mimo mobile radio systems · 10/12/07 weber: mimo mobile radio systems 22 optimization task...
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Institut fürNachrichtentechnik
MIMO Mobile Radio SystemsProf. Tobias WeberUniversity of RostockEmail: [email protected]
Institut fürNachrichtentechnik
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 310/12/07
1. Introduction
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Weber: MIMO Mobile Radio Systems 409/03/13
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 510/12/07
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
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Weber: MIMO Mobile Radio Systems 610/12/07
Cable Binder
insufficient shielding cross couplings
R parallel wires (R, R) MIMO systemcapacity proportional to R (for fixed transmitted power per input)
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 710/12/07
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
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Weber: MIMO Mobile Radio Systems 810/12/07
2. System Modelling
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Weber: MIMO Mobile Radio Systems 922/02/13
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
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Weber: MIMO Mobile Radio Systems 1022/02/13
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
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Weber: MIMO Mobile Radio Systems 1122/11/08
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
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Weber: MIMO Mobile Radio Systems 1210/12/07
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
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Weber: MIMO Mobile Radio Systems 1302/02/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 1410/12/07
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
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Weber: MIMO Mobile Radio Systems 1513/07/09
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 1622/02/13
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
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Weber: MIMO Mobile Radio Systems 1710/12/07
Multi Carrier Transmission, OFDM
h(t)
H(f)
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Weber: MIMO Mobile Radio Systems 1810/12/07
3. Channel Capacity
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Weber: MIMO Mobile Radio Systems 1922/02/13
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2010/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2110/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2210/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2328/10/09
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2428/10/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2510/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2610/12/07
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
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Weber: MIMO Mobile Radio Systems 2710/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2804/10/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 2910/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 3022/11/08
Singular Value Decomposition (2)
*T *T Σ U H V H U Σ V
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 3110/12/07
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
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Weber: MIMO Mobile Radio Systems 3205/11/09
Matrix Structures (2)
*T *T Σ U H V H U Σ V
example: M > N
H
VU*T
1
q
Q
0
0=
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Weber: MIMO Mobile Radio Systems 3310/12/07
Matrix Structures (3)
*T *T Σ U H V H U Σ V
example: M < N
H
VU*T
=
1
q
Q
0
0
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 3422/02/13
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 3522/02/13
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 3622/11/08
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
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0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Weber: MIMO Mobile Radio Systems 3713/07/09
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
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Weber: MIMO Mobile Radio Systems 3822/02/13
4. Channel Models
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Weber: MIMO Mobile Radio Systems 3922/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4022/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4111/11/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4222/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4314/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4422/02/13
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4514/12/07
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:
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4605/03/08
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:
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4702/01/08
-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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4822/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 4922/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5025/03/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5122/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5206/04/09
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5310/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5402/01/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5530/11/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5622/02/13
Flat Ribbon Cable Channel Model
1
N
1
N
1 0E
0 1
H
singular values:
no crosscouplings
rank NH
1q
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5710/12/07
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
Institut fürNachrichtentechnik
0 5 10 15 200
5
10
15
20
25
30
Multiplexing Gain of theFlat Ribbon Cable Channel
Weber: MIMO Mobile Radio Systems 5819/01/11
2ld 1 SC N
N
210log dBS
1 10N
N
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 5919/01/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6014/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6127/10/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6222/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6310/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6424/09/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6510/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6610/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6710/12/07
Ergodic Channel Capacity
0 5 10 15 200
5
10
15
20
25
N M
ergC with TxCSI
without TxCSI2
2h
1
1
S
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6810/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 6924/09/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7010/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7110/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7202/01/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7302/01/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7402/01/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7510/12/07
5. Canonical System Implementation
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7610/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7725/03/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7810/12/07
-10 0 10 20 3010
-4
10-3
10-2
10-1
100
Symbol Error Performance of PAM
10log dB
SP2, 4, 8,16M
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 7910/12/07
-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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8010/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8110/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8210/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8310/12/07
-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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8410/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8527/10/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8610/12/07
6. Signal Processing with non Cooperative Inputs (BLAST)
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8731/05/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8831/05/13
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 8927/06/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9022/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9105/01/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9210/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9310/12/07
7. Signal Processing with non Cooperative Outputs
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9404/10/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9527/06/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9602/02/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9722/11/08
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9810/12/07
8. Diversity
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 9922/02/13
Diversity
transmission paths are unreliable transmit information in parallel on several (independent)
transmission pathsexamples: time diversity frequency diversity antenna diversity
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10020/01/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10113/07/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10213/07/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10318/08/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10410/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10510/12/07
Diversity Degree
consider the negative asymptoticslope of the bit error probability curve
diversity degree
here: D = M
bloglim
log
PD
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10613/07/11
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)
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10713/07/11
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10810/12/07
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 10930/01/13
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 11027/06/10
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
Institut fürNachrichtentechnik
Weber: MIMO Mobile Radio Systems 11110/12/07
End