IEEE C802.16m-09_1344
Project
IEEE 802.16 Broadband Wireless Access Working Group <http://ieee802.org/16>
Title
Principal Singular Vector Calculation for 4xN Channels
Date
Submitted
2009-07-06
Source(s)
Ron Porat, Phil Orlik
Voice:
E-mail: poratron@yahoo.com; porlik@merl.com
Mitsubishi Electric Research Labs
Toshiyuki Kuze
E-mail :
kuze.Toshiyuki@ah.MitsubishiElectric.co.jp
Mitsubishi Electric Corporation
5-1-1 Ofuna Kamakura, Kanagawa
2478501, JAPAN
*<http://standards.ieee.org/faqs/affiliationFAQ.html>
Re:
.
Abstract
This document describes a proposal for 802.16m channel feedback calculation
Purpose
To be discussed and adopted by 802.16m AWD.
Notice
Release
Patent
Policy
This document does not represent the agreed views of the IEEE 802.16 Working Group or any of its subgroups. It
represents only the views of the participants listed in the “Source(s)” field above. It is offered as a basis for
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contained herein.
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Further information is located at <http://standards.ieee.org/board/pat/pat-material.html> and
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Rank-1 Calculation for 4xN channels
Ron Porat
Mitsubishi Electric Research Labs
1. Introduction
This contribution extends the results of a previous contribution [1] whereby the SVD of a general 2xN channel
was shown to have a simple closed form solution.
We provide an extension to 4xN channels. The closed form solution is not mathematically optimal but provides
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IEEE C802.16m-09_1344
practically close to optimal performance.
The idea can be used in the same way for larger size channels.
A typical usage scenario is for an MS with 4 antennas estimating a joint rank-1 precoder for multi-BS scenario.
2. Rank-1 Calculation
The main idea is to concentrate on H H which is of size Nx4 and find the optimal gain and phase combining
weights of the two right singular vectors of H1  H (1: 2,:) H and H 2  H (3 : 4,:) H . The optimal solution for the
right singular vector of an Nx2 channel was given in [1].
Denote by U1 and U 2 the two right singular vectors of the two Nx2 channels.
We would like to solve
rU1
rU1

 2

 2
r ,   arg max || H H 
||

arg
max
||
[
H
H
]
|| 


1
2
 1  r 2 e j U 
 1  r 2 e jU 
r ,
r ,

2

2
r 2 || H1U1 ||2 (1  r 2 ) || H 2U 2 ||2 2r 1  r 2 Re{e  j ( H 2U 2 )* ( H1U1 )}
The solution for  is straightforward and given by
x
where x  ( H 2U 2 )* ( H1U1 )
|| x ||
The solution for r can now be found by solving arg max f (r )  Ar 2  Br 1  r 2  C where
r
A || H1U1 ||  || H 2U 2 || and B  2 | x | .
2
2
Differentiating f ( r ) and equating to zero we can arrive at a quadratic equation in r 2 , the solution of which
(skipping a few stages) is r  0.5 
A/ 2
A2  B 2
Finally the rank-1 solution to H is given by V 
rU1


H HU
U

where
.

 1  r 2 e jU 
|| H H U ||

2
3. Simulation Results
The following plot shows the rank-1 SU-MIMO effective channel-norm loss using this formula as compared to
the optimal SVD based rank-1 precoding. Uncorrelated flat fading 4x4, 4x8 and 4x12 channels were simulated.
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IEEE C802.16m-09_1344
It can be seen that the loss assuming perfect channel knowledge is minimal.
Effective Channel Norm Relative to Optimum
0
10
4x8
4x12
4x4
-1
10
Pr(Loss>|x|)
-2
10
-3
10
-4
10
-5
10
-1
-0.9
-0.8
-0.7
-0.6
-0.5
[dB]
-0.4
-0.3
-0.2
-0.1
0
We have also simulated this formula for MU-MIMO using regularized ZF for 4x8 channel in SCM Suburban
Macro.
The BS was assumed to have 4 closely spaced cross-pols XXXX or two clusters [XX XX] and the MS 2 closely
spaced cross-pols ++ (V-H).
Other simulation assumptions were 3kmph, 5mS feedback delay and 4PRB band size and perfect channel
estimation and feedback.
The spectral efficiency loss as compared to perfect SVD based was found to be 0.5% in all SNR ranges
Conclusion
The closed form formula provides practically close to perfect solution for rank-1 calculation for any 4xN
channel.
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IEEE C802.16m-09_1344
References
1. C802.16m-08/522r1 – “Analog Feedback Complexity Clarification”
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