mimo system (potential candidate for 4g system)

An Overview of MIMO Systems inWireless Communications

Lecture in “Communication Theory for Wireless Channels”

Sebastien de la Kethulle — September 27, 2004

An Overview of MIMO Systems in Wireless Communications 1

Future Broadband Wireless Systems

• Desired attributes

– Significant increase in spectral efficiency and data rates

– High Quality–of–Service (QoS) — bit error rate, outage, . . .

– Wide coverage

– Low deployment, maintenance and operation costs

• The wireless channel is very hostile

– Severe fluctuations in signal level (fading)

– Co–channel interference

– Signal power falls off with distance (path loss)

– Scarce available bandwidth

– . . .

[1]An Overview of MIMO Systems in Wireless Communications 2

The Wireless Channel

• Multipath propagation causes signal fading


MIMO System


Performance Improvements Using MIMO Systems

• Array gain =⇒ increase coverage and QoS

• Diversity gain =⇒ increase coverage and QoS

• Multiplexing gain =⇒ increase spectral efficiency

• Co–channel interference reduction =⇒ increase cellular capacity


Array Gain

• Increase in average received SNR obtained by coherently combiningthe incoming / outgoing signals

• Requires channel knowledge at the transmitter / receiver

[2, 3]An Overview of MIMO Systems in Wireless Communications 6

Array Gain

λ1, . . . , λm =

(eig

`HH†

´if M < N

eig`H†H

´if M ≥ N

y = Hx + n

• H ∈ CM×N (E|Hik|2 = 1). x ∈ CN , y ∈ CM

• n ∈ CM : zero–mean complex Gaussian noise

• Principle: To obtain the full array gain, one should transmit using themaximum eigenmode of the channel

• The singular value decomposition (SVD) H = UDV†, withD = diag(

√λ1, . . . ,

√λm, 0, . . . , 0) and m = min{N,M}, yields

m equivalent SISO channels

y = Dx + n,

where y = U†y, x = V†x and n = U†n (U,V unitary)


Array Gain

y = Dx + n

• If λi = λmax = max{λ1, . . . , λm}, (maximum eigenmode)

yi =√

λmax xi + ni

• Known results

– For N × 1 and 1×M arrays, the array gain (increase in averageSNR) is respectively of 10 log10 N and 10 log10 M dB

– In the asymptotic limit, with M large:

λmax < (√

c + 1)2M c = NM ≥ 1

λmin > (√

c− 1)2M c = NM > 1

• For maximum

– Capacity: waterfilling (later in this presentation)

– Array gain: use only the maximum eigenchannel


Diversity Gain

• Principle: provide the receiver with multiple identical copies of agiven signal to combat fading =⇒ gain in instantaneous SNR


Diversity Gain

• Intuitively, the more independently fading, identical copies of agiven signal the receiver is provided with, the faster the bit error rate(BER) decreases as a function of the per signal SNR. At high SNRvalues, it has been shown that

Pe ≈ (Gc · SNR)−d

where d represents the diversity gain and Gc the coding gain

• Definition: For a given transmission rate R, the diversity gain is

d(R) = − limSNR→∞

log(Pe(R,SNR))log SNR

, (1)

where Pe(R,SNR) is the BER at the given rate and SNR

• Independent versus correlated fading

• Diminishing return for each extra signal copy

[3, 5, 6]An Overview of MIMO Systems in Wireless Communications 10

Diversity Gain

L , d

←− per receive antenna

• The diversity gain is the magnitude of the slope of the BER Pe(R, SNR) plotted

as a function of SNR on a log–log scale


Multiplexing Gain

• Principle: Transmit independent data signals from differentantennas to increase the throughput


Co–Channel Interference


Co–Channel Interference Reduction

• N − 1 interferees can be cancelled with N transmit antennas

• M − 1 interferers can be cancelled with M receive antennas


Capacity of MIMO Systems — The Gaussian Channel

y = Hx + n,

with:

• H ∈ CM×N with uniform phase and Rayleigh magnitude (Rayleighfading environment)—i.i.d. Gaussian, zero–mean, independent realand imaginary parts, variance 1/2

• x ∈ CN , y ∈ CM

• n: zero–mean complex Gaussian noise. Independent and equalvariance real and imaginary parts. E [nn†] = IM

• Transmitter power constraint: E [x†x] = tr(E [xx†]

)≤ P


Circularly Symmetric Random Vectors

Definition: A complex Gaussian random vector x ∈ Cn is said to becircularly symmetric if the corresponding vector

x ∈ R2n =[

Re(x)Im(x)

]has the structure

E[(x− E [x])(x− E [x])†

]=

12

[Re(Q) −Im(Q)Im(Q) Re(Q)

]

for some Hermitian non–negative definite Q ∈ Cn×n


Circularly Symmetric Random Vectors

The pdf of a CSCG random vector x with mean µ and covariance matrixQ is given by

fµ,Q(x) =1

det πQexp

[− (x− µ)†Q−1(x− µ)

]

and has differential entropy

h(X) = −∫

Cnfµ,Q(x) log fµ,Q(x) dx

= log det πeQ


The Deterministic Gaussian Channel — Capacity

y = Hx + n, E [x†x] ≤ P

Idea: Maximize the mutual information between x and y

I(X;Y) = h(Y)− h(Y|X)

= h(Y)− h(N)

=⇒ Maximize h(Y)


Maximizing h(Y)

It can be shown that:

• If x satisfies E [x†x] ≤ P , then so does x− E [x]

• For all y ∈ CM , h(Y) is maximized if y is Circularly SymmetricComplex Gaussian (CSCG)

• If x ∈ CN is CSCG with covariance Q, then y = Hx + n ∈ CM is alsoCSCG

=⇒ I(X;Y) = log detπe(IM + HQH†)− log det πe

= log det(IM + HQH†)

• A non–negative definite Q such that I(X;Y) is maximum andtr(Q) ≤ P remains to be found


Deterministic Gaussian MIMO Channel

• H known at the transmitter (“waterfilling solution”): Choose Qdiagonal, such that

Qii = (α− λ−1i )+, i = 1, . . . , N

with (·)+ , max(·, 0), (λ1, . . . , λN) the eigenvalues of H†H and α suchthat

∑i Qii = P . The capacity is given by:

CWF =N∑

i=1

(log(αλi)

)+ [bits/s/Hz]

• H unknown at the transmitter: Choose Q = PN IN (equal power).

Then,CEP = log det(IM + P

NHH†) [bits/s/Hz]


Waterfilling Solution


Rayleigh Fading MIMO Channel

• Memoryless Rayleigh fading Gaussian channel (unknown at thetransmitter)

• Choose x CSCG and Q = PN IN . The ergodic capacity is given by:

CEP = EH[log det(IM + P

NHH†)]

[bits/s/Hz]

= EH[ m∑

i=1

log(1 + P

Nλi

)],

where m = min(N,M) and λ1, . . . , λm are the eigenvalues of theWishart matrix

W ={

HH† M < NH†H M ≥ N

• For large SNR, CEP = min(N,M) log P +O(1), i.e. the capacitygrows linearly with min(N,M)!


Capacity of Fading Channels

• Rayleigh fading: the capacity grows linearly with min(N,M)

• Ricean channels: Increasing the line–of–sight (LOS) strength at fixedSNR reduces the capacity

• If the gains in H become highly correlated, there is a capacity loss

• Waterfilling (WF) capacity gains over Equal Power (EP) capacityare significant at low SNR but converge to zero as the SNR increases

=⇒ Question: Is it beneficial to feed the channel state back to thetransmitter ?

• Many exact capacity results are known for i.i.d. Rayleigh channels.For other channels (Rice, etc.), we have many limiting results


Ergodic Capacity of Ideal MIMO Systems

MT , NMR , M

Channel unknown at the transmitter, i.i.d. Rayleigh fading


Outage Capacity

• The capacity of a fading channel is a random variable

• Definition: The q% outage capacity Cout,q of a fading channel is theinformation rate that is guaranteed for (100− q)% of the channelrealizations, i.e.

P (I(X;Y) ≤ Cout,q) = q%

• Since, for large SNR and i.i.d. Rayleigh fading,

C = min(N,M) log SNR +O(1),

we can define the multiplexing gain r as

r = limSNR→∞

C(SNR)log SNR

,

which comes at no extra bandwidth or power


Outage Capacity of Ideal MIMO Systems

MT , NMR , M

Channel unknown at the transmitter, i.i.d. Rayleigh fading


Transmission over MIMO channels

We can use the advantages provided by MIMO channels to:

• Maximize diversity to combat channel fading and decrease the biterror rate (BER) =⇒ space–time codes (STC)

• Maximize the throughput =⇒ spatial multiplexing, V–BLAST (Belllaboratories layered space–time)

• Try to do both at the same time =⇒ trade–off between increasing thethroughput and increasing diversity


Maximizing Diversity with Space–Time Codes

• Space–Time Trellis Codes (STTC) ←− often better performanceat the cost of increased complexity

– Complex decoding (vector version of the Viterbi algorithm) —increases exponentially with the transmission rate

– Full diversity. Coding gain

• Space–Time Block Codes (STBC)

– Simple maximum–likelihood (ML) decoding based on linearprocessing

– Full diversity. Minimal or no coding gain


Alamouti Scheme for Transmit Diversity (STBC)

{r1 = h1c1 + h2c2 + n1 [time t]r2 = −h1c

∗2 + h2c

∗1 + n2 [time t + T ]

=⇒{

r1 = h∗1r1 + h2r∗2 = (|h1|2 + |h2|2)c1 + h∗1n1 + h2n

∗2 −→ c1

r2 = h∗2r1 − h1r∗2 = (|h1|2 + |h2|2)c2 − h1n

∗2 + h∗2n1 −→ c2

• Assumption: the channel remains unchanged over two consecutivesymbols

• Rate = 1 — Diversity order = 2 — Simple decoding


STBC Receiver Structure


STBCs from Complex Orthogonal Designs

• Alamouti’s scheme works only when N = 2 =⇒ Generalization

• Definition: A complex orthogonal design Oc of size N is anorthogonal matrix with entries in the indeterminates±x1,±x2, . . . ,±xN , their conjugates ±x∗1,±x∗2, . . . ,±x∗N or multiplesof these indeterminates by ±

√−1

• Example (2× 2): Oc(x1, x2) =(

x1 x2

−x∗2 x∗1

)space −→ time

↓

• Coding scheme (using a constellation A with 2b elements):

1. At time slot t, Nb bits arrive at the encoder. Select constellationsignals c1, . . . , cN

2. Set xi = ci to obtain a matrix C = Oc(c1, . . . , cN)3. At each time slot t = 1, . . . , N , the entries Cti, i = 1, . . . , N are

transmitted simultaneously from transmit antennas 1, 2, . . . , N


STBCs from Complex Orthogonal Designs

• The maximum–likelihood detection rule reduces to simple linearprocessing for STBCs

• One can obtain the maximum possible diversity order MN attransmission rate R = 1 using STBCs based on orthogonal designs

• However: complex orthogonal designs exist only if n = 2. . . !


Generalized Complex Orthogonal Designs (GCOD)

• Definition: Let Gc be a p×N matrix with entries in the indeterminates±x1,±x2, . . . ,±xk, their conjugates ±x∗1,±x∗2, . . . ,±x∗k or multiples ofthese indeterminates by ±

√−1 or 0. If G†cGc = (|x1|2 + · · ·+ |xk|2)I,

then Gc is referred to as a generalized complex orthogonal design of sizeN and rate R = k/p

• Definition: Generalized complex linear processing orthogonal design(GCLPOD) Lc: exactly like above, but the entries can be linearcombinations of x1, . . . , xk and their conjugates

• One can obtain a diversity order of MN at rate R using a STBCbased on a GCOD or a GCLPOD of size N and rate R


Generalized Complex Orthogonal Designs

• Generalized complex linear processing orthogonal designs of rates:

– R = 1 exist for N = 2– R = 3/4 exist for N = 3 and N = 4– R = 1/2 exist for N ≥ 5

• For N ≥ 3, it is not known whether GCLPODs with higher rates exist

• Example (GCLPOD, R = 34, N = 3 and GCOD, R = 1

2, N = 3):

L3c =

x1 x2

x3√2

−x∗2 x∗1x3√

2x∗3√

2

x∗3√2

−x1−x∗1+x2−x∗22

x∗3√2− x∗3√

2

x2+x∗2+x1−x∗12

G3c =

x1 x2 x3

−x2 x1 −x4

−x3 x4 x1

−x4 −x3 x2

x∗1 x∗2 x∗3−x∗2 x∗1 −x∗4−x∗3 x∗4 x∗1−x∗4 −x∗3 x∗2


Capacity and Space–Time Block Codes

• Space–time block codes

– have extremely low encoder/decoder complexity

– provide full diversity

• However

– For the i.i.d. Rayleigh channel, STBCs result in a capacity loss inthe presence of multiple receive antennas

– STBCs are only optimal with respect to capacity when they haverate R = 1 and there is one receive antenna


Maximizing the Throughput with V–BLAST


Maximizing the Throughput with V–BLAST

Description

• Transmitters operate co–channel, symbol synchronized

• Substreams are exactly independent (no coding across the transmitantennas — each substream can be individually coded)

• Individual transmit powers scaled by 1N so the total power is kept

constant

• Channel estimation burst by burst using a training sequence

• Requires near–independent channel coefficients


Receivers for Spatial Multiplexing

y = Hx + n, i.e.

y1

y2...

yM

=

h11 h12 · · · h1N

h21. . . ...

... . . . ...hM1 · · · · · · hMN

x1

x2...

xN

+

n1

n2...

nM

• If we transmit a block of N × T symbols, we have Y = HX + N, with

Y,N ∈ CM×T and X ∈ CN×T

• Optimal (ML) Receiver: x = arg minx

∥∥y −Hx∥∥

– Exhaustive search (often prohibitive complexity)

– Diversity order for each data stream: M (N ≤M)



y = Hx + n

• Zero–forcing (ZF) Receiver:

x = H#y

with H# = (H†H)−1H† (pseudo–inverse)

– Significantly reduced receiver complexity

– Noise enhancement problem

– Diversity order for each data stream: M −N + 1 (N ≤M)

[3, 4, 6, 12]An Overview of MIMO Systems in Wireless Communications 39


y = Hx + n

• Minimum mean–square error (MMSE) Receiver:

x = W · y, where W = arg minWE[∥∥Wy − x

∥∥2].

We obtain:

x = H†(HH† + E

[nn†

])−1

· y

– Minimizes the overall error due to noise and mutual interference

– Equivalent to the zero–forcing receiver at high SNR

– Diversity order for each data stream: approximately M −N + 1(N ≤M)

[3, 4, 6, 12]An Overview of MIMO Systems in Wireless Communications 40


y = Hx + n, H =[

h1 h2 · · · hN

]• V–BLAST receiver — successive interference cancellation (SIC):

x1 = wT1 y

x1 = Q(x1) (quantization)

y2 = y − x1h1 (interference cancellation)

x2 = wT2 y2, etc.

• The ith ZF–nulling vector wi is defined as the unique minimum–normvector satisfying

wTi hj =

{0 j > i1 j = i,

is orthogonal to the subspace spanned by the contributions to yi dueto the symbols not yet estimated and cancelled and is given by the ithrow of H# = (H†H)−1H† (N ≤M)



y = Hx + n, H =[

h1 h2 · · · hN

]• V–BLAST receiver

– The SNR of xi is proportional to 1/‖wi‖2– Idea: detect the components xi in order of decreasing SNR =⇒

ordered successive interference cancellation (OSIC)

initialization: G1 = H# Gi =ˆ

g1i g2

i · · · gNi

˜T

i = 1

y1 = y

recursion: ki = arg minj /∈{k1,...,ki−1}‚‚gj

i

‚‚2

wki= gki

iexki= wT

kiyi

xki= Q(exki

)

yi+1 = yi − xkihki

Gi+1 = H#

kiHki

, H with columns k1, · · · , ki set to 0

i = i + 1



• The V–BLAST SIC receiver:

– Provides a reasonable trade–off between complexity and performance(between MMSE and ML receivers)

– Achieves a diversity order of approximately M −N + 1 per datastream (N ≤M)

• The V–BLAST OSIC receiver:

– Provides a reasonable trade–off between complexity and performance(between MMSE and ML receivers)

– Achieves a diversity order which lies between M −N + 1 and M foreach data stream (N ≤M)


Performance Comparison

←− diversity receiver

←− SIC←− OSIC

↓N

↓M


Performance Comparison


D–BLAST


Linear Dispersion Codes

• V–BLAST

– is unable to work with fewer receive than transmit antennas

– doesn’t have any built–in spatial coding

• Space–time codes do not perform well at high data rates

• Linear dispersion codes

– include V–BLAST and the orthogonal design STBCs as special cases

– can be used for any number of transmit and receive antennas

– can be decoded with V–BLAST like algorithms

– satisfy an information–theoretic optimality criterion



• A linear dispersion code of rate R = kp b is one for which

X =k∑

i=1

(ciCi + c∗iDi), X =

x1

x2

...xp

where ci, . . . , ck belong to a constellation A with 2b symbols andCi,Di ∈ Cp×N

Number of transmit antennas: NNumber of receive antennas: M



• If Y = XHT +N, it can be shown that: (H ∈ CM×N ; Y,N ∈ Cp×M) y1...

yM

︸︷︷︸

η

= H

c1...ck

︸︷︷︸

ξ

+

n1...

nM

,Y =

[y1 · · · yM

]N =

[n1 · · · nM

]where yi ,

[Re(yi)Im(yi)

], ni ,

[Re(ni)Im(ni)

], ci ,

[Re(ci)Im(ci)

]and

H ∈ C2Mp×2k = f(H,C1, . . .Ck,D1, . . .Dk)

• V–BLAST like techniques can thus be used to decode lineardispersion codes

• {C1, . . . ,Ck,D1, . . . ,Dk} are dispersion matrices designed to optimizegiven criteria (e.g. maximum mutual information between η and ξ)


Diversity vs. Multiplexing Trade–off

C = min{N,M} log SNR +O(1)

• Definition: A scheme {C(SNR)} is a family of codes of block lengthl, one for each SNR level. R(SNR) [b/symbol] denotes the rate of thecode C(SNR)

• Definition: A scheme {C(SNR)} is said to achieve spatialmultiplexing gain r and diversity gain d if the data rate

limSNR→∞

R(SNR)log SNR

= r

and the average error probability

limSNR→∞

log Pe(SNR)log SNR

= −d (2)



• For each r, d∗(r) is the supremum of the diversity gains achievedover all schemes

• We also define:

– d∗max , d∗(0), the maximal diversity gain

– r∗max , sup{r|d∗(r) > 0}, the maximal spatial multiplexing gain

• Theorem: Assume l ≥ N + M − 1. The optimal trade–off curved∗(r) is given by the piecewise–linear function connecting the points(k, d∗(k)), k = 0, 1, . . . ,min{N,M}, where

d∗(k) = (N − k)(M − k).

In particular, d∗max = NM and r∗max = min{N,M}.


Diversity vs. Multiplexing: Optimal Trade–off

m , Nn , M


Diversity vs. Multiplexing Trade–off: V–BLAST

n , N = M


Diversity vs. Multiplexing Trade–off: Alamouti Scheme

m , Nn , M



• Definitions (1) and (2) for the diversity gain are not equivalent: inthe former one, a fixed data rate is assumed for all SNRs, whereas inthe latter one, the data rate is a fraction of C(SNR), and henceincreases with the SNR

• Definition (1) is the most widely used in the literature

• Definition (2) allows to quantify the diversity vs. multiplexingtrade–off


MIMO Channel Modeling

• A good MIMO channel model must include:

– Path loss

– Shadowing

– Doppler and delay spread profiles

– Ricean K factor distribution

– Joint antenna correlation at transmit and receive ends

– Channel matrix singular value distribution


Ricean K factor distribution

H = HLOS + HNLOS

• The higher the Ricean K factor, the more dominant HLOS

(line–of–sight)

• HLOS is a time–invariant, often low rank matrix =⇒ high K factorchannels often exhibit a low capacity

• In a near–LOS link, the improvement in link budget often more thancompensates for the loss of MIMO capacity =⇒ usually, the LOScomponent is not intentionally reduced

• Experimental measurements show that, in general:

– K increases with antenna height– K decreases with transmitter–receiver distance =⇒ MIMO

substantially increases throughput in areas far away from the basestation


Correlation Model for HNLOS

“One–ring” model

• Base Station (BS) usually elevated and unobstructed by local scatterers

• Subscriber Unit (SU) often surrounded by local scatterers — assumedhere uniformly distributed in θ

TAl : lth transmitting antenna element Θ : angle of arrivalRAl : lth receiving antenna element ∆ : angle spreadS(θ) : scatterer located at angle θ



• Correlation from one BS antenna element to two SU antenna elements:

E [Hl,pH∗m,p] ≈ J0

(2π

λd(l,m)

)

• Correlation from two BS antenna elements to one SU antenna elementin the broadside direction (Θ = 0):

E [Hm,pH∗m,q] ≈ J0

(∆

2π

λd(p, q)

)

• Correlation from two BS antenna elements to one SU antenna elementin the inline direction (Θ = π

2):

E [Hm,pH∗m,q] ≈ e−j2πλ d(p,q)

(1−∆2

4

)· J0

((∆2

)22π

λd(p, q)

)

↑distance between antennas l and m

↑distance between antennas p and q



←− J0(x)

• The mobiles have to be in the broadside direction to obtain the highestdiversity

• Interelement spacing has to be high to have low correlation =⇒beamforming and MIMO yield conflicting criteria

• Using the above results, one can obtain upper bounds for the MIMOcapacity


Decoupling Between Rank and Correlation

Pinhole channel

• Uncorrelated fading at both ends doesn’t necessarily imply ahigh–rank channel


MIMO Channel Modeling

• Time–varying wideband MIMO channel:

H(τ) =L∑

i=1

Hiδ(τ − τi)

where H(τ) ∈ CM×N and only H1 contains a LOS component

• Typical interelement spacing:

– Base station: 10λ (due to the absence of local scatterers)

– Subscriber unit: 12λ (rich scattering)


MIMO–OFDM Systems

SISO OFDM Transmitter SISO OFDM Receiver

N , K, l = OFDM symbol number N , K

• Net result: The frequency selective fading channel of bandwidth B isdecomposed into K parallel frequency-flat fading channels, eachhaving bandwidth B

K . (Condition: The impulse response of thechannel is shorter than the length of the cyclic prefix)


MIMO–OFDM Systems

• OFDM can be extended to MIMO systems by performing theIDFT/DFT and CP operations at each of the transmit and receiveantennas (with the appropriate condition on the length of the cyclicprefix)

• Diversity systems: (Ex: Alamouti scheme)

– Send c1 and c2 over OFDM tone i over antennas 1 and 2

– Send −c∗2 and c∗1 over OFDM tone i + 1 over antennas 1 and 2within the same OFDM symbol

– Alternative technique: Code on a per–tone basis across OFDMsymbols in time


MIMO–OFDM Systems

• Spatial multiplexing: Maximize spatial rate (r = min{N,M}) bytransmitting independent data streams over different antennas =⇒spatial multiplexing over each tone

• Space–frequency coded MIMO–OFDM

– OFDM tones with spacing larger than the coherence bandwidthBC experience independent fading

– If Deff = BBC

, the total diversity gain that can be realized is ofNMDeff


Throughput in MIMO Cellular Systems


Conclusions

• MIMO channels offer multiplexing gain, diversity gain, array gainand a co–channel interference cancellation gain

• Careful balancing between those gains is required

• MIMO systems offer a promising solution for future generationwireless networks

• Ongoing research

– Space–time coding (orthogonal designs, etc.)

– Receiver design (ML receiver is too complex)

– Channel modeling

– Capacity of non–ideal MIMO channels

– . . .


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’ ’

[15] B. Hassibi and B. M. Hochwald, “High–rate codes that are linear in space and time,” IEEE Trans. Inform. Theory, vol. 48,no. 7, pp. 1804–1824, July 2002.

[16] M. J. Gans D. Shiu, G. J. Foschini and J. M. Kahn, “Fading correlation and its effect on the capacity of multielementantenna systems,” IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000.

[17] M. Sandell, Design and analysis of estimators for multicarrier modulation and ultrasonic imaging, Ph.D. thesis, LuleaUniversity, Sweden, 1996.


mimo system (potential candidate for 4g system)

Education

overview of mimo systems

mimo systems array gain

capacity of mimo systems

array gain y

array gain increase

log10 n

signal snr

qos diversity gain