1
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MultipleAntenna Systems Jan Mietzner (
[email protected], Room: Kaiser 4110)
1. Introduction •
Multiple−antenna techniques
How is it possible to build (digital) wireless communication systems offering high data rates and small error rates ?
...
Tx
Tradeoff between spectral efficiency (high data rates) and power effi
...
•
Rx
ciency (small error rates), given fixed bandwidth & transmission power •
Example: Increase cardinality of modulation scheme ⇒ Data rate ↑, error rate ↑ Decrease rate of channel code ⇒ Error rate ↓, data rate ↓
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Spatial multiplexing techniques
Smart antennas (Beamforming)
Conventional transmitter & receiver techniques operate in time domain and/ or in frequency domain
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Spatial diversity techniques (Space−time coding & diversity reception)
Trade−off
Idea: Utilize multiple antennas at the transmitter and/ or the receiver
Multiplexing gain
Trade−off Diversity gain
Antenna gain
Coding gain
Interference suppression
Smaller error rates
Higher bit rates/ Smaller error rates
– Multipleinput multipleoutput (MIMO) system – Singleinput multipleoutput (SIMO) system – Multipleinput singleoutput (MISO) system
⇒ Exploit spatial domain (in addition to time/ frequency domain)
⇒ Better tradeoff between spectral efficiency and power efficiency
•
Benefits of multiple antennas: – Increased data rates by means of spatial multiplexing techniques – Decreased error rates by means of spatial diversity techniques – Improved signaltonoise ratios (SNRs)/ signaltointerferenceplusnoise ratios (SINRs) by means of beamforming techniques
Higher bit rates
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2. Basic Principles
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•
Improved SNRs: Focus antenna patterns on desired angles of reception/ transmission, e.g., towards lineofsight (LoS) or significant scatterers ⇒ Antenna gain
2.1 Beamforming Techniques •
Goal: Improved SNRs or SINRs in multiuser scenarios
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Beamforming can be interpreted as linear filtering in the spatial domain
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Consider antenna array with N elements and directional antenna pat
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Steer nulls towards cochannel users ⇒ Interference suppression •
frequencydivision multiple access (TDMA/ FDMA)
Due to antenna array geometry, impinging RF signal reaches antenna elements at different times (underlying baseband signal does not change)
⇒ Adjust phases of RF signals to achieve constructive superposition
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SNR/ SINR gains can be utilized to decrease error rates or to increase data rates (by switching to a higherorder modulation scheme)
⇒ Corresponds to steering of antenna pattern towards desired direction ⇒ Additional weighting of RF signals can shape antenna pattern
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Principle can also be utilized at the transmitter (reciprocity)
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In practical systems directions of significant scatterers must be estimated (e.g., MUSIC or ESPRIT algorithm); SINR can also be optimized without knowing the directions of all cochannel users (Capon beamformer)
(N −1 degrees of freedom for placing maxima or nulls)
•
Beamforming/ smart antenna techniques thus enable spacedivision multiple access (SDMA), as an alternative to timedivision or
tern receiving a radiofrequency (RF) signal from a certain direction •
Improved SINRs:
Beamforming techniques are well established since the 1960’s (origins are in the field of radar technology); however, intensive research for wireless communication systems started only in the 1990’s
Receiver
1
...
M
N Desired directions of transmission/reception
Phased array
Phased array
• Beamformer
1
...
Information bit sequence
Beamformer
Transmitter
Literature: An exhaustive overview on smart antenna techniques for wireless communications can be found in [Godara’97]
to detector •
Final remark: Beamforming can also be performed in baseband domain, if channel is known at transmitter and receiver (eigenbeamforming)
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2.2 Spatial Multiplexing Techniques
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2.3 Spatial Diversity Techniques
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Goal: Increased data rates compared to singleantenna system
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Goal: Decreased error rates compared to singleantenna system
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Capacity of MIMO systems grows linearly with min{M, N }
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Send/ receive multiple redundant versions of the same data sequence
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At the transmitter, the data sequence is split into M subsequences that
and perform appropriate combining (in baseband domain)
⇒ If the redundant signals undergo statistically independent fading,
are transmitted simultaneously using the same frequency band
it is unlikely that all signals simultaneously experience a deep fade
⇒ Data rate increased by factor M (multiplexing gain) •
⇒ Spatial diversity gain (typically, small antenna spacings sufficient)
At the receiver, the subsequences are separated by means of interferencecancellation algorithm, e.g., linear zeroforcing (ZF)/ minimummean
•
combining of the received signals
squarederror (MMSE) detector, maximumlikelihood (ML) detector, suc
– Various combining strategies, e.g., equalgain combining (EGC),
cessive interference cancellation (SIC) detector, ... •
Typically, channel knowledge required solely at the receiver
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For a good error performance, typically N ≥ M required
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Intensive research started at the end of the 1990’s
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Literature: [Foschini’96]
Receive diversity: SIMO system with N receive antennas and linear
selection combining (SC), maximumratio combining (MRC), ... – Wellestablished since the 1950’s, see [Brennan’59] •
Transmit diversity: MISO system with M transmit antennas – Appropriate preprocessing of transmitted redundant signals to enable coherent combining at receiver (spacetime encoder/ decoder) – Optionally, N > 1 receive antennas for enhanced performance
(Tutorials can be found in [Gesbert et al.’03], [Paulraj et al.’04]) Transmitter
– Typically, channel knowledge required solely at the receiver – Intensive research started at the end of the 1990’s
Receiver
– Wellknown techniques are Alamouti’s scheme for M = 2 transmit 1 2 M
M sub−sequences
N
...
...
Information bit sequence
Demultiplexing
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antennas [Alamouti’98], spacetime trellis codes [Tarokh et al.’98], Detection Algorithm
and orthogonal spacetime block codes [Tarokh et al.’99] Estimated bit sequence
– An abundance of transmitter/ receiver structures has been proposed (some offer additional coding gain) •
Literature: An exhaustive overview of the benefits of spatial diversity in wireless communication systems can be found in [Diggavi et al.’04]
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Transmitter
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•
Receiver
Discretetime channel model (cont’d): – xµ [k]: Transmitted symbol of transmit antenna µ, time index k ,
1 Space−Time Decoder
...
Information bit sequence
Space−Time Encoder
M
E{xµ [k]} = 0, Estimated bit sequence
E{xµ[k]2} =: σx2µ
(Underlying information symbols are denoted as a[k]) – hν,µ : Channel gain between µth transmit & ν th receive antenna,
hν,µ ∼ CN (0, σh2 ) (i.i.d)
Redundant signals
(Amplitude hν,µ  is Rayleigh distributed)
– nν [k]: Additive white Gaussian noise (AWGN) sample at receive
3. Mathematical Details
antenna ν , time index k ,
nν [k] ∼ CN (0, σn2 ) (i.i.d)
3.1 System Model
– yν [k]: Received symbol at receive antenna ν , time index k •
Consider a MIMO system with M transmit and N receive antennas
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Assumptions: – Frequency nonselective fading & squareroot Nyquist filters at
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Matrixvector model – Transmitted vector: x[k] := [ x1[k], ..., xM [k] ]T
transmitter and receiver (pulse energy Eg := 1)
– Noise vector: n[k] := [ n1[k], ..., nN [k] ]T
⇒ No intersymbol interference (ISI)
– Received vector: y[k] := [ y1[k], ..., yN [k] ]T
– Rayleigh fading (no LoS component), i.e., channel gains are
– Channel matrix:
– Block fading, i.e., channel gains are invariant over complete data block and change randomly from one block to the next •
Discretetime channel model: – k : Discrete time index (1 ≤ k ≤ NB, NB block length)
– µ: Transmit antenna index (1 ≤ µ ≤ M ) – ν : Receive antenna index (1 ≤ ν ≤ N )
h1,1 · · · h1,M ... . . . ... H := hN,1 · · · hN,M
zeromean complex Gaussian random variables
⇒ System model: y[k] = H x[k] + n[k]
(1)
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3.2 EigenBeamforming
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Transmit power allocation: In addition, the transmit power allocated to the parallel channels can be
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Consider a quadratic MIMO system with M = N > 1 antennas
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Assume that the instantaneous realization of the channel matrix is
optimized, based on the instantaneous SNRs
λν 2 σx2µ σn2
(ν = 1, ..., N ) and a
certain optimization criterion
perfectly known both at the transmitter and at the receiver •
Eigenvalue decomposition of H:
3.3 Spatial Multiplexing H
H := UΛU
(2)
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U: Unitary (N×N )matrix, i.e., UHU = IN
Consider a MIMO system with N ≥ M > 1 antennas (For N < M , the system is inherently rankdeficient)
Λ: Diagonal (N×N )matrix containing eigenvalues λ1, ..., λN of H:
λ1 · · · 0 Λ = diag(λ1, ..., λN ) = ... . . . ...
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•
0 · · · λN
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Assume that the instantaneous realization of the channel matrix is known solely at the receiver
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Linear ZF detection: Received vector y[k] is postprocessed as
zZF[k] := (HHH)−1 HHy[k] =: H+y[k]
Since H is perfectly known, transmitter and receiver can calculate the
(4)
matrix U (e.g., using the Jacobian algorithm [Golub et al.’96, Ch. 8.4])
(H+: Lefthand pseudoinverse of H; for M = N and full rank use H−1 )
Eigenbeamforming:
⇒
– Instead of x[k], transmitter sends preprocessed vector x′ [k] := Ux[k] ′
i.e., spatial interference completely removed; however, variance of the
H ′
– The received vector y [k] is postprocessed as U y [k] =: y[k]
⇒
H y[k] = UHy′ [k] = UH(Hx′ [k] + n[k]) = UHHUx[k] + U n[k]} {z 
¯ [k] = Λx[k] + n ¯ [k] = UHUΛUH Ux[k] + n
⇒
yν [k] = λν xµ [k] + n ¯ ν [k] for all µ, ν = 1, ..., N
¯ [k] =: n
resulting noise samples may be significantly enhanced •
Linear MMSE detection: (assume σx21 = ... = σx2M =: σx2 ) Received vector y[k] is postprocessed as
zMMSE [k] := (HH H + σn2 /σx2 · IM )−1 HHy[k]
(3)
(5)
– Usually better performance than ZF detection, since better tradeoff
– Thus, assuming full rank (λ1 6= 0, ..., λN 6= 0) we have N parallel
between spatial interference mitigation & noise enhancement
scalar channels without spatial interference (i.e., data rate enhanced
– For high SNR values (σn2 → 0), both detectors become equivalent
by factor N compared to singleantenna system) – Noise samples n ¯ ν [k] are still i.i.d. ∼ CN (0, σn2 ), due to unitarity of U
zZF[k] = H+y[k] = H+ (Hx[k] + n[k]) = x[k] + H+n[k],
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Performance of ZF/ MMSE detection often quite poor, unless N ≫ M
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•
ML detection:
3.4 Receive Diversity
ˆ ML [k] := argminx˜ [k] y[k] − H˜ x[k] x
2
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˜ [k] – For example, bruteforce search over all possible hypotheses x for the transmitted vector x[k]
⇒ For Qary modulation scheme, there are QM possibilities ⇒ Optimal detection strategy (w.r.t. ML criterion), but very complex •
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Consider a SIMO system with N receive antennas
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Assume that the instantaneous realization of the (N×1)channel matrix is perfectly known at the receiver
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Received sample at receive antenna ν , time index k :
yν [k] = hν,1 x1[k] + nν [k]
SIC detection:
– hν,1 ∼ CN (0, σh2 ) ⇒ Amplitude hν,1 =: αν Rayleigh distributed
– Good tradeoff between complexity and performance – Originally proposed in [Foschini’96] for the wellknown BLAST
p(αν ) =
scheme (‘BellLabs Layered SpaceTime Architecture’) – QR decomposition of H: (assume N = M )
H := QR
– Instantaneous SNR
(7)
p(γν ) =
Q: Unitary (N×N )matrix, i.e., QHQ = IN R: Upper triangular (N×N )matrix:
r1,1 · · · r1,N R = ... . . . ...
where γ¯ :=
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0 · · · rN,N (There are various algorithms for calculating the QR decomposition)
be detected – Assuming that the detection of xN [k] was correct, the influence of
xˆN [k] can be subtracted from the (N −1)th row of (8); then symbol
(9)
=: γν Chisquare (χ2) distributed
1 γν exp − γ¯ γ¯
(γν ≥ 0),
(10)
⇒ Large probability of small instantaneous SNRs
favorable SNR distribution at combiner output (γcomb ) – Equalgain combining (EGC): Add up all samples
zcomb [k] :=
N X
ν=1
⇒
yν [k] =
(8)
– Symbol xN [k] is not affected by spatial interference and can directly
xN−1 [k] can directly be detected, and so on ...
hν,1 2 σx21 σn2
(αν ≥ 0),
Idea: Combine received samples y1[k], ..., yN [k] to obtain more
– Received vector y[k] is first postprocessed as zSIC [k] := QH y[k]
¯ [k] ⇒ zSIC [k] := QHy[k] = QH (Hx[k]+n[k]) = Rx[k]+ n
σh2 σx21 σn2
αν2 2αν − exp 2 σh σh2
hcomb ∼ CN (0, N σh2 ),

N X
ν=1{z
=: hcomb
N X
ν=1
}
N X
nν [k] ν=1 {z } =: ncomb [k]

ncomb [k] ∼ CN (0, N σn2 ), i.e., no gain!
⇒ Do it coherently (hν,1 := αν ejφν ) ′ zcomb [k] :=
hν,1 x1[k] +
e−jφν yν [k] =
N X
αν x1[k] +
ν=1{z  } =: h′comb
Combineroutput SNR: γcomb =
N X
ν=1 
e−jφν nν [k] {z
=: n′comb [k] P ( ν αν )2σx21 /(N σn2 )
}
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– Selection combining (SC): Select branch with largest instant. SNR Combineroutput SNR: γcomb =
maxν {αν2 }σx21 /σn2
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Example: BPSK, N = 1, ..., 4 receive branches
= maxν {γν }
0
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N=1 receive branches N=2 receive branches N=3 receive branches N=4 receive branches Alamouti‘s scheme (M=2, N=1)
– Maximumratio combining (MRC): ν=1
h∗ν,1 yν [k]
N X

Combineroutput SNR:
N X
2
−1
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h∗ν,1 nν [k]
hν,1 x1[k] + ν=1 {z ν=1 {z } }  =: hcomb =: ncomb [k] P P γcomb = ( ν hν,12)σx21 /σn2 = ν γν =
−2
10 SER
zcomb [k] :=
N X
−3
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⇒ Maximizes combineroutput SNR; optimal w.r.t. ML criterion •
Symbol error rates (SERs) with MRC: (without derivation ;) )
−4
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γ¯ : Average SNR per receive branch – Binary PhaseShift Keying (BPSK)
SER(¯ γ) =
N
v u u u t
1 γ¯ 1 − 2N 1+ γ¯
N−1 X i=0
[Proakis’01, Ch. 14]
N −1 + i 1 γ¯ 1 + 2i 1+ γ¯ i
– Qary PhaseShift Keying (QPSK)
1 SER(¯ γ) = π
(Q−1)π ZQ
0
N
– Qary AmplitudeShift Keying (QASK)
[Simon et al.’00]
2(Q−1) Z2 (Q2 −1) sin2ϕ dϕ SER(¯ γ) = Qπ 0 (Q2 −1) sin2ϕ + 3¯ γ
2
π
π
Z4
0
16
18
20
N
N
•
Consider a MISO system with M transmit antennas
•
Assume that the instantaneous realization of the (1×M )channel matrix is perfectly known at the receiver, but not at the transmitter
1 Z2 2(Q−1) sin2ϕ 4 1− √ dϕ SER(¯ γ) = π Q 0 2(Q−1) sin2ϕ + 3¯ γ 4 1 1− √ − π Q
6 8 10 12 14 Average SNR per branch (in dB)
Asymptotic slope (i.e., γ¯ → ∞) of the curves is −N (‘diversity order N ’)
(13)
– Qary QuadratureAmplitude Modulation (QQAM) [Simon et al.’00]
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3.5 Transmit Diversity
(12)
N
(11)
2
[Simon et al.’00]
sin2ϕ dϕ sin2ϕ + γ¯ sin2(π/Q)
π
0
i
v u u u t
2(Q−1) sin2ϕ dϕ 2(Q−1) sin2ϕ + 3¯ γ
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Transmit Diversity: Suitable preprocessing of transmitted data sequence required to allow for coherent combining at the receiver – Example: Send identical signals over all transmit antennas
(14)
⇒ No diversity gain! (corresponds to EGC without cophasing)
– Instead: Perform appropriate twodimensional mapping/ encoding in time and space (i.e., over the transmit antennas)
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Example: Alamouti’s scheme for M = 2 transmit antennas (N = 1 receive antennas considered; can be extended to N > 1) – Spacetime mapping: Information symbols to be transmitted are processed in pairs [ a[k], a[k + 1] ]; at time index k , symbol a[k] is
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⇒ Two parallel scalar channels for the symbols a[k] and a[k+1] (no spatial interference)
⇒ Corresponds to MRC with M = 1 transmit and N = 2 receive antennas; however, using the same average transmit power, Alamouti’s scheme
transmitted via the first antenna and symbol a[k + 1] via the second
exhibits a 3 dB loss compared to MRC
antenna; at time index k+1, symbol −a∗[k+1] is transmitted via the first antenna and symbol a∗[k] via the second antenna
A =
4. Literature
a[k] a[k+1] ←− time index k ∗ −a [k+1] a∗[k] ←− time index k+1 ↑ ↑
antenna 1
4.1 Cited References •
antenna 2
L. C. Godara, “Application of antenna arrays to mobile communications – Part I: Performance improvement, feasibility, and system consid
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erations; Part II: Beamforming and directionofarrival considerations,”
(In terms of prior system model: A =: [ xT[k], xT [k+1] ]T )
Proc. IEEE, vol. 85, no. 7/8, pp. 1031–1060, 1195–1245, July/Aug. 1997.
– Received samples (time index k, k+1): •
y1[k] = h1,1 a[k] + h1,2 a[k+1] + n1[k]
cation in a fading environment when using multielement antennas,” Bell
y1[k+1] = −h1,1 a∗[k+1] + h1,2 a∗[k] + n1[k+1]
Syst. Tech. J., pp. 41–59, Autumn 1996.
∗
– Equivalent matrixvector model (by taking the (.) of y1 [k+1])

y1[k] y1∗ [k+1] {z
=: yeq [k]
}
= 
h1,1 h1,2 h∗1,2 −h∗1,1 {z
=: Heq
}
a[k] a[k+1] {z
=: a[k]
}
+
n1[k] n∗1 [k+1]

{z
=: neq [k]
G. J. Foschini, “Layered spacetime architecture for wireless communi
•
to practice: An overview of MIMO spacetime coded wireless systems,”
IEEE J. Select. Areas Commun., vol. 21, no. 3, pp. 281–302, Apr. 2003.
}
•
– Detection step at the receiver:
D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory
A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Boelcskei, “An overview of MIMO communications – A key to gigabit wireless,” Proc. IEEE,
2 2 Heq is always orthogonal (!), while HH eq Heq = (h1,1  + h1,2  ) I2 H H ⇒ zcomb [k] := HH eq yeq [k] = Heq Heq a[k] + Heq neq [k] 
{z
=: n′eq [k]
}
= (h1,12 + h1,22) a[k] + n′eq[k]
vol. 92, no. 2, pp. 198–218, Feb. 2004. •
D. G. Brennan, “Linear diversity combining techniques,” Proc. IRE, vol. 47, pp. 1075–1102, June 1959, Reprint: Proc. IEEE, vol. 91, no. 2, pp. 331356, Feb. 2003.
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S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1451– 1458, Oct. 1998.
•
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4.2 Books on MultipleAntenna Systems •
Hall, 1985.
V. Tarokh, N. Seshadri, and A. R. Calderbank, “Spacetime codes for high data rate wireless communication: Performance criterion and code
•
Mar. 1998.
•
B. Vucetic and J. Yuan, SpaceTime Coding. John Wiley & Sons, 2003.
V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Spacetime block codes
•
E. G. Larsson and P. Stoica, SpaceTime Block Coding for Wireless Communications. John Wiley & Sons, 2003.
from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. •
•
and W. Utschick, Eds., Smart Antennas – State of the Art.
expectations: The value of spatial diversity in wireless networks,” Proc.
Hindawi Publishing Corp., 2004. •
G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed.
J. G. Proakis, Digital Communications, 4th ed.
New York: McGraw
Hill, 2001. •
M. K. Simon and M.S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. John Wiley & Sons, 2000.
New York:
E. Biglieri and G. Taricco, Transmission and Reception with Multiple Antennas: Theoretical Foundations.
Balti
Hanover (MA)  Delft: now Pub
lishers Inc., 2004.
more  London: The Johns Hopkins University Press, 1996. •
T. Kaiser, A. Bourdoux, H. Boche, J. R. Fonollosa, J. Bach Andersen,
S. N. Diggavi, N. AlDhahir, A. Stamoulis, and A. R. Calderbank, “Great IEEE, vol. 92, no. 2, pp. 219–270, Feb. 2004.
•
A. Paulraj, R. Nabar, and D. Gore, Introduction to SpaceTime Wireless Communications. Cambridge University Press, 2003.
construction,” IEEE Trans. Inform. Theory, vol. 44, no. 2, pp. 744–765,
•
S. Haykin, Ed., Array Signal Processing. Englewood Cliffs (NJ): Prentice
•
H. Jafarkhani, SpaceTime Coding – Theory and Practice. University Press, 2005.
Cambridge