mi mo lecture

8/11/2019 Mi Mo Lecture

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15/11/02

Capacity of multiple-input multiple-output

(MIMO) systems in wireless communications

Bengt Holter

Department of TelecommunicationsNorwegian University of Science and Technology

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Outline 15/11/02

Introduction

Channel capacity

Single-Input Single-Output (SISO)

Single-Input Multiple-Output (SIMO)

Multiple-Input Multiple-Output (MIMO)

MIMO capacity employing space-time block coding (STBC)

Outage capacity SISO

SIMO

MIMO employing STBC

Summary

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Introduction 15/11/02

MIMO = Multiple-Input Multiple-Output

Initial MIMO papers

I. Telatar, Capacity of multi-antenna gaussian channels, AT&T TechnicalMemorandum, jun. 1995

G. J. Foschini, Layered space-time architecture for wireless communication ina fading environment when using multi-element antennas, Bell Labs TechnicalJournal, 1996

MIMO systems are used to (dramatically) increase the capacity andquality of a wireless transmission.

Increased capacity obtained with spatial multiplexing of transmitteddata.

Increased quality obtained by using space-time coding at the trans-mitter.

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Entropy 15/11/02

For a discrete random variable X with alphabet X and distributedaccording to the probability mass function p(x), the entropy is defined

as

H(X) =xX

log21

p(x)p(x) =

xX

log2p(x) p(x) =E

log21

p(x)

.

(1)

The entropy of a random variable is a measure of the uncertainty ofthe random variable; it is a measure of the amount of informationrequired on the average to describe the random variable.

With a base 2 logarithm, entropy is measured in bits.

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Gamma distribution 15/11/02

Xfollows a gamma distribution with shape parameter >0 and scaleparameter >0 when the probability density function (PDF) of X isgiven by

fX(x) =x1ex/

() . (2)

where () is the gamma function (() =

0

ett1dt [()>0]).

The short hand notation X G(, ) is used to denote that X followsa gamma distribution with shape parameter and scale parameter .

Mean: x=E{X}= .

Variance: 2x =E{X2} 2x = 2.

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SISO contd 15/11/02

The mutual information between X and Y can also be written asI(X; Y) =H(Y) H(Y|X). (4)

From the equation above, it can be seen that mutual informationcan be described as the reduction in the uncertainty of one randomvariable due to the knowledge of the other.

The mutual information between X and Y will depend on the prop-

erties of the wireless channel used to convey information from thetransmitter to the receiver.

For a SISO flat fading wireless channel, the input/output relations(per channel use) can be modelled by the complex baseband notation

y =hx + n (5)

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SISO contd 15/11/02

y represents a single realization of the random variable Y (per channeluse).

h represents the complex channel between the transmitter and the

receiver.

x represents the transmitted complex symbol.

n represents complex additive white gaussian noise (AWGN).

Note that in previous lectures by Prof. Alouini, the channel gain|h|was denoted . In this presentation, is used as the shape parameterof a gamma distributed random variable.

Based on different communication scenarios,|h| may be modelled byvarious statistical distributions.

Common multipath fading models are Rayleigh, Nakagami-q (Hoyt),Nakagami-n (Rice), and Nakagami-m.

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SISO contd 15/11/02

Capacity with a transmit power constraint

With an average transmit power constraint PT, the channel capacityis defined as

C= maxp(x):PPTI(X; Y). (6)

If each symbol per channel use at the transmitter is denoted by x, theaverage power constraint can be expressed as P =E{|x|2} PT.

Compared to the original definition in (3), the capacity of the channelis now defined as the maximum of the mutual information between theinput random variable X and the output random variable Y over allstatistical distributions on the input that satisfy the power constraint.

Since both x and y are continuous upon transmission and reception,the channel is modelled as an amplitude continuous but time discretechannel.

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SISO contd 15/11/02

Noise differential entropy

Since Nalready is assumed to be a complex gaussian random variable,

i.e., the noise PDF is given by

fN(n) = 1

2nen2

2n (11)

Differential entropy

hd(N) =

fN(n)log2 fN(n)dn (12)

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SISO contd 15/11/02

Inserting the noise PDF into (12)

hd(N) = fN(n) n2 log2 e

2n log22n dn

= log2 e

2n

n2fN(n)dn + log2

2n

fN(n)dn

=E

N2

2n log2 e + log2 2n

= log2 e + log2

2n

= log2

e2n

,

where

E{N2

}=2n.

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SISO contd 15/11/02

Received signal power

Since hd(N) is given, the mutual informationI(X; Y) =hd(Y) hd(N)is maximized by maximizing h

d(Y).

Since the normal distribution maximizes the entropy over all distri-butions with the same covariance,I(X; Y) is maximized when Y isassumed gaussian, i.e., hd(Y) = log2(e

2y ), whereE{Y2}=2y .

Assuming the optimal gaussian distribution for X, the received averagesignal power 2y may be expressed as

E{Y2} = E{(hX+ N)(hX+ N)} (13)=

2

x|h|2

+ 2

n. (14)

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SISO contd 15/11/02

SISO fading channel capacity

C = hd(Y) hd(N) (15)= log2(e(

2x|h|2 + 2n)) log2(e2n) (16)

= log2 1 +2x

2n |h|2 (17)

= log2

1 +

PT

2n|h|2

, (18)

where it is assumed that 2x =PT.

Denoting the total received signal-to-noise ratio (SNR) t = PT2n |h|2,

the SISO fadig channel capacity is given by

C= log2(1 + t)

Note that since t is a random variable, the capacity also becomes arandom variable.

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SISO contd 15/11/02

Nakagami-m fading Gamma distributed SNR

With the assumption that the fading amplitude|h| is a Nakagami-mdistributed random variable, the PDF is given by

f|h|(|h|) =2mm|h|2m1m(m)

expm|h|2

(19)

where =E{|h|2} and m is the Nakagami-m fading parameter whichranges from 1/2 (half Gaussian model) to (AWGN channel).

Using transformation of random variables, it can be shown that theoverall received SNRt is a gamma distributed random variableG(, ),

ft(t) =m1t et/

m(m) , (20)

where =m and =t/m. In short t G

m, tm

where t =

PT2n

.

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SISO contd 15/11/02

0 3 6 9 12 15 18 21 240

1

2

3

4

5

6

7

Ergodic channel capacity of SISO channel with Rayleigh fading

Capac

ity

[bit

/s

/Hz

]

SNR [dB]

Ergodic capacity of a Rayleigh fading SISO channel (dotted line) compared to theShannon capacity of a SISO channel (solid line)

3dB increase in SNR 1 bit/s/Hz capacity increaseNTNU

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SIMO 15/11/02

SIMO = Single-Input Multiple-Output

For a SIMO flat fading wireless channel, the input/output relations(per channel use) can be modelled by the complex baseband notation

y = hx + n (21)

y represents a single realization of the multivariate random variable Y(array repsonse per channel use).

h represents the complex channel vector between a single transmitantenna and nR receive antennas, i.e., h = [h11, h21, . . . , hnR1]

T.

x represents the transmitted complex symbol per channel use.

n represents a complex additive white gaussian noise (AWGN) vector.

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SIMO contd 15/11/02

Desired signal covariance matrix

For a complex gaussian vector Y, the differential entropy is less than

or equal to log2 det(eKy), with equality if and only ifY is a circularlysymmetric complex Gaussian withE{YYH}= Ky.

With the assumption that the signalXis uncorrelated with all elementsin N, the received covariance matrix Ky may be expressed as

E{YYH} = E{(hX+N)(hX+ N)H} (26)= 2xhh

H +Kn (27)

where 2x =E{X2}.

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SIMO contd 15/11/02

SIMO fading channel capacity

C = hd(Y) hd(N) (28)= log2[det(e(

2xhh

H +Kn))] log2[det(eKn)] (29)

= log2[det(

2

xhh

H

+ K

n

)] log2[detKn

] (30)= log2[det((

2xhh

H +Kn)(Kn)1)] (31)= log2[det(

2xhh

H(Kn)1 + InR)] (32)= log2[det(InR+

2x(K

n)1hHh)] (33)

= log2 1 +

PT

2n ||h||

2 det(InR) (34)= log2

1 +

PT

2n||h||2

(35)

where it is assumed that Kn =2nInR and 2x =PT.

Note that for the SISO fading channel, Kn =2n.

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SIMO contd 15/11/02

The capacity formula for the SIMO fading channel could also havebeen found by assuming maximum ratio combining at the receiver.

With perfect channel knowledge at the receiver, the optimal weightsare given by

wopt= (Kn)1h. (36)

Using these weights together with the assumption that Kn = 2nInR,the overall (instantaneous) SNR t for the current observed channel h

is equal to

t=PT

2n||h||2. (37)

Thus, since t in this case represents the maximum available SNR, thecapacity can be written as

C= log2(1 + t) = log2(1 +PT

2n||h||2). (38)

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SIMO contd 15/11/02

Nakagami-m fading Gamma distributed SNR

With the assumption that all channel gains in the channel vector h areindependent and indentically distributed (i.i.d.) Nakagami-m random

variables (i.e. ml =m), then the overall SNR t is a gamma distributedrandom variable with shape parameter =nR m and scale parameter=l/m)

In short, t G(nR m, l/m).

l represents the average SNR per receiver branch (assumed equal forall branches in this case)

Coefficient of variation = tt

= 1nRm.

Effective diversity order [Nabar,02]: Ndiv= 12 =nR m.

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MIMO 15/11/02

MIMO = Multiple-Input Multiple-Output

For a MIMO flat fading wireless channel, the input/output relations(per channel use) can be modelled by the complex baseband notation

y = Hx+ n (39)

x is the (nT 1) transmit vector.

y is the (nR 1) (array response) receive vector.

H is the (nR nT) channel matrix. n is the (nR 1) additive white Gaussian noise (AWGN) vector.

H = h11 h1nTh21 h2nT... . . . ...hnR1 hnRnT

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MIMO contd 15/11/02

Mutual information

With hd() denoting differential entropy (entropy of a continuous ran-dom variable), the mutual information may be expressed as

I(X;Y) = hd(Y) hd(Y|X) (40)= hd(Y) hd(HX +N|X) (41)= hd(Y) hd(N|X) (42)= hd(Y)

hd(N) (43)

Assuming N N(0,Kn).

Since the normal distribution maxmizes the entropy over all distribu-tions with the same covariance (i.e. the power constraint), the mutual

information is maxmized when Y represents a multivariate Gaussianrandom variable.

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MIMO contd 15/11/02

Desired signal covariance matrix

With the assumption that X and N are uncorrelated, the receivedcovariance matrix Ky may be expressed as

E{YYH} = E{(HX +N)(HX+ N)H} (44)= HKxHH + Kn (45)

where Kx =E{XXH}.

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MIMO contd 15/11/02

MIMO fading channel capacity

C = hd(Y) hd(N) (46)= log2[det(e(HK

xHH +Kn))] log2[det(eKn)] (47)= log2[det(HK

xHH + Kn)] log2[detKn] (48)= log

2[det((HKxHH + Kn)(Kn)

1)] (49)

= log2[det(HKxHH(Kn)1 + InR)] (50)

= log2[det(InR+ (Kn)1HKxHH)] (51)

When the transmitter has no knowledge of the channel, it is optimal toevenly distribute the available power PTamong the transmit antennas,

i.e., Kx

= PTnTInT.

Assuming that the noise is uncorrelated between branches, the noisecovariance matrix Kn =2nInR.

The MIMO fading channel capacity can then be written as

C= log2

det

InR+

PT

nT2nHHH

. (52)

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MIMO contd 15/11/02

By the law of large numbers, the term 1nTHHH InR as nT gets largeand nR is fixed. Thus the capacity in the limit of large nT is

C=nR log2

1 +PT

2n

SI SO capacity(53)

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MIMO contd 15/11/02

Further analysis of the MIMO channel capacity is possible by diago-nalizing the product matrix HHH either by eigenvalue decompositionor singular value decomposition.

Eigenvalue decomposition of the matrix product HHH

= EEH

:

C= log2

det

InR+

PT

2nnTEEH

(54)

where E is the eigenvector matrix with orthonormal columns and isa diagonal matrix with the eigenvalues on the main diagonal.

Singular value decomposition of the channel matrix H = UVH:

C= log2

det

InR+

PT

2nnTUHUH

(55)

where U and V are unitary matrices of left and right singular vectorsrespectively, and is a diagonal matrix with singular values on themain diagonal.

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MIMO contd 15/11/02

Using the singular value decomposition approach, the capacity cannow be expressed as

C = log2 detInR+ PT

2nnTUHUH (56)

= log2

det

InT+

PT

2nnTUHU2

(57)

= log2

det

InT+

PT

2nnT2

(58)

= log2

1 + PT

2nnT21

1 + PT

2nnT22

1 + PT

2nnT2k

(59)

=

ki=1

log2

1 +

PT

2nnT2i

(60)

where k = rank{H} min{nT, nR}, is a real matrix, and det(IAB+AB) = det(IBA + BA)

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MIMO contd 15/11/02

Using the same approach with an eigenvalue decomposition of thematrix product HHH, the capacity can also be expressed as

C=

ki=1

log2

1 +

PT

2nnTi

(61)

where i are the eigenvalues of the matrix .

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MIMO contd 15/11/02

9 6 3 0 3 6 9 12 15 18 21 24 27 30 33 36 390

6

12

18

24

30

36

42

48

54

60

66

Ergodic channel capacity of a MIMO fading channel

Capac

ity

[bit

/s

/Hz

]

SNR [dB]

The Shannon capacity of a SISO channel (dotted line) compared to the ergodiccapacity of a Rayleigh fading MIMO channel (solid line) with nT =nR = 6

3dB increase in SNR 6 bits/s/Hz capacity increase!

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MIMO with STBC 15/11/02

Transmit diversity

Antenna diversity techniques are commonly utilized at the base sta-tions due to less constraints on both antenna space and power. Inaddition, it is more economical to add more complex equipment to

the base stations rather than at the remote units.

To increase the quality of the transmission and reduce multipath fadingat the remote unit, it would be beneficial if space diversity also couldbe utilized at the remote units.

In 1998, S. M. Alamouti published a paper entitled A simple transmitdiversity technique for wireless communications. This paper showedthat it was possible to generate the same diversity order tradition-ally obtained with SIMO system with a Multiple-Input Single-Output(MISO) system.

The generalized transmission scheme introduced by Alamouti has laterbeen known as Space-Time Block Codes (STBC).

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Alamouti STBC

With the Alamouti space-time code [Alamouti,1998], two consecutivesymbols{s0, s1} are mapped into a matrix codeword S according tothe following mapping:

S =

s1 s2s2 s1

, (62)

The individual rows represent time diversity and the individual columnsspace (antenna) diversity.

Assuming a block fading model, i.e., the channel remains constantfor at least T channel uses, the received signal vector x (array re-sponse/per channel use) may be expressed as

xk = Hsk+ nk, k= 1, . . . , T. (63)

[Alamouti,1998] S. M. Alamouti, A simple transmit diversity technique for wireless com-

munications, IEEE J. Select. Areas Comm., Vol.16, No.8, October 1998

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xk = Hsk+ nk, k = 1, . . . , T.

xk CnR

denotes the received signal vector per channel use.

sk CnT denotes the transmitted signal vector (a single row from thematrix codeword S transposed into a column vector).

H

CnRnT denotes the channel matrix with (possibly correlated) zero-

mean complex Gaussian random variable entries.

nk CnR denotes the additive white Gaussian noise where each entryof the vector is a zero-mean complex Gaussian random variable.

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For T consecutive uses of the channel, the received signal may beexpressed as

X = HS +N, (64)

where X = [x1, x2, ,xT] (T consecutive array responses time re-sponses in nR branches), S = [s1, s2, , sT], and N = [n1,n2, ,nT].

For notational simplicity [Hassibi,2001], the already introduced matri-ces X, S, and N may be redefined as X = [x1,x2, ,xT]T, S =[s1, s2,

, sT]

T, and N = [n1,n2,

,nT]

T.

With this new definition of the matrices X, S, and N, time runs verti-cally and space runs horizontally and the received signal for T channeluses may now be expressed as

X = SHT +N. (65)

[Hassibi,2001] B. Hassibi, B. M. Hochwald, High-rate codes that are linear in space and

time, 2001

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In [Hassibi,2001], the transpose notation on H is omitted and H is justredefined to have dimension nT nR.

For a 2 2 MIMO channel, equation (65) becomes [Hassibi,2001] x11 x12

x21 x22

=

s1 s2s2 s1

h11 h12h21 h22

+

n11 n12n21 n22

(66)

x11 and x12 represent the received symbols at antenna element no.1and 2 at time index t and likewise x21 and x22 represent the received

symbols at antenna element no.1 and 2 at time index t + Ts

This can be reorganized [Alamouti,1998] and written as

x11x21

x12x22

x

=

h11 h21h21 h11h12 h22h22 h12

H

s1

s2 s

+

n11n21

n12n22

n

(67)

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With matched filtering at the receiver (perfect channel knowledge):y = HHx

= H

H

Hs +

HHn

= ||H||2Fs + HHn. (68)where||H||2F represents the squared Frobenius norm of the matrix H.

HHH = h11 h21 h12 h22h21 h11 h22 h12

h11 h21h21 h11h12 h22h22 h12

=

|h11|2 + |h12|2 + |h21|2 + |h22|2 00

|h11

|2 +

|h12

|2 +

|h21

|2 +

|h22

|2

= ||H||2F I2.

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The overall SNR before detection of each symbol is equal to

mimot = ||H||4F|sk|2E{||HHn||2} =

||H||4FPTnT||H||2F2n

=P||H||2F. (72)

where P = PT2nnT .

For each transmitted symbol, the effective channel is a scaled AWGNchannel with SNR=P||H||2F.

The capacity of a MIMO fading channel using STBC can then be

written as

C=K

T log2

1 +

PT

2nnT||H||2F

. (73)

where KT

in front of the equation denotes the rate of the STBC.

With the Alamouti STBC, two symbols (K = 2) are transmitted intwo time slots (T= 2), i.e., the Alamouti code is a full rate STBC.

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Assuming uncorrelated channels and that all channel envelopes arei.i.d. Nakagami-m distributed random variables with equal averagepowerE{|hij|2} = , the overall SNR may be expressed as a gammadistributed random variable:

mimot = PT

nT2n ||H||2F (74)

||H

||2F

G(nT

nR, ) (75)

mimot G(N m, l/m) (76)

where N =nT nR and l = PT

2nnT .

Effective diversity order Ndiv = 12 =N m.

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Capacity summary

Note that the capacity formulas given below are obtained with theassumption of an average power constraint PT at the transmitter, un-correlated equal noise power 2n in all branches, perfect channel knowl-

edge at the receiver and no channel knowledge at the transmitter.

SISO: C= log2

1 + PT2n

|h|2

.

SIMO: C= log2

1 + PT2n

||h||2. MIMO: C= log2

InR+

PT2nnT

HHH

.

MIMO with STBC: C= log2

1 + PT2nnT

||H||2F

.

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STBC - a capacity perspective

STBC arec useful since they are able to provide full diversity over thecoherent, flat-fading channel.

In addition, they require simple encoding and decoding.

Although STBC provide full diversity at a low computational cost, itcan be shown that they incur a loss in capacity because they convert

the matrix channel into a scalar AWGN channel whose capacity issmaller than the true channel capacity.

S. Sandu, A. Paulraj,Space-time block codes: A capacity perspective, IEEE Commu-

nications Letters, Vol.4, No.12, December 2000.

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C = log2

InR+

PT

2nnTHHH

= log2

ki=1

1 +

PT

2nnT2i

= log2

1 + P k

i=1

2i + P2

i1


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When the channel matrix is a rank one matrix, there is only a singlenon-zero singular value, i.e., a space-time block code is optimal (withrespect to capacity) when it is rate one (K =T) and it is used overa channel of rank one [Sandhu,2000].

For the i.i.d. Rayleigh channel with nR > 1, the rank of the channelmatrix is greater than one, thus a space- time block code of any rateused over the i.i.d. Rayleigh channel with multiple receive antennasalways incurs a loss in capacity.

A full rate space-time block code used over any channel with onereceive antenna is always optimal with respect to capacity.

Essentially, STBC trades off capacity benefits for low complexity en-coding and decoding.

Note that with spatial multiplexing, the simplification is opposite ofSTBC. It trades of diversity benefits for lower complexity.

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Outage capacity 15/11/02


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Outage capacity

Defined as the probability that the instantaneous capacity falls belowa certain threshold or target capacity Cth

Pout(Cth) = Prob[CCth] = Cth

0

fC(C)dC=PC(Cth) (77)

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SISO capacity

C= log2

1 +

PT

2n |h|2

= log2

1 + sisot

. (78)

Assuming that|h| is Nakagami-m distributed random variable,

sisot is a Gamma distributed random variable with shape parameter=m and scale parameter =l/m.

l =E{sisot }=E{PT|h|22n }= PT

2n.

E{|h|2}= .

siso

t G(m, l/m).

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Outage capacity SISO 15/11/02

Transformation of random variables

Let X and Y be continuous random variables with Y = g(X). Sup-pose g is one-to-one, and both g and its inverse function, g1, arecontinuously differentiable. Then

fY(y) =fX[g1(y)]

dg1(y)dy

. (79) Let C=g(sisot ) = log2(1 + sisot ).

Then sisot =g1(C) = 2C 1.

Capacity PDF

fC(C) =fsisot (2C

1) 2C

ln 2 =

(2C

1)m1e(2C1)/

m(m) 2C

ln 2 (80)

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Outage capacity SISO 15/11/02

The SISO outage capacity can be obtained by solving the integral

Pout(Cth) = Cth

0

(2C 1)m1e(2C1)/

m(m) 2C ln 2

dC (81)

= 1 Q

m,(2Cth 1)m

l

(82)

Q(, ) is the normalized complementary incomplete gamma functiondefined as

Q(a, b) =(a, b)

(a) (83)

(a, b) =

b

etta1dt.

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Outage capacity - SIMO 15/11/02


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Outage capacity SIMO 15/11/02

SIMO capacity

C= log2 1 +PT

2n ||

h

||2

= log2 1 + simot . (84)

Assuming that every channel gain in the vector h,|hl|, is a Nakagami-mdistributed random variable with the same m parameter.

simo

t

is a Gamma distributed random variable with shape parameter=nR m and scale parameter =l/m.

simot G(nR m, l/m).

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Outage capacity - MIMO with STBC 15/11/02


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MIMO with STBC

C=K

T

log2 1 +PT

2

n ||H

||2F=

K

T

log2 1 + mimot . (88)

Assuming that every channel gain in the matrix H,|hij|, is a Nakagami-m distributed random variable with the same m parameter.

mimot is a Gamma distributed random variable with shape parameter

=N m (N=nT nR) and scale parameter =l/(nTm).

mimot G(N m, l/(nTm)).

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Outage capacity - MIMO with STBC 15/11/02


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Transformation of random variables

Let C=g(mimot ) = KT log2(1 + mimot ).

Then mimo

t =g1

(C) = 2(C

T)/K

1. Capacity PDF

fC(C) =fmimot (2(CT)/K 1) 2(CT)/KK

Tln 2 (89)

The MIMO outage capacity can be obtained by solving the integral

Pout(Cth) =

Cth0

(2(CT)/K 1)N m1e(2(CT)/K1)/Nm(N m) 2

(CT)/K ln 2 dC

= 1 QN m,(2(CthT)/K

1)m

nT

l (90)

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Outage capacity - MIMO 15/11/02


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MIMO capacity

Recall that C=ki=1 log2

1 + PT2n

i

.

With the assumption that all eigenvalues are i.i.d random variables

and nT = nR, the maximum capacity can be expressed as C = nTlog2(1 +

PT2n

).

Let C=g() =nT log2(1 + PT2n ).

Then =g1(C) = 2C/nT1PT/2n .

Capacity PDF

fC(C) =f 2C/nT 1

PT/2n

2C/nTnT

2n

PT

ln 2. (91)

Need to know the PDF of to obtain the capacity PDF.

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0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity in bits/s/Hz

Prob.capacity

abscissa

Capacity CDF at 10dB SNR

1x13x31x810x10

Outage capacity of i.i.d. Rayleigh fading channels at 10dB branch SNR

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity in bits/s/Hz

Prob.capacity

absciss

a

Capacity CDF at 1dB SNR

2x22x2(STBC)

Outage capacity of a 2x2 MIMO Rayleigh fading channel using the Alamouti STBC atthe transmitter at 1dB branch SNR

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Summary 15/11/02


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The capacity formulas of SISO, SIMO and MIMO fading channels havebeen derived based on maximizing the mutual information between the

transmitted and received signal.

The Alamouti space-time block code has been presented. Althoughcapable of increasing the diversity benefits, the use of STBC tradesoff capacity for low complexity encoding and decoding.

By using transformation of random variables, closed-form expressionsfor the outage capacity for SISO, SIMO and MIMO (STBC at thetransmitter) i.i.d. Nakagami-m fading channels were derived.

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Eigenvalues 15/11/02


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1/2

t + 2

t (t + 8)

(r 1)

1/2

t + 2 +

t (t + 8)

(r 1)

(r 1) (t 1)

1/4 tr+ 1 + 1/2 r+ 1/2 t + 1/4

r (t + 2)2 (8 + r) + 1/4

2t(t+8)10 tr2+16 tr+tr3+32 r+20 r2+6 rr(t

1/4 tr+ 1 + 1/2 r+ 1/2 t + 1/4

r (t + 2)2 (8 + r) 1/4

2

t(t+8)

10 tr2+16 tr+tr3+32 r+20 r2+6 r

r(t

1/4 tr+ 1 + 1/2 r+ 1/2 t 1/4r (t + 2)2 (8 + r) + 1/42 t(t+8)10 tr2+16 tr+tr3+32 r+20 r26 rr(t1/4 tr+ 1 + 1/2 r+ 1/2 t 1/4

r (t + 2)2 (8 + r) 1/4

2

t(t+8)

10 tr2+16 tr+tr3+32 r+20 r26 r

r(t

1/2 r+ 2 r (8 + r) (t 1)

1/2

r+ 2 +

r (8 + r)

(t 1)

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mi mo lecture

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