2007-09-10
IEEE C802.16m-07/187r2
Project
IEEE 802.16 Broadband Wireless Access Working Group <http://ieee802.org/16>
Title
Link Performance Abstraction for ML Receivers based on RBIR Metrics
Date
Submitted
2007-09-10
Source(s)
Hongming Zheng, Intel Corporation
hongming.zheng@intel.com
may.wu@intel.com
yang-seok.choi@intel.com
nageen.himayat@intel.com
jingbao.zhang@intel.com
senjie.zhang@intel.com
May Wu, Intel Corporation
Yang-seok Choi, Intel Corporation
Nageen Himayat, Intel Corporation
Jingbao Zhang, Intel Corporation
Senjie Zhang, Intel Corporation
Louay Jalloul, Beceem Communications
Jalloul@beceem.com
Re:
IEEE 802.16m-07/031 – Call for Comments on Draft 802.16m Evaluation Methodology
Document
Abstract
This contribution provides a link abstraction methodology for ML receivers based on RBIR
metrics.
Purpose
For discussion and approval by TGm
Notice
Release
Patent
Policy
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Link Performance Abstraction for ML Receivers based on RBIR Metrics
Hongming Zheng, May Wu, Yang-seok Choi,
Nageen Himayat, Jingbao Zhang, Senjie Zhang, Intel Corporation
1
2007-09-10
IEEE C802.16m-07/187r2
Louay Jalloul, Beceem Communications
1.0 Purpose
This contribution provides a detailed description of a link evaluation methodology for MIMO Maximum
likelihood (ML) receivers. With the proposed modeling technique, accurate link abstraction can be obtained
based on a mean RBIR (Received Bit Information Rate) between the transmitted symbols and their LLR values
under symbol-level ML detection.
2.0 Introduction
In order to reduce complexity from real link level simulations to system level simulations, an accurate block
error rate (BLER) prediction method is required to map the performance between the link and the system for the
system capacity evaluation.
A well-known approach to link performance prediction is the Effective Exponential SINR Metric (EESM)
method. This approach has been widely applied to OFDM link layers [1][2][3] and MMSE detection for
receiver algorithms, but this approach is only one of many possible methods of computing an ‘effective SINR’
metric.
One of the disadvantages of the EESM approach is that a normalization parameter (usually represented by a
scalar, β) must be computed for each modulation and coding (MCS) scheme for many scenarios. In particular,
for broader link-system mapping applications, it can be inconvenient to use EESM for adaptive modulation
when HARQ is used in the system, where the codewords in different modulation types will be combined in the
different transmission/retransmissions. In addition, it is difficult to extend this method to MLD detection in the
SISO/MIMO case because EESM uses the post-processing SINR.
In order to overcome the shortcomings of EESM as described above, in this contribution we focus on the
conventional Mutual Information method (RBIR) for the phy abstraction/ link performance prediction in MLD
receivers. It is shown in this contribution that link abstraction can be achieved by using the RBIR metrics
exclusively, i.e., by mapping RBIR directly to BLER. The procedure for modeling MIMO-ML only requires
obtaining the RBIR metric for the matrix channel and then mapping the BLER for the performance of ML
receiver, which is not much more complex.
We develop a solution that computes the RBIR metric in an ML receiver given by a channel matrix under
MIMO 2x2 antenna configuration. We split the channel matrix into different ranges (different qualities of H)
which means that there will be different combining parameters for the mapping from the symbol-level LLR
value to RBIR metric. This RBIR method for ML receivers can be applied to both “vertical” encoding and
“horizontal” encoding system profiles in the WiMAX system.
The first part of the contribution will provide an overview of RBIR PHY metric using symbol-level ML
detection; the second part of this contribution presents the theory derviation/approximation and simulation
results of symbol LLR distribution from an ML receiver in both SISO /MIMO systems; the third part provides
detailed solutions for RBIR PHY mapping for SISO/MIMO system for an ML Receiver which includes the
general symbol LLR PDF model, procedure for RBIR PHY Mapping for SISO/MIMO System in an ML
Receiver and parameter ‘a’ for RBIR MLD PHY Mapping for ML Receiver and the parameter ‘a’ searching
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procedure, etc. Finally this contribution gives out the proposed text section for .16m EVM document on RBIR
in section 4.3.1.1.
3.0 RBIR Mapping for SISO/MIMO System
This section describes RBIR definition for SISO system, focusing on the theoretical concepts and notations. The
numerical expression/approximations for the actual RBIR from symbol-level LLR values will be derived in
detail.
The symbol-level LLR given xi is transmitted for ML receiver can be computed as
LLRi  log e (
P( y | x  xi )
N
 P( y | x  x )

)  log e (
k
k 1
k i
e
N
M i2
2
e

M k2

)
i  1, 2,..., N
(1.1)
2
k 1
k i
Mi, (i=1, 2, …, N), indicates the ith distances for the current received symbol which is output from MLD
detector, so there is M k  y  Hxk  ( y  Hxk )( y  Hxk ) H , where x k represents kth symbol.
Due to the different QAM mappings, the RBIR over one constellation can be represented as
I ( x, LLR) 
1 m
 I ( xi , LLR( xi ))
m i 1
(1.2)
where I ( xi , LLR ( xi )) is the RBIR metric of the transmission symbol xi over the whole constellations.
For a WiMAX system the mutual information – RBIR metric will be considered at all N c -subcarriers as
RBIR 
1 N m
 I n ( xi , LLR( xi ))
mN c n1 i 1
c
(1.3)
3.1 RBIR Computation – MQAM
According to Mutual Information definition, we have
MI  I ( X ; Y )    p ( y xi ) P(xi ) log 2
i
p ( y xi )
p( y)
dy

N
 xi  xk  w 2  w 2  
1 N 
 log 2 N   E log 2 1   exp 
 
2

N i 1 

 
 k 1,k i 

1 N
 log 2 N   Elog 2 1  exp(  LLRi ) 
N i 1
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IEEE C802.16m-07/187r2
1 N
  Elog 2 N  log 2 1  exp(  LLRi ) 
N i 1

1 N 
N
  E log 2

N i 1 
1  exp(  LLRi ) 
(1.4)
And the mutual information can be calculated as:
MI 
1 N
N


p
LLR
log
dLLRi

i
2

N i 1 LLR
1

exp(

LLR
)
i
i
(1.5)
In QPSK, LLRi and LLRk have the same pdf but not in QAM in general. However, since the Euclidean
distance around the first tier constellation is dominant (i.e. first 3 or 4 neighboring constellation points), in
QAM we can approximately calculate the LLR around the 3 or 4 constellations as following








2
M
 2i



e
.
(1.6)
LLRi  ln 
M k2 

 2
 e  
 k i

indices
 ofk{dominant

Euclidean
 distance}



For example, in 16 and 64 QAM, the outer constellation point will have 3 dominant Euclidean distances while
the inner constellation points will have 4 dominant Euclidean distances. Note that the inner and outer
constellation may have different pdf of the LLR. For simplicity, we can choose one LLR among N possibilities
to represent the signal quality.
Define RBIR as
RBIR  MI / log 2 N
(1.7)
If symbol-level LLR satisfies the distribution of Gaussian then the above formula can continuously be derived as
RBIR 
1 1 N
N
p  LLRi  log 2
dLLRi


log 2 N N i 1 LLR
1  exp( LLRi )
i

1 1 N
1

e


log 2 N N i 1 LLR 2  VARi
 LLRi  AVEi 2
2VARi
i
log 2
N
dLLRi
1  exp( LLRi )
(1.8)
where it is assumed that symbol LLRi under ML detection satisfies the Gaussian distribution; its mean is AVEi
and the variance is VARi .
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In the following we will see if the symbol LLR satisfies the Gaussian distribution or not from the theory
derivation and real simulation results.
3.2 LLR Distribution of Symbol-Level ML Detection (SISO) – Theory Derivation/
Simulation
1) Theory Derivation for Symbol LLR (SISO QPSK as Example)
Firstly we will make the theory derivation from QPSK modulation for SISO system.
In the following we have the parameter setting for the different modulation. For example, QPSK: d  2 ;
16QAM: d  2 / 10 ; 64QAM: d  2 / 42 . ‘d’ indicates the minimum distance in QAM constellation.
For the ith symbol:

hxi  n  hxi

e
LLRi  log e
N
e
2

2
hxi  n  hxk


e
 log e
2
2
dh  n


e
2
2
e

n

djh  n

2
2
2
2
e

( d  dj ) h  n

k 1
k i

d2 h
2
2
(1.9)
2
2
K
where
K  log e (e

2 d ( hr nr  hi ni )

2
e

2 d ( hr ni  hi nr )

e
2

d2 h

2
2

2 d ( hr ni  hi nr )
e
2

2 d ( hr nr  hi ni )
e
2
(1.10)
)
and
LLRi 
d2 h
2
K
2
(1.11)
From the above formula we can see that for QPSK the symbol LLRi can be approximated as Gaussian
distribution.
Average of LLRi is:
AVEi  E{LLRi } 
d2 h
2
2
 E{K }
(1.12)
The variance of LLRi is
VARi  E{ LLRi  E ( LLRi ) }  E{K 2 }  E 2{K }
2
For that:
5
(1.13)
2007-09-10
IEEE C802.16m-07/187r2
nr
ni
are Gaussian , and : nr
1
N (0,  2 )
2
ni
(1.14)
Here:
E{K }  E{log e (e
3


2 d ( hr nr  hi ni )


3
h

1
2
2
h
e



3
h

1
2
2 d ( hr ni  hi nr )

2
e

d2 h

2
2

2 d ( hr ni  hi nr )
2
e

2 d ( hr nr  hi ni )
2
e
)}
2
h
2
2
log e (2e
2 d ( hr nr  hi ni )
2
2 dx
e

d2 h
2
2
e 4 dx )dx
2
h
e
e

2 d ( hr ni  hi nr )
2
e

d2 h
2
2

2 d ( hr ni  hi nr )
e
2

(1.15)
2 d ( hr nr  hi ni )
e
2
)]2 }
x2

h

e


E{K 2 }  E{[log e (e
3
x

h

2
h

2
2
[log e (2e
2 dx
e

d2 h

2
2
e 4 dx )]2 dx

Then LLRi is distributed as:
p( LLRi )  N ( AVEi ,VARi )
(1.16)
We can also get the similar theory approximation for 16QAM/64QAM. All these two modulations also can be
approximated as Gaussian.
2) Simulation Results for Symbol LLR (SISO) – QPSK/16QAM/64QAM
Assuming that the transmitted symbol is ’11 …1 (M ‘1’)’, here ‘M’ is the number of bits in the QAM order, the
LLR distributions under different normalized fading factor ‘h’ are simulated as in Figure 1a, 1b and 1c for the
different modulation. In Figure 1a-1b-1c the curve in black color is the standard Gaussian curve generated by
Matlab function which is used to approximate the real LLR value shown in Red color. It is testified that the
mean and variance can meet the derivation of LLR distribution in the previous section.
So from the figure below it is easy to see that the symbol level LLR from ML detection satisfies the Gaussian
distribution, which also satisfies the theoretical derivation of symbol LLR distribution as the previous section.
We now provide an example for QPSK for a clear explanation of the relationship between the theoretical
1/ 2
derivation and simulation results. For QPSK SISO, according the formula, let h=1, AVE and VAR can be
1/ 2
computed: when SNR = 5dB, AVE = 4.2147 and VAR = 2.8290; when SNR = 10dB, AVE = 16.3990
1/ 2
and VAR =
5.0956. We see that there is a close relationship for 16QAM and 64QAM between the
theoretical derivation and simulation results.
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IEEE C802.16m-07/187r2
SISO LLR Distribution
16QAM LLR Distribution
0.16
0.35
0.14
0.3
SNR=5dB
0.12
SNR=5dB
0.25
SNR=10dB
0.1
Prob
Prob.
0.2
0.08
SNR=10dB
0.15
0.06
0.1
0.04
0.05
0.02
0
-20
-10
0
10
LLR Value
20
30
0
-15
40
-10
Figure 1a QPSK LLR Distribution (SISO)
-5
0
LLR Value
5
10
15
Figure 1b 16QAM LLR Distribution (SISO)
64QAM LLR Distribution
0.4
0.35
SNR=5dB
0.3
Prob
0.25
SNR=10dB
0.2
0.15
0.1
0.05
0
-15
-10
-5
LLR Value
0
5
Figure 1c 64QAM LLR Distribution (SISO)
3.3 LLR Distribution of Symbol-Level ML Detection (MIMO) – Theory Derivation/
Simulation
1) Theory Derivation for Symbol LLR (MIMO QPSK as Example)
For the 1st stream:
LLR1i  log e
P( y | x  xi1 )
N
 P( y | x  x
k1
k 11
k 1 i1
(1.17)
In 2x2 SM combined MLD, there are
7
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t ransmitted
x 
x :  i1 
 xi 2 
h
H   11
 h21
h12 
h22 
y  Hx  n
(1.18)
P( y | x  xk1 ) 
N
1
e


 x  n 
x 
H  i1    1   H  k 1 
 xi 2   n2 
 xk 2 
2

2
k 2 1
N
1
e


 h11 ( xi 1  xk 1 )  h12 ( xi 2  xk 2 )  n1 
h ( x  x ) h ( x  x ) n 
22 i 2
k2
2
 21 i 1 k 1
2
2
k 2 1
The LLR for the first stream of 2x2 Matrix B is
P ( y | x  xi1 )
LLR1i  log e N
 P( y | x  xk1 )
k 11
k 1 i1
2
d 2 ( h11  h21 )
2


2
2
d 2 ( h11  h21 )
2

2
 log e (1  e

2
d 2 ( h12  h22 )
2 d [ h12 r n1 r  h12 i n1i ]
2 d [ h22 r n2 r  h22 i n2 i ]
2
2
)  log e (1  e

2
d 2 ( h12  h22 )
2 d [ h12 r n1i  h12 i n1r ]
2 d [ h22 r n2 i  h22 i n2 r ]
2
)  o()
(1.19)
2
 K1
Where:
2
K1   log e (1  e

2
d 2 ( h12  h22 )
2 d [ h12 r n1 r  h12 i n1i ]
2 d [ h22 r n2 r  h22 i n2 i ]
2
2
)  log e (1  e

2
d 2 ( h12  h22 )
2 d [ h12 r n1i  h12 i n1 r ]
2 d [ h22 r n2 i  h22 i n2 r ]
2
)
(1.20)
From the above we can see that the symbol LLR for the first stream can still be approximated as a Gaussian
distribution. The distribution is given by
p ( LLR1i )  N ( AVE1i , VAR1i )
(1.21)
where
d 2 ( h11  h21 )
2
AVE1i 
2
 K1  AVE1
2
(1.22)
VAR1i  E{K }  E {K1}  VAR1
2
1
2
For simplicity, the different conditional LLR1i distributions can be approximated by the same Gaussian because
we used the dominant constellation points for LLR calculation.
p( LLR1i )  N ( AVE1 ,VAR1 )
i  1, 2,..., N
(1.23)
And
d 2 ( h11  h21 )
2
AVE1 
2
2
 E{K1}
VAR1  E{K }  E [ K1 ]
8
2
1
2
(1.24)
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For high SNR we will have
2
[ dh11 ]
[ dh21 ]* n2  dh21n2*
*
E{K1}  E{log e (e
3
d
2
h11  h21

2
2

3
2
h11  h21
d
3
11
d
2

)}

2
log e (2e  x  e

2
d 2 ( h11  h21 )

2
e 2 x )dx
(1.25)

x2

2
d
2
2d 2
e
2
2
2
21
1
h11  h21

2
2
2
2

E{K } 
2
h11  h21
e
 h
e
2
d 2 ( h11  h21 )
[ h11 ( d  dj )]* n1  h11 ( d  dj ) n1*
[ h21 ( d  dj )]* n2  h21 ( d  dj ) n2*
x2
2d 2
h

2
1
n1  djh11n1*
2

2
h11  h21
d

2

d
e
1
2
[ djh11 ]
[ djh21 ]* n2  djh21n2*
*
2


3
n1  dh11n1*
h
11
 h
2
h11  h21
2
2
2
x
[log e (2e  e

2
d 2 ( h11  h21 )
2
e 2 x )]2 dx
21

2) Simulation Results for Symbol LLR (MIMO) – QPSK/16QAM/64QAM
Assuming that the transmitted symbol is ’11 …1 (M ‘1’)’ for each of the 2 transmit antennas, the LLR
distributions under different fading factors ‘H’ are simulated as in Figure 2a, 2b and 2c for the different
modulation.
The channel matrix used in the example is H = [-0.1753 + 0.1819i 0.1402 + 0.5974i;
0.4019 + 0.3107i] and the figures give the LLR distribution obtained from H and SNR.
0.4829 - 0.2616i
In Figure 2a-2b-2c the curve in black color is the standard Gaussian curve generated by the Matlab function
which is used to approximate the real LLR value shown in Red color. For a MIMO system, the figures simulated
the ‘horizontal’ encoder and there are two streams in the system which has two LLRs, each corresponding to
different stream.
So from the figure below it can be seen that the symbol level LLR from ML detection satisfies the Gaussian
distribution, which also meets the theoretical derivation of symbol LLR distribution as described in the previous
section.
In the example with 2x2 SM QPSK, let H=[ -0.1753 + 0.1819i 0.1402 + 0.5974i;
0.4829 1/ 2
0.4019 + 0.3107i], AVE and SE can be computed: when SNR = 5dB, AVE1 = 0.8848; VAR1 =
1/ 2
AVE2 = 2.2740; VAR2
1/ 2
2
VAR
1/ 2
= 2.2347; when SNR = 10dB, AVE1 = 5.0586; VAR1
=
0.2616i
1.6756;
3.0481; AVE2 = 9.7909;
= 4.0439.
According to the computed AVE and VAR, plot the Gaussian distribution; this makes good approximation to
LLR distribution.
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MIMO 2x2 QPSK
MIMO 2x2 16QAM
0.35
0.45
0.4
0.3
0.35
0.25
5dB
0.3
5dB
Prob.
Prob.
0.2
0.15
0.25
0.2
10dB
10dB
0.15
0.1
0.1
0.05
0
-15
0.05
-10
-5
0
5
10
LLR Value
15
20
25
0
-15
30
-10
-5
0
5
10
LLR Value
Figure 2a QPSK LLR Distribution (Matrix B 2x2)
Figure 2b 16QAM LLR Distribution (Matrix B 2x2)
MIMO 2x2 64QAM
0.5
0.45
5dB
0.4
0.35
10dB
Prob.
0.3
0.25
0.2
0.15
0.1
0.05
0
-12
-10
-8
-6
-4
LLR Value
-2
0
2
4
Figure 2c 64QAM LLR Distribution (Matrix B 2x2)
4.0 Solutions on RBIR PHY for SISO/MIMO System under ML Receiver
4.1 Generalized Symbol LLR PDF Model – Gaussian Approximation
As shown in the previous section the conditional PDF of symbol LLR can be approximated as single Gaussian
curve for SISO under the three modulations; For SISO the distribution of LLR from ML receiver can be written
as p( LLRSISO )  N ( AVE,VAR) .
For MIMO Matrix B 2x2 system the conditional PDF of symbol LLR output can be approximated by two
Gaussian curves for two streams of each of three modulations for the ‘horizontal’ encoding system. The
distribution
of
LLR
for
one
stream
from
ML
receiver
can
be
written
as
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p( LLRMIMO,stream )  N ( AVEstream ,VARstream ) .
For MIMO Matrix B 2x2 and ‘vertical’ encoding system the distribution of LLR from ML receiver can be
written as p( LLRMIMO )  p1  N ( AVEstream1 ,VARstream1 )  p2  N ( AVEstream 2 ,VARstream 2 ) .
The simplified Gaussian approximation on the symbol LLR is beneficial for different ‘encoding’ schemes and
antenna configurations (for example, 4x4, etc). This approach can reduce the offline optimal parameter
searching complexity greatly and make the search practical.
The single approximation of Gaussian for different modulations shows reduced complexity compared to the
MMIB method. In the case of MMIB for QPSK, there are twoLLR Gaussian distributions; for 16QAM there are
four LLR Gaussian distributions for ‘horizontal’ encoding system and for 64QAM there are six LLR Gaussian
distributions for a ‘horizontal’ encoding system. Many LLR distributions for the bit-level LLR output over the
different modulation schemes increases the complexity for the offline parameter search and it is also difficult for
the realization of phy abstraction of 4x4 antenna configuration system.
4.2 Procedure for RBIR PHY Mapping for SISO/MIMO System under ML
Receiver
The principle of RBIR PHY on ML Receiver is the fixed relationship between the LLR distribution and BLER.
Given the channel matrix ‘H’ and SNR, the system can have the fixed symbol LLR distribution. This implies
we can have the fixed predicted PER/BLER, which is the mapping principle for RBIR PHY mapping for ML
Receiver. RBIR MLD Metric is required for the Integral/Average of all LLR values for one resource block
between LLR distribution for each subcarrier and PER/BLER for one block.
As shown in section 3.2 the real symbol LLR distribution given channel matrix ‘H’ and SNR can be
approximated as formula (1.11 – 1.13 and 1.19 – 1.21). So we can set up the fixed mapping function between
the parameter-bin (H, SNR) and PER/BLER (from real LLR distribution) which is our RBIR PHY Mapping
function for ML symbol-level detection.
Procedure for RBIR PHY Mapping on symbol-level ML detection:
1. Given the channel matrix ‘H’ and SNR for each subcarrier, the fixed LLR distribution parameter pair
(AVE, VAR ) can be computed from formulas in equations (1.11 – 1.13 and 1.19 – 1.21). The detailed
formula is also given in the proposed Text section below.
2. Calculate the RBIR metric based on RBIR definition (formula 1.8) and LLR distribution as Step 1.
Some approximation is introduced in previous deduction and the conditional distribution of ‘LLRi’ can
be modified as ‘ p( LLRi )  N (a  AVE,VAR) ’. RBIR can be then computed as:
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1
1
RBIR 
log 2 N N

1
log 2 N

LLR

1
e
2  VAR
N

i 1 LLRi
1
e
2  VAR

 LLRi  AVEi 2
2VARi
 LLR  a AVE 2
2VAR
log 2
log 2
N
dLLRi
1  exp( LLRi )
(1.26)
N
dLLR
1  exp( LLR)
We also can use the proposed method [6, Beceem] to simplify the numerical integration (1.26).
3. Average the RBIR values over the multiple subcarriers for an OFDM system
4. Convert the averaged RBIR for one resource block to one single ‘effective SINR’ from the SNR-to-MI
table.
5. Lookup the WiMAX AWGN table to get the predicted PER/BLER.
4.3 PHY Abstraction Results on RBIR PHY Mapping for Matrix B 2x2 system
under ML Receiver
This simulation is done for the WiMAX downlink with AMC permutation and Matrix B 2x2 MIMO
configuration. The channel is ITU PedB 3 kmph. Some main parameters for simulation are given in the Table
1.0.
Table 1.0 Simulation Parameters for RBIR MLD PHY Abstraction
Parameter
Description
MIMO Scheme
2by2 SM, MCW/SCW
Frame Duration
5 ms
Band Width / Number OFDM
10 MHz / 1024
Subcarrier
Channel Estimation
Ideal
Channel Model
ITU PedB 3kmph
Channel Correlation
BS_Corr = 0.25; SS_Corr = 0;
MCS
QPSK ½; 16QAM ½; 64QAM ½
Resource Block Size
16 Subcarriers by 6 Symbols
Reciever
MLD Receiver
From the simulation we stored the PER values and channel matrix ‘H’. With the channel matrix of ‘H’ and the
given SNR we can get the RBIR metric from the LLR distribution and then average the RBIR MLD metric over
multiple subcarriers. In the final step, convert the averaged RBIR metric to one effective SNR. The PHY figure
is to map the measured PER vs. the effective SNR from LLR and RBIR MLD metric.
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MLD RBIR WiMax SM 2by2 MCW PHY Abstraction
0
Per
10
QPSK 1/2 AWGN
16QAM 1/2 AWGN
64QAM 1/2 AWGN
QPSK 1/2 1st stream
QPSK 1/2 2nd stream
16QAM 1/2 1st stream
16QAM 1/2 2nd stream
64QAM 1/2 1st stream
64QAM 1/2 2nd stream
-1
10
-2
10
0
5
10
15
Effctive SNR
20
25
30
Figure 3 RBIR PHY Abstraction for ML Detection
From the above PHY Abstraction result we can see that our proposed RBIR mapping method can work very
well for the ‘horizontal’ encoding system of WiMAX when ML detection is used for the MIMO receiver.
Reference
[1] 3GPP TSG-RAN-1, Nortel Networks, "Effective SIR Computation for OFDM System-Level Simulations,"
Document R1-03-1370, Meeting #35, Lisbon, Portugal, November 2003
[2] Ericsson, “System-level Evaluation of OFDM – Further Considerations”, R1-031303
[3] Lei Wan, “A Fading-Insensitive Performance Metric Unified Link Quality Model”, VTC paper, 2006
[4] Mot, “Link Performance Abstraction for ML Receivers based on MMIB Metrics”, IEEE C802.16m-07/142
[5] Mot, “”Link Performance Abstraction based on Mean Mutual Information per Bit (MMIB) of the LLR
Channel”, IEEE C802.16m-07/097
[6] Beceem, Louay Jalloul, “On the Expected Value of the Received Bit Information Rate”, Sept. 2007
Proposed Text
Include Section 4.3.1.1: RBIR ML Receiver Abstraction for SISO/MIMO
-----------------------------Begin Proposed Text ----------------------------------------------------------------------
4.3.1.1 RBIR ML Receiver Abstraction for SISO/MIMO
1) RBIR Mapping for ML Receiver
The symbol-level LLR for ML receiver can be computed as
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LLRi  log e (
P( y | x  xi )
N
 P( y | x  x )
k 1
k i

)  log e (
k
e
N
M i2
2
e

M k2
)
i  1, 2,..., N
2
k 1
k i
Mi, (i=1, 2, …, N), indicates the ith distances for the current received symbol which is output from MLD
detector, so there is M k  y  Hxk  ( y  Hxk )( y  Hxk ) H .
Due to the different QAM mappings, the RBIR over one constellation can be represented as
I ( x, LLR) 
1 m
 I ( xi , LLR( xi ))
m i 1
where I ( xi , LLR ( xi )) is the RBIR metric of the transmission symbol xi over the whole constellation.
For a WiMAX system, the mutual information – RBIR metric will be considered at all N c -subcarriers as
1 N m
RBIR 
 I n ( xi , LLR( xi ))
mN c n1 i 1
c
2) Generalized Symbol LLR PDF Model – Gaussian Approximation
As shown in the previous section, the conditional PDF of symbol LLR can be approximated as a single
Gaussian curve for SISO under the three modulations; For SISO, the distribution of LLR for an ML receiver can
be written as p( LLRSISO )  N ( AVE,VAR) .
For a MIMO Matrix B 2x2 system, the conditional PDF of symbol LLR output can be approximated as two
Gaussian curves for two streams for each of three modulations for the ‘horizontal’ encoding system. The
distribution
of
LLR
for
one
stream
from
ML
receiver
can
be
written
as
p( LLRMIMO,stream )  N ( AVEstream ,VARstream ) .
For a MIMO Matrix B 2x2 and ‘vertical’ encoding system, the distribution of LLR from ML receiver can be
written as p( LLRMIMO )  p1  N ( AVEstream1 ,VARstream1 )  p2  N ( AVEstream 2 ,VARstream 2 ) .
3) Procedure for RBIR PHY Mapping for SISO/MIMO System under ML Receiver
The principle of RBIR PHY Mapping in an ML Receiver is the fixed relationship between the LLR distribution
and BLER. Given the channel matrix ‘H’ and SNR, the system can have the fixed symbol LLR distribution.
This implies that we can have a fixed predicted PER/BLER, which is the principle for RBIR PHY mapping for
an ML Receiver. RBIR MLD Metric is required for the Integral/Average of all LLR values for one resource
block between LLR distribution for each subcarrier and PER/BLER for one block.
Given channel matrix ‘H’ and SNR, the real symbol LLR distribution ( AVE , VAR ) can be approximated by
the formula provided in the procedure below. This enables us to set up the fixed mapping function between the
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parameter-bin (H, SNR) and PER/BLER (from real LLR distribution) which is our RBIR PHY Mapping
function for ML symbol-level detection.
Procedure for RBIR PHY Mapping on symbol-level ML detection:
1.
Given the channel matrix ‘H’ and SNR for each subcarrier, the fixed LLR distribution parameter pair
( AVE , VAR ) can be obtained as specified below. Here in the LLR calculation, we used the given
constellation point (1,1) for QPSK, (1,1,1,1) for 16QAM and (1,1,1,1,1,1) for 64QAM. Also in the
LLR distribution calculation we only considered the impact from the neighboring dominant
constellation points so that all modulations will have the same theoretical formulation as below. For
example, for all modulations, the neighboring 3 constellation points is considered.
1) For SISO System
Average of LLR is:
AVE  E{LLR} 
d2 h
2
 E{K }
2
The square error (Variance) of LLR is
2
VAR  E{ LLR  E ( LLR) }  E{K 2 }  E 2{K }
where
3

3
E{K } 
2
h
2
h

3
2
1

E{K } 
x2

h
e

3
h


2
2
log e (2e 2 dx  e

d2 h

2
2
e 4 dx )dx

x2

h

h
1
2
2
h
e
h
2
2
[log e (2e
2 dx
e

d2 h
2
2
e 4 dx )]2 dx

The three modulations will have the same formula for the real LLR distribution ( AVE,VAR) . The
only difference is the minimum distance ‘d’. Here for QPSK: d  2 ; 16QAM: d  2 / 10 ;
64QAM: d  2 / 42 . ‘d’ indicates the minimum distance in QAM constellation.
2) For MIMO Matrix B 2x2 System
The mean and variance for 1st stream are
d 2 ( h11  h21 )
2
AVE1 
2
2
VAR  E{K12 }  E 2 [ K1 ]
Where
1
5
 E{K1}
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3


E{K1} 
3
3
2d 2
1
d H1

2
d H
e

E{K } 
3
d H1

x
log e (2e  e

d 2 H1

2
2
e 2 x )dx
x2
2d 2
1
2

2
2



H1
1
d H1
2
1
x2

d H1
d H
e
H1
2
2
x
[log e (2e  e

d 2 H1
2
2
e 2 x )]2 dx
1

h 
2
2
h
H 2    11 12  , H1  h11  h21
 h21 h22 
A similar formula could be used for the second stream of ‘horizontal’ encoding system of WiMAX.
and
H   H1
In a practical implementation, the numerical integral can be realized by the look-up table to reduce the
complexity and running time.
2.
Calculate the RBIR metric based on RBIR definition as below and LLR distribution as Step 1.
Introducing some approximation in the previous derivation, the distribution of ‘LLR’ can be modified as
‘ p( LLR)  N (a  AVE ,VAR) ’, The RBIR can then be computed as:
1
1
RBIR 
log 2 N N
1

log 2 N

LLR
N

i 1 LLRi

1
e
2  VAR

1
e
2  VAR
 LLRi  AVEi 2
2VARi
 LLR  a AVE 2
2VAR
log 2
log 2
N
dLLRi
1  exp( LLRi )
N
dLLR
1  exp( LLR)
The parameter ‘a’ is used to close the gap between measured PER and RBIR MLD PHY.
For QPSK, 16QAM and 64QAM, the LLR mean value will be optimized as AVE  a  AVEcomputed for
the RBIR calculation;
The LLR variance will be optimized for 64QAM as, VAR  2  VARcomputed . Here only 64QAM
modulation needs to be tuned because in the LLR calculation, the 3 dominant constellation points are
not enough for 64QAM from the simulation results on the LLR variance. For 64QAM, the 8
constellation points need to be considered for the LLR variance, which can use the variance tune
parameter ‘2’ to realize the impact from 8 constellation point from the simulation.
The ‘a’ values will be presented in next sub-section for the different H ranges.
3.
4.
5.
Average the RBIR values over the multiple subcarriers for OFDM system
Convert the averaged RBIR for one resource block to one single ‘effective SINR’ from the SNR-to-MI
Table.
Lookup the WiMAX AWGN table to get the predicted PER/BLER.
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4) Parameter ‘a’ for RBIR MLD PHY Mapping for ML Receiver
In the WiMax II 2by2 SM simulation, it is not easy to divide ‘H’ into several classes based on independent
hij for the different qualities of ‘H’ as shown in the Symbol LLR calculation for the mean and variance. Since
the ‘H’ separated into independent hij will result in many scenarios, it is not reasonable for practical
implementation.
We therefore rely on the eigenvalue decomposition method to divide ‘H’ into several ranges for the different
qualities of ‘H’ as shown in the table below.
From simulation, it is concluded that ‘H’ can be classified into scenarios by the following eigenvalue
decomposition as

 H

H H H  V  max
V  k  max

min 
min

min dB  10 log10(min /  2 )
According to the simulation, the optimized parameters ‘a’ for a certain channel model (Here PedB 3kmph is
used.) are searched in the following Tables.
Parameters ‘a’
1<=k<10
10<=k<100
k=>100
Parameters ‘a’
1<=k<10
10<=k<100
k=>100
Parameters ‘a’
1<=k<10
10<=k<100
k=>100
Parameters ‘a’
1<=k<10
10<=k<100
k=>100
Parameters ‘a’
1<=k<10
Table 1a. QPSK ½ case for the 1st stream
min dB <=-10
-10<= min dB <8
0.9000
2.8372
1.9264
0.8833
0.8000
0.9736
Table 1b. QPSK ½ case for the 2nd stream
min dB <=-10
-10<= min dB <8
0.9000
1.6172
0.8111
1.4801
0.8857
2.6241
Table 2a. 16QAM ½ case for the 1st stream
min dB <=-10
-10<= min dB <8
1.0000
0.4343
1.0000
0.5000
0.4000
1.7303
Table 2b. 16QAM ½ case for the 2nd stream
min dB <=-10
-10<= min dB <8
1.0000
1.0000
0.4000
0.6389
0.5000
0.4667
Table 3a. 64QAM ½ case for the 1st stream
min dB <=-10
-10<= min dB <8
1.0000
0.7872
1
7
min dB =>8
1.2000
1.1000
0.9000
min dB =>8
1.2000
1.1000
0.9000
min dB =>8
0.6000
0.5500
0.4500
min dB =>8
0.6000
0.5500
0.4500
min dB =>8
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10<=k<100
k=>100
2.0000
0.6611
10.0000
1.4895
Table 3b. 64QAM ½ case for the 2nd stream
-10<= min dB <8
min dB <=-10
Parameters ‘a’
1<=k<10
10<=k<100
k=>100
1.0000
2.0000
8.9000
1.1000
0.6500
0.6444
0.7310
0.9000
min dB =>8
0.3111
0.9111
0.9000
Then RBIR can be computed from numerical integration.
5) Optimized Parameter ‘a’ Searching
These parameters are optimized to minimize the difference between effective SNR and AWGN SNR for every
definite PER. The searching procedure can be decrypted as follows.
SNRAWGN  WiMaxAWGN _ Per _ to _ SNR( Per )
SNReff  AWGN _ RBIR _ to _ SNR( RBIR( H , SNR, a))
a  min SNRAWGN  SNReff
a
Since there are 9 parameters, if the search is based on combined optimization, it will be too complex to achieve.
The practical suboptimal searching program can be used here to get the optimal parameter ‘a’.
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