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Multiple-Antenna 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

Trade-off 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 ↓

•

Spatial multiplexing techniques

Smart antennas (Beamforming)

Conventional transmitter & receiver techniques operate in time domain and/ or in frequency domain

•

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

– Multiple-input multiple-output (MIMO) system – Single-input multiple-output (SIMO) system – Multiple-input single-output (MISO) system

⇒ Exploit spatial domain (in addition to time/ frequency domain)

⇒ Better trade-off 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 signal-to-noise ratios (SNRs)/ signal-to-interference-plusnoise 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 line-of-sight (LoS) or significant scatterers ⇒ Antenna gain

2.1 Beamforming Techniques •

Goal: Improved SNRs or SINRs in multiuser scenarios

•

Beamforming can be interpreted as linear filtering in the spatial domain

•

Consider antenna array with N elements and directional antenna pat-

•

Steer nulls towards co-channel users ⇒ Interference suppression •

frequency-division 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

•

SNR/ SINR gains can be utilized to decrease error rates or to increase data rates (by switching to a higher-order modulation scheme)

⇒ Corresponds to steering of antenna pattern towards desired direction ⇒ Additional weighting of RF signals can shape antenna pattern

•

Principle can also be utilized at the transmitter (reciprocity)

•

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 co-channel users (Capon beamformer)

(N −1 degrees of freedom for placing maxima or nulls)

•

Beamforming/ smart antenna techniques thus enable space-division multiple access (SDMA), as an alternative to time-division or

tern receiving a radio-frequency (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 (eigen-beamforming)

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2.2 Spatial Multiplexing Techniques

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2.3 Spatial Diversity Techniques

•

Goal: Increased data rates compared to single-antenna system

•

Goal: Decreased error rates compared to single-antenna system

•

Capacity of MIMO systems grows linearly with min{M, N }

•

Send/ receive multiple redundant versions of the same data sequence

•

At the transmitter, the data sequence is split into M sub-sequences 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 sub-sequences are separated by means of interferencecancellation algorithm, e.g., linear zero-forcing (ZF)/ minimum-mean-

•

combining of the received signals

squared-error (MMSE) detector, maximum-likelihood (ML) detector, suc-

– Various combining strategies, e.g., equal-gain combining (EGC),

cessive interference cancellation (SIC) detector, ... •

Typically, channel knowledge required solely at the receiver

•

For a good error performance, typically N ≥ M required

•

Intensive research started at the end of the 1990’s

•

Literature: [Foschini’96]

Receive diversity: SIMO system with N receive antennas and linear

selection combining (SC), maximum-ratio combining (MRC), ... – Well-established since the 1950’s, see [Brennan’59] •

Transmit diversity: MISO system with M transmit antennas – Appropriate pre-processing of transmitted redundant signals to enable coherent combining at receiver (space-time 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

– Well-known techniques are Alamouti’s scheme for M = 2 transmit 1 2 M

M sub−sequences

N

...

...

Information bit sequence

Demultiplexing

1

antennas [Alamouti’98], space-time trellis codes [Tarokh et al.’98], Detection Algorithm

and orthogonal space-time 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

8

•

Receiver

Discrete-time 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

•

Assumptions: – Frequency non-selective fading & square-root Nyquist filters at

•

Matrix-vector 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 •

Discrete-time 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

zero-mean complex Gaussian random variables

        

⇒ System model: y[k] = H x[k] + n[k]

(1)

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3.2 Eigen-Beamforming

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Transmit power allocation: In addition, the transmit power allocated to the parallel channels can be

•

Consider a quadratic MIMO system with M = N > 1 antennas

•

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)

•

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 rank-deficient)

Λ: Diagonal (N×N )-matrix containing eigenvalues λ1, ..., λN of H: 

λ1 · · · 0 Λ = diag(λ1, ..., λN ) = ... . . . ...        

•

•

0 · · · λN



•

       

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 post-processed as

zZF[k] := (HHH)−1 HHy[k] =: H+y[k]

Since H is perfectly known, transmitter and receiver can calculate the

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matrix U (e.g., using the Jacobian algorithm [Golub et al.’96, Ch. 8.4])

(H+: Left-hand pseudo-inverse of H; for M = N and full rank use H−1 )

Eigen-beamforming:

⇒

– Instead of x[k], transmitter sends pre-processed vector x′ [k] := Ux[k] ′

i.e., spatial interference completely removed; however, variance of the

H ′

– The received vector y [k] is post-processed 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 post-processed as

zMMSE [k] := (HH H + σn2 /σx2 · IM )−1 HHy[k]

(3)

(5)

– Usually better performance than ZF detection, since better trade-off

– 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 single-antenna 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],

•

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, brute-force search over all possible hypotheses x for the transmitted vector x[k]

⇒ For Q-ary 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

•

Assume that the instantaneous realization of the (N×1)-channel matrix is perfectly known at the receiver

•

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 trade-off between complexity and performance – Originally proposed in [Foschini’96] for the well-known BLAST

p(αν ) =

scheme (‘Bell-Labs Layered Space-Time Architecture’) – QR decomposition of H: (assume N = M )

H := QR

– Instantaneous SNR

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p(γν ) =

Q: Unitary (N×N )-matrix, i.e., QHQ = IN R: Upper triangular (N×N )-matrix: 

r1,1 · · · r1,N R = ... . . . ...        

where γ¯ :=

        

•

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

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=: γν Chi-square (χ2) distributed 

1 γν exp −  γ¯ γ¯

(γν ≥ 0),

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⇒ Large probability of small instantaneous SNRs

favorable SNR distribution at combiner output (γcomb ) – Equal-gain 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 post-processed 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

Combiner-output 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 Combiner-output SNR: γcomb =

maxν {αν2 }σx21 /σn2

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Example: BPSK, N = 1, ..., 4 receive branches

= maxν {γν }

0

10

N=1 receive branches N=2 receive branches N=3 receive branches N=4 receive branches Alamouti‘s scheme (M=2, N=1)

– Maximum-ratio combining (MRC): ν=1

h∗ν,1 yν [k]





N X

|

Combiner-output SNR:

N X

2

−1

10

h∗ν,1 nν [k]

|hν,1| x1[k] + ν=1 {z ν=1 {z } } | =: hcomb =: ncomb [k] P P γcomb = ( ν |hν,1|2)σx21 /σn2 = ν γν =



−2

10 SER

zcomb [k] :=

N X

−3

10

⇒ Maximizes combiner-output SNR; optimal w.r.t. ML criterion •

Symbol error rates (SERs) with MRC: (without derivation ;-) )

−4

10

γ¯ : Average SNR per receive branch – Binary Phase-Shift 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

– Q-ary Phase-Shift Keying (Q-PSK)

1 SER(¯ γ) = π

(Q−1)π  ZQ

0

 

N

– Q-ary Amplitude-Shift Keying (Q-ASK)

[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 ’)

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– Q-ary Quadrature-Amplitude Modulation (Q-QAM) [Simon et al.’00] 

4

3.5 Transmit Diversity

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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

•

Transmit Diversity: Suitable pre-processing of transmitted data sequence required to allow for coherent combining at the receiver – Example: Send identical signals over all transmit antennas

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⇒ No diversity gain! (corresponds to EGC without co-phasing)

– Instead: Perform appropriate two-dimensional 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) – Space-time 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: Beam-forming and direction-of-arrival 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 multi-element 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 matrix-vector 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 space-time architecture for wireless communi-



•

to practice: An overview of MIMO space-time 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]

}

= (|h1,1|2 + |h1,2|2) 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. 331-356, Feb. 2003.

17

•

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.

•

18

4.2 Books on Multiple-Antenna Systems •

Hall, 1985.

V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code

•

Mar. 1998.

•

B. Vucetic and J. Yuan, Space-Time Coding. John Wiley & Sons, 2003.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes

•

E. G. Larsson and P. Stoica, Space-Time 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. Al-Dhahir, 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 Space-Time 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, Space-Time Coding – Theory and Practice. University Press, 2005.

Cambridge