Channel Capacity of MIMO Channel Capacity of MIMO ChannelsChannels

Channel Capacity of MIMO Channel Capacity of MIMO ChannelsChannels

指導教授：黃文傑 老師指導教授：黃文傑 老師 學　　生：曾凱霖學　　生：曾凱霖 學　　號：學　　號： M9121014M9121014

無線通訊實驗室無線通訊實驗室

Outline1 、 Introduction2 、 Shannon capacity of MIMO systems 3 、 The ”pipe” interpretation4 、 To exploit the MIMO

channel– BLAST– Space Time Coding

5 、 Conclusion

Why multiple Why multiple antennas ????antennas ????

• Frequency and time processing are at limits.

• Space processing is interesting because it does not increase bandwidth.

Initial Assumptions

• Flat fading channel (Bcoh>> 1/ Tsymb)• Slowly fading channel (Tcoh>> Tsymb)• receive and nt transmit antennas• Receiver estimates the channel

perfectly• We consider space diversity only

rn

SISO Systems

y(t) = h • x(t) + n(t)

x(t): transmitted signaly(t): received signalh(t): channel transfer functionn(t): noise (AWGN, 2)

Signal to noise ratio : Capacity : C = log2(1+)

x(t)y(t)

h

2T

σ

Pρ

Receive DiversityH11

H21

= log2[1+(PT2)·|H|2] [bit/(Hz·s)]

H = [ H11 H21]

Capacity increases logarithmically with number of receive antennas...

*

22 detlog HHIt

T

nσ

PC

Transmit Diversity / Beamforming

H11

H12

Cdiversity = log2(1+(PT2)·|H|2) [bit/(Hz·s)]

•Capacity increases logarithmically with nt

MIMO SystemsH11

H22

H12

H21

2221

1211

HH

HHH

Cdiversity = log2det[I +(PT2 )·HH†]=

222122 2

1log2

1log

TT PP

Where the i are the eigenvalues to HH†

m=min(nr, nt) parallel channels, equal power allocated to each ”pipe”

Interpretation:

ReceiverTransmitter

MIMO Capacity in General

m

ii

t

T

t

T

n

P

HHn

PIC

122

*22

1log

detlog

H unknown at TX H known at TX

m

i

iipC1

22 1log

Where the power distribution over”pipes” are given by a water filling solution

p1

p2

p3

p4

),min( tr nnm

The Channel Eigenvalues

Orthogonal channels HH† =I, 1= 2= …= m= 1

)/1(log),min(1log 22

122 tTrt

m

ii

t

T nPnnn

PC

diversity

• Capacity increases linearly with min( nr , nt )• An equal amount of power PT/nt is allocated to each ”pipe”

Transmitter Receiver

To Exploit the MIMO Channel

Time

s0

s0

s0

s0

s0

s0

s1

s1

s1

s1

s1

s2

s2

s2

s2

V-BLAST

D-BLAST

Ante

nna

s1 s1 s1 s1 s1 s1

s2 s2 s2 s2 s2 s2

s3 s3 s3 s3 s3 s3

• nr nt required• Symbol by symbol detection. Using nulling and symbol cancellation• V-BLAST implemented -98 by Bell Labs (40 bps/Hz)

Bell Labs Layered Space Time Architecture

{G.J.Foschini, Bell Labs Technical Journal 1996 }

Space Time Coding

• Use parallel channel to obtain diversitydiversity not spectral efficiency as in BLAST• Space-Time trellistrellis codes : coding and and diversity gain (require Viterbi detector)• Space-Time blockblock codes : diversity gain

(use outer code to get coding gain)• nr= 1 is possible• Properly designed codes acheive diversity of nr nt

Orthogonal Space-time Block Codes

STBC

Block of K symbols

• K input symbols, T output symbols T K• R=K/T is the code rate code rate • If R=1 the STBC has full rate full rate • If T= If T= nt the code has minimum delayminimum delay• Detector is Detector is linearlinear !!! !!!

Block of T symbols

nt transmit antennas

Constellation mapper

Data in

STBC for 2 Transmit Antennas

[ c0 c1 ]

*01

*10

cc

cc

Time

Antenna

Full rateFull rate andminimum delayminimum delay

1*02

*111

012010

nchchrnchchr

Assume 1 RX antenna:

Received signal at time 0

Received signal at time 1

ncHr

1

0*1

0*1

*2

21*

1

0 ,,,c

c

n

n

hh

hh

r

rcnHr

ncHnHcHHrHr ~~ 2*** F

Diagonal matrix due to orthogonality

The MIMO/ MISO system is in fact transformed to an equivalent SISO system with SNR

SNReq = || H ||F2 SNR/nt

|| H ||F2 =

Conclusion

• MIMO systems are a promising technique for high data rates.

• Their efficiency depends on the channel between the transmitters and the receivers (power and correlation).

• Practical issues need to be resolved.