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