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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
[1]An Overview of MIMO Systems in Wireless Communications 3
MIMO System
An Overview of MIMO Systems in Wireless Communications 4
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
[1]An Overview of MIMO Systems in Wireless Communications 5
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)
[2, 3]An Overview of MIMO Systems in Wireless Communications 7
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
[2, 3]An Overview of MIMO Systems in Wireless Communications 8
Diversity Gain
• Principle: provide the receiver with multiple identical copies of agiven signal to combat fading =⇒ gain in instantaneous SNR
[4]An Overview of MIMO Systems in Wireless Communications 9
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
[4, 6]An Overview of MIMO Systems in Wireless Communications 11
Multiplexing Gain
• Principle: Transmit independent data signals from differentantennas to increase the throughput
[1]An Overview of MIMO Systems in Wireless Communications 12
Co–Channel Interference
[1]An Overview of MIMO Systems in Wireless Communications 13
Co–Channel Interference Reduction
• N − 1 interferees can be cancelled with N transmit antennas
• M − 1 interferers can be cancelled with M receive antennas
[1]An Overview of MIMO Systems in Wireless Communications 14
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
[7]An Overview of MIMO Systems in Wireless Communications 15
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
[7]An Overview of MIMO Systems in Wireless Communications 16
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
[7]An Overview of MIMO Systems in Wireless Communications 17
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)
[7]An Overview of MIMO Systems in Wireless Communications 18
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
[7]An Overview of MIMO Systems in Wireless Communications 19
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]
[3, 7]An Overview of MIMO Systems in Wireless Communications 20
Waterfilling Solution
An Overview of MIMO Systems in Wireless Communications 21
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)!
[3, 7]An Overview of MIMO Systems in Wireless Communications 22
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
[3]An Overview of MIMO Systems in Wireless Communications 23
Ergodic Capacity of Ideal MIMO Systems
MT , NMR , M
Channel unknown at the transmitter, i.i.d. Rayleigh fading
[6]An Overview of MIMO Systems in Wireless Communications 24
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
[1, 3, 6]An Overview of MIMO Systems in Wireless Communications 25
Outage Capacity of Ideal MIMO Systems
MT , NMR , M
Channel unknown at the transmitter, i.i.d. Rayleigh fading
[6]An Overview of MIMO Systems in Wireless Communications 26
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
[3, 6, 8]An Overview of MIMO Systems in Wireless Communications 27
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
[3]An Overview of MIMO Systems in Wireless Communications 28
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
[9]An Overview of MIMO Systems in Wireless Communications 29
STBC Receiver Structure
[3]An Overview of MIMO Systems in Wireless Communications 30
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
[10]An Overview of MIMO Systems in Wireless Communications 31
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. . . !
[10]An Overview of MIMO Systems in Wireless Communications 32
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
[10]An Overview of MIMO Systems in Wireless Communications 33
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
[10]An Overview of MIMO Systems in Wireless Communications 34
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
[11]An Overview of MIMO Systems in Wireless Communications 35
Maximizing the Throughput with V–BLAST
[1]An Overview of MIMO Systems in Wireless Communications 36
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
[4]An Overview of MIMO Systems in Wireless Communications 37
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)
[3, 4, 6]An Overview of MIMO Systems in Wireless Communications 38
Receivers for Spatial Multiplexing
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
Receivers for Spatial Multiplexing
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
Receivers for Spatial Multiplexing
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)
[13]An Overview of MIMO Systems in Wireless Communications 41
Receivers for Spatial Multiplexing
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
[13]An Overview of MIMO Systems in Wireless Communications 42
Receivers for Spatial Multiplexing
• 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)
[3, 6]An Overview of MIMO Systems in Wireless Communications 43
Performance Comparison
←− diversity receiver
←− SIC←− OSIC
↓N
↓M
[6]An Overview of MIMO Systems in Wireless Communications 44
Performance Comparison
[4]An Overview of MIMO Systems in Wireless Communications 45
D–BLAST
[4, 14]An Overview of MIMO Systems in Wireless Communications 46
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
[4, 15]An Overview of MIMO Systems in Wireless Communications 47
Linear Dispersion Codes
• 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
[15]An Overview of MIMO Systems in Wireless Communications 48
Linear Dispersion Codes
• 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)
[15]An Overview of MIMO Systems in Wireless Communications 49
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)
[8]An Overview of MIMO Systems in Wireless Communications 50
Diversity vs. Multiplexing Trade–off
• 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}.
[8]An Overview of MIMO Systems in Wireless Communications 51
Diversity vs. Multiplexing: Optimal Trade–off
m , Nn , M
[8]An Overview of MIMO Systems in Wireless Communications 52
Diversity vs. Multiplexing Trade–off: V–BLAST
n , N = M
[8]An Overview of MIMO Systems in Wireless Communications 53
Diversity vs. Multiplexing Trade–off: Alamouti Scheme
m , Nn , M
[8]An Overview of MIMO Systems in Wireless Communications 54
Diversity vs. Multiplexing Trade–off: Alamouti Scheme
m , Nn , M
[8]An Overview of MIMO Systems in Wireless Communications 55
Diversity vs. Multiplexing Trade–off
• 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
[6, 8]An Overview of MIMO Systems in Wireless Communications 56
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
[3]An Overview of MIMO Systems in Wireless Communications 57
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
[3]An Overview of MIMO Systems in Wireless Communications 58
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
[16]An Overview of MIMO Systems in Wireless Communications 59
Correlation Model for HNLOS
• 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
[3, 16]An Overview of MIMO Systems in Wireless Communications 60
Correlation Model for HNLOS
←− 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
[3, 16]An Overview of MIMO Systems in Wireless Communications 61
Decoupling Between Rank and Correlation
Pinhole channel
• Uncorrelated fading at both ends doesn’t necessarily imply ahigh–rank channel
[3, 4]An Overview of MIMO Systems in Wireless Communications 62
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)
[3]An Overview of MIMO Systems in Wireless Communications 63
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)
[6, 17]An Overview of MIMO Systems in Wireless Communications 64
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
[6]An Overview of MIMO Systems in Wireless Communications 65
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
[6]An Overview of MIMO Systems in Wireless Communications 66
Throughput in MIMO Cellular Systems
[1, 4]An Overview of MIMO Systems in Wireless Communications 67
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
– . . .
[1, 4]An Overview of MIMO Systems in Wireless Communications 68
References
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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.
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