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1
IERG 4100 Wireless Communications
Part IIX: Multiple antenna systems
2
Motivation
Current wireless systems Cellular mobile phone systems, WLAN, Bluetooth,
Mobile LEO satellite systems, …
Increasing demand Higher data rate ( > 100Mbps) IEEE802.11n Higher transmission reliability (comparable to wire
lines) 4G
Physical limitations in wireless systems Multipath fading Limited spectrum resources Limited battery life of mobile devices …
3
Motivation
Time and frequency processing Coding Adaptive modulation Equalization Dynamic bandwidth and power allocation …
Multiple antenna open a new signaling dimension: space Higher transmission rate (Multiplexing gain) Higher link reliability (Diversity gain) Wider coverage …
4
Multiple antenna systems
SU-MISO, TX diversity SU-SIMO, RX diversity
SU-MIMO, Diversity vs. Multiplexing
5
Multiple antenna systems
MS
MS
BS
MS
MS
MS
BS
MS
MS
MS
BS
MS
MIMO Broadcast MIMO Multi-access
MS
MS
BS
MS
MISO Broadcast SIMO Multi-access
6
Multiplexing gain
Multiple antennas at both Tx and Rx Can create multiple parallel channels Multiplexing order = min(M, N) Transmission rate increases linearly
7
Diversity gain
Multiple Tx or multiple Rx or both Can create multiple independently faded
branches Diversity order = MN Link reliability improved exponentially
Today’s Lecture
Diversity schemes Beamforming Space time coding
8
9
Achieving diversity:Maximum ratio combining
Recall Fading flattens BER curves
Space-domain diveristy Improve BER from
~(SNR)-1 to (SNR)-n
Assume N=1 or M=1 for the time being
0 5 10 15 20 25 3010
-6
10-5
10-4
10-3
10-2
10-1
100
SNR (dB)B
it E
rro
r R
ate
Rayleigh fading channel
AWGN channel
BER~exp(SNR) BER~(SNR)1
Diversity order n
10
Maximum ratio combining (SIMO)
Ways to combine the received signals Equal gain combining
All paths co-phased and summed with equal weighting Maximum ratio combining
All paths co-phased and summed with optimal weighting
Tx Rx
h1h2
hN
1 2, ,T
Nh h hh
1 1
2 2
11 11 NN
N N
h x
h xx
h x
y h
11
Maximum ratio combining
Maximal Ratio Combining (MRC) Optimal technique (maximizes output SNR) Combiner SNR is the sum of the branch SNRs. Achieve diversity order of N
r =hH hx+( ) = hn* hnx+n( )
n=1
N
∑ = hn
2
n=1
N
∑ x+ hn*n
n=1
N
∑
SNR=signal powernoise power
=
E hn
2
n=1
N
∑ x⎛
⎝⎜⎞
⎠⎟
2⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
E hn*n
n=1
N
∑⎛
⎝⎜⎞
⎠⎟
2⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
=hn
2
n=1
N
∑ E x2( )
σ 2
Variance of noise
Distribution of SNR in Rayleigh Fading Channel
: exponential distribution
: chi-square distribution with degree of freedom
2N when hn are independent for different n
Recall the BER Calculation:
Through simple calculation, it can be seen that
12
h
2
hn
2
n=1
N
∑
BER ~ SNR( )
−NDiversity order
Average SNR
E x2( )
σ 2
13
Diversity
0 5 10 15 20 25 3010
-6
10-5
10-4
10-3
10-2
10-1
100
SNR (dB)
Bit
Err
or
Ra
te
Rayleigh fading channelwith 1 receive antenna
Diversity order 2
Diversity order 4
Diversity Gain
14
15
Maximum ratio transmission (MISO)
1 11 1 M My
h sRxTx
h1
h2
hM
1 2, , Mh h hh
The signal transmitted by M antennas
2
21
12 2
1 1
2 2
12
,
M
mH H Mm
mM Mm
m mm m
M
mm
hx x
y x h x
h h
h E x
SNR
σ
h hs
h
Transmitter must know the channel!
What if it does not know?
16
Achieving diversity without CSIT:Space time coding
Core idea: complement traditional time with added space
Without channel knowledge at the transmitter ST trellis codes (Tarokh’98), ST block codes
(Alamouti’98) Coding techniques designed for multiple antenna
transmission. Coding is performed by adding properly designed
redundancy in both spatial and temporal domains which introduces correlation into the transmitted signal.
17
Space time coding
The ST encoder maps a block of information symbols X to coded symbols S
1,TM
t ts s
Information source
S-T Encoder S/P
Receiver
1
M
1
N
X S
1 1, , , ,t t t S s s s
18
An Introductory Example
Two transmit antennas and 1 receive antenna If two time intervals for the transmission of 1 symbol is
allowed
Received signal
Equal to 1 by 2 MIMO systems Diversity gain = 2 Data rate is reduced!!
1 1 2 2( ) ( 1)y t h s n y t h s n
19
Space time block code: Alamouti code
Two transmit antennas and 1 receive antenna Assume channel does not change across two
consecutive symbols
A1 A2
t x1 x2
t+1 x2* x1
*
Space
time
1 1 2 2 1
* *1 2 2 1 2
Received signal
( )
( 1)
y t h x h x n
y t h x h x n
20
Alamouti code
The combining scheme
The decision statistics
Maximum-likelihood estimates of the transmitted symbols Choose xi if
* *1 1 2
* *2 2 1
( ) ( 1)
( ) ( 1)
x h y t h y t
x h y t h y t
%x1 = h12+ h2
2⎛⎝⎜
⎞⎠⎟
x1 + h1*n1 + h2n2
%x2 = h12+ h2
2⎛⎝⎜
⎞⎠⎟
x2 −h1n2 + h2*n1
The combined signals are equivalent to that obtained from two-branch MRC!
Diversity gain =2
Data rate is not reduced!
21
Alamouti code
Full-rate complex code Is the only complex S-T block code with a code rate of
unity.
Optimality of capacity For 2 transmit antennas and a single receive
antenna, the Alamouti code is the only optimal S-T block code in terms of capacity
22
Alamouti Code Performance
From Alamouti, A simple transmit diversity technique for wireless communications
23
Alamouti Code
The performance of Alamouti code with two transmitters and a single receiver is 3 dB worse than two-branch MRC.
The 3-dB penalty is incurred because is assumed that each transmit antenna radiates half the energy in order to ensure the same total radiated power as with one transmit antenna.
If each transmit antenna was to radiate the same energy as the single transmit antenna for MRC, the performance would be identical.
24
Space time block code
Alamouti code can be generalized to an arbitrary number of antennas
A S-T code is defined by an k x M transmission matrix M – number of TX antennas k – number of time periods for transmission of one
block of coded symbols Fractional code rate Reduced Spectral efficiency Non-square transmission matrix
Ref.: V. Tarokh, et al. “Space-time block codes from orthogonal designs.”
25
SVD
SVD-Singular value decomposition Allows H to be decomposed into parallel channels
as follows
where S is a N-by-M diagonal matrix with elements only along the diagonal n=m that are real and non-negative
U is a unitary N-by-N matrix and V is a unitary M-by-M matrix
A Matrix is Unitary if AH=A-1 so that AHA= I For example
HH USV
10 5 0.628 0.683 0.374 16.491 00.660 0.751
2 9 0.490 0.720 0.492 0 6.16720.751 0.660
6 8 0.605 0.126 0.787 0 0
26
What are Singular Values?
Note we can generate a square M-by-M matrix asHHH= (USVH)H(USVH)=V(SHS)VH
Alternatively we can generate a square N-by-N matrix asHHH= (USVH)(USVH)H= UH (SSH)U
We can see that the square of the singular values are the eigenvalues of HHH and HHH
27
SVD—What does it mean?
Implies that UHHV=S is a diagonal matrix Therefore if we pre-process the signals by V at
the transmitter and then post-process them with UH we will produce an equivalent diagonal matrix
This is a channel without any interference and channel gains s11 and s22 for example
28
Water-Filling
When we have parallel multiple channels each with different attenuations we can use water filling to optimize the capacity by modifying the transmit powers
The capacity of multiple channels is given by
The question is how to find the distribution of powers to maximize the capacity under the constraint that
C = log2 1+
Piα i
N0B
⎛
⎝⎜⎞
⎠⎟i=1
N
∑ = log2 1+ P∞iα i( )i=1
N
∑ b/s/Hz
P∞i
i=1
N
∑ P∞T
29
Water-Filling
Use Lagrangian multiplier to find the solution Write
Take partial derivatives wrt to power allocations
∂f∂Pi
=0⇒α i
1+α i P∞
i=+λ
P∞i =1λ−
1α i
∀α i > λ
0 elsewhere
⎧
⎨⎪
⎩⎪
f =log2 1+ P∞iα i( )−λ P∞i
i=1
N
∑ −P∞T⎛
⎝⎜⎞
⎠⎟
30
Water-filling
Know as water filling
Good channels get more power than poor channels
Channel index
Adaptive Modulation in Fading Channels
31
6 bps/HZ
4 bps/HZ
2 bps/HZ
0 bps/HZ
Adaptive Modulation in Fading Channels
32
0 5 10 15 20 25 3010
-5
10-4
10-3
10-2
10-1
100
Average Received Eb/N
o (dB)
Bit
Err
or
Ra
te2 bps/Hz4 bps/HZ6 bps/Hz
Adaptive
Non-Adaptive
Adaptive Modulation
Data rate varies with channel fading amplitude Variable data-rate transmission can also be achieved by
adapting the code rate Adaptive coding and modulation are often combined Coding and modulation schemes can be chosen
according to several criteria Maximize average data rate given a fixed BER (bit error
rate) Minimize average BER given a fixed average data rate In practice, need to consider the modulation types being
discrete
Example
An adaptive modulation system can choose to use QPSK or 8-PSK for a target BER of 10-3. The channel is Rayleigh fading with average SNR
Adaptation rule The BER should always be smaller than 10-3
If the target BER cannot be met with either scheme, then no data is transmitted
γ 20dB
35
Apply to MIMO with SVD
Decompose MIMO channels with SVD
Allocate the power according to water-filling principle and adaptive modulation
Can transmit same (achieve diversity gain) or different (achieve multiplexing gain) data streams on the parallel channels
36
Achieving diversity using SVD
1 2 3 4 5 6 7 8 9 10 1110
-6
10-5
10-4
10-3
10-2
SNR in dB
Bit
err
or
rate
2 by 2 MIMO, SVD
1 by 4 SIMO, MRC
Achieve diversity order of 4!
0 5 10 15 20 25 3010
-6
10-5
10-4
10-3
10-2
10-1
100
SNR (dB)
Bit
Err
or
Ra
te
Rayleigh fading channelwith 1 receive antenna
Diversity order 2
Diversity order 4
37
Achieving multiplexing gain using SVD
Transmit different data streams on parallel channels
Use water filling to distribute power on the channels
Transmission rate on each channel is adapted to the effective channel gain
38
MIMO Detection
SVD requires channel state information at both the transmitter and receiver
When the transmitter doesn’t have knowledge of the channel, each antenna transmits independent data streams
The received signal
Our target is to detect original signals x from the received signal y
1 11 N M M NN y H x n
39
MIMO MLD
Let’s first consider optimum receivers in the sense of maximum likelihood detection (MLD)
In MLD we wish to maximize the probability of p(y|x) To calculate p(y|x) we observe that the distribution
must be jointly Gaussian
We need to find an optimal x from the set of all possible transmit vectors
Complexity grows exponentially!
2
00
1| exp
( )Np
NN
y Hxy x
2arg min
xy Hx
40
MIMO Zero-Forcing
We still use the idea of
Instead of minimizing only over the constellation points of x we minimize over all possible complex numbers (this is why it is sub-optimum)
In other words, we want to force
We then quantize the complex number to the nearest constellation point of x
2min y Hx
0 y Hx
41
MIMO Zero-Forcing
The decision statistics is given by
where is the pseudo-inverse of H
Requirement: N>=M
x H y
1H HH H H H
42
MIMO Zero-Forcing
Similar to passing the signal through a Gaussian channel, but with a different noise variance
The variance of the noise added to xi is given by
Problem: Noise enhancement The diversity order achieved by each stream is
given by NM+1
x H y x H n
1
0 0
H H
ii ii ii
Cov N N
H n H H H H
43
V-BLAST
The performance of ZF is not good enough, while the complexity of MLD is too high
Motivate different sub-optimal approaches V-BLAST (Vertical-Bell laboratories layered space
time) Information stream is split into M sub-streams, each of
them is modulated and transmitted from an antenna Only applicable when N>=M Based on interference cancellation Ref.: P. W. Wolniansky, G. J. Foschini, G. D. Golden, R.
A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel”, 1998
44
VBLAST
45
V-BLAST: Key idea
Successive interference cancellation Select the best bit stream and output its result
using ZF Use this result to remove the interference of
the detected bit stream from the other received signals
Then detect the best of the remaining signals and continue until all signals are detected
It is a non-linear process
46
V-BLAST receiver
V-BLAST successive interference cancelling (SIC)
The ith ZF-nulling vector wi is defined as the unique minimum-norm vector satisfying
1 1
1 1
2 1 1
2 2 2
ˆ ( ) (quantization)
ˆ (interference cancellation)
,
......
T
T
x
x Q x
x
x
w y
y y h
w y
1
0Ti j
i j
i j
w h
Tiw is the ith row of H+
47
V-BLAST optimal ordering
Problem in SIC: error propagation If the first decode channel is in low SNR, may
decode in error and propagate to subsequent decoding process
Ordered Successive Interference Cancellation (OSIC) Idea: Detect the symbols in the order of
decreasing SNR Provides a reasonable trade–off between
complexity and performance (between MMSE and ML receivers)
Achieves a diversity order which lies between N − M + 1 and N for each data stream
48
V-BLAST Performance
M N