CSI Feedback for Closed-Loop MIMO-OFDM Systems based on B-splines

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Ren- Shian Chen , Ming-Xian Chang

Institute of Computer and Communication Engineering

National Cheng Kung University 2010/12/06

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Outline



System and Channel Model



Parameterization of CSI

 The B-splines

 Model the CRs Variation across Subchannels



Coefficient Analyse and Gaussian Quantization

 _{The Analysis of Coefficient Variance}  Gaussian Quantization



Numerical Results



Conclusion

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Outline



System and Channel Model



Parameterization of CSI

 The B-splines



Coefficient Analyse and Gaussian Quantization



Numerical Results



Conclusion

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System Model

Parameterization Parameterization Based on B-spline Based on B-spline Gaussian quantization • Forward channel:

 Model B of IEEE 802.11 TGn channel models: Rayleigh fading channel with 9 taps and we assume f_dT_s=0.01

 Cyclic prefix is longer than the channel taps.

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Channel Model

 In this paper, our simulation channel model is formed by modified Jackes model.  We assume that the channel is constant during one block period.

 We use IEEE 802.11 TGn model B (9 taps) as our power delay profile.  We assume that for simplification here.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Power-delay profile P ow er s T _  _

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Outline



System and Channel Model



Parameterization of CSI

 The B-splines



Coefficient Analyse and Gaussian Quantization



Numerical Results



Conclusion

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The B-splines

 A B-spline of order n, denoted by , is an n-fold convolution of the B-spline

of zero order .

 For n = 0, 1, 2 we have the following B-splines

( ) n B t 0( ) B t 0 1, | | , 2 2 1 ( ) , | | , 2 2 0, otherwise. T T t T B t t  _{  }     _     1 | | 1 , | | , ( ) 2 0, otherwise. t T t B t    T   2 2 2 2 2 3 , | | , 4 2 3 | | 9 3 ( ) , | | 2 2 8 2 2 0, otherwise. t T t T t t T T B t t T T        _      

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Outline



System and Channel Model



Parameterization of CSI

 The B-splines



Coefficient Analyse and Gaussian Quantization



Numerical Results



Conclusion

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Model the CRs Variation across Subchannels

 In the modeling process, we transform estimated CRs of subchannels into

B-spline coefficients at the receiver.

 Let , where is the CR of the subchannels at some symbol

time slot.

c

N

k

H

_k

_th

1. We partition into segments, with the segment

2. In this paper, we choose as the fitting curve, the receiver uses m to fit each . p N 1 [ ,..., ] c T N H H  h 1 [ T ,..., T_p]T N  h h h ith 2( ) B t B t₂( ) 1 ( 1) [ ,..., ] , 1 c c p p T i _i N _iN p N N H H i N      h i h h

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Model the CRs Variation across Subchannels

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Model the CRs Variation across Subchannels

3. On each , we sample points, where the parameter is determined by

4. Let , where is the sampling point from , or

2( ) B t q q ( 1) . 2 c p N q q m N    1 [ ,..., ]T q b b  b b_ B t2( ) 2 3 3 ( ( )), =1,..., . 2 1 T T b B q q       

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Model the CRs Variation across Subchannels

5. Define a matrix of size

6. Then the approximation of can be expressed as . 7. By the least-squares-fitting principle :

Q c p N m N  , 1,..., , 1,..., , (( 1) , ) 0, otherwise. 2

for is even, and

, 1,..., , 1,..., , 1 (( 1) , ) 0, otherwise. 2 if is odd b j m q q j j q b j m q q j j q      _{ }        _{ }  Q Q       i h ˆ , [ ,...,₁ ]T i  i i  ci cim h Qc c 2 2 ˆ min ||h_i h_i ||  min ||h_i Qc_i || ( ( , ) : ( , )th element of matrix )Q i j i j Q

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Model the CRs Variation across Subchannels

 After are determined, the receiver can feed back these to the transmitter

with the quantization process.

 The number of feedbak coefficients for each OFDM block is , which is

usually much smaller than .

 For an MIMO-OFDM systems with transmit antennas and receive

antennas, the number of feedback coefficients is .

's i c c_i's

h

N mp c N t N N_r t r p N N N m

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Outline



System and Channel Model



Parameterization of CSI

 The B-splines



Coefficient Analyse and Gaussian Quantization



Numerical Results



Conclusion

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Coefficient Analyse and Gaussian Quantization

 To implement efficient feedback, we need to quantize these coefficients.  The variations of coefficients have a great impact on the feedback load .  Through the constant matrix , each element of also has

complex Gaussian distribution ( ) with zero mean and variance

1 ( T ) T  P Q Q Q i  i c Ph i c ( ) 1 1 2 0 0 1 1 1 2 0 0 0 ( , ) ( , ) { } ( , ) ( , ) cos(2 ( ) )

denotes the path gain of the th path, and is the number of paths in the channel.

c c i m p _c _c N N c x y x y L N N z z x y c z p P m x P m y E H H z P m x P m y x y N z L          _ _      

 

  

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Coefficient Analyse and Gaussian Quantization

 By formula and with simulations, we can find out the coefficient’s variances.

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Coefficient Analyse and Gaussian Quantization

 We observe that the coefficients of B-splines have smaller variation than the

coefficients of Polynomial.

 Since the coefficients are complex Gaussian distribution, the Gaussian

quantization (GQ) algorithm can be applied before the feedback.

 The GQ algorithm is based on two primary condition, the nearest neighborhood

condition (NCC) and the centroid condition (CC) .

Gaussian Quantization

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Outline



System and Channel Model



Parameterization of CSI

 The B-splines



Coefficient Analyse and Gaussian Quantization



Numerical Results

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Numerical Results

It is clear that the Polynomial model has smaller MSE. For 2 and deg =5,

the resulted MSEs(at the receiver) are 0.076 and 0.1845 for the Polynomial and

p

N   m

0.076 0.1845

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Numerical Results

0.076 

0.1845



Fig. 4 shows the MSEs(parametric + quantized distortion) of the rebuilt CRs at the transmitter.

We observe that in Fig. 4 that for lower the B-spline model

has smaller reconstructed MSEs at the transmitter.

c B

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Numerical Results

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Numerical Results

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Outline



System and Channel Model



Parameterization of CSI

 The B-splines



Coefficient Analyse and Gaussian Quantization



Numerical Results



Conclusion

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Conclusion

 We propose an efficient CRs feedback approach based on the B-spline model.  We compare the performance with Polynomial model.

 The proposed algorithm has better performance when we use smaller number of

fed-back bits.

 The proposed algorithm can attain the upper bound of system capacity with low

feedback load.

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sampling points q

ˆ

i



h

N 1 m

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1 10 19 28 37 46 55 64 channel frequency response

37 46

channel frequency response

Polynomial Model

2 0 1 2 0 1 2 ˆ _{( , )} _, _{( , , )}T n H  f c n  c n c n c c  c c c L=10 0 H 1 H 2 H H₃ 4 H 5 H 6 H 7 H 8 H H₉ 0 1 2 3 4 5 6 7 8 9 ( , , , , , , , , , )_{H H H H H H H H H H} T  h           ˆ ( ) _LSF f c  h