P O Z N A N U N I V E R S I T Y O F T E C H N O L O G Y A C A D E M I C J O U R N A L S No 1 Electronics & Communications 2007
Maciej KRASICKI *
AN INTUITIVE SPACE-TIME DIVERSITY SCHEME FOR BI-STCM-ID
In this paper the performance of Bit-Interleaved Space-Time Coded Modulation with Iterative Decoding (BI-STCM-ID) used in the Wireless Local Area Networks is investi- gated. A new ‘intuitive’ space-time diversity scheme that can ensure full compatibility with today WLAN specifications and minimize the number of iterative decoder’s passes is described. The advantages of the proposed scheme are verified by coding gain calcula- tion and distance spectrum analyzing. Simulation results (Bit Error Rate vs. Eb/N0) vali- date the accuracy of the analysis.
Keywords — Multiple-input multiple-output channels, bit-interleaved space- time coded modulation, Alamouti scheme, constellation labeling, block fading, distance spectrum, coding gain.
1. INTRODUCTION
Wireless Local Area Networks today seem to be common Internet access technique. Almost each notebook is equipped with 802.11a/b/g card. Expecta- tions for WLAN throughput are still growing. A new specification 802.11n [1], which is going to be ratified soon, provides some promising techniques such as multi-antenna transmission and space-time block code (STBC). The key issue is to make use of higher Multiple-Input Multiple-Output channel capacity. The idea of BI-STCM-ID [2] seems to be an excellent solution. Unfortunately, 802.11n specification – as a legacy of today’s WLAN – uses Gray constellation mapping that makes iterative processing useless [3]. On the opposite there are known some constellation labeling maps, optimized for the smallest bit error rate (BER) in case of ‘error-free feedback’. The author is an advocate of a new ‘in- tuitive’ approach to overall mapping (constellation labeling and space-time cod- ing) described in Section 4. The author claims that the proposed space-time di- versity scheme minimizes the number of passes while reasonable BER is kept.
Theoretical analysis and simulation results are presented in Section 5.
* Poznan University of Technology
2007
Poznańskie Warsztaty Telekomunikacyjne Poznań 6 - 7 grudnia 2007 POZNAN UNIVERSITY OF TECHNOLOGY ACADEMIC JOURNALS
2. BI-STCM-ID OVERVIEW
The system using BI-STCM-ID is shown in Fig. 1. In the first instance, in- formation bits are encoded by a convolutional encoder of rate RC =1/kc. Next, K interleaved encoded bits choose q constellation points, which form a space-time (ST) symbol Xt∈ℵ. ST symbols define the signals transmitted by Nt antennas in L time slots. As can be noticed, each ST symbol is unequivocally associated with K encoded bits, so an overall mapping rule :
{ }
,K →ℵcan be defined. If the orthogonal Alamouti scheme is used, then( ) ( )
= − * *
t t
t t
t x x
x
X x . (1)
Fig. 1. BI-STCM-ID system: a) transmiter, b) receiver
The signals received by Nr antennas in L time slots are expressed by
t t t
t X H W
Y = + (2)
Matrix Ht, describes the channel, where hi,j is the gain of the path between ith transmit- and jth receive antenna. Wt represents the Gaussian noise.
Space-time demapper evaluates its output log-likelihood ratios (LLRs)
( )
v Otk; using a priori LLRs
( )
vtk;I and channel information. SISO decoder [5] increases LLR’s reliability with max-log-MAP rule.3. ANALYSIS OF BI-STCM-ID PERFORMANCE
When ideal interleaving is assumed, the union bound of bit error probability in first pass of decoding is given by [3]:
( )
∑
∞=
ℵ
≤
df
d I c
b W d f d
P k ( ) , , (3)
where df is the free distance of the convolutional code, and WI (d) denotes the total input weight of error events at Hamming distance d. Finally, f
(
d, ,ℵ)
is the pairwise error probability (PEP). As stated in [4], for BI-STCM systems with Gray labeling ‘expurgated’ PEP can be used instead:( ) ( ) ( )
≤
ℵ
∑ ∑ ∑
= = ∈ℵ
K
k b s
ex K s
K d f
k
b
Z X X
,ˆ
-û
min ,
, , (4) where Zˆ is the nearest neighbor of X, that has kth bit different. -û(X,Zˆ)(s) is the Laplace transform of probability density function of the metric difference
2 2
) ˆ ,ˆ
(X Z = Y−ZH − Y−XH
∆ . (5)
Following [4], it can be written that
( )
( )
r( )
Nri
s s i N
−
=
+
=
∏
ˆ,ˆ
û ˆ /
- min
Z
X , (6)
where ˆ are the nonzero eigenvalues of the matrix i Aˆ =(X−Zˆ)H(X−Zˆ), and rˆ is the rank of Aˆ . As N→
( )
d N r ex
r
N d f
ˆ
ˆ /
~ ,
,
ℵ
, (7)
where
( )
( )r
k b
r rN
K
k b
r N
i K i
r
N K
ˆ / ˆ
, ˆ ˆ ,
−
= = ∈ℵ
−
=
=
ℵ
∑∑ ∑ ∏
X
(8) can be interpreted as coding gain associated with space-time coding and con- stellation labeling. Taking in (3) only the first term (for d =df ) and assuming that energy per information bit Eb =1/R, where R is the overall information rate, the asymptotic BER for BI-STCM system (without feedback chain) can be ap- proximated in logarithmic scale as [4]
( ) ( )
[
R E N]
constd N
Pb ≈−r r f Ω dB+ b 0 dB +
2
10 ˆ /
10 ˆ ˆ
log (9)
This equation is accurate for Gray mapping, but for other labeling maps is an overoptimistic approximation.
If an iterative decoding is implemented, one can look for asymptotic per- formance of BI-STCM-ID (ideal ‘error-free feedback’ is assumed then). In this
approach all bits are assumed to be perfectly known at the demapper, except one, for which LLR is actually evaluated. In such case asymptotic BER can be cal- culated as
( ) ( )
[
R E N]
constd N
Pb ≈−r r f Ω dB+ b 0 dB +
2
10 ~ /
10
~ ~
log (10)
where ~ ℵ
(
, ,Nr)
is similar to ˆ ℵ(
, ,Nr)
in Eq. (9), but ˆ and rˆ must be i replaced with ~i and r~, that are respectively the nonzero eigenvalues and the rank of the matrix A~=(
X−Z~) (
H X−Z~)
. Signals X and Z~ differ only on the kth position.An accurate way to characterize labeling maps in Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) is their Euclidean distance spec- trum [6]. The idea is briefly depicted below. For each symbol x and each kth label position, all neighbors z with the opposite kth bit are found on the constel- lation. For each x− value, the number of its occurrences is counted. Simi-z larly the ‘expurgated’ spectrum Dex for x− distances between x and its the zˆ nearest neighbor zˆ with the opposite kth bit can be used. Next, for x−~z dis- tances, where z~ differs from x only on kth position ‘error-free feedback’ (EF) spectrum Def can be defined. The smaller number and frequency of short dis- tances in Dex and Def spectrum, the better system performance in the first pass and after many iterations, respectively.
Let us extend the idea of distance spectrum for BI-STCM-ID. Of course, if orthogonal space-time codes are used, the problem of ‘optimal’ overall mapping rule ξ is reduced to constellation optimization [4]. As a more general approach, the author proposes to associate the spectrum Dex with
∏
= r
i
i ˆ
ˆ
values for X∈ℵ
symbols. In the same manner Def should involve
∏
= r
i
i
~ ~
values. The correspon- dence between the meaning of the Euclidean distance for 2-dimensional space and the product of eigenvalues for matrices makes this approach justified. The idea of distance spectrum is used to verify performance of the proposed scheme in Section 5.
4. BI-STCM-ID DESIGNING FOR WLAN SYSTEMS
A ‘typical’ approach to BI-STCM-ID is to use orthogonal STBC and find the
‘optimal’ constellation mapping rule, which can ensure maximal coding gain ~
. As a cost of good ‘error free’ bound, the ‘optimal’ labeling does not guaran-
tee fast convergence of iterative decoding. In the region of BI-STCM-ID’s po- tential applications, like WLAN systems, decoding time is the key problem. The solution is a new mapping rule ξ, that enables achieving demanded BER in at most a few iterations. Moreover, the compatibility with IEEE WLAN standards would be invaluable.
The author proposed in [7] an ‘intuitive’ space-time coding scheme for BI- STCM-ID to be used in WLAN systems, which is shown in Fig. 2. The convolu- tional encoder is taken from 802.11a/g/n specifications ([171 133]OCT). The idea is to take advantages of both Gray and ‘optimal’ [4] labeling maps for 16-QAM.
There are two transmit chains in the transmitter. The first one, which uses Gray mapper, is expected to ensure good performance at the first pass. The second one is responsible for asymptotic gain. The shaded blocks shown in Fig. 2 are op- tional, and should be turned off when other devices in the network run in ‘stan- dard’ mode. The block denoted by Π is the symbol interleaver with single depth.
So the proposed space-time codeword is
( ) ( )
( ) ( )
=
. . x opt opt
x
Gray x Gray x
t t
t t
t
X . (11)
Fig. 2. The proposed space-time coded modulation scheme: a) transmitter, b) receiver
Decoding routine at the receiver is the same as for typical scheme of BI- STCM-ID. Of course, the proposed space-time code suffers from non- orthogonality (detection and decoding are more complex).
5. EVALUATION OF PROPOSED SCHEME
In Table 1 Dex spectrum for the proposed mapping rule is compared with dis- tances for Gray and optimally [4] mapped space-time coded modulation using the Alamouti scheme. Furthermore the contribution of
−
=
∏
ri
i ˆ
ˆ values to the
coding gain in Eq. (8) are listed. In the same manner, Table 2 lists features of
‘error-free feedback’ spectrum (Def). It is evident that the shortest distance has the biggest contribute to bad performance of particular mapping.
Distances
0,16 0,32 0,64 0,8 2,56 Table 1.
D
exOverall mapping rule ξ
Number of occurrence The contribution of particular distances
to the sum in Eq. (8)
Coding gain
ˆ
Gray labeling with Alamouti scheme
1536 60000
512
78,125 0,4297 optimal labeling with
Alamouti scheme
1920 75000
32 78,125
96
14,648 0,4064 proposed scheme 1682
65703 318 3105,4
42 102,54
4 6,25
2
0,3052 0,4152
The STBC Alamouti code with Gray constellation mapping has the smallest frequency of the neighbors with the smallest distance. On the opposite side is the scheme with ‘optimal’ labeling for which the poorest BER in the first pass can be predicted. The proposed scheme should offer better performance in the first pass than optimally mapped STBC coded system due to smaller number of neighbors with the shortest distance.
Distances
0,16 0,64 0,8 1,28 1,6 2,08 2,56 4 7,2 Table 2.
Def
ξ Number of occurrence
The contribution of particular distances to the sum 1536
Gray
60000
64 896 optimal
9,766 56
64 704 128 192 448 192
proposed
156,25 1100 78,125 75 103,55 3,704
Distances and their (-2)th powers
10,24 11,52 12,96 14,4 16 27,04 ξ Number of occurrence and the contribution of particular dist.
Coding gain
~
Gray 512
3,048 0,4298
320 64 256 448
optimal
3,052 0,381 1 0,613 2,319
192 64 64
proposed
1,447 0,381 0,3087 1,078
The fact that Gray mapping is inappropriate for iteratively decoded system is well known [4] but Def enables a quantitative analysis. The number of neighbors with the shortest distance is the same as in Dex spectrum. The difference is only in the second distance value (12,96 instead of 2,56), but its influence on coding gain is negligible. On the opposite is the system with optimal mapping. Its ideal performance in EF case is reinforced by high value of the smallest distance be- tween X and Z~
. Furthermore, the frequency of that distance occurrence is very small. The most entries in the spectrum belong to the second distance with rela- tively large value. The convergence of iterative decoding can be observed while analyzing Dex and Def for optimal mapping. As mentioned before, spectrum for the first pass was the poorest one (most of entries belonged to very small dis- tance value).
The proposed scheme can be recognized as a compromising solution. Its EF performance is much better than for Gray mapped constellation with Alamouti’s STBC due to small frequency of entries for the shortest distance. It is worth mentioning that iterative progress is possible even though Gray mapping is still used in one of transmit chains.
To verify good performance of proposed transmission scheme, it was com- pared with Gray mapped BI-STCM-ID with Alamouti’s STBC in exhaustive BER simulations. Data frames of 10 000 bits, [171 133]OCT encoder and flat fading MIMO channel were taken. BER curves are shown in Fig. 3. For statisti- cal reliability, 5×108 bits were transmitted. It can be noticed that proposed scheme ensures reasonable BER in small number of iterations.
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3URSRVHGVFKHPH *UD\PDSSHGFRQVWHOODWLRQ$ODPRXWLVFKHPH 3DVV
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Fig. 3. BER for proposed scheme and Gray mapped STBC transmission
6. CONCLUSION AND FUTURE WORK
The ‘intuitive’ space-time coded transmission scheme which uses advantages from both Gray and ‘optimal’ constellation labeling has been analyzed. Good performance of presented technique has been proved by evaluation of coding gain and distance spectrum, generalized for space-time symbols. It seems that the proposed solution can be used in parallel with WLAN specifications. As orthogonality of space-time code has been lost in the proposed scheme, detection and decoding are more complex and further research on their simplification is necessary. The author proposes to look for a new ‘optimal’ labeling map that can ensure sufficient BER in at most a few iterations.
REFERENCES
[1] HT PHY Specification, V.1.27, www.enhancedwirelessconsortium.org, 2005 [2] Y. Huang, J.A. Ritcey, “Tight BER Bounds for Iteratively Decoded Bit-Inter-
leaved Space-Time Coded Modulation”, IEEE Comm. Letters, vol. 8, No. 3, March 2004
[3] G. Caire, G. Taricco, “Bit-Interleaved Coded Modulation”, IEEE Trans. on Inf.
Theory, vol. 44, No. 3. May 1998
[4] Y. Huang, J.A. Ritcey, “Optimal Constellation Labeling for Iteratively Decoded Bit-Interleaved Space-Time Coded Modulation”, IEEE Trans. on Inf. Theory, vol. 51, No. 5, May 2005
[5] S. Benedetto, D. Divsalar, G. Montorsi, F. Pollara, “A Soft-input soft-output APP module for iterative decoding of concatenated codes”, IEEE Comm.
Letters, vol. 1, January 1997
[6] F. Schreckenbach, P. Henkel, „Analysis and design of mappings for iterative decoding of BICM“, URSI Symposium, Poznan 2005
[7] M. Krasicki, “Modyfikacja BI-STCM-ID dla zastosowania w bezprzewodo- wych sieciach komputerowych”, Krajowa Konferencja Radiokomunikacji, Ra- diofonii i7HOHZL]ML*GDVN
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