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Transmit Beamforming Method Based on Maximum-Norm Combining for MIMO Systems Andy Wang
1
Paper • Heunchul Lee, Seokhwan Park, Inkyu Lee, “Transmit Beamforming Method Based on Maximum-Norm Combining for MIMO Systems”, IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO 4, pp2067-2075, APRIL 2009 •
Manuscript received April 30, 2008; accepted Dec. 4, 2008.
2
Key Concept • Reduce complexity by rotation transformation method to derive the unitnorm weight vector for maximum-norm combining (MNC).
3
Outline • Background • Vector Orthogonalization and Vector-norm Maximization • Rotation Transformation • Proposed TX BF Method • Simulation Results • Conclusions
4
Background • MIMO antennas system, coupled with space-time processing and diversity gain, increase system capacity and improve quality of service. • The capacity further improved if CSI known at the TX side. • Many method proposed based on SVD of channel matrix. The dominant singular vector information obtained through iterative algorithm is sufficient for optimizing transmit BF performance.
*CSI: Channel State Information *SVD: Singular Value Decomposition
5
System Model Tx=Mt
Rx=Mr
•
Tx transmit Data symbol s with unit-norm transmit weight vector t (|| t ||=1)
•
Rx receive and recover symbols by the receive weight vector r (|| r || =1) +
Estimated symbol
+
z = r Hts + r n
(.)+: complex conjugate transpose
effective channel gain
• Through properly choosing t and r, one can maximum the output SNR 6
The Iterative Power Method • Assume an MRC receiver given by r = * MRC: Maximum-Ratio Combining
Ht Ht
• It can be shown that the channel gain can be maximized when the dominant right singular vector of H is employed as t. +
r H t = H t = λmax (the largest singular value of H )
(*)
• The optimal solution requires the computation of the dominant singular vector of the Mr-by-Mt channel matrix. The transmit vector t can be determined by iterative power method started with an initial vector. r
( i +1)
=
Ht Ht
(i ) (i )
and
t
( i +1)
=
+
( i +1)
+
( i +1)
H r H r
(*) P. A. Dighe, R. K. Malik, and S. S. Jamuar, “Analysis of transmit-receive diversity in Rayleigh fading,” IEEE Trans. Commun., vol. 51, pp. 694-703, Apr. 2003 7
Issue • Although the Iterative Power method can provide a near-optimal result, the complexity of computation is high. – Number of iterations also increase the complexity directly. – (8MtMr+4Mt+4Mr)Nit real multiplications involved for the Mr-by-Mt channel matrix, where Nit indicates the number of iterations.
8
Target and Method • Reduce complexity by proposed method rather than a Iterative Power method. – Propose a rotation transformation method to derive the unit-norm weight vector for maximum-norm combining (MNC). And employs successive MNC to the MIMO channel matrix.
9
Vector Orthogonalization Process and Vector-norm Maximization • It was shown given two real-value vectors the norm of one column is maximized and the norm of the other is minimized when the two columns become orthogonal to each other by a rotation transformation.
*J. C. Nash, “A one-sided transformation method for the singular value decomposition and algebraic eigenproblem,” The Computer J., vol. 18, no. 1, pp. 74-76, Feb. 1975.
10
Defined Complex Vector Orthogonality (1/2) • Given two complex vectors h1 and h2, the Hermitian product is h1 , h2 = h1+ h2 = h1 , h2
R
+ j h1 , h2
I
• To give complex orthogonality a stronger condition by defining h1 , h2 = 0 – inner orthogonal in CMr : R
– outer orthogonal in CMr :
h1 , h2
– complex orthogonal in CMr :
h1 , h2
I R
=0 = 0 and
h1 , h2
I
=0
11
Defined Complex Vector Orthogonality (2/2) • For any complex vector v, define two real-value vectors as
[] []
[] []
⎡ R v ⎤ • ⎡− I v ⎤ v=⎢ ⎥, v = ⎢ ⎥ I v R v ⎣ ⎦ ⎣ ⎦
It’s easy to show
•
•
v = v , v •v = 0
,where (dot) means inner product
• With the defined orthogonality one can have
12
Rotation Transformation and MNC • Consider a rotation matrix F, which will be used to find MNC weight vector, to establish orthogonality between h1’and h2’ and maximum the norm of h1’ Recall: • The real-valued representation is [] []
[] []
⎡ R v ⎤ • ⎡− I v ⎤ v=⎢ ⎥, v = ⎢ ⎥ I v ⎣ ⎦ ⎣ Rv ⎦ •
•
v = v , v •v = 0
re-write as *MNC: Maximum-Norm Combining 13
Inner Orthogonality • '
• ' 2
and h ⊥ h • To make h1 , h2 R = 0 i.e. h ⊥ h • Applying the following rotation transformation where complex domain presentation is where • The inner orthogonality (maximize ) can be achieved if '
1
' 2
1
Recall that
(*maximize
solution is also computable)
and the Frobenius norm remain un-changed
14
Outer Orthogonality • ' 2
•
and h ⊥ h1' • To make h1 , h2 I = 0 i.e. h ⊥ h • Applying the following rotation transformation ' 1
' 2
complex domain presentation is where • The outer orthogonality (maximize
) can be achieved if Recall that
(*maximize
solution is also computable)
15
Complex Orthogonality '
'
• To make h1 , h 2 = 0 ' ' h , h • Inner rotation force the real part Hermitian product 1 2 R ' ' to zero will not affect the imaginary part (i.e. h1 , h 2 = h1 , h 2 I ) I • Outer rotation force the imaginary part Hermitian product ' ' ' ' h1 , h 2 to zero will not affect the real part (i.e. h1 , h 2 = h1 , h 2 ) R I R • Therefore one can achieve complex orthogonality through
16
Comparison of Results If maximum • Inner Orthogonality • Outer Orthogonality
• Complex Orthogonality
• The Inner Orthogonality and Outer Orthogonality lead to small decrease in the norm. Therefore one can select Inner or Outer weight for implementation. 17
Proposed TX BF Method: Mt=2 (1/2) • MNC technique which combines two vectors by the unit-norm weight vector
• Goal: To choose output norm
which maximizes the
or
or if
h1 , h2
R
≥ h1 , h2
I
if
h1 , h2
R
< h1 , h2
I
18
Proposed TX BF Method: Mt=2 (2/2) • Figure shows CDF of the squared vector norm achieved by the MNC process
( assume are i.i.d. complex Gaussian distribution with zero mean and unit variance)
• The square norm value achieved by the complex weight is equal to the square of the maximum singular value of the matrix • The norm value is close if properly Inner/Outer weight chosen 19
Proposed TX BF Method: Mt>=2 (1/3) • Determine weight vector by applying (Mt-1) successive MNC processes between column vectors of the MIMO channel matrix. Select Inner/Outer Orthogonality (TYPE 1) or Complex Orthogonality (TYPE 2)
20
Proposed TX BF Method: Mt>=2 (2/3)
MRC: 1-by-4 SIMO system Antenna Selection: reference [21] Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2)
• Figure shows CDF of relative channel gain (normalized by the maximum channel gain) for Mt=Mr=4 MIMO channels 21
Proposed TX BF Method: Mt>=2 (3/3) • Proposed Algorithm is optimal only for Mt=2 • Further repeat the proposed algorithm, which increase the computational complexity, may obtain the optimum solution. But simulation results show that the performance improvement is marginal.
22
Complexity • The weight vector can be generated through simple computation of θI , θo and θc . • The main computational cost is in the evaluation of
• Inner Orthogonality
• Outer Orthogonality
• Complex Orthogonality
23
Simulations • Flat fading channels • Assume TX has perfect knowledge of the MIMO channel. • Performance comparison between proposed method and optimal BF scheme. – Optimal BF vector is obtained from the dominant right singular vector of Mt-by-Mr channel matrix – Mt=2 with 4 QAM – Mt={3,4} with 16 QAM
• Performance/Complexity comparison between proposed method and iterative power method. – Mt=Mr, 16 QAM and iteration={1,2,3} 24
Mt=2 with 4 QAM Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2)
• Type 2 provides the same performance as the optimal BF 25
Mt={3,4} with 16 QAM Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2)
• Performance slightly degraded for Mt>2 26
Mt=Mr, 16 QAM and iteration={1,2,3} Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2) SNR=10dB
*Assume the square root operation is performed by a lookup table 27
Conclusions • A new form of complex matrices has been derived as a general expression of rotation transformations for the complex vector orthogonalization. • Three different weight vectors have been proposed for maximum-norm combining between two vectors. • Simulations show the good performance and low complexity achieved.
28
1
Paper • Heunchul Lee, Seokhwan Park, Inkyu Lee, “Transmit Beamforming Method Based on Maximum-Norm Combining for MIMO Systems”, IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO 4, pp2067-2075, APRIL 2009 •
Manuscript received April 30, 2008; accepted Dec. 4, 2008.
2
Key Concept • Reduce complexity by rotation transformation method to derive the unitnorm weight vector for maximum-norm combining (MNC).
3
Outline • Background • Vector Orthogonalization and Vector-norm Maximization • Rotation Transformation • Proposed TX BF Method • Simulation Results • Conclusions
4
Background • MIMO antennas system, coupled with space-time processing and diversity gain, increase system capacity and improve quality of service. • The capacity further improved if CSI known at the TX side. • Many method proposed based on SVD of channel matrix. The dominant singular vector information obtained through iterative algorithm is sufficient for optimizing transmit BF performance.
*CSI: Channel State Information *SVD: Singular Value Decomposition
5
System Model Tx=Mt
Rx=Mr
•
Tx transmit Data symbol s with unit-norm transmit weight vector t (|| t ||=1)
•
Rx receive and recover symbols by the receive weight vector r (|| r || =1) +
Estimated symbol
+
z = r Hts + r n
(.)+: complex conjugate transpose
effective channel gain
• Through properly choosing t and r, one can maximum the output SNR 6
The Iterative Power Method • Assume an MRC receiver given by r = * MRC: Maximum-Ratio Combining
Ht Ht
• It can be shown that the channel gain can be maximized when the dominant right singular vector of H is employed as t. +
r H t = H t = λmax (the largest singular value of H )
(*)
• The optimal solution requires the computation of the dominant singular vector of the Mr-by-Mt channel matrix. The transmit vector t can be determined by iterative power method started with an initial vector. r
( i +1)
=
Ht Ht
(i ) (i )
and
t
( i +1)
=
+
( i +1)
+
( i +1)
H r H r
(*) P. A. Dighe, R. K. Malik, and S. S. Jamuar, “Analysis of transmit-receive diversity in Rayleigh fading,” IEEE Trans. Commun., vol. 51, pp. 694-703, Apr. 2003 7
Issue • Although the Iterative Power method can provide a near-optimal result, the complexity of computation is high. – Number of iterations also increase the complexity directly. – (8MtMr+4Mt+4Mr)Nit real multiplications involved for the Mr-by-Mt channel matrix, where Nit indicates the number of iterations.
8
Target and Method • Reduce complexity by proposed method rather than a Iterative Power method. – Propose a rotation transformation method to derive the unit-norm weight vector for maximum-norm combining (MNC). And employs successive MNC to the MIMO channel matrix.
9
Vector Orthogonalization Process and Vector-norm Maximization • It was shown given two real-value vectors the norm of one column is maximized and the norm of the other is minimized when the two columns become orthogonal to each other by a rotation transformation.
*J. C. Nash, “A one-sided transformation method for the singular value decomposition and algebraic eigenproblem,” The Computer J., vol. 18, no. 1, pp. 74-76, Feb. 1975.
10
Defined Complex Vector Orthogonality (1/2) • Given two complex vectors h1 and h2, the Hermitian product is h1 , h2 = h1+ h2 = h1 , h2
R
+ j h1 , h2
I
• To give complex orthogonality a stronger condition by defining h1 , h2 = 0 – inner orthogonal in CMr : R
– outer orthogonal in CMr :
h1 , h2
– complex orthogonal in CMr :
h1 , h2
I R
=0 = 0 and
h1 , h2
I
=0
11
Defined Complex Vector Orthogonality (2/2) • For any complex vector v, define two real-value vectors as
[] []
[] []
⎡ R v ⎤ • ⎡− I v ⎤ v=⎢ ⎥, v = ⎢ ⎥ I v R v ⎣ ⎦ ⎣ ⎦
It’s easy to show
•
•
v = v , v •v = 0
,where (dot) means inner product
• With the defined orthogonality one can have
12
Rotation Transformation and MNC • Consider a rotation matrix F, which will be used to find MNC weight vector, to establish orthogonality between h1’and h2’ and maximum the norm of h1’ Recall: • The real-valued representation is [] []
[] []
⎡ R v ⎤ • ⎡− I v ⎤ v=⎢ ⎥, v = ⎢ ⎥ I v ⎣ ⎦ ⎣ Rv ⎦ •
•
v = v , v •v = 0
re-write as *MNC: Maximum-Norm Combining 13
Inner Orthogonality • '
• ' 2
and h ⊥ h • To make h1 , h2 R = 0 i.e. h ⊥ h • Applying the following rotation transformation where complex domain presentation is where • The inner orthogonality (maximize ) can be achieved if '
1
' 2
1
Recall that
(*maximize
solution is also computable)
and the Frobenius norm remain un-changed
14
Outer Orthogonality • ' 2
•
and h ⊥ h1' • To make h1 , h2 I = 0 i.e. h ⊥ h • Applying the following rotation transformation ' 1
' 2
complex domain presentation is where • The outer orthogonality (maximize
) can be achieved if Recall that
(*maximize
solution is also computable)
15
Complex Orthogonality '
'
• To make h1 , h 2 = 0 ' ' h , h • Inner rotation force the real part Hermitian product 1 2 R ' ' to zero will not affect the imaginary part (i.e. h1 , h 2 = h1 , h 2 I ) I • Outer rotation force the imaginary part Hermitian product ' ' ' ' h1 , h 2 to zero will not affect the real part (i.e. h1 , h 2 = h1 , h 2 ) R I R • Therefore one can achieve complex orthogonality through
16
Comparison of Results If maximum • Inner Orthogonality • Outer Orthogonality
• Complex Orthogonality
• The Inner Orthogonality and Outer Orthogonality lead to small decrease in the norm. Therefore one can select Inner or Outer weight for implementation. 17
Proposed TX BF Method: Mt=2 (1/2) • MNC technique which combines two vectors by the unit-norm weight vector
• Goal: To choose output norm
which maximizes the
or
or if
h1 , h2
R
≥ h1 , h2
I
if
h1 , h2
R
< h1 , h2
I
18
Proposed TX BF Method: Mt=2 (2/2) • Figure shows CDF of the squared vector norm achieved by the MNC process
( assume are i.i.d. complex Gaussian distribution with zero mean and unit variance)
• The square norm value achieved by the complex weight is equal to the square of the maximum singular value of the matrix • The norm value is close if properly Inner/Outer weight chosen 19
Proposed TX BF Method: Mt>=2 (1/3) • Determine weight vector by applying (Mt-1) successive MNC processes between column vectors of the MIMO channel matrix. Select Inner/Outer Orthogonality (TYPE 1) or Complex Orthogonality (TYPE 2)
20
Proposed TX BF Method: Mt>=2 (2/3)
MRC: 1-by-4 SIMO system Antenna Selection: reference [21] Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2)
• Figure shows CDF of relative channel gain (normalized by the maximum channel gain) for Mt=Mr=4 MIMO channels 21
Proposed TX BF Method: Mt>=2 (3/3) • Proposed Algorithm is optimal only for Mt=2 • Further repeat the proposed algorithm, which increase the computational complexity, may obtain the optimum solution. But simulation results show that the performance improvement is marginal.
22
Complexity • The weight vector can be generated through simple computation of θI , θo and θc . • The main computational cost is in the evaluation of
• Inner Orthogonality
• Outer Orthogonality
• Complex Orthogonality
23
Simulations • Flat fading channels • Assume TX has perfect knowledge of the MIMO channel. • Performance comparison between proposed method and optimal BF scheme. – Optimal BF vector is obtained from the dominant right singular vector of Mt-by-Mr channel matrix – Mt=2 with 4 QAM – Mt={3,4} with 16 QAM
• Performance/Complexity comparison between proposed method and iterative power method. – Mt=Mr, 16 QAM and iteration={1,2,3} 24
Mt=2 with 4 QAM Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2)
• Type 2 provides the same performance as the optimal BF 25
Mt={3,4} with 16 QAM Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2)
• Performance slightly degraded for Mt>2 26
Mt=Mr, 16 QAM and iteration={1,2,3} Inner/Outer Orthogonality (TYPE 1) Complex Orthogonality (TYPE 2) SNR=10dB
*Assume the square root operation is performed by a lookup table 27
Conclusions • A new form of complex matrices has been derived as a general expression of rotation transformations for the complex vector orthogonalization. • Three different weight vectors have been proposed for maximum-norm combining between two vectors. • Simulations show the good performance and low complexity achieved.
28
