mimo mobile radio systemsmimo mobile radio systems

Microsoft PowerPoint - tutorial_public.ppt [Kompatibilitätsmodus]Institut für Nachrichtentechnik
MIMO Mobile Radio SystemsMIMO Mobile Radio Systems Prof. Tobias Weber University of RostockUniversity of Rostock Email: [email protected]
Institut für NachrichtentechnikTopics
2nd lesson: channel models
3 d l3rd lesson: canonical system implementation signal processing with non cooperative inputs (BLAST) signal processing with non cooperative inputs (BLAST) signal processing with non cooperative outputs diversity
Weber: MIMO Mobile Radio Systems 210/12/07
diversity
Institut für NachrichtentechnikApplication of MIMO: 802.11n
characteristics: • 20 MHz to 40 MHz bandwidth • OFDM • 2 to 4 antennas per station • spatial multiplexingspatial multiplexing • up to 540 Mbit/s
Institut für NachrichtentechnikTerms
N = 1 N > 1
MIMO
Institut für NachrichtentechnikCable Binder
R parallel wires (R R) MIMO systemR parallel wires (R, R) MIMO system capacity proportional to R (for fixed transmitted power per input)
insufficient shielding cross couplings
Institut für NachrichtentechnikMultiple Antenna System
1 1
RxTx N M
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
alternative: use spectrum more efficiently
Institut für Nachrichtentechnik2. System Modelling
Institut für NachrichtentechnikQuadrature Modulator
f ta t u t
u t f t u t f t

R 0 I 02 cos 2 2 sin 2u t f t u t f t
t in-phase component quadrature component
Ru t
a t
Iu t
I
: equivalent lowpass signalu t
Institut für NachrichtentechnikQuadrature Demodulator

0j2 R 0 I 02 2 cos 2 2 sin 2
2 2 j 2 i 2
f ta t e u t f t u t f t
f f
2 cos 2 j 2 sin 2f t f t
u t f t u t f tu t

R
I j cos 4 j sin 4j u t f t u t f tu t high frequency
Ru t
u t
Iu t
sinc tu t u lT l T

l l
T u

/t T-1 1
the signal lies in a L-dimensional vector space spanned by the basis functions
0 1L
basis functions

T

2Re e ef f t
a t
u t

Small time shifts correspond to phase rotations by 0j2e f Small time shifts correspond to phase rotations by in equivalent lowpass domain.
0je
1W

e h s h T h wT

00 0e h h

He p
SISO b t Tx1,Kh
1 1 Rx ,1Kh
SISO subsystem: Rx Rx Tx Tx,k k k k e H s
KTx KRx
MIMO system: 1 11 1 1K

e H H s Rx Rx Rx Tx Tx,1 ,K K K K K
e H H s e H s

Institut für NachrichtentechnikMIMO System, Single Tap Channel
1 1
KTx KRx Rx Tx,K Kh
Rx TxRx Tx

Institut für NachrichtentechnikWhite Multivariate Gaussian Noise
0 0np p Re p Imn n n
2 2
0 0
1 1
Re Im

independent identically distributed, i.i.d.
h(t)
H(f)
Institut für Nachrichtentechnik3. Channel Capacity
Institut für NachrichtentechnikSISO Channel Capacity
channel coefficient h
Shannon (1948): A Mathematical Theory of Communication
h l (N i t t )per channel use (Nyquist rate):
2 bith S
Institut für NachrichtentechnikUncoupled (R, R) MIMO Channel
1 2h1
Sor power

Total Channel Capacity without Transmitter Side Channel State Information
All R parallel channels get the same transmitted power:All R parallel channels get the same transmitted power:
SS
rr r r

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

S S
Institut für NachrichtentechnikOptimization (1)

S SS 0.2
S SS
Lagrange: grad f grad g S S 0 0 0.2 0.4 0.6 0.8 1
0

f g S S

R RS S
1
r

S h
Institut für NachrichtentechnikOptimization (3)
R
h
S
Institut für NachrichtentechnikWaterfilling
R
Holsinger (1964): Digital communication over fixed time-continous channels with ith i l li ti t t l h h l
memory - with special application to telephone channels
Institut für NachrichtentechnikTotal Channel Capacity with Waterfilling

R R r rrh h S
C S h
r
h
S
special case: all R channels used 2 2 21R RS 2 2 2
W W2 2 2 1 1
1R R r r r
r W r rr r r
SS S S S S R Rh h h

C
1 1r rr r
t V s H e U*T r
s should have 2
m, m R E
m, m should be white:
*T2R U U*T *T *TV V Σt r
diagonal matrix
T T T
t V Vt t t V V E
find unitary matrices U, V and a diagonal matrix such that
diagonal matrix
*T *TΣ U H V H U Σ V
Σ U H V H U Σ V
Institut für NachrichtentechnikSingular 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
*TΣ U H V
is a M N diagonal matrix with nonnegative real diagonal
TΣ U H V
*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 U V: 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
*T *Teigenvalues of or q HH*T H*TH *T :HH Grammian of the row vectorsG a a o t e o ecto s
*T :H H Grammian of the column vectors
Institut für NachrichtentechnikMatrix Structures (1)
*T *T Σ U H V H U Σ V example: M = N
1
VU*T U
rank minR Q N M Hrank of the channel: 1 2 1 0R R Q
rank min ,R Q N M Hrank of the channel:
example: M > N
example: M < N
Institut für NachrichtentechnikMIMO Channel Capacity
without transmitter side channel state information (Foschini, 1996): all inputs with same powerall inputs with same power
2 21 ld 1 ld 1
RR r rS SC
*T *T *T *T *T 2 2ld det ld det
N

N N
S S
H 2 2ld det det det ld det N N
V E H H V E H
*Tld det S
2ld det N
with transmitter side channel statewith transmitter side channel state information (Telatar, 1995, 1999):

W 1 1
instantaneous channel capacity:
rS W
2 1
ld 1 without TxCSI
r N
C
g
inst out outPr 1C C P
Institut für NachrichtentechnikGraphical Evaluation
0 6
RayleighC }
C
Cerg = 1,9415
„you can always find an environment that fits your model“
Institut für NachrichtentechnikDeterministic SISO Channel Model
LOS NLOS LOS: line of sight,
NLOS: non line of sight s t e t
g
P
P
in general time dispersive i e in general frequency selective
1
j2
in general time dispersive, i.e., impulse response spread in time
in general frequency selective, i.e., frequency dependent transfer function
Institut für NachrichtentechnikSingle Tap Channel
ti d i f d itime domain single tap channel:
frequency domain flat fading channel:
1 für alle 'p p constH f
impulse response: transfer function: ' für alle , 'p p p p
B constH f

Institut für Nachrichtentechnik
neglect access delay P
ph a
if the ap are independent it follows for P→ (central limit theorem):
1 p
h 2 h
0.2
0 1 2 3 4 5 h
Institut für NachrichtentechnikCapacity of Rayleigh Channels
average SNR: 2 h
2
0 5
0.5
erg 1ln2
1 1
direction of departure, DOD: direction of arrival, DOA: directional channel coefficient: ph
Tx
directional channel coefficient: RPh
wave front
Rx Rxexp jm p m pa
RP RL
, , Rx Rx
T1, , Rx Rx Rx p p M pa aa
due to reciprocity dual results hold for transmitter side
Institut für NachrichtentechnikWeighting Network
Mw ww
Nw ww
Rx 1wTx 1w
Institut für NachrichtentechnikAntenna Gain
antenna gain:
Rx RxRx
g
Institut für NachrichtentechnikAntenna Diagram
R Rg
090 example:
270
Schwarz inequality: 2 2 2*T *T
Rx RxRx RxRxg w a w a
equality for
Rx RxRx
M matched filtering
1 Txa 1
RPh T RPRx Txh H a a
spatial channel coefficient: ,
Institut für NachrichtentechnikSingular Value Decomposition

h
MN
TV
RPWmax 0 ld ld 1 R
r h SSC MN
SNR gain due to transmitter and receiver side beam forming
RPW 2 2
r C MN
2 RPld 1 ld 1
R r h SSC M
SNR gain due to receiver side beam forming,
2 2 1 ld 1 ld 1r
r C M

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)
Institut für NachrichtentechnikExample: (2, 4) MIMO Channel
(2, 4) MIMO channel
RP
RP
1 1 1 1 1 8 0 1 1 j 1 j 1 11 10 01 1
21 1 1 1 1 1 120 0
h
h
RP Tx *T

Rxa U Σ
s 1e
1 1 1 1
11 1 t 1r
j 1 1 1
2t
3
4
Σ

antenna array antenna array
superpose channels of individual paths
1 1 T RP Tx0
P h
RP T0
p P P
aA

RxA A
receiver side steering matrix: t itt id t i t i TxAtransmitter side steering matrix:
rank min , ,N M PH
both beam forming and multiplexing gains possible
rich scattering: , rank not limited by number of pathsP
both beam forming and multiplexing gains possible
Institut für NachrichtentechnikFlat Ribbon Cable Channel Model
1 1no crossno cross couplings
N N
g q
all subchannels identical equal power allocation is optimal
h l it ith d ith t T CSI i th
2.5
ld 1 SC N

0.5
2
2
SNC
0 5 10 15 20 N
l (li i d) i b i l l i l i
N
30
25
30
5 1 10N
210log dBS
Institut für NachrichtentechnikMultiplexing Gain, Degrees of Freedom
consider the asymptotic slope of theconsider the asymptotic slope of the
capacity as a function of the PSNR !2 S

lim
Institut für NachrichtentechnikKeyhole Channel Model
Chizhik (2002): Keyholes, Correlations, and Capacities of Multielement Transmit and Receive Antennas
1 1 1 1h 2
1h
1
Huygen‘s elementary source

1 2 2 1 2 1
, Nh h h h h h
h h h h h h


rank 1 rank deficient! H

Institut für NachrichtentechnikCapacity of the Keyhole Channel
with transmitter side channel state information: 2 21 2S h h
2ld 1C

2 1S
10
12

g g N M
1 11 1 1
,k mh K
k k k k k k
h h

N M K
H
Institut für NachrichtentechnikChannel Model with Independent Fading
1,1 1,Nh h
channel coefficients independent identically normally distributed
2 , ,h0, , here: 0,1m n m nh h
Channel capacity is a function of the eigenvalues of theChannel capacity is a function of the eigenvalues of the Wishart-Matrix:
*T M N HH
*T M N
a do at t eo y, see et a a do at ces eigenvalues are Wishart distributed
1
C 2
1
C h 1
Foschini, Gans (1998): On Limits of Wireless Communications in a Fading
Foschini, Gans (1998): On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas
Institut für NachrichtentechnikErgodic Channel Capacity
25
20
15
N M
Institut für NachrichtentechnikOutage Channel Capacity
25
20
N M
Institut für NachrichtentechnikCorrelated 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 fully characterized by the correlations ! *E i k j lh hy
channel correlation matrix
*T HH E vec vec R H H
,
1 2 1 2 *T 1 2 HH HH HH HH HH
: i.i.d. normally distributed, 0,1
g

HH HH HH HH HH, e.g. by Cholesky decomposition of R R R R R
Institut für NachrichtentechnikReceiver Antenna Correlations
*If the receiver antenna correlations are independent of the transmitter antenna k:
* , ,E i k j kh h
*E h h , ,Rx, , E i k j ki j
h h
Rx,1,1 Rx,1,
Institut für NachrichtentechnikTransmitter Antenna Correlations
*If the transmitter antenna correlations are independent of the receiver antenna i:
, ,E i k i lh h
*E h h , ,Tx, , E i k i lk l
h h
Tx,1,1 Tx,1,
Institut für NachrichtentechnikEnergy of the Channel Coefficients
If both the receiver antenna correlations are independent of the transmitter antenna k and the transmitter antenna correlations are i d d t f th i t iindependent of the receiver antenna i:
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
All channel coefficients have the same energy Eh!
, , , , , , , , hi k i k i k i l i l j l j l j l
Rx Tx htrace trace M N E R R
Institut für NachrichtentechnikKronecker 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 ofindependent of the receiver antenna i and for the correlation of two arbitrary channel coefficients holds:
* , , Rx, , Tx, ,
h
1E i k j l i j k l h h
E
Institut für NachrichtentechnikKronecker Channel Model
1 2vec vec H R G

trace
Institut für Nachrichtentechnik5. Canonical System Implementation
Institut für NachrichtentechnikBlock Diagram
d d
t itt id h l t t i id h l t ttransmitter side channel state information e g by signaling back CSI or
receiver side channel state information e g by training signal basede.g. by signaling back CSI or
exploiting channel reciprocity in TDD systems
iti l
uncritical
critical
2-PAM = BPSK 10
4-PAM 0 1-1

2 2E
erfc 2 1
P M M
bit error probability (Gray coded): b SP P B
Institut für NachrichtentechnikSymbol Error Performance of PAM
10 0
10 -1
-4
Institut für NachrichtentechnikBit Error Performance of PAM
10 0
-4
2-RQAM = 2-PAM = BPSK 4-RQAM = 4-QAM = QPSK 0 1 10 11
00 01 1, 1
000 001 011 010
0100 0101 0111 0110
1100 1101 1111 1110
0000 0001 0011 0010
0100 0101 0111 0110
one unit
Institut für NachrichtentechnikAnalysis of RQAM
RQAM consists of a MR-ary PAM for the real part and a MI-ary PAM for the imaginary part, R I 2BM M average symbol energy is the sum of the symbol energies in
real and imaginary part
1 1 1 E
no symbol error occurs if there is neither a symbol error in
2 2 2 2 S S R I R I 2
1 1 11 1 2 , 3 3 3
EE M M M M

no symbol error occurs if there is neither a symbol error in the real part nor in the imaginary part
1 1 1P P P S S,R S,I
R I S 2 2 2 2
1 1 1
P P P

R R I I R I
b S
2 2
P P
b SB
10 0
10 -1
10 -3
4
Institut für NachrichtentechnikBit Error Performance of RQAM
10 0
10 -1
bP simulation
10 -2
10 -3
4
Institut für NachrichtentechnikAdaptive Modulation
For given maximum acceptable bit error probability Pbmax the number of bits which can be transmitted depends on p the SNR !
1 2 3 4 5 6 7 8 9 10B
3 bmaxe.g. 10P
1 2 3 4 5 6 7 8 9 10
2 4 8 16 32 64 128 256 512 1024
B
2BM
6,8 9,8 14 17 21 23 26 28 32 34 10log dB
Transmit power increments required for transmitting one additional bit depend on the channel quality and the n mber B of alread transmitted bits!
2 r rS
the number B of already transmitted bits!
Institut für NachrichtentechnikHughes Hartogs Algorithm
max
min
S B : argmin rr S
min
S B r R
: 1, : : 1, :
r

Institut für NachrichtentechnikSystem Model
for good performance: M Nfor good performance: M N
no transmitter side cooperation, only receiver side cooperation
only receiver side cooperation
exploit discrete nature of the modulation alphabet
maximum a posteriori criterion (MAP): argmax Pr argmax p Pr d d e e d d
ma im m likelihood criterion (ML)
argmax Pr argmax p Pr N N d d
d d e e d d
maximum likelihood criterion (ML):

Institut für NachrichtentechnikLinear Data Estimation
d s H D d
n
H D quantizer
: -th row of , receiver filter : -th column of , channel signature
n
n
n

d d d
noiseuseful interference
Institut für NachrichtentechnikReceive Zero Forcing (ZF) (1)
do not restrict the search to discrete elements of the modulation alphabet
2argmin N
N
d

Institut für NachrichtentechnikWirtinger Calculus
R Ijx x x Definition: Let x be a complex vector with the elements R, I,jn n nx x xLet 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 to
th N di i l tx as the N dimensional vector
df
f x x x
Rules:
x
Institut für NachrichtentechnikReceive Zero Forcing (ZF) (2)
*T *T *T *T *T *T d
T * T * *
e e d H e e Hd d H Hd
T * T * *
H e H H d
d H H H e
1*T *T D H H H ZF D H H H
left pseudoinverse
filter
Institut für NachrichtentechnikSystem Model
for good performance:N Mfor good performance:N M
only transmitter side cooperation, no receiver side cooperation
no receiver side cooperation
Institut für NachrichtentechnikLinear Transmitter
M quantizer
: -th column of , transmitter filter : -th row of , channel signature
m
m
m m m m mm m m
m
d d d n
noiseuseful interference
Institut für NachrichtentechnikTransmit Zero Forcing (ZF) (1)
Design a transmitted signal of minimum energy resulting in interference free data estimates!resulting in interference free data estimates!
2 *T 2 *Tminimize f subject to the constraintsM
s s s s

d H s s d H s s d H s
Lagrangian multipliers:
m m m
Institut für NachrichtentechnikTransmit Zero Forcing (ZF) (2)
* T
matched filter

1 1*T *T *T *T ZFs H HH d M H HH right pseudoinverse

Institut für NachrichtentechnikDiversity
transmission paths are unreliable transmit information in parallel on several (independent)
transmission paths examples:
antenna diversity antenna diversity
Institut für NachrichtentechnikAntenna Diversity Techniques
receive diversity T Rreceive diversity (SIMO) Tx Rx
transmit diversity T Ry (MISO)
Simultaneous transmission of the same signal over several
Tx Rx
micro diversity: antennas close to each other antenna
Simultaneous transmission of the same signal over several antennas does not yield any diversity gain! micro diversity: antennas close to each other, antenna
arrays, same transmission paths macro diversity: antennas far apart from each other,
y p , different propagation environments
Institut für NachrichtentechnikReceive Diversity
Mh
Institut für NachrichtentechnikPerformance Analysis
channel energy is chi square distributed with 2M degrees of freedom:
2 hE H
SNR of the estimated data symbol:

M M M E
M E M SNR of the estimated data symbol:

2 2 2
E E E ,

P P P
ergodic bit error probability:
M
M
m
b 2 1 1
Institut für NachrichtentechnikBit Error Performance
10 0
10 -2
8
Institut für NachrichtentechnikDiversity Degree
id th ti t ticonsider the negative asymptotic slope of the bit error probability curve
diversity degree diversity degree
Institut für NachrichtentechnikTransmit Diversity
d
Nh
* Nh
H
d
N
1 Nh hH 2 h : variance of the channel coefficients nh

Institut für NachrichtentechnikPerformance Analysis
channel energy is chi square distributed with 2N degrees of freedom:
2 hE H

1 1 2h h
hh h2 h h
NEN M N E
N E N
2 22
bit error probability QPSK modulation:

bit error probability, QPSK modulation:
bb b 1 erfc , p d 2 2
P P P
diversity degree D = N
diversity degree D = N
Institut für NachrichtentechnikAlamouti Scheme
Transmit Diversity can be exploited without transmitter side channel state information! Alamouti (1998): A Simple Transmit Diversity Technique forAlamouti (1998): A Simple Transmit Diversity Technique for Wireless Communications
h

d ne H
Institut für NachrichtentechnikMaximum Likelihood Receiver
the columns of the system matrix H are orthogonal optimum receiver consists in a matched filter followed
b tiby a quantizer:
2 2* **
2 21 2 2 1
* * 1 1 2 21

d h h h n h n
SNR of the estimates:
h h d

p there is no beam forming gain!
Institut für NachrichtentechnikRate
Similar to the concept of the rate used in coding theory one defines the rate of a spatio temporal code:
number of data symbols number of channel uses
R
examples:
number of channel uses

min , 1R N M

mimo mobile radio systemsmimo mobile radio systems

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