An Overview of MIMO Systems in Wireless Communications
Lecture in “Communication Theory for Wireless Channels” S? ebastien de la Kethulle — September 27, 2004
An Overview of MIMO Systems in Wireless Communications
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Future Broadband Wireless Systems
? Desired attributes – – – – Signi?cant increase in spectral e?ciency 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 ?uctuations in signal level (fading) Co–channel interference Signal power falls o? with distance (path loss) Scarce available bandwidth ...
An Overview of MIMO Systems in Wireless Communications 2
[1]
The Wireless Channel
? Multipath propagation causes signal fading
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MIMO System
An Overview of MIMO Systems in Wireless Communications
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Performance Improvements Using MIMO Systems
? Array gain
=?
increase coverage and QoS
? Diversity gain
=?
increase coverage and QoS
? Multiplexing gain
=?
increase spectral e?ciency
? Co–channel interference reduction
=?
increase cellular capacity
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Array Gain
? Increase in average received SNR obtained by coherently combining the incoming / outgoing signals ? Requires channel knowledge at the transmitter / receiver
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Array Gain
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 the maximum eigenmode of the channel
? ? The singular value decomposition (SVD) H = UDV , with √ √ D = diag( λ1, . . . , λm, 0, . . . , 0) and m = min{N, M }, yields ( m equivalent SISO channels ` ??
λ1 , . . . , λm =
eig HH if M < N ` ? ? eig H H if M ≥ N
y = Dx + n, where y = U?y, x = V?x and n = U?n
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(U, V unitary)
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Array Gain
y = Dx + n ? If λi = λmax = max{λ1, . . . , λm}, (maximum eigenmode) yi = ? Known results – For N × 1 and 1 × M arrays, the array gain (increase in average SNR) is respectively of 10 log10 N and 10 log10 M dB – In the asymptotic limit, with M large: √ λmax < ( c + 1)2M c= √ λmin > ( c ? 1)2M c= λmax xi + ni
N M N M
≥1 >1
? For maximum – Capacity: water?lling (later in this presentation) – Array gain: use only the maximum eigenchannel
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Diversity Gain
? Principle: provide the receiver with multiple identical copies of a given signal to combat fading =? gain in instantaneous SNR
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Diversity Gain
? Intuitively, the more independently fading, identical copies of a given signal the receiver is provided with, the faster the bit error rate (BER) decreases as a function of the per signal SNR. At high SNR values, it has been shown that Pe ≈ (Gc · SNR)?d where d represents the diversity gain and Gc the coding gain ? De?nition: For a given transmission rate R, the diversity gain is log(Pe(R, SNR)) d(R) = ? lim , SN R→∞ log SNR where Pe(R, SNR) is the BER at the given rate and SNR ? Independent versus correlated fading ? Diminishing return for each extra signal copy
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(1)
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
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Multiplexing Gain
? Principle: Transmit independent data signals from di?erent antennas to increase the throughput
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Co–Channel Interference
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Co–Channel Interference Reduction
? N ? 1 interferees can be cancelled with N transmit antennas ? M ? 1 interferers can be cancelled with M receive antennas
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Capacity of MIMO Systems — The Gaussian Channel
y = Hx + n, with: ? H ∈ CM ×N with uniform phase and Rayleigh magnitude (Rayleigh fading environment)—i.i.d. Gaussian, zero–mean, independent real and imaginary parts, variance 1/2 ? x ∈ CN , y ∈ CM ? n: zero–mean complex Gaussian noise. Independent and equal variance real and imaginary parts. E [nn?] = IM ? Transmitter power constraint: E [x?x] = tr E [xx?] ≤ P
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Circularly Symmetric Random Vectors
De?nition: A complex Gaussian random vector x ∈ Cn is said to be circularly symmetric if the corresponding vector ? ∈ R2 n = x has the structure E (? x ? E [? x])(? x ? E [? x])? = 1 2 Re(Q) ?Im(Q) Im(Q) Re(Q) Re(x) Im(x)
for some Hermitian non–negative de?nite Q ∈ Cn×n
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Circularly Symmetric Random Vectors
The pdf of a CSCG random vector x with mean ? and covariance matrix Q is given by 1 exp ? (x ? ?)?Q?1(x ? ?) det π Q
f?,Q(x) =
and has di?erential entropy h(X) = ?
Cn
f?,Q(x) log f?,Q(x) dx
= log det πeQ
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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)
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Maximizing h(Y)
It can be shown that: ? If x satis?es E [x?x] ≤ P , then so does x ? E [x] ? For all y ∈ CM , h(Y) is maximized if y is Circularly Symmetric Complex Gaussian (CSCG) ? If x ∈ CN is CSCG with covariance Q, then y = Hx + n ∈ CM is also CSCG =? I (X; Y) = log det πe(IM + HQH?) ? log det πe = log det(IM + HQH?) ? A non–negative de?nite Q such that I (X; Y) is maximum and tr(Q) ≤ P remains to be found
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Deterministic Gaussian MIMO Channel
? H known at the transmitter (“water?lling solution”): Choose Q diagonal, such that
1 + Qii = (α ? λ? i ) ,
i = 1, . . . , N
with (·)+ max(·, 0), (λ1, . . . , λN ) the eigenvalues of H?H and α such that i Qii = P . The capacity is given by:
N
CWF =
i=1
log(αλi)
+
[bits/s/Hz]
? H unknown at the transmitter: Choose Q = Then, P CEP = log det(IM + N HH?)
P N IN
(equal power).
[bits/s/Hz]
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Water?lling Solution
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Rayleigh Fading MIMO Channel
? Memoryless Rayleigh fading Gaussian channel (unknown at the transmitter) ? Choose x CSCG and Q =
P N IN .
The ergodic capacity is given by: [bits/s/Hz]
P CEP = EH log det(IM + N HH?) m
= EH
i=1
P log 1 + N λi
,
where m = min(N, M ) and λ1, . . . , λm are the eigenvalues of the Wishart matrix HH? M <N W= H?H M ≥N ? For large SNR, CEP = min(N, M ) log P + O(1), i.e. the capacity grows linearly with min(N, M )!
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Capacity of Fading Channels
? Rayleigh fading: the capacity grows linearly with min(N, M ) ? Ricean channels: Increasing the line–of–sight (LOS) strength at ?xed SNR reduces the capacity ? If the gains in H become highly correlated, there is a capacity loss ? Water?lling (WF) capacity gains over Equal Power (EP) capacity are signi?cant at low SNR but converge to zero as the SNR increases =? Question: Is it bene?cial to feed the channel state back to the transmitter ? ? Many exact capacity results are known for i.i.d. Rayleigh channels. For other channels (Rice, etc.), we have many limiting results
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Ergodic Capacity of Ideal MIMO Systems
Channel unknown at the transmitter, i.i.d. Rayleigh fading
MT MR
N M
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Outage Capacity
? The capacity of a fading channel is a random variable ? De?nition: The q % outage capacity Cout,q of a fading channel is the information rate that is guaranteed for (100 ? q )% of the channel realizations, 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 de?ne the multiplexing gain r as C (SNR) r = lim , SNR→∞ log SNR which comes at no extra bandwidth or power
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Outage Capacity of Ideal MIMO Systems
Channel unknown at the transmitter, i.i.d. Rayleigh fading
MT MR
N M
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Transmission over MIMO channels
We can use the advantages provided by MIMO channels to: ? Maximize diversity to combat channel fading and decrease the bit error rate (BER) =? space–time codes (STC)
? Maximize the throughput =? spatial multiplexing, V–BLAST (Bell laboratories layered space–time)
? Try to do both at the same time =? trade–o? between increasing the throughput and increasing diversity
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Maximizing Diversity with Space–Time Codes
? Space–Time Trellis Codes (STTC)
←?
often better performance at 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 linear processing – Full diversity. Minimal or no coding gain
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Alamouti Scheme for Transmit Diversity (STBC)
r1 = h1c1 + h2c2 + n1 ? r2 = ?h1c? 2 + h2 c1 + n2
[time t] [time t + T ]
=?
? 2 2 ? ? r1 = h? r + h r = ( | h | + | h | ) c + h n + h n 1 2 1 2 1 1 2 1 2 1 2 ?→ c1 ? 2 2 ? ? r2 = h? 2 r1 ? h1 r2 = (|h1 | + |h2 | )c2 ? h1 n2 + h2 n1 ?→ c2
? Assumption: the channel remains unchanged over two consecutive symbols ? Rate = 1 — Diversity order = 2 — Simple decoding
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STBC Receiver Structure
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STBCs from Complex Orthogonal Designs
? Alamouti’s scheme works only when N = 2 =? Generalization ? De?nition: A complex orthogonal design Oc of size N is an orthogonal matrix with entries in the indeterminates ? ? ? , . . . , ± x , ± x ±x1, ±x2, . . . , ±xN , their conjugates ± x 2 1 N or multiples √ of these indeterminates by ± ?1
space ?→
? Example (2 × 2):
Oc(x1, x2) =
x1 x2 ? ?x? 2 x1
time
↓
? Coding scheme (using a constellation A with 2b elements): 1. At time slot t, N b bits arrive at the encoder. Select constellation signals 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
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STBCs from Complex Orthogonal Designs
? The maximum–likelihood detection rule reduces to simple linear processing for STBCs
? One can obtain the maximum possible diversity order M N at transmission rate R = 1 using STBCs based on orthogonal designs
? However: complex orthogonal designs exist only if n = 2. . . !
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Generalized Complex Orthogonal Designs (GCOD)
? De?nition: Let Gc be a p × N matrix with entries in the indeterminates ? ? ? , . . . , ± x , ± x ±x1, ±x2, . . . , ±xk , their conjugates ± x 2 1 k or multiples of √ ? these indeterminates by ± ?1 or 0. If Gc Gc = (|x1|2 + · · · + |xk |2)I , then Gc is referred to as a generalized complex orthogonal design of size N and rate R = k/p
? De?nition: Generalized complex linear processing orthogonal design (GCLPOD) Lc: exactly like above, but the entries can be linear combinations of x1, . . . , xk and their conjugates
? One can obtain a diversity order of M N at rate R using a STBC based on a GCOD or a GCLPOD of size N and rate R
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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
3 ? Example (GCLPOD, R = 4 , N = 3 and GCOD, R = 1 2 , N = 3): ? x1 x2 x3 ? ? x1 ?x4 x ? ?x2 √3 x1 x2 ? 2 ?x3 x4 x1 ? x3 ? ? ? ?x2 ? √ x1 ? ?x4 ?x3 2 x2 ? ? 3 3 ? ? ? ? ? x3 x3 ?x1 ?x1 +x2 ?x2 ? = G Lc = ? √ ? ? ? c ? √ x x x ? ? 1 2 3 2 2 2 ? ? ? ? ? ? ? ? x3 x3 x2 +x2 +x1 ?x1 ? ? x x ? x 2 1 4 √ ? √2 ? ?x? ? ? 2 2 x x
3 ? ?x4 4 ?x? 3 1 x? 2
? ? ? ? ? ? ? ? ? ?
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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 in the presence of multiple receive antennas – STBCs are only optimal with respect to capacity when they have rate R = 1 and there is one receive antenna
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Maximizing the Throughput with V–BLAST
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Maximizing the Throughput with V–BLAST
Description ? Transmitters operate co–channel, symbol synchronized ? Substreams are exactly independent (no coding across the transmit antennas — each substream can be individually coded) ? Individual transmit powers scaled by constant
1 N
so the total power is kept
? Channel estimation burst by burst using a training sequence ? Requires near–independent channel coe?cients
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Receivers for Spatial Multiplexing
y = Hx + n, ? y1 h11 h12 · · · ? y2 ? ? h21 . . . ? . ?=? . ... ? . ? ? . yM hM 1 · · · · · · ? ? i.e. ?? ? ? ?
h1 N x1 n1 . ? x2 ? ? n2 ? . ? ? ? . ?+? . ? . . ?? . ? ? . ? hM N xN nM
? If we transmit a block of N × T symbols, we have Y = HX + N, with Y, N ∈ CM ×T and X ∈ CN ×T ? = arg min y ? Hx ? Optimal (ML) Receiver: x
x
– Exhaustive search (often prohibitive complexity) – Diversity order for each data stream: M (N ≤ M )
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Receivers for Spatial Multiplexing
y = Hx + n ? Zero–forcing (ZF) Receiver: ? = H# y x with H# = (H?H)?1H? (pseudo–inverse)
– Signi?cantly reduced receiver complexity – Noise enhancement problem – Diversity order for each data stream: M ? N + 1 (N ≤ M )
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Receivers for Spatial Multiplexing
y = Hx + n ? Minimum mean–square error (MMSE) Receiver: ? = W · y, x We obtain: ? = H HH + E nn x
? ? ?
where W = arg min E
W
Wy ? x
2
.
?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 )
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Receivers for Spatial Multiplexing
y = Hx + n, H= h1 h2 · · · hN
? V–BLAST receiver — successive interference cancellation (SIC):
T x1 = w1 y
x ?1 = Q(x1) y2 = y ? x ?1h1
T x2 = w2 y2,
(quantization) (interference cancellation) etc.
? The ith ZF–nulling vector wi is de?ned as the unique minimum–norm vector satisfying 0 j>i T wi hj = 1 j = i, is orthogonal to the subspace spanned by the contributions to yi due to the symbols not yet estimated and cancelled and is given by the ith row of H# = (H?H)?1H? (N ≤ M )
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Receivers for Spatial Multiplexing
y = Hx + n, ? 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 i y1
H=
h1 h2 · · ·
hN
= = = = = = = = = =
H 1 y
#
Gi =
?
1 gi
2 gi
···
N gi
?T
recursion:
ki
wki x eki x ? ki yi+1 Gi+1
? j ?2 ? ? arg minj ∈{ / k1 ,...,ki?1 } gi
gi i T wk yi i Q(x eki ) yi ? x ?ki hki H# Hk
ki k
i
H with columns k1, · · · , ki set to 0
i
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i+1
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An Overview of MIMO Systems in Wireless Communications
Receivers for Spatial Multiplexing
? The V–BLAST SIC receiver: – Provides a reasonable trade–o? between complexity and performance (between MMSE and ML receivers) – Achieves a diversity order of approximately M ? N + 1 per data stream (N ≤ M )
? The V–BLAST OSIC receiver: – Provides a reasonable trade–o? between complexity and performance (between MMSE and ML receivers) – Achieves a diversity order which lies between M ? N + 1 and M for each data stream (N ≤ M )
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Performance Comparison
N ↓
M ↓
←? diversity receiver ←? SIC ←? OSIC
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Performance Comparison
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D–BLAST
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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
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Linear Dispersion Codes
? A linear dispersion code of rate R =
k p
b is one for which
k
X=
i=1
(ciCi + c? i Di ),
x ? x2 ? ? X=? ? . . ? xp
?
1
?
where ci, . . . , ck belong to a constellation A with 2b symbols and Ci, Di ∈ Cp×N Number of transmit antennas: N Number of receive antennas: M
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Linear Dispersion Codes
? If Y = XHT + N, it can be shown that: (H ∈ CM ×N ; Y, N ∈ Cp×M ) ? ?1 y ? . . ? = H? ?M y
η
Re(yi ) Im(yi )
?
?
?1 c ?1 n . . ?+? . . ?, ?M c ?k n
ξ
Re(ni ) Im(ni )
?
?
?
Y= N=
y1 · · · n1 · · ·
yM nM
?i where y
?i ,n
,c ?i
Re(ci ) Im(ci )
and
H ∈ C2M p×2k = f (H, C1, . . . Ck , D1, . . . Dk ) ? V–BLAST like techniques can thus be used to decode linear dispersion codes ? {C1, . . . , Ck , D1, . . . , Dk } are dispersion matrices designed to optimize given criteria (e.g. maximum mutual information between η and ξ )
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Diversity vs. Multiplexing Trade–o?
C = min{N, M } log SNR + O(1) ? De?nition: A scheme {C (SNR)} is a family of codes of block length l, one for each SNR level. R(SNR) [b/symbol] denotes the rate of the code C (SNR) ? De?nition: A scheme {C (SNR)} is said to achieve spatial multiplexing gain r and diversity gain d if the data rate R(SNR) lim =r SNR→∞ log SNR and the average error probability log Pe(SNR) lim = ?d SNR→∞ log SNR
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50
Diversity vs. Multiplexing Trade–o?
? For each r, d?(r) is the supremum of the diversity gains achieved over all schemes ? We also de?ne: – d? max
? – rmax
d?(0), the maximal diversity gain sup{r|d?(r) > 0}, the maximal spatial multiplexing gain
? Theorem: Assume l ≥ N + M ? 1. The optimal trade–o? curve d?(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 = N M and rmax = min{N, M }.
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Diversity vs. Multiplexing: Optimal Trade–o?
m n N M
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Diversity vs. Multiplexing Trade–o?: V–BLAST
n
N =M
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Diversity vs. Multiplexing Trade–o?: Alamouti Scheme
m n
N M
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Diversity vs. Multiplexing Trade–o?: Alamouti Scheme
m n
N M
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Diversity vs. Multiplexing Trade–o?
? De?nitions (1) and (2) for the diversity gain are not equivalent: in the former one, a ?xed data rate is assumed for all SNRs, whereas in the latter one, the data rate is a fraction of C (SNR), and hence increases with the SNR
? De?nition (1) is the most widely used in the literature
? De?nition (2) allows to quantify the diversity vs. multiplexing trade–o?
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MIMO Channel Modeling
? A good MIMO channel model must include: – Path loss – Shadowing – Doppler and delay spread pro?les – Ricean K factor distribution – Joint antenna correlation at transmit and receive ends – Channel matrix singular value distribution
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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 factor channels often exhibit a low capacity ? In a near–LOS link, the improvement in link budget often more than compensates for the loss of MIMO capacity =? usually, the LOS component 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 base station
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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 — assumed here uniformly distributed in θ TAl : lth transmitting antenna element RAl : lth receiving antenna element S (θ) : scatterer located at angle θ
[16]
Θ : angle of arrival ? : angle spread
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Correlation Model for HNLOS
? Correlation from one BS antenna element to two SU antenna elements: E [Hl,pH? m,p ] ≈ J0 2π d(l, m) λ ↑
distance between antennas l and m
? Correlation from two BS antenna elements to one SU antenna element in the broadside direction (Θ = 0): E [Hm,pH? m,q ] ≈ J0 ? 2π d(p, q ) λ ↑
distance between antennas p and q
? Correlation from two BS antenna elements to one SU antenna element in the inline direction (Θ = π 2 ): E [Hm,pH? m,q ] ≈e
π ?j 2 λ d(p,q ) 2 1? ? 4
· J0
? 2
2
2π d(p, q ) λ
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Correlation Model for HNLOS
←? J0 (x)
? The mobiles have to be in the broadside direction to obtain the highest diversity ? Interelement spacing has to be high to have low correlation =? beamforming and MIMO yield con?icting criteria ? Using the above results, one can obtain upper bounds for the MIMO capacity
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Decoupling Between Rank and Correlation
Pinhole channel
? Uncorrelated fading at both ends doesn’t necessarily imply a high–rank channel
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MIMO Channel Modeling
? Time–varying wideband MIMO channel:
L
H(τ ) =
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: 1 2 λ (rich scattering)
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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 is decomposed into K parallel frequency-?at fading channels, each B having bandwidth K . (Condition: The impulse response of the channel is shorter than the length of the cyclic pre?x)
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MIMO–OFDM Systems
? OFDM can be extended to MIMO systems by performing the IDFT/DFT and CP operations at each of the transmit and receive antennas (with the appropriate condition on the length of the cyclic pre?x)
? Diversity systems: (Ex: Alamouti scheme) – Send c1 and c2 over OFDM tone i over antennas 1 and 2
? – Send ?c? 2 and c1 over OFDM tone i + 1 over antennas 1 and 2 within the same OFDM symbol
– Alternative technique: Code on a per–tone basis across OFDM symbols in time
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MIMO–OFDM Systems
? Spatial multiplexing: Maximize spatial rate (r = min{N, M }) by transmitting independent data streams over di?erent antennas =? spatial multiplexing over each tone
? Space–frequency coded MIMO–OFDM – OFDM tones with spacing larger than the coherence bandwidth BC experience independent fading – If De? = N M De?
B BC ,
the total diversity gain that can be realized is of
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Throughput in MIMO Cellular Systems
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Conclusions
? MIMO channels o?er multiplexing gain, diversity gain, array gain and a co–channel interference cancellation gain ? Careful balancing between those gains is required ? MIMO systems o?er a promising solution for future generation wireless 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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