of the DOA

example:

RP

2

Rx

1 * 2 *Rx Rx

212

K

w w

Rx

Rx Rx

Rx Rx

10 log dBmax

gg



Conventional Beam Forming

maximize antenna gain

Schwarz inequality:

equality for

choose

due to reciprocity dual results hold for transmitter side

2 2 2*T *TRx RxRx RxRxg w a w a

Rx Rxw a

Rx Rx1M

w a corresponds to maximal ratio combining,matched filtering



Single Path Channel Model

SISO subsystem directional channel coefficient:

spatial channel coefficient:

MIMO system total channel matrix:

RPh

1Txa

TxNa

RxMa

1Rxa

1

M

1

N

RPh

,RPRx Tx

m n m nh a h a

TRPRx Txh H a a



Singular Value Decomposition

RP

*T

jargRx

orthonormalcolumns

RP

TTx

orthonormal rows

1 e

0 00 0 0

0 0 0

1

h

M

h

MN

N

U

Σ

V

H a

a



Capacity

with transmitter side channel state information:

SNR gain due to transmitter and receiver side beam forming without transmitter side channel state information:

SNR gain due to receiver side beam forming,no gain due to increased number of transmitter antennas

double number of antennas double SNR capacity gain of 1 Bit (at large SNRs)

2RPW

2 21max 0, ld ld 1

Rr

r

h SSC MN

2RP

2 21ld 1 ld 1

Rr

r

h SSC MN



Example: (2, 4) MIMO Channel

RP

RP

TRP Tx *T

Rx

1 1 1 1 1 8 01 1 j 1 j 1 11 10 01 1

21 1 1 1 1 1 120 01 1 j 1 j 0 0

h

h

HH a

Va U Σ

(2, 4) MIMO channel

RPh

2



Example, System Architecture

Σ

1

2

ss

H1

2

3

4

eeee

1

2

tt

1

2

3

4

rrrr

RP8 00 00 00 0

h

1

1

j 1

1 j 1 j1 1

2

1

1j

1 1

1

1

11

1 12 1

1

2

tt

1

2

3

4

rrrr

V

*TU



Example: Antenna Diagrams

-20

0

45

225

90

270

135

315

180 0

-20

0

45

225

90

270

135

315

180 0

transmitterantenna array

receiverantenna array

RxTx

Tx Tx

Tx Tx

10 log dBmax

gg

Rx Rx

Rx Rx

10 log dBmax

gg



Multiple Path Channel Model

superpose channels of individual paths

1 1 TRP Tx

T 1RPRx Tx Rx Rx

1 TRP TxRx

TRP Tx

0

0

Pp p p P

p P P

hh

h

a

H a a a a

aAH A

both beam forming and multiplexing gains possible

RxATxA

receiver side steering matrix:transmitter side steering matrix:

rank min , ,rich scattering: , rank not limited by number of paths

N M PP

H



Flat Ribbon Cable Channel Model

1

N

1

N

1 0E

0 1

H

singular values:

no crosscouplings

rank NH

1q



0 5 10 15 200

0.5

1

1.5

2

2.5

Capacity of the Flat Ribbon Cable Channel

all subchannels identical equal power allocation is optimal channel capacity with and without TxCSI is the same

2

2

2

ld 1

limiting value :

ld 1lim ld e1N

SC NN

N

SSNC

N

only (limited) gains by spatial multiplexing

N

C

2 1S

2 ld eS


0 5 10 15 200

5

10

15

20

25

30

Multiplexing Gain of theFlat Ribbon Cable Channel


2ld 1 SC N

N

210log dBS

1 10N

N



Multiplexing Gain, Degrees of Freedom

degrees of freedom

limld

here:

CR

R N

consider the asymptotic slope of the

capacity as a function of the PSNR !2S



Keyhole Channel Model

1

N

1

M

11h

1Nh

21h

2Mh

Huygen‘selementary source

1 2 2 1 2 11 1 1 1 1

1 2 2 1 T

1 2 2 1 2 11

,N

N M M M N

h h h h h h

h h h h h h

h h H h h

Chizhik (2002): Keyholes, Correlations, and Capacities of Multielement Transmit and Receive Antennas



Singular Value Decomposition of theKeyhole Channel

1 21 T

21

2-1 orthonormal

columns - 1 orthonormal rows

*T

0 0

0 0 0

0 0 0M

N

h h hh hHh

U VΣ

1 21 2, 0

rank 1 rank deficient!

optimum signal processing strategy consists in transmitter and receiver side matched filtering

Q

h h

H



Capacity of the Keyhole Channel

with transmitter side channel state information:

both transmitter and receiverside beam forming gains

without transmitter sidechannel state information:

only receiver sidebeam forming gains

2 21 2

2ld 1S

C

h h

2 21 2

2ld 1S

CN

h h

1 2

2

1, 1

1

n mh hS

100 101 1020

2

4

6

8

10

12

14

N M

Cwith TxCSI

without TxCSI



Channel Model with Multiple Keyholes

1

N

1

M 1,n kh 2

,k mh

1

K

1 21, ,1

1, 2, 2, 1, T

11 2, ,

,

rank min , ,

k k Kk k k k k k

k

N k k M

h h

h h

N M K

h h H h h H H

H



Channel Model with Independent Fading

channel coefficients independent identically normally distributed

2, ,h0, , here: 0,1m n m nh h

Channel capacity is a function of the eigenvalues of theWishart-Matrix:

random matrix theory, see : Metha: Random Matriceseigenvalues are Wishart distributed

*T

*T

M N

M N

HHW

H H

1,1 1,

,1 ,

N

M M N

h h

h h

H



Instantaneous Channel Capacity with Transmitter Side Channel State Information

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1 20N M

instPr C C

C2

2h

1

1

S



Instantaneous Channel Capacity without Transmitter Side Channel State Information

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1 20N M

instPr C C

C

2

2h

1

1

S

Foschini, Gans (1998): On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas



Ergodic Channel Capacity

0 5 10 15 200

5

10

15

20

25

N M

ergC with TxCSI

without TxCSI2

2h

1

1

S



Outage Channel Capacity

0 5 10 15 200

5

10

15

20

25

N M

outC with TxCSI

without TxCSI

out 0,1P

2

2h

1

1

S



Correlated Channel Coefficients

In general the channels between two different antenna pairs are statistically dependent due to the common environment!

For normal distributions the joint statistics are fullycharacterized by the correlations !

channel correlation matrix

generating a channel matrix:

*, ,E i k j lh h

*THH E vec vec R H H

1 2HHvec vec H R G

,

1 2 1 2 *T 1 2HH HH HH HH HH

: i.i.d. normally distributed, 0,1

, e.g. by Cholesky decomposition of m n

g

G

R R R R R



Receiver Antenna Correlations

If the receiver antenna correlationsare independent of the transmitter antenna k:

*, ,E i k j kh h

*, ,Rx, ,

Rx,1,1 Rx,1,*T

Rx

Rx, ,1 Rx, ,

E

receiver side correlation matrix:

E

i k j ki j

M

M M M

h h

N

R HH



Transmitter Antenna Correlations

If the transmitter antenna correlationsare independent of the receiver antenna i:

*, ,E i k i lh h

*, ,Tx, ,

Tx,1,1 Tx,1,*T

Tx

Tx, ,1 Tx, ,

E

transmitter side correlation matrix:

E

i k i lk l

N

N N N

h h

M

R H H



Energy of the Channel Coefficients

If both the receiver antenna correlations are independent of the transmitter antenna k and the transmitter antenna correlations are independent of the receiver antenna i:

All channel coefficients have the same energy Eh!

22 * * *, , , , , , , , hE E E E Ei k i k i k i l i l j l j l j lh h h h h h h h E

Rx Tx htrace trace M N E R R



Kronecker Channel Model, Assumptions

Both the receiver antenna correlations are independent of the transmitter antenna k and the transmitter antenna correlations are independent of the receiver antenna i and for the correlation of two arbitrary channel coefficients holds:

*, , Rx, , Tx, ,

h

HH Tx RxRx

1E

1trace

i k j l i j k lh h

E

R R RR



Kronecker Channel Model

1 2HH

1 2Tx Rx

Rx

1 21 2RxTx

Rx

T1 2 1 2Rx Tx

Rx

vec vec1 vec

trace

1 vectrace

1 vectrace

H R G

R R GR

R R GR

R G RR

T1 2 1 2

Rx TxRx

1trace

H R G RR

Kermoal etal. (2002): A Stochastic MIMO Radio Channel Model With Experimental Validation



5. Canonical System Implementation



Block Diagram

transmitter side channel stateinformation e.g. by signaling back CSI or

exploiting channel reciprocity in TDD systems

critical

receiver side channel stateinformation e.g. by training signal based

channel estimation uncritical

HV *TUs

n

emod. demod.

d d



Pulse Amplitude Modulation (PAM)

2-PAM = BPSK

4-PAM

in general: M-ary PAM,

symbol error probability:

bit error probability (Gray coded):

0 1-1

10

0 1-1

1101

3

10

-3

00

2S

1

2

1 2 1

1 13

M

mE m M

M

M

average symbol energy:

noise power density:average SNR of real part:

S2

2E

2BM

S 2

1 3erfc2 1

MPM M

b SP P B

1, 1

3, 1, 1, 3

20N



-10 0 10 20 3010

-4

10-3

10-2

10-1

100

Symbol Error Performance of PAM

10log dB

SP2, 4, 8,16M



-10 0 10 20 3010

-4

10-3

10-2

10-1

100

Bit Error Performance of PAM

10log dB

bP

2, 4, 8,16M

theory(approximate)

simulation



Rectangular Quadrature Amplitude Modulation (RQAM), Constellations

2-RQAM = 2-PAM = BPSK

8-RQAM

4-RQAM = 4-QAM = QPSK

16-RQAM = 16-QAM

0 1

00 01

10 11

000 001 011 010

100 101 111 110

0000 0001 0011 0010

0100 0101 0111 0110

1100 1101 1111 1110

1000 1001 1011 1010

one unit

1, 1

1 j, 1 j, 1 j, 1 j



Analysis of RQAM

RQAM consists of a MR-ary PAM for the real part and a MI-ary PAM for the imaginary part, average symbol energy is the sum of the symbol energies in

real and imaginary part

no symbol error occurs if there is neither a symbol error inthe real part nor in the imaginary part

R I 2BM M

2 2 2 2 SS R I R I 2

1 1 11 1 2 ,3 3 3

EE M M M M

S S,R S,I

R IS 2 2 2 2

R R I I R I

b S

1 1 1

1 13 31 1 erfc 1 erfc2 2

1

P P P

M MPM M M M M M

P PB



Symbol Error Performance of RQAM

-10 0 10 20 3010

-4

10-3

10-2

10-1

100

10log dB

SP

2, 4, 8,16, 32, 64,128, 256M



-10 0 10 20 3010

-4

10-3

10-2

10-1

100

Bit Error Performance of RQAM

10log dB

bP

2, 4, 8,16, 32, 64,128, 256M

theory(approximate)

simulation



Adaptive Modulation

For given maximum acceptable bit error probability Pbmaxthe number of bits which can be transmitted depends on the SNR !

1 2 3 4 5 6 7 8 9 10

2 4 8 16 32 64 128 256 512 1024

6,8 9,8 14 17 21 23 26 28 32 34

B

2BM

10log dB

3bmaxe.g. 10P

Transmit power increments required for transmitting one additional bit depend on the channel quality and the number B of already transmitted bits!

2r rS



Hughes Hartogs Algorithm

max

min

: available transmit power: bits to be transmitted

SB

min

max

and

r

B B

S S S

: argmin rr S

: 1, :: 1, :

r r r r r

r

B B S S SB B S S S

: 0: 0: 0, 1: 0, 1

r

r

BS

B r RS r R

Begin

End

no

yes



6. Signal Processing with non Cooperative Inputs (BLAST)



System Model

e H s n H d n

for good performance: M N

no transmitter side cooperation,only receiver side cooperation

joint datadetectionHd s e d

2nn,

(white)n R E

N M N



Optimum Data Detection

maximum a posteriori criterion (MAP):

maximum likelihood criterion (ML):

exploit discrete nature of themodulation alphabet

argmax Pr argmax p PrN N

d d

d d e e d d

2

argmax p

argmin for white Gaussian noise

N

N

d

d

d e d

e H d n



Linear Data Estimation

linear data estimator described by demodulator matrix D:

d s H Dd

n

equantizer

d D e D H d D n

' '' 1

' '' noiseuseful

interference

: -th row of , receiver filter: -th column of , channel signature

n

nN

n n n nn nn

n n n n nn n nn n

nn

d d

d d d

D DH H

D e D H D n

D H D H D n



Receive Zero Forcing (ZF) (1)

2

*T *T *T *T *T *T

argmin

argmin

N

N

d

d

d e H d

e e d H e e Hd d H Hd

do not restrict the search to discrete elements ofthe modulation alphabet



Wirtinger Calculus

R, I,jn n nx x x Definition:Let x be a complex vector with the elements and f(x) be a scalar complex valued function of x.One defines the generalized derivative of f(x) with respect tox as the N dimensional vector

with

Rules:

1

x

dd

fd

d N

fx

fx

R, I,

d 1 f fjd 2n n n

fx x x

x

*T *x

0c

a x a

*Tx

*T T *x

0

x a

x Ax A x



Receive Zero Forcing (ZF) (2)

*T *T *T *T *T *Td

T * T * *

1*T *T

0

e e d H e e Hd d H Hd

H e H H d

d H H H e

1*T *TZF

D H H H

*T

matchedfilter

H 1*T

decorrelator

H He d

left pseudoinverse



7. Signal Processing with non Cooperative Outputs



System Model

e d H s n

for good performance:N M

only transmitter side cooperation,no receiver side cooperation

jointtransmission Hs e dd

2nn,

(white)n R E

N MM



Linear Transmitter

linear transmitter described by themodulator matrix M:

d H s n H M d n

' '' 1

' '' noiseuseful

interference

: -th column of , transmitter filter: -th row of , channel signature

m

mM

m m m m mm mm

m m m m mm m mm m

mm

d n d n

d d d n

M MH H

H s H M

H M H M

s HM

ne d

quantizerd



Transmit Zero Forcing (ZF) (1)

Design a transmitted signal of minimum energyresulting in interference free data estimates!

2 *T

1

minimize fsubject to the constraints

g 0

g 0

Lagrangian multipliers:

grad f grad g 0

mmm

M

m mm

M

s s s s

d H s s d H ss d H s

s s



Transmit Zero Forcing (ZF) (2)

* T

1

T1

*T * *T *

1*T * * *T

1 1*T *T *T *TZF

with Wirtinger calculus:

0

using :

0

substitute in g :

0

M

m mm

M

s H

λ

s H λ s H λ

s

d Hs d HH λ λ HH d

s H HH d M H HH right pseudoinverse

*T

matchedfilter

H 1*T

decorrelator

HHd s



8. Diversity



Diversity

transmission paths are unreliable transmit information in parallel on several (independent)

transmission pathsexamples: time diversity frequency diversity antenna diversity



Antenna Diversity Techniques

micro diversity: antennas close to each other, antenna arrays, same transmission paths

macro diversity: antennas far apart from each other, different propagation environments

transmit diversity(MISO)

receive diversity(SIMO)

Simultaneous transmission of the same signal over several antennas does not yield any diversity gain!

Tx Rx

Tx Rx



Receive Diversity

1

M

h

h

H 2h : variance of the channel coefficients mh

Rayleighchannel

d

1h

Mh

21, powern

2, powerMn

*1

2hH

*

2Mh

H

d

maximalratiocombining



Performance Analysis

channel energy is chi square distributed with 2Mdegrees of freedom:

SNR of the estimated data symbol:

bit error probability, QPSK modulation:

2hE H

h h2h h

1 12h h

hh h2h h

p e e ,1 ! 1 !

E MEM M ME

MME E ME E M

M E M

2 2 22hh h

2 2 2

1

E E E,

p e1 !

MM M

M

E d E d M d

MM

bb b0

1 erfc , p d2 2

P P P



Ergodic Bit Error Probability

ergodic bit error probability:

for large average SNR :

1

b1

1

0

1 3 2 11 1 12 2 2

!2 12

11 1 1 12 2 2 22 2

M

mm m

M mM

m

mP

Mm

M

M mmM M

b2 1 1

2

M

M

M MPM



Bit Error Performance

-5 0 5 10 15 20 25 3010

-8

10-6

10-4

10-2

100

bP

10log dB

staticchannel

1 10M



Diversity Degree

consider the negative asymptoticslope of the bit error probability curve

diversity degree

here: D = M

bloglim

log

PD



Transmit Diversity

1 Nh hH 2h : variance of the channel coefficients nh

d

1h

Nh

2, powern

*1h

H

*Nh

H

d

Rayleighchannel

maximalratiocombining(constanttransmittedenergy)



Performance Analysis

channel energy is chi square distributed with 2Ndegrees of freedom:

SNR of the estimated data symbol:

bit error probability, QPSK modulation:

same performance as receive diversity,diversity degree D = N

2hE H

h h2h h

1 12h h

hh h2h h

p e e ,1 ! 1 !

E NEN M NE

NNE E NE E N

N E N

2 221h h

2 2

E E, , p e

1 !

NN N

N

E d N d NN

bb b0

1 erfc , p d2 2

P P P



Alamouti Scheme

Transmit Diversity can be exploited without transmitter side channel state information!Alamouti (1998): A Simple Transmit Diversity Technique for Wireless Communications

*1 2,d d

*2 1,d d

1

2

dd

d

Tx Rxd

1h

2h e

1 2 11 1* * * *

22 1 22

e h h ndde h h n

d ne H



Maximum Likelihood Receiver

the columns of the system matrix H are orthogonal optimum receiver consists in a matched filter followed

by a quantizer:

SNR of the estimates:

Same SNR as for transmit diversity with transmitter side channel state information but twice the transmit power because there is no beam forming gain!

*1 1 1*T *T 1 2

2 2* **2 21 2 2 1

* *1 1 2 21

2 2 * *2 1 2 2 1 1 2

1diag

1

e eh he eh h h h

d h n h nd h h h n h n

d H H H

2 2

2 1 22E n

h hd



Rate

Similar to the concept of the rate used in coding theory onedefines the rate of a spatio temporal code:

examples: spread spectrum system, spreading factor Q (repetition code):

Alamouti code:

spatial multiplexing:

number of data symbolsnumber of channel uses

R

1 1RQ

2 12

R

min , 1R N M



End

Documents

MIMO Mobile Radio Systems · 10/12/07 Weber: MIMO Mobile Radio Systems 22 Optimization Task Question: How large is the total channel capacity C for limited total transmitted power