2007-11-12
IEEE C802.16m-07/187r4
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
2007-11-12
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
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2007-11-12
IEEE C802.16m-07/187r4
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

k 1, k  i
P( y | x  xk )
)  log e (
e
di2
2
N


e
d k2

i  1, 2,..., M
)
(1.1)
2
k 1, k  i
th
di, (i=1, 2, …, M), indicates the i distances for the current received symbol which is output from MLD
detector, so there is dk  y  Hxk  ( y  Hxk )( y  Hxk ) H , where x k represents kth symbol.
According to the definition of the mutual information per symbol as symbol information (SI), we have
p( y xi )
SI  I ( X ; Y )    p ( y xi ) P (xi ) log 2
dy
p( y )
i
1
 log 2 M 
M
x x w
M




E log 2 1   e

 k 1,k i
i 1


M
i
k
2
w
2

 
 
 

1 M
 log 2 M   E log 2 1  e LLR 
M i 1
1 M 
M 
  E log 2

M i 1 
1  e LLR 
i
2
i
(1.2)
Furthermore the mutual information per symbol (SI) can be calculated as:
1 M
M
SI    p  LLRi  log 2
dLLRi
M i 1 LLR
1  e LLR
i
(1.3)
i
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
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IEEE C802.16m-07/187r4








2
M
 2i



e
.

(1.4)
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
N
RBIR 
 SI
n 1
N
n
(1.5)
 m( n)
n 1
where SIn is the mutual information over the n-th subcarrier and m(n) is the information bit per symbol over the
n-th subcarrier.
If symbol-level LLR satisfies the distribution of Gaussian then the SI over the n-th subcarrier can continuously
be derived as
SI 
1
M
1

M
M
  p  LLR  log
i 1 LLRi
M

i 1 LLRi
i
2
M
dLLRi
1  e  LLR

1
e
2  VARi
i
 LLRi  AVEi 2
2VARi
(1.6)
M
log 2
dLLRi
1  e  LLR
i
where it is assumed that symbol LLRi under ML detection satisfies the Gaussian distribution; its mean is AVEi
and the variance is VARi.
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.
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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:

M


2
e
LLRi  log e
2
hxi  n  hxi

hxi  n  hxk
e

e
 log e
2
2

dh  n
e

2
e
2

n
djh  n

2
2
2
2
e

( d  dj ) h  n

2
2
(1.7)
k 1, k  i

d2 h
2
2
K
where
K  log e (e

2 d ( hr nr  hi ni )
2
e

2 d ( hr ni  hi nr )
2
e

d2 h
2
2

2 d ( hr ni  hi nr )
e
2

e
2 d ( hr nr  hi ni )
2
)
(1.8)
and
LLRi 
d2 h
2
K
2
(1.9)
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 }
(1.10)
The variance of LLRi is
VARi  E{ LLRi  E ( LLRi ) }  E{K 2 }  E 2{K }
2
(1.11)
For that:
nr
ni
are Gaussian , and : nr
ni
1
N (0,  2 )
2
Here:
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(1.12)
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IEEE C802.16m-07/187r4
E{K }  E{log e (e
3


2 d ( hr nr  hi ni )


3
h

1
2
2
h
e


3
h

h

2

2
2
log e (2e
2
e

d2 h

2
2

2 d ( hr ni  hi nr )
2
e

2 d ( hr nr  hi ni )
2
e
)}
2 dx
e

d2 h

2
2
e 4 dx )dx
1
2
h
e
e

2 d ( hr ni  hi nr )
2
e

d2 h
2
2

2 d ( hr ni  hi nr )
e
2

e
(1.13)
2 d ( hr nr  hi ni )
2
)]2 }
x2

2
2 d ( hr ni  hi nr )
2
2 d ( hr nr  hi ni )

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 )
(1.14)
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’, 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
derivation and simulation results. For QPSK SISO, according the formula, let h=1, AVE and VAR1/2 can be
computed: when SNR = 5dB, AVE = 4.2147 and VAR1/2 = 2.8290; when SNR = 10dB, AVE = 16.3990 and
VAR1/2= 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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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 )
M

k 11, k 1 i1
(1.15)
P( y | x  xk1 )
In 2x2 SM combined MLD, there are
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t ransmitted
x 
x :  i1 
 xi 2 
h
H   11
 h21
h12 
h22 
y  Hx  n
(1.16)
1
P( y | x  xk1 ) 

M
e

 x  n 
x 
H  i1    1   H  k 1 
x
n
i
2
   2
 xk 2 
2

2
k 2 1
M
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 M
 P( y | x  xk1 )
k 11
k 1 i1
2
d 2 ( h11  h21 )
2


2
2
d 2 ( h11  h21 )
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()
2
2
 K1
(1.17)
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
)
(1.18)
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 )
(1.19)
where
d 2 ( h11  h21 )
2
AVE1i 
2
 K1  AVE1
2
(1.20)
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
(1.21)
And
d 2 ( h11  h21 )
2
AVE1 
2
2
 E{K1}
(1.22)
VAR1  E{K }  E [ K1 ]
2
1
2
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For high SNR we will have
2
E{K1}  E{log e (e
3
d
2
h11  h21

[ dh11 ]* n1  dh11n1*
[ dh21 ]* n2  dh21n2*
2
2
2

3
d
2
h11  h21
d
d
2
h11  h21

2
)}
2
2
x
log e (2e  e
2

2
d 2 ( h11  h21 )
2
e 2 x )dx
21

x2

e
2
2
d
2
2d 2
1
2
h11  h21
2
2

E{K } 
3
11

2
1
 h
h

2
d 2 ( h11  h21 )
[ h11 ( d  dj )]* n1  h11 ( d  dj ) n1*
[ h21 ( d  dj )]* n2  h21 ( d  dj ) n2*
2
2
2
e
2
h11  h21
d
x
2d
1
e
2


3

2


e
2
[ djh11 ]* n1  djh11n1*
[ djh21 ]* n2  djh21n2*
h
11
 h
h11  h21

2
2
x
[log e (2e  e
2
2

2
d 2 ( h11  h21 )

2
e 2 x )]2 dx
21

(1.23)
2) Simulation Results for Symbol LLR (MIMO) – QPSK/16QAM/64QAM
Assuming that the transmitted symbol is ’11 …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 - 0.2616i 0.4019 +
0.3107i], AVE and SE can be computed: when SNR = 5dB, AVE1 = 0.8848; VAR11/2 = 1.6756; AVE2 =
2.2740; VAR21/2 = 2.2347; when SNR = 10dB, AVE1 = 5.0586; VAR11/2 = 3.0481; AVE2 = 9.7909; VAR21/2
= 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 Gaussian; 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
p( LLRMIMO,stream )  N ( AVEstream ,VARstream ) .
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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 two LLR 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 (1.10 – 1.13 and 1.20 – 1.23). 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. Calculate the Symbol-Level LLR distribution (AVE, VAR) given the channel matrix ‘H’ and SNR
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 (1.10 – 1.13 and 1.20 – 1.23). The detailed
formula is also given in the proposed Text section below.
2. Calculate the RBIR metric based on RBIR definition (formula 1.5) and LLR distribution as Step 1.
After calculating the mean and variance of LLR (AVE, VAR) at given subcarrier, the SI can then be
computed as:
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1
SI  I  X ; Y  
M


LLR
M

i 1 LLRi
1
e
2 VAR


1
e
2  VAR
 LLR  AVE 2
2VAR
log 2
 LLRi  AVEi 2
2VARi
log 2
M
dLLRi
1  e LLRi
(1.24)
M
dLLR
1  e  LLR
We also can use the proposed method [6, Beceem] to simplify the numerical integration (1.24).
In MIMO systems, an eigenvalue spread depending parameter “a” is introduced. Then, the distribution of
‘LLR’ can be modified as
 N (a  AVE ,VAR) for QPSK ,16QAM
.
p( LLRi )  
 N (a  AVE , 2 VAR) for 64QAM
3. Sum the SI values over the multiple subcarriers for OFDM system.
4. Divide the SI sum by the sum of bits per symbol to get RBIR.
5. Convert the RBIR for one resource block to one single ‘effective SINR’ from the SNR-to-RBIR Table
given by later section.
6. Lookup the 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, horizontal/vertical
Frame Duration
5 ms
Band Width / Number OFDM
10 MHz / 1024
Subcarrier
Channel Estimation
Ideal
Channel Model
ITU PedB 3kmph/VA 30kmph
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. Figure 3, 4 show the
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simulation results for the horizontal/vertical encoding under ITU PedB 3km/hr, and Figure 5 shows the PHY
result under ITU VA 30kmph by using the same parameter ‘a’ as ITU PedB 3km/hr. From the above PHY
results we can get that our optimization parameter ‘a’ is not sensitive to the different channel profile.
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 for ML Detection – Horizontal Encoding (PedB 3km/hr)
MLD RBIR WiMax SM 2by2 SCW PHY Abstraction
0
10
AWGN QPSK 1/2
AWGN 16QAM 1/2
AWGN 64QAM 1/2
PHY QPSK 1/2
PHY 16QAM 1/2
PHY 64QAM 1/2
-1
Per
10
-2
10
-3
10
-4
10
0
5
10
15
Effective SNR
20
25
30
Figure 4 RBIR PHY for ML Detection – Vertical Encoding (PedB 3km/hr)
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MLD RBIR WiMax SM 2by2 MCW PHY Abstraction
0
Per
10
QPSK 1/2 1st stream
QPSK 1/2 2nd stream
16QAM 1/2 1st stream
16QAM 1/2 2nd stream
AWGN QPSK 1/2
AWGN 16QAM 1/2
-1
10
-2
10
0
5
10
15
Effective SNR
20
25
30
Figure 5 RBIR PHY for ML Detection – Horizontal Encoding (VA 30km/hr)
From the above PHY Abstraction result we can see that our proposed RBIR mapping method can work very
well for the ‘horizontal/vertical’ 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) Generalized Symbol LLR PDF Model – Gaussian Approximation
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The symbol-level log-likelihood ratio (LLR) can be obtained by


 P( y | x  x ) 
i

LLRi  log e  M
i  1, 2,..., M


  P ( y | x  xk ) 
 k 1,k i

where y is the received symbol and xi is the i-th constellation. Then, the mutual information per symbol (SI) in
equation (39) can be written as
SI  I ( X ;Y ) 

1
M
1
M
M

 E log
i 1

2

M

1  exp( LLRi ) 
M

i 1 LLRi
p ( LLRi )log 2
M
dLLRi
1  exp( LLRi )
The conditional PDF of symbol LLR from ML receiver can be approximated as Gaussian. For SISO, the
distribution of LLR for an 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 as two
Gaussian curves for two streams in the ‘horizontal’ encoding system. The distribution of LLR for one stream
from ML receiver can be written as p( LLRMIMO,stream )  N ( AVEstream ,VARstream ) .
In MIMO Matrix B 2x2 and ‘vertical’ encoding system, the distribution of LLR from ML receiver can be
approximated as a Gaussian mixture. Thus, the PDF of LLR can be written as
p( LLRMIMO )  p1  N ( AVEstream1 , VARstream1 )  p2  N ( AVEstream 2 ,VARstream 2 ) . The parameters p1 and p2 are the
optimized parameter for the ‘vertical’ encoding system to make the gap smaller between the effective SNR and
AWGN SNR.
2) Procedure for RBIR PHY Mapping for SISO/MIMO System under ML Receiver
Given channel matrix ‘H’ and SNR, the mean and variance of Gaussian LLR can be approximated by the
formula provided in the procedure below.
Procedure for RBIR PHY Mapping on Symbol-Level ML detection:
1. Calculate the Symbol-Level LLR distribution (AVE, VAR) given the channel matrix ‘H’ and SNR
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, the constellation point
(1,1) for QPSK, (1,1,1,1) for 16QAM, and (1,1,1,1,1,1) for 64QAM are used. Also in the LLR
distribution calculation the impact from the neighboring dominant constellation points is considered so
that all modulations will have the same theoretical formulation as below. For example, for all
modulations, the neighboring 3 constellation points are considered.
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1) For SISO System
Average of LLR over the channel ‘h’ of SISO system is:
AVE  E{LLR} 
d2 h
2
 E{K }
2
where ‘d’ indicates the minimum distance in QAM constellation, for example, QPSK: d  2 ;
16QAM: d  2 / 10 ; 64QAM: d  2 / 42 and the mean of K is defined below.
The variance of LLR is
2
VAR  E{ LLR  E ( LLR) }  E{K 2 }  E 2{K }
where
3
 x2
h

1

E{K } 
3
3
E{K } 
2
e
h
2
h

2

3
h


2
d 2  h
log e (2e 2 d  x  e
2

2
2
e 4 d  x )dx

.
 x2
h

h
1
2
2
h
e
h
2
2
d 2  h
[log e (2e
2 d  x
e
2
2
e 4 d  x )]2 dx

Here the simplified numerical integration can be applied for the above calculation of the mean and
variance of symbol level LLR as
f ( 3VARt ) f 2 ( 3VARt )
2
E{K}  log e (2)  [ f 2 (0)  2

]
3
6
6
where f 2 ( x)  log e (1  ak e
2 x
),
1 
ak  e
2
d2 h

2
2
, and VARt 
d2 h
2
2
.
E{K 2 }  4VARt  2 log e (2) E{K }  log e 2 (2)
[
f ( 3VARt ) f3 ( 3VARt )
f ( 3VARt ) f 4 ( 3VARt )
2
2
f3 (0)  3

]  4[ f 4 (0)  4

]
3
6
6
3
6
6
where the functions f 3 () and f 4 () have the following definitions
f3 ( x)  log e 2 (1  ak e 2 x )
f 4 ( x)  x log e (1  ak e 2 x )
.
The three modulations will have the same formula for the LLR distribution ( AVE,VAR) . The only
difference is the minimum distance ‘d’.
2) For MIMO Matrix B 2x2 System
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The mean and variance for 1st stream are
d 2 ( h11  h21 )
2
AVE1 
2
2
 E{K1}
VAR1  E{K12 }  E 2 [ K1 ]
where
3
 x2
d  H1


E{K1} 
3
d  H1
3
2d 2 
1

2
d H
e
3
log e (2e  e
 d 2  H1

2
2
e 2 x )dx

 x2


E{K } 

x
2
1
d  H1
2
1
2
H1
2d 2 
1
d  H1
2

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 
The same simplified function as in SISO section can be used for above integration for the expectation
2
over the K 1 and K1 .
and
H   H1
Similarly, AVE2 and VAR2 can be defined for the second stream in ‘horizontal’ encoding system.
2.
Calculate the RBIR metric for each stream based on RBIR definition as below and LLR distribution
as Step 1.
.
After calculating the mean and variance of LLR (AVE, VAR) at given subcarrier, the mutual information
per symbol (SI) can then be computed as:
1
SI  I  X ; Y  
M


LLR
M

i 1 LLRi

1
e
2 VAR

1
e
2  VAR
 LLR  AVE 2
2VAR
log 2
 LLRi  AVEi 2
2VARi
log 2
M
dLLRi
1  e LLRi
M
dLLR
1  e  LLR
The simplified numerical integration for the above SI can be written as
1 2
f ( AVE  3VAR ) f1 ( AVE  3VAR )
SI  1 
[ f1 ( AVE )  1

]
log e 2 3
6
6
where the function f1 () has the following definition
f1 ( x)  loge (1  e x ) .
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In MIMO systems, an eigenvalue spread depending parameter “a” is introduced. Then, the distribution of
‘LLR’ can be modified as
 N (a  AVE ,VAR) for QPSK ,16QAM
.
p( LLRi )  
 N (a  AVE , 2 VAR) for 64QAM
From the simulation results on the LLR variance, the three dominant constellation points are not
enough for 64QAM. For 64QAM, the eight constellation points need to be considered for the LLR
variance calculation, which increases the variance twice to realize the impact from 8 constellation
point.
The ‘a’ values will be presented in next sub-section for the different ranges of channel matrix H.
For a MIMO Matrix B 2x2 and ‘vertical’ encoding system, the SI metric can be computed by
combination of the two SI metric for both streams as SI  p1  SI stream1  p2  SI stream 2 . Here RBIR metric
for each stream is computed by the method as mentioned for horizontal encoding.
3.
4.
5.
6.
Sum the SI values over the multiple subcarriers for OFDM system (eq. (41))
Divide the sum by the sum of bits per symbol (eq. (42)) to get RBIR
Convert the RBIR for one resource block to one single ‘effective SINR’ from the SNR-to-RBIR
Table given by later section.
Lookup the AWGN table to get the predicted PER/BLER.
4) Parameter ‘a’ for Horizontal RBIR MLD PHY Mapping for ML Receiver
The parameter ‘a’ is introduced to cover the different scenarios, for example, different inter-stream interference
impact on the performance, the accuracy of LLR distribution, and etc due to different eigenvalue spread.
The channel matrix ‘H’ is classified into several classes to represent the different qualities of ‘H’ as shown in
the table below using eigenvalue decomposition.
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’ are searched in the following Tables.
Table 1 Optimization Parameter ‘a’
k  10
min dB  10
k  10
10  min dB  8
1st Stream
QPSK
1/2
0.9000
QPSK
3/4
1.0000
16QAM
1/2
1.0000
16QAM
3/4
1.0000
64QAM
1/2
1.0000
64QAM
2/3
1.0000
64QAM
3/4
1.0000
64QAM
5/6
1.0000
2nd Stream
0.9000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1st
2.8372
1.4444
0.4343
1.5737
0.7872
1.0000
1.0000
1.0000
Stream
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k  10
min dB  8
10  k  100
min dB  10
10  k  100
10  min dB  8
10  k  100
min dB  8
k  100
min dB  10
k  100
10  min dB  8
k  100
min dB  8
IEEE C802.16m-07/187r4
2nd Stream
1st Stream
2nd Stream
1st Stream
2nd Stream
1st Stream
2nd Stream
1st Stream
2nd Stream
1st Stream
2nd Stream
1st Stream
2nd Stream
1st Stream
2nd Stream
1.4801
1.2000
1.2000
1.9264
1.6172
0.8833
0.8857
1.1000
1.1000
0.8000
0.8111
0.9736
2.6241
0.9000
0.9000
1.4859
1.0000
1.2000
1.1731
1.3444
1.1900
1.3000
1.0000
1.1000
0.9737
1.2456
0.9573
1.0222
1.0000
1.0000
0.6389
0.6000
0.6000
1.0000
1.0000
0.5000
0.5000
0.5500
0.5500
0.4000
0.4000
1.7303
0.4667
0.4500
0.4500
1.1526
0.9889
1.3632
1.0000
1.0000
1.1246
0.8532
1.0000
1.0000
1.0000
0.7479
0.8532
0.8310
1.0000
1.0000
1.1000
0.4695
0.3111
2.0000
2.0000
0.6611
0.6500
0.7310
0.9111
10.000
8.9000
1.4895
0.6444
0.9000
0.9000
1.0000
1.5889
2.0667
1.0000
1.0000
0.8556
0.8333
1.0778
1.0778
1.0000
1.0000
0.8889
0.9445
0.9000
1.0000
1.1000
1.5000
1.0667
1.0000
1.0000
1.0111
1.1556
1.1111
1.1667
1.0000
1.0000
0.8889
1.0555
0.8889
1.0000
1.0000
0.9222
0.9333
1.0000
1.0000
1.0000
1.0111
0.8333
0.8333
0.6889
1.0000
0.7556
0.8445
0.7556
0.7667
Then RBIR can be computed from numerical integration.
5) Optimized Parameter ‘a’ Searching
The parameter is optimized to minimize the difference between effective SNR and AWGN SNR for every
definite PER. The searching procedure can be described as follows
Step 1: Get the interpolated SNR value from the measured PER by using the AWGN curve;
Step 2: Get the effective SNReff value from the calculated RBIR under given channel matrix ‘H’ , SNR and
parameter ‘a’ using the Table SNR-to-RBIR below;
Step 3: Find parameter ‘a’ which has the smallest gap over all PER between the interpolated SNR (step 1) and
effective SNR (step 2).
a  min SNRAWGN ( PER)  SNReff ( PER)
2
a
for  PER .
6) Parameter ‘p’ for vertical RBIR MLD PHY Mapping for ML Receiver
The optimized parameters ‘p’ are tabulated in the following Table.
Table 2 Parameter ‘p’
QPSK
1/2
k  10
min dB  10
k  10
10  min dB  8
k  10
min dB  8
10  k  100
min dB  10
10  k  100
10  min dB  8
p1
p2
p1
p2
p1
p2
p1
p2
p1
p2
16QAM
1/2
0.5000
0.5000
0.5445
0.6556
0.5000
0.5000
1.0000
0.1111
0.6667
0.3333
0.5000
0.5000
0.6667
0.4444
0.5000
0.5000
0.5000
0.5000
0.1111
0.8889
1
9
64QAM
1/2
0.5000
0.5000
1.0000
0.3333
0.5667
0.5223
0.5000
0.5000
0.5333
0.5111
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10  k  100
min dB  8
k  100
min dB  10
k  100
10  min dB  8
k  100
min dB  8
p1
p2
p1
p2
p1
p2
p1
p2
0.5000
0.5000
0.2222
0.7778
0.6667
0.0000
0.5000
0.5000
0.5000
0.5000
0.0000
0.5556
0.4778
0.6556
0.5000
0.5000
0.8889
0.1111
0.5000
0.5000
0.0000
1.0000
0.5000
0.5000
Then RBIR can be computed from numerical computation.
7) Optimized Parameter ‘p’ Searching
The parameter is optimized to minimize the difference between effective SNR and AWGN SNR for every
definite PER. The searching procedure can be described as follows
Step 1: Get the interpolated SNR value from the measured PER by using the AWGN curve;
Step 2: Calculate the corresponding RBIR metrics over the two streams under given channel matrix ‘H’ , SNR
and parameter ‘a’.
Step 3: Get the average RBIR metric by the multiply of ‘p1’ and ‘p2’ and calculate the effective SNReff value
from the averaged RBIR using the Table SNR-to-RBIR below;
Step 4: Find parameter ‘p1’ and ‘p2’ which provide the smallest gap over all PER between the interpolated
SNR (step 1) and effective SNR (step 3).
p  min SNRAWGN  SNReff
( p1 , p2 )
7) SNR-to-RBIR Table
The SNR-to-RBIR Table is obtained from Monte Carlo simulation using (40) and (42).
Table 3 SNR-to-MI Table
SNR Span
(dB)
RBIR Value
QPSK
[-20:1:27]
0.0000
0.0114
0.0225
0.0442
0.0855
0.1615
0.2910
0.4859
0.7207
0.9119
0.9901
0.9999
1.0000
0.0072
0.0143
0.0282
0.0551
0.1061
0.1978
0.3489
0.5628
0.7944
0.9507
0.9968
1.0000
1.0000
16QAM
[-20:1:27]
0.0090
0.0179
0.0352
0.0688
0.1311
0.2407
0.4141
0.6422
0.8592
0.9760
0.9992
1.0000
1.0000
0.0000
0.0057
0.0112
0.0221
0.0428
0.0808
0.1461
0.2474
0.3852
0.5509
0.7317
0.8949
0.9821
64QAM
[-20:1:27]
0.0036
0.0071
0.0141
0.0276
0.0531
0.0990
0.1756
0.2896
0.4379
0.6103
0.7910
0.9343
0.9927
2
0
0.0045
0.0089
0.0176
0.0344
0.0656
0.1206
0.2094
0.3357
0.4933
0.6709
0.8463
0.9633
0.9976
0.0000
0.0038
0.0075
0.0147
0.0285
0.0539
0.0974
0.1653
0.2583
0.3718
0.4997
0.6374
0.7802
0.0024
0.0047
0.0094
0.0184
0.0354
0.0660
0.1172
0.1937
0.2942
0.4131
0.5448
0.6848
0.8269
0.0030
0.0060
0.0117
0.0229
0.0437
0.0805
0.1398
0.2247
0.3321
0.4558
0.5907
0.7325
0.8708
2007-11-12
IEEE C802.16m-07/187r4
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9994
1.0000
1.0000
0.9999
1.0000
1.0000
2
1
1.0000
1.0000
1.0000
0.9100
0.9796
0.9971
0.9425
0.9883
0.9995
0.9668
0.9937
1.0000