Capacity of multiple-input multiple-output (MIMO) … · Capacity of multiple-input multiple-output...

15/11/02

Capacity of multiple-input multiple-output

(MIMO) systems in wireless communications

Bengt Holter

Department of TelecommunicationsNorwegian University of Science and Technology

NTNU

1

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 definedas

H(X) =∑x∈X

log21

p(x)· p(x) = −

∑x∈X

log2 p(x) · p(x) = E[log2

1

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

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

fX(x) =xα−1e−x/β

βαΓ(α). (2)

where Γ(·) is the gamma function (Γ(α) =∫ ∞0 e−ttα−1dt [(α) > 0]).

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

• Mean: EX = µx = α · β.

• Variance: EX2 = σ2x − µ2

x = α · β2.

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SISO 15/11/02

SISO = Single-input Single-output

• Representing the input and and output of a memoryless wireless chan-nel with the random variables X and Y respectively, the channel ca-pacity is defined as

C = maxp(x)

I(X;Y ). (3)

• I(X;Y ) denotes the mutual information between X and Y and itis measure of the amount of information that one random variablecontains about another random variable.

• According to the definition in (3), the mutual information is maximizedwith respect to all possible transmitter statistical distributions p(x).

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SISO cont’d 15/11/02

• The mutual information between X and Y can also be written as

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

• h represents the complex channel between the transmitter and thereceiver.

• 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 α. In this presentation, α 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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Capacity with a transmit power constraint

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

C = maxp(x):P≤PT

I(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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Assumptions

• Perfect channel knowledge at the receiver.

• X is independent of N .

• N ∼ N (0, σ2n)

Mutual information: With hd(·) denoting differential entropy (entropy ofa continuous random variable), the mutual information may be expressedas

I(X;Y ) = hd(Y ) − hd(Y |X) (7)= hd(Y ) − hd(hX + N |X) (8)= hd(Y ) − hd(N |X) (9)= hd(Y ) − hd(N) (10)

• (9) follows from the fact that since h is assumed perfectly known bythe receiver, there is no uncertainty in hX conditioned on X.

• (10) follows from the fact that N is assumed independent of X, i.e.,there is no information in X which reduces the uncertainty of N .

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Noise differential entropy

• Since N already is assumed to be a complex gaussian random variable,i.e., the noise PDF is given by

fN(n) =1

πσ2n

e−n2

σ2n (11)

• Differential entropy

hd(N) = −∫

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

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• Inserting the noise PDF into (12)

hd(N) = −∫

fN(n)

[−n2 log2 e

σ2n

− log2

(πσ2

n

)]dn

=log2 e

σ2n

∫n2f(n)dn + log2

(πσ2

n

) ∫f(n)dn

=E

N2

σ2n

log2 e + log2

(πσ2

n

)= log2 e + log2

(πσ2

n

)= log2

(πeσ2

n

),

where EN2 = σ2n.

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Received signal power

• Since hd(N) is given, the mutual information I(X;Y ) = hd(Y )−hd(N)is maximized by maximizing hd(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(πeσ2

y), where EY 2 = σ2y .

• Assuming the optimal gaussian distribution for X, the received averagesignal power σ2

y may be expressed as

EY 2 = E(hX + N)(h∗X∗ + N∗) (13)= σ2

x|h|2 + σ2n. (14)

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SISO fading channel capacity

C = hd(Y ) − hd(N) (15)= log2(πe(σ2

x|h|2 + σ2n)) − log2(πeσ2

n) (16)

= log2

(1 +

σ2x

σ2n

|h|2)

(17)

= log2

(1 +

PT

σ2n

|h|2)

, (18)

where it is assumed that σ2x = PT .

• Denoting the total received signal-to-noise ratio (SNR) γt = PT

σ2n

|h|2,the SISO fadig channel capacity is given by

C = log2(1 + γt)

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

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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

fα(α) =2mm|h|2m−1

ΩmΓ(m)exp

(m|h|2

Ω

)(19)

where Ω = 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 SNR γt is a gamma distributed random variable G(α, β),

fγt(γt) =γm−1

t e−γt/β

βmΓ(m), (20)

where α = m and β = γt/m. In short γt ∼ G(m, γt

m

)where γt = PTΩ

σ2n

.

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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

Ca

pa

city

[b

it/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 increase

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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 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) (22)= hd(Y) − hd(hX + N|X) (23)= hd(Y) − hd(N|X) (24)= hd(Y) − hd(N) (25)

• It will be assumed that N ∼ N (0,Kn), where Kn = ENNH is thenoise covariance matrix.

• Since the normal distribution maxmizes the entropy over all distribu-tions with the same covariance (i.e. the power constraint), the mutualinformation is maximized when Y represents a multivariate Gaussianrandom variable, i.e., Y = N (0,Ky) where Ky = EYYH is the co-variance matrix of the desired signal.

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SIMO cont’d 15/11/02

Desired signal covariance matrix

• For a complex gaussian vector Y, the differential entropy is less thanor equal to log2 det(πeKy), with equality if and only if y is a circularlysymmetric complex Gaussian with EYYH = Ky.

• With the assumption that the signal X is uncorrelated with all elementsin N, the received covariance matrix Ky may be expressed as

EYYH = E(hX + N)(hX + N)H (26)

= σ2xhhH + Kn (27)

where σ2x = EX2.

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SIMO fading channel capacity

C = hd(Y) − hd(N) (28)= log2[det(πe(σ2

xhhH + Kn))] − log2[det(πeKn)] (29)

= log2[det(σ2xhhH + Kn)] − log2[detKn] (30)

= log2[det((σ2xhhH + Kn)(Kn)−1)] (31)

= log2[det(σ2xhhH(Kn)−1 + InR)] (32)

= log2[det(InR + σ2x(K

n)−1hHh)] (33)

= log2

[(1 +

PT

σ2n

||h||2)

· det(InR)

](34)

= log2

(1 +

PT

σ2n

||h||2)

(35)

where it is assumed that Kn = σ2nInR and σ2

x = PT .

• Note that for the SISO fading channel, Kn = σ2n.

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• 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 = σ2nInR

,the overall (instantaneous) SNR γt for the current observed channel his equal to

γt =PT

σ2n

||h||2. (37)

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

C = log2(1 + γt) = log2(1 +PT

σ2n

||h||2). (38)

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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 randomvariables (i.e. ml = m), then the overall SNR γt is a gamma distributedrandom variable with shape parameter α = nR ·m and scale parameterβ = γl/m)

• In short, γt ∼ G(nR · m, γl/m).

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

• Coefficient of variation τ = σγt

µγt

= 1√nR·m.

• Effective diversity order [Nabar,02]: Ndiv = 1τ 2 = 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 · · · h1nT

h21 · · · h2nT

... . . . ...hnR1 · · · hnRnT

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MIMO cont’d 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 mutualinformation is maxmized when Y represents a multivariate Gaussianrandom variable.

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Desired signal covariance matrix

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

EYYH = E(HX + N)(HX + N)H (44)

= HKxHH + Kn (45)

where Kx = EXXH.

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MIMO fading channel capacity

C = hd(Y) − hd(N) (46)= log2[det(πe(HKxHH + Kn))] − log2[det(πeKn)] (47)= log2[det(HKxHH + Kn)] − log2[detKn] (48)= log2[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 PT among the transmit antennas,i.e., Kx = PT

nTInT .

• Assuming that the noise is uncorrelated between branches, the noisecovariance matrix Kn = σ2

nInR.

• The MIMO fading channel capacity can then be written as

C = log2

[det

(InR +

PT

nTσ2n

HHH

)]. (52)

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• By the law of large numbers, the term 1nT

HHH ⇒ InR as nT gets largeand nR is fixed. Thus the capacity in the limit of large nT is

C = nR · log2

(1 +

PT

σ2n

)︸︷︷︸

SISO capacity

(53)

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• 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 = EΛEH:

C = log2

[det

(InR +

PT

σ2nnT

EΛEH

)](54)

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

• Singular value decomposition of the channel matrix H = UΣVH:

C = log2

[det

(InR

+PT

σ2nnT

UΣΣHUH

)](55)

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

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

C = log2

[det

(InR +

PT

σ2nnT

UΣΣHUH

)](56)

= log2

[det

(InT

+PT

σ2nnT

UHUΣ2

)](57)

= log2

[det

(InT +

PT

σ2nnT

Σ2

)](58)

= log2

[(1 +

PT

σ2nnT

σ21

) (1 +

PT

σ2nnT

σ22

)· · ·

(1 +

PT

σ2nnT

σ2k

)](59)

=k∑

i=1

(1 +

PT

σ2nnT

σ2i

)(60)

where k = rankH ≤ minnT , nR, Σ is a real matrix, and det(IAB +AB) = det(IBA + BA)

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

C =k∑

i=1

(1 +

PT

σ2nnT

λi

)(61)

where λi are the eigenvalues of the matrix Λ.

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−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

Ca

pa

city

[b

it/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 tothe 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 s2

−s∗2 s∗1

], (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 ∈ CnR×nT 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 s2

−s∗2 −s1

] [h11 h12

h21 h22

]+

[n11 n12

n21 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 receivedsymbols at antenna element no.1 and 2 at time index t + Ts

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

x11

x∗21

x12

x∗22

︸︷︷︸x

=

h11 h21

h∗21 −h∗

11h12 h22

h∗22 −h∗

12

︸︷︷︸H

[s1

s2

]︸︷︷︸

s

+

n11

n∗21

n12

n∗22

︸︷︷︸n

(67)

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• With matched filtering at the receiver (perfect channel knowledge):

y = HHx

= HHHs + HHn

= ||H||2Fs + HHn. (68)

where ||H||2F represents the squared Frobenius norm of the matrix H.

HHH =

[h∗

11 h21 h∗12 h22

h∗21 −h11 h∗

22 −h12

]

h11 h21

h∗21 −h∗

11h12 h22

h∗22 −h∗

12

=

[|h11|2 + |h12|2 + |h21|2 + |h22|2 0

0 |h11|2 + |h12|2 + |h21|2 + |h22|2]

= ||H||2F · I2.

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• This means that the received signals after matched filtering are de-coupled and they can be written individually as

y1 = ||H||2Fs1 + HHn (69)

y2 = ||H||2Fs2 + HHn (70)

• In general, the effective channel induced by space-time block coding ofcomplex symbols (before detection) can be represented as [Sandhu,2000]

yk = ||H||2Fsk + HHn. (71)

[Sandhu,2000] S. Sandhu, A. Paulraj, ”Space-Time Block Codes: A Capacity Perspec-

tive,” IEEE Comm. Letter, Vol.4, No.12, December 2000.

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• The overall SNR before detection of each symbol is equal to

γmimot =

||H||4F |sk|2E|HHn|2 =

||H||4F PT

nT

||H||2Fσ2n

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

where P = PT

σ2nnT

.

• 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 bewritten as

C =K

T· log2

(1 +

PT

σ2nnT

||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 averagepower E|hij|2 = Ω, the overall SNR may be expressed as a gammadistributed random variable:

γmimot =

PT

nTσ2n

· ||H||2F (74)

||H||2F ∼ G(nT · nR,Ω) (75)

γmimot ∼ G(N · m, γl/m) (76)

where N = nT · nR and γl =PTΩσ2

nnT.

• Effective diversity order Ndiv = 1τ 2 = 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 σ2

n in all branches, perfect channel knowl-edge at the receiver and no channel knowledge at the transmitter.

• SISO: C = log2

(1 + PT

σ2n

|h|2).

• SIMO: C = log2

(1 + PT

σ2n

||h||2).

• MIMO: C = log2

(InR

+ PT

σ2nnT

HHH).

• MIMO with STBC: C = log2

(1 + PT

σ2nnT

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

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

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

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C = log2

(InR +

PT

σ2nnT

HHH

)

= log2

k∏i=1

(1 +

PT

σ2nnT

σ2i

)

= log2

1 + P

k∑i=1

σ2i + P 2

i1<i2∑i1 =i2

σ2i1σ

2i2 + P 3

i1<i2<i3∑i1 =i2 =i3

σ2i1σ

2i2σ

2i3 + · · · + P k

k∏i=1

σ2i

= log2

1 + P ||H||2F + P 2

i1<i2∑i1 =i2

σ2i1σ

2i2 + P 3

i1<i2<i3∑i1 =i2 =i3

σ2i1σ

2i2σ

2i3 + · · · + P k

k∏i=1

σ2i

≥ log2

(1 + P ||H||2F

)≥ K

T· log2

(1 + P ||H||2F

)• The capacity difference is a function of the channel singular values.

This can used to determine under which conditions STBC is optimalin terms of capacity.

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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

Outage capacity

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

Pout(Cth) = Prob[C ≤ Cth] =

∫ Cth

0fC(C)dC = PC(Cth) (77)

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Outage capacity - SISO 15/11/02

SISO capacity

C = log2

(1 +

PT

σ2n

· |h|2)

= log2

(1 + γsiso

t

). (78)

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

• γsisot is a Gamma distributed random variable with shape parameter

α = m and scale parameter β = γl/m.

• γl = Eγsisot = EPT |h|2

σ2n

= PTΩσ2

n

.

• E|h|2 = Ω.

• γsisot ∼ G(m, γl/m).

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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, g−1, arecontinuously differentiable. Then

fY (y) = fX[g−1(y)]

∣∣∣∣dg−1(y)

dy

∣∣∣∣ . (79)

• Let C = g(γsisot ) = log2(1 + γsiso

t ).

• Then γsisot = g−1(C) = 2C − 1.

• Capacity PDF

fC(C) = fγsisot

(2C − 1) · 2C ln 2 =(2C − 1)m−1e−(2C−1)/β

βmΓ(m)· 2C ln 2 (80)

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• The SISO outage capacity can be obtained by solving the integral

Pout(Cth) =

∫ Cth

0

(2C − 1)m−1e−(2C−1)/β

βmΓ(m)· 2C ln 2 · dC (81)

= 1 − Q

(m,

(2Cth − 1)m

γl

)(82)

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

Q(a, b) =Γ(a, b)

Γ(a)(83)

• Γ(a, b) =∫ ∞

be−tta−1dt.

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Outage capacity - SIMO 15/11/02

SIMO capacity

C = log2

(1 +

PT

σ2n

· ||h||2)

= log2

(1 + γsimo

t

). (84)

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

• γsimot is a Gamma distributed random variable with shape parameter

α = nR · m and scale parameter β = γl/m.

• γsimot ∼ G(nR · m, γl/m).

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Outage capacity - SIMO 15/11/02


• Let C = g(γsimot ) = log2(1 + γsimo

t ).

• Then γsimot = g−1(C) = 2C − 1.

• Capacity PDF

fC(C) = fγsimot

(2C − 1) · 2C ln 2 =(2C − 1)nRm−1e−(2C−1)/β

βnR·mΓ(nR · m)· 2C ln 2 (85)

• The SIMO outage capacity can be obtained by solving the integral

Pout(Cth) =

∫ Cth

0

(2C − 1)nRm−1e−(2C−1)/β

βnR·mΓ(nR · m)· 2C ln 2 · dC (86)

= 1 − Q

(nR · m,

(2Cth − 1)m

γl

)(87)

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Outage capacity - MIMO with STBC 15/11/02

MIMO with STBC

C =K

Tlog2

(1 +

PT

σ2n

· ||H||2F)

=K

Tlog2

(1 + γmimo

t

). (88)

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

• γmimot is a Gamma distributed random variable with shape parameter

α = N · m (N = nT · nR) and scale parameter β = γl/(nTm).

• γmimot ∼ G(N · m, γl/(nTm)).

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Outage capacity - MIMO with STBC 15/11/02


• Let C = g(γmimot ) = K

Tlog2(1 + γmimo

t ).

• Then γmimot = g−1(C) = 2(C·T )/K − 1.

• Capacity PDF

fC(C) = fγmimot

(2(C·T )/K − 1) · 2(C·T )/K K

Tln 2 (89)

• The MIMO outage capacity can be obtained by solving the integral

Pout(Cth) =

∫ Cth

0

(2(C·T )/K − 1)Nm−1e−(2(C·T )/K−1)/β

βN ·mΓ(N · m)· 2(C·T )/K ln 2 · dC

= 1 − Q

(N · m,

(2(Cth·T )/K − 1)m · nT

γl

)(90)

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Outage capacity - MIMO 15/11/02

MIMO capacity

• Recall that C =∑k

i=1 log2

(1 + PT

σ2n

λi

).

• With the assumption that all eigenvalues are i.i.d random variablesand nT = nR, the maximum capacity can be expressed as C = nT ·log2(1 + PT

σ2n

λ).

• Let C = g(λ) = nT · log2(1 + PT

σ2n

λ).

• Then λ = g−1(C) = 2C/nT−1PT/σ2

n

.

• Capacity PDF

fC(C) = fλ

(2C/nT − 1

PT/σ2n

)· 2C/nT

nTσ2n

PTln 2. (91)

• Need to know the PDF of λ 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

Capacit y in bits/s/Hz

Pro

b. ca

pa

city

≤

ab

scis

sa

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

Capacit y in bits/s/Hz

Pro

b.

cap

aci

ty

≤ a

bsc

issa

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

• The capacity formulas of SISO, SIMO and MIMO fading channels havebeen derived based on maximizing the mutual information between thetransmitted 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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Capacity of multiple-input multiple-output (MIMO) … · Capacity of multiple-input multiple-output...

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