Low-Density Lattice Codes for Inter-Symbol ... - Semantic Scholar
JOURNAL OF NETWORKS, VOL. 8, NO. 8, AUGUST 2013
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Low-Density Lattice Codes for Inter-Symbol Interference Channels Yang Li, Zhisong Bie, and Jiaru Lin University of Posts and Telecommunications, Beijing, People's Republic of China Email: [email protected]
Kim Hyok Hui Huichon University of Technology, Democratic People's Republic of Korea
Abstract—Low-density lattice codes (LDLC) designed for inter-symbol interference (ISI) channels are proposed in this paper, including the node-by-node belief propagation (BP) decoding algorithm and the construction of LDLC codes for ISI channels. A theorem is presented to illustrate that the nominal coding gain of LDLC codes can be increased by introducing ISI into the primary codes. Shaping operation must be used to minimize the power of transmitted lattice points and to utilize the potential nominal coding gain. By using a semi-analytical density evolution method based on all-zero lattice assumption elements of parity check matrix are optimized. Simulation results show that error performance of LDLC code at dimension 10,000 is 1.1 dB from the capacity when the spectral efficiency is 1 bps/Hz. Index Terms—LDLC Codes; Node-By-Node Decoding; Density Evolution; Voronoi Shaping; ISI Channel
I.
INTRODUCTION
It has been proved that lattice codes can achieve the capacity of AWGN channel with [1] and without power restrictions [2]. LDLC codes [3] are lattice codes defined by a sparse parity check matrix in the Euclidean space. With a linear-time decoder, error performance of LDLC codes for AWGN channel without power restrictions is 0.6 dB from the capacity at dimension 10,000. Passing messages in the decoding are continuous real probability density functions (PDF). They were quantized into discrete sequences for implementation. To reduce the computational and storage complexity passing messages were approximated as Gaussian mixtures [4-5] or single Gaussian distribution [6]. Single Gaussian approximation was also used to search the BP threshold of LDLC codes for AWGN channel [6-7]. As to the power-constrained AWGN channel, encoding must be accompanied with shaping operation to prevent the power of transmitted lattice points from being too large [8-9]. The shaping operation can also realize the extra shaping gain if the shaping region is properly chosen. Voronoi shaping is a method to use the Voronoi region of a sub-lattice as the shaping region. Convolutional lattice codes [10] are another lattice codes constructed directly from the Euclidean space. They provide the desire analogy to finite alphabet © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.8.1899-1905
convolutional codes, and are attractive to ISI channels. The code filter was designed to generate lattice codes with minimum distance maximized. Practical sequential decoders were employed to decode convolutional lattice codes. In this paper we investigate LDLC codes designed for ISI channels. Instead of regarding ISI as interference, we introduce the ISI into the primary LDLC codes so that the nominal coding gain of lattice codes is increased. Voronoi shaping is used to utilize the coding gain. A node-by-node BP decoder for LDLC codes based on factor graph is derived. Apart from variable nodes and check nodes of LDLC code, channel nodes are introduced to update passing messages iteratively. The updating rule of channel nodes is formulated. To construct LDLC codes for ISI channels a simplified single Gaussian decoder with all-zero lattice assumption is used to search the BP threshold and elements of parity check matrix. This optimization method can be generalized to construct irregular LDLC codes. This paper is organized as follows. In Section II we introduce the basic concepts and system structure. Also a shaping method is proposed in this section. The node-bynode decoding algorithm is proposed in Section III. Section IV gives a density evolution method with all-zero assumption. Section V describes the Voronoi shaping to LDLC codes for ISI channel. II.
BASIS CONCEPTS AND SYSTEM STRUCTURE
A. Lattice Codes Lattice codes can be regarded as the Euclidean space analogue of linear binary codes. An n dimensional lattice in real field n is defined by the set of all linear combinations of n independent basis vectors in n . Coefficients of these linear combinations are integers. The Voronoi region of a lattice point is the set of points in n that are closest to this lattice point. The volume of Voronoi region equals to det(G ' G) in which G is the generator matrix. Denote lattice point by x ( x0 , x1 ,..., xn 1 ) n [11] with
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x Gb where b (b0 , b1 ,..., bk 1 )
k
s0 s 1 S sL
(1)
. Clearly if xi , x j then
xi x j . The minimum distance of a lattice is defined as the minimal squared Euclidean distance between any pair of lattice points, 2 dmin () min || x1 x2 ||2 min || x ||2 x1 , x2
x , x 0
(2)
and x is called the shortest vector. B. LDLC Codes for ISI Channels An LDLC code is a lattice whose parity check matrix (the inverse generator matrix) H is sparse and realvalued. Particularly H is assumed to be a Latin square matrix [3], in which each row and column has the same D non-zero elements except for a possible change of order and random signs. Denote elements of H by Convergence condition {h : h1 h2 ... hD 0}. D
| hi |2 / | h1 |2 1
can
ensure
the
exponential
i 2
convergence of the BP decoding algorithm. An LDLC code transmitted over ISI channels can be modeled as L
yi s0 xi sl xi l zi , i 0,..., n 1
(3)
l 1
where L is the channel memory length and sl is the discrete-time channel impulse response, which is supposed to be known to both the transmitter and the receiver. zi is the AWGN with variance 2 . The capacity of ISI channel in Eq. (3) is [12]
s0 s1
s0
sL s1
s0
sL
s1
s0
(6)
The system diagram is as shown in Fig.1. At the transmitter side, the information vector b is first shaped by a lattice s to generate vector b ' . Then b ' is encoded by the generator matrix G to obtain the shaped lattice point x , which is further transmitted over the ISI channel. It is worth mentioning that the encoding procedures can be altered without changing the power of shaped lattice points. The information vector b can be first encoded by generator matrix G . Then the coded lattice point is shaped by lattice s . This equivalence is interpreted in the following section, as is shown in Eq. (10). At the receiver side, the received lattice point y is first decoded by a BP decoding algorithm based on the factor graph as shown in Fig. 3. Then the decoded information vector bˆ ' is inversely shaped by lattice s . b
b'
shaping mod
x
x = Gb' S
ISI channel
bˆ
mod
bˆ' S
y BP decoding
Figure 1. System of LDLC codes for ISI channels
1 C lim sup I (x; y) n n p ( x x x ... x ) 1, 2, n
(4)
where I (x; y ) is the mutual information between channel input symbols and output symbols. Combining Eq. (1) and (3) can get L
L
yi sl Gi l , j b j zi b j ( sl Gi l , j ) zi (5) l 0
j
j
l 0
where Gi , j is the (i,j)th element of G. Eq. (5) defines a
It can be transformed into the equivalent system below. The receiver is the same with the LDLC codes for ISI channels while in the transmitter side, the generator matrix is C . The information vector is first shaped by a lattice s and then encoded by generator matrix C . We can observe that designing LDLC codes directly in the Euclidean space can naturally match the continuous ISI channels. b
shaping mod
b'
x
x = Cb ' S
new generator matrix C whose (i,j)th element is AWGN channel
L
sl Gi l , j . We can see that LDLC code for ISI channels l 0
with generator matrix G is transformed into lattice code for AWGN channel with generator C. Lattice code (denoted as ' ) defined by C is constructed by concatenating the primary LDLC code with a convolutional lattice code (denoted as * ) defined by S ( viz, C SG ):
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bˆ
mod
bˆ' S
y BP decoding
Figure 2. Equivalent system of Fig. 1
The nominal coding gain of lattice code is defined as [10] 2 c () dmin () / V ()2/ n
(7)
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Theorem 1: Introducing ISI into LDLC code in Eq. (5) provides extra nominal coding gain since the minimum distance is increased. Proof: See Appendix I. According to Theorem 1 we can see that ISI is not regarded as the interference but a useful concatenated convolutional lattice code which can realize the extra nominal coding gain. However, the power of equivalent lattice code ' is increased simultaneously. Shaping operation must be employed to prevent it from being too large so that the potential nominal coding gain is utilized. If the shaping region is well chosen, extra shaping gain can also be obtain compared to the n-cube shaping region. To LDLC codes used for ISI channels, shaping “whiten” the ISI channel. The shaping gain of the power-constrained lattice code is defined as [13]
E ( c ) s ( s ) E ( s )
(8)
generator matrix Cs MC. Lattice point shaped by s is
x' Cb mod s
(9)
Eq. (9) can also be expressed as
x Cb Cs c C(b Mc) ( Cs MC, c
k
)
(10)
The purpose of shaping operation is to find the shaping factor c that minimizes || x' ||2 . This is essentially finding the nearest lattice point of sub-lattice s to the non-shaped lattice point x Cb. The lattice searching algorithms have been studied [14-15]. The purpose of Voronoi shaping in this paper is not to propose a fast lattice searching algorithm but to present an idea to transform the shaping problem for ISI channels into Voronoi shaping problem for AWGN channel. We introduce here a shaping method based on hypercube shaping, which indicates all the shaped lattice
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x' Gb LGk x LGk Gb'
(11)
where L is the side-length of Voronoi region of the hypercube. Without loss of generality, assume that each dimension length of this hypercube is Li . The shaping operation is to find k to ensure that xi' Li / 2 . First decompose the equivalent generator matrix C by using QR decomposition method, (12)
C = TQ Then the encoding can be expressed as
x' = Cb' = RQb' Rb'
(13)
where R lower-triangular matrix and Q orthonormal matrix. Thus we have
b' QT b'
where E (c ) is the average power of lattice codes shaped by c ( Voronoi region is n-cube) and E ( s ) is the average power of lattice codes shaped by s . Extra 1.53 dB shaping gain can be achieved if the Voronoi region of shaping lattice is n-sphere. Voronoi shaping is a shaping method to choose the Voronoi region of a sub-lattice ( s ) of lattice as the shaping region. Voronoi shaping is also called nested shaping. The shaping lattice s is referred as the coarse lattice while is the fine lattice [1]. A good fine lattice (channel codes) combined with a good coarse lattice (quantizer of lattices) achieves the capacity of AWGN channel. In this paper we show that a good fine lattice combined with a good coarse lattice also achieves the capacity of ISI channel by transforming ISI channel system into equivalent AWGN channel system. Suppose sub-lattice s defined by equivalent
'
points are located in a hypercube. Eq. (10) can also be written as
is the (14)
Let b Qb , Eq. (13) can be rewritten as
x' = R(b Lk ) Rb'
(15)
in which b' (b Lk ) . Further there has i 1
' i
x 1 ki (bi L Ri ,i
R
' i ,l l
l 1
b
)
Ri ,i
(16)
Since xi' L / 2 and L Li Ri ,i , then we have i 1
1 ki (bi Li
R l 1
' i ,l l
Ri ,i
b
)
(17)
When we get b ' , the shaped information vector can be obtained as b' QT b' . Further the shaped lattice point is calculated as x' = Cb' . Although the parity check matrix H is sparse, the equivalent generator matrix C is unnecessarily a sparse matrix, and so is the R lower-triangular matrix. As a result, the computational complexity and storage requirements of are O(n2 ) . To reduce the complexity we can keep only the J nonzero values of R that have the largest absolute values, and nullify all the other elements of R . This makes a trade-off between the shaping operation complexity and the shaping gain. A larger number of nonzero elements of R will ensure a better shaping gain but at the cost of higher complexity. This shaping method can be regarded as a generalization of Tomlinson-Harashima precoding scheme for inter-symbol interference (ISI) channels [1617]. The ISI here is the contribution of the xi' components that were already calculated. Therefore the lattice point components will be uniformly distributed.
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III.
BP DECODING ALGORITHM
f l i (
A. Factor Graph Representation The factor graph [18] of LDLC codes for ISI channels is as shown in Fig. 3. There are three kinds of nodes: channel nodes, variable nodes and check nodes. The solid squares represent channel nodes, the solid circles represent variable nodes and the empty squares represent check nodes. Messages are iterated and updated among those nodes. In LDLC code for ISI channels message passing follows a node-by-node strategy. First channel nodes update messages and send them to variable nodes. Then messages are updated at variable nodes and are further sent to check nodes. Instead of making a decision directly, messages from check nodes to variable nodes are updated at variable nodes and then are sent back to channel nodes served as the priori information. This strategy can avoid the excess iteration of initial unreliable channel information and fully utilize the messages generated by the nodes in the factor graph. h3
h1
h1
h3
h2
h2
h2
h1
h2 h3
h3
s1
( x mk ) ). 2Vk 2 V k The decoding algorithm of LDLC codes for ISI channels is described as follows. The variable nodes processing and check nodes processing are the same with that in LDLC codes for AWGN channel. For completion we present all the updating rules here. Channel nodes updating rule: Message from the lth(l =0,...,n-1) channel node to the ith(i =0,...,n-1) variable node is the convolution of initial channel message (the received symbol) and messages from variable nodes in l \ i where l \ i denotes set of all variable nodes connecting to the lth channel node except the ith variable node:
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exp(
(19)
nodes connecting to the ith variable node except the lth channel node:
f i l ( x ) f j i ( x ) f l i ' ( x ) ji
(20)
l ' i \ l
Message from the ith variable node to the jth(j 0,..., k 1) check node is the product of messages from check nodes in i \ j and messages from channel nodes in i where i \ j denotes set of all check nodes connecting to the ith variable node except the jth check node and i denotes set of all channel nodes connecting to the ith variable node:
1
-1
3
Figure 4. Decision of bˆiR when M 4
f i j ( x)
j ' i \ j
(18)
k 1
ck
sl i'
'
)i l \i
where is convolution operation and 2 f ( yl ) ~ N ( yl , ). The derivation of channel nodes updating rule is as seen in Appendix II. Variable nodes updating rules: Message from the ith variable node to the lth channel node is the product of messages from check nodes in i and messages from channel nodes in i \ l where i denotes set of all check nodes connecting to the ith variable node, and i \ l denotes set of all channel
-3
K
2
x
s0
B. BP Decoding Algorithm Messages in LDLC codes for ISI channels are real PDFs, which can be represented by Gaussian mixture distributions [5]
in which ck N (mk ,Vk )
sl i
) f ( yl ) fi' l (
decision region
Figure 3. Factor graph of LDLC codes for ISI channels
fGM ( x) ck N (mk ,Vk )
x
f j' i ( x) fl i ( x)
(21)
li
Check nodes updating rule: First convolve all message from variable nodes in j \ i where j \ i denotes set of all variable nodes connecting to the jth check node except the ith variable node:
f j*i (
x x i'j \i ) fi' j ( ) Hi, j H i' , j
(22)
where H i , j is the (i, j )th element of H. Then f j*i ( x) is periodically extended since we need to search bˆ in the i
constellation M :
f j i ( x) f j*i ( x tM
t ) Hi, j
(23)
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f j i ( x) is the message from the jth check node to
1903
mi j (
the ith variable node. Messages after periodically extended can be approximated as Gaussian mixtures. Decision: After L iterations bˆi can be estimated:
( M 1), f ( M 2) ( M 3), ( M 2) f ( M 4) (24) bˆi ... ... M 3, ( M 4) f ( M 2) f ( M 2). M 1, in which f arg max( fi j ( x
IV.
x i j ) ). Hi, j
CODES CONSTRUCTION
Codes construction is to design an LDLC code for certain ISI channel on the criterion that the BP threshold of decoding algorithm is maximized. The BP threshold of LDLC codes signifies the worst channel parameter (denoted as 2 ) for which BP decoding algorithm of a large dimension LDLC codes can converge. A semianalytical density evolution is used to optimize elements of H as well as to search the BP threshold. A. Single Gaussian Decoder with All-zero Lattice Point Assumption Due to the linearity of lattice codes, error performance of LDLC codes does not depend on the specific transmitted lattice points. For simplicity we can use allzero lattice point as the transmitted codeword when optimizing h. In this case the periodical extension of messages at check nodes is unnecessary so that all passing messages are instinctively single Gaussian, which can be represented by a mean and a variance. By using Claim 1 and Claim 2 in [3] to the BP decoding algorithm in Section III.B we can derive the following simplified decoding algorithm. Channel nodes updating rule: Message from the lth channel node to the ith variable node is single Gaussian fl i ( x) ~ N (ml i ,Vl i ) with
ml i ( yl Vl i ( 2
s
i 'l \ i
l i'
s
i 'l \ i
2 l i'
mi' l ) / sl i and Vi' l ) / sl2i .
Variable nodes updating rules: Message from the ith variable node to the lth channel node is single Gaussian fi l ( x) N (mi l ,Vi l ) with mi l ( m j iV j1 i ml ' iVl '1 i )Vi l and ji
Vi l ( V ji
l ' i \ l
1 j i
V
l i \ l '
1 1 l ' i
) .
Message from the ith variable node to the jth check node is fi j ( x) N (mi j ,Vi j ) with
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j i \ j '
Vi j (
1 1 m j' iV j' ml iVl i )Vi j and i li
V
j ' i \ j
1 j ' i
V li
1 1 l i
) .
Check nodes updating rule: Message from the jth check node to the ith variable node is f j i ( x) ~ N (m j i ,V j i ) with
m j i (
H i' , j mi' j ) / H i , j and
H i2' , jVi' j ) / H i2, j .
i ' j \ i
V j i (
i j \ i '
Decision: Decision rule for single Gaussian decoder is the same with Eq. (17) except that here we have f max( H i , j mi j ). i j
By using this simplified decoding algorithm to received symbols we can statistically get the error probability of large dimension LDLC code whose parity check matrix elements are h under certain channel condition 2 and {s0 ,..., sL }. The cost function can be modeled as
Pe density _ evolution( 2 ; h1 , h2 ,..., hD ; s0 ,..., sL )
(25)
With all-zero lattice point assumption we can optimize h by differential evolution method [11] given {s0 ,..., sL }. The optimization procedures are described as below. Step 1: Set initial channel conditions 2 and {s0 ,..., sL } . Set dimension of LDLC codes (e.g., 104). Step 2: Under certain channel conditions construct an LDLC code via spatial coupling based on a randomly initialized h restricted by convergence condition 1. Generate symbols y from N (0, 2 ) under all-zero lattice assumption. Carry out density evolution in Eq. (17) by using the single Gaussian approximation decoding algorithm. Use differential evolution algorithm [19] to adjust h until certain h that ensures Pe converges is optimized. Step 3: Record 2 under which h has been optimized. Increase 2 by a small step. Return to Step 2 to optimize new h at new 2 . If convergence is satisfied, continue to increase 2 ; otherwise the last recorded 2 is BP threshold *2 . The corresponding h is the optimized elements of H. Differential evolution method can adjust h that maximize 2 while guaranteeing the convergence of decoding algorithm. The maximal 2 is the BP threshold *2 we intend to search. V.
SIMULATION RESULTS
Fig. 5 gives optimized BP threshold measured in SNR (with channel impulse response 1 S ( z) 0.7071 0.7071z ).
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We also give the performances of LDLC codes designed for power-constrained ISI channel at dimension 10,000 by using the node-by-node BP decoding algorithm. The degree of H is 7. The optimized h is {1, 0.8056, 0.3274, 0.1542, 0.1285, 0.0887, 0.0747}. The spectral efficiency is 1 bps/Hz. The performance loss measured in SER (symbol error rate) is about 1.1 dB, which is about 0.3 dB better than LDPC coded modulation for ISI channels in [20]. -1
10
-2
SER
x , x 0
i'
x , x 0
i
(30) Then the nominal coding gain of ' is
c ( ' )
2 2 () d min (' ) s02 d min c () ' 2/ n 2 V ( ) s0 V ()2/ n
(31)
VIII. APPENDIX II CHANNEL NODES UPDATING RULE DERIVATION
[0.7071 0.7071] ISI channel capacity BP threshold SER performance at dimension = 1,000 SER performance at dimension = 10,000
10
2 2 dmin (' ) ' min ( xi''2 ) min ( ( s0 xi )2 ) s02 dmin ( ) ' '
The check equation at channel nodes is as shown in Eq. (3). Consider the lth(l 0,..., n -1) channel node, then Eq. (3) can also be written as
-3
10
sl i xi yl -4
s
i ' l \ i
10
(32)
x
l i ' i'
Then we get the marginal probability density function: -5
10
2.5
3
3.5
4
4.5
5
SNR (dB)
f l i (
Figure 5. BP threshold and the simulation performances
VI.
x sl i
) f ( yl
s
i ' l \ i
x | xi' , y)
l i ' i'
(33)
Since we assume that xi' are independent with each
CONCLUSION
In this paper we propose LDLC codes used for ISI channels, including a node-by-node BP decoding algorithm and a semi-analytical density evolution method to design LDLC codes with all-zero lattice assumption. By introducing ISI into the primary LDLC codes extra nominal coding gain is obtained. Voronoi shaping to the equivalent lattice codes that are generated by concatenating LDLC codes with convolutional lattice codes is described. Further study will focus on the construction of irregular LDLC codes for ISI channels.
other, the probability density functions of linear combination equal to the convolution of probability density functions of its components,
f l i (
x sl i
) f ( yl | xi' , y) f (sl i ' xi' | xi' , y)i l '
(34)
Thus
f l i (
x sl i
) f ( yl ) fi' l (
x sl i'
'
)i l \i
(35)
VII. APPENDIX I PROOF OF THEOREM 1 We will prove that lattice ' has a better nominal coding gain than lattice by introducing ISI into the original LDLC codes, viz, c (' ) c (). Clearly there is
V (' )2/ n (det(C'C))1/ n = det(G'G)1/ n det(S'S)1/ n When L
n and n we have det(S'S)
1/ n
V (' )2/ n s02 V ()2/ n
(26)
s . So 2 0
(27)
It is left to consider the minimum distance of ' . To x' ' , L
xi' s0 xi sl xi l , i 0,..., n 1
(28)
l 1
The minimum distance of ' is 2 dmin (' ) ' min (|| x' ||2 ) ' min ( xi' 2 ) (29) ' ' ' ' x , x 0
x , x 0
i
Notice that since the filtering operations of ISI are linear the shortest vector of generates the shortest vector of ' . Combining Eq. (21) with (22) gives © 2013 ACADEMY PUBLISHER
ACKNOWLEDGMENT This work is supported by National Science Foundation of China (60902048) and Chinese Universities Scientific Fund (BUPT 2013RC0102). REFERENCES [1] Erez U, Zamir R. Achieving 1/2log (1+SNR) on the AWGN channel with lattice encoding and decoding. IEEE Transactions on Information Theory, 2004, 50(10) pp. 2293-2314. [2] Poltyrev G. On coding without restrictions for AWGN channel. IEEE Transactions on Information Theory, 1994, 40(2) pp. 409-417. [3] Sommer N, Feder M, Shalvi O. Low-density lattice codes. IEEE Transactions on Information Theory, 2008, 54(4) pp. 1561-1585. [4] Yoda Y, Feder M. Efficient parametric decoder of lowdensity lattice codes. IEEE International Symposium on Information Theory: June 28-July 3, 2009, Seoul, Korea. New York, NY, USA: IEEE, 2009, 8 pp. 744-748. [5] Kurkoski B, Dauwels J. Message-passing decoding of lattices using Gaussian mixtures. IEEE International Symposium on Information Theory: June 6-11, 2008, Toronto, Canada. New York, NY, USA: IEEE, 2008, 8 pp. 2489-2493.
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[6] Kurkoski B, Yamaguchi K, Kobayashi K. Single-Gaussian messages and noise thresholds for decoding low-density lattice codes. IEEE International Symposium on Information Theory: June 28-July 3, 2009, Seoul, Korea. New York, NY, USA: IEEE, 2009, 8 pp. 734-738. [7] Uchikawa H, Kurkoski B, Dauwels J. Threshold improvement of low-density lattice codes via spatial coupling. IEEE International Conference on Computing, Networking and Communications (ICNC): Jan. 30-Feb. 2, 2012, Maui, HI. New York, NY, USA: IEEE, 2012, 3 pp. 1036-1040. [8] Fischer R. Precoding and signal shaping for digital transmission. New York: John Wiley & Sons, 2002. [9] N. Sommer, M. Feder, O. Shalvi, “Shaping methods for low-density lattice code,” IEEE Information workshop: Oct 11-16, 2009, Taormina, Italy. New York, NY, USA: IEEE, 2009, 12 pp. 238-242. [10] Shalvi O, Sommer N, Feder M. Signal codes: convolutional lattice codes. IEEE Transactions on Information Theory, 2011, 57(8) pp. 2503-5226. [11] Conway J, Sloane N. Sphere packings lattices and groups. New York: Springer-verlag, 1988. [12] Xia X. Modulated coding for Inter-symbol interference channels, New York: McGraw-Hill, 2001 [13] Forney G, Ungerboeck G. Modulation and coding forlinear Gaussian channels. IEEE Transactions on Information Theory, 1998, 44(6) pp. 2384-2415. [14] Conway J, Sloane N. A fast encoding method for lattice codes and quantizers. Transactions on Information Theory, 1983, 29(6) pp. 820-824.
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[15] E. Agrell, T. Eriksson, A. Vardy. Closest point search in lattices. IEEE Transactions on Information Theory, 2002, 48(8): 2201-2214. [16] M. Tomlinson, “New automatic equlizer empoying modulo arithmetic”, Electronics Letters, pp. 138-139, 1971 [17] G. D. Forney and M. V. Eyuboglu, “Combined equalization and coding using precoding”, IEEE Commnun. Magzine, 1991, 29(12): 25-34 [18] F. R. Kschischang, B. J. Frey, H. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Transactions on Information Theory 2001, 47(2), 498-519 [19] Price K, Storn R, Lampinen J et al. Differential evolution—a practical approach to global optimization. New York: Springer, 2005. [20] Franceschini M, Ferarri G, Raheli R et al. LDPC coded modulation, Berlin: Springer, 2009.
Yang Li is a doctoral candidate in the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications. He received a BA degree with major in English Literature and minor in communication engineering from Beijing University of Posts and Telecommunications, and a PhD in Information science from Beijing University of Posts and Telecommunications. Her current research is in coding theory, signal processing in communication system.
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Low-Density Lattice Codes for Inter-Symbol Interference Channels Yang Li, Zhisong Bie, and Jiaru Lin University of Posts and Telecommunications, Beijing, People's Republic of China Email: [email protected]
Kim Hyok Hui Huichon University of Technology, Democratic People's Republic of Korea
Abstract—Low-density lattice codes (LDLC) designed for inter-symbol interference (ISI) channels are proposed in this paper, including the node-by-node belief propagation (BP) decoding algorithm and the construction of LDLC codes for ISI channels. A theorem is presented to illustrate that the nominal coding gain of LDLC codes can be increased by introducing ISI into the primary codes. Shaping operation must be used to minimize the power of transmitted lattice points and to utilize the potential nominal coding gain. By using a semi-analytical density evolution method based on all-zero lattice assumption elements of parity check matrix are optimized. Simulation results show that error performance of LDLC code at dimension 10,000 is 1.1 dB from the capacity when the spectral efficiency is 1 bps/Hz. Index Terms—LDLC Codes; Node-By-Node Decoding; Density Evolution; Voronoi Shaping; ISI Channel
I.
INTRODUCTION
It has been proved that lattice codes can achieve the capacity of AWGN channel with [1] and without power restrictions [2]. LDLC codes [3] are lattice codes defined by a sparse parity check matrix in the Euclidean space. With a linear-time decoder, error performance of LDLC codes for AWGN channel without power restrictions is 0.6 dB from the capacity at dimension 10,000. Passing messages in the decoding are continuous real probability density functions (PDF). They were quantized into discrete sequences for implementation. To reduce the computational and storage complexity passing messages were approximated as Gaussian mixtures [4-5] or single Gaussian distribution [6]. Single Gaussian approximation was also used to search the BP threshold of LDLC codes for AWGN channel [6-7]. As to the power-constrained AWGN channel, encoding must be accompanied with shaping operation to prevent the power of transmitted lattice points from being too large [8-9]. The shaping operation can also realize the extra shaping gain if the shaping region is properly chosen. Voronoi shaping is a method to use the Voronoi region of a sub-lattice as the shaping region. Convolutional lattice codes [10] are another lattice codes constructed directly from the Euclidean space. They provide the desire analogy to finite alphabet © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.8.1899-1905
convolutional codes, and are attractive to ISI channels. The code filter was designed to generate lattice codes with minimum distance maximized. Practical sequential decoders were employed to decode convolutional lattice codes. In this paper we investigate LDLC codes designed for ISI channels. Instead of regarding ISI as interference, we introduce the ISI into the primary LDLC codes so that the nominal coding gain of lattice codes is increased. Voronoi shaping is used to utilize the coding gain. A node-by-node BP decoder for LDLC codes based on factor graph is derived. Apart from variable nodes and check nodes of LDLC code, channel nodes are introduced to update passing messages iteratively. The updating rule of channel nodes is formulated. To construct LDLC codes for ISI channels a simplified single Gaussian decoder with all-zero lattice assumption is used to search the BP threshold and elements of parity check matrix. This optimization method can be generalized to construct irregular LDLC codes. This paper is organized as follows. In Section II we introduce the basic concepts and system structure. Also a shaping method is proposed in this section. The node-bynode decoding algorithm is proposed in Section III. Section IV gives a density evolution method with all-zero assumption. Section V describes the Voronoi shaping to LDLC codes for ISI channel. II.
BASIS CONCEPTS AND SYSTEM STRUCTURE
A. Lattice Codes Lattice codes can be regarded as the Euclidean space analogue of linear binary codes. An n dimensional lattice in real field n is defined by the set of all linear combinations of n independent basis vectors in n . Coefficients of these linear combinations are integers. The Voronoi region of a lattice point is the set of points in n that are closest to this lattice point. The volume of Voronoi region equals to det(G ' G) in which G is the generator matrix. Denote lattice point by x ( x0 , x1 ,..., xn 1 ) n [11] with
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x Gb where b (b0 , b1 ,..., bk 1 )
k
s0 s 1 S sL
(1)
. Clearly if xi , x j then
xi x j . The minimum distance of a lattice is defined as the minimal squared Euclidean distance between any pair of lattice points, 2 dmin () min || x1 x2 ||2 min || x ||2 x1 , x2
x , x 0
(2)
and x is called the shortest vector. B. LDLC Codes for ISI Channels An LDLC code is a lattice whose parity check matrix (the inverse generator matrix) H is sparse and realvalued. Particularly H is assumed to be a Latin square matrix [3], in which each row and column has the same D non-zero elements except for a possible change of order and random signs. Denote elements of H by Convergence condition {h : h1 h2 ... hD 0}. D
| hi |2 / | h1 |2 1
can
ensure
the
exponential
i 2
convergence of the BP decoding algorithm. An LDLC code transmitted over ISI channels can be modeled as L
yi s0 xi sl xi l zi , i 0,..., n 1
(3)
l 1
where L is the channel memory length and sl is the discrete-time channel impulse response, which is supposed to be known to both the transmitter and the receiver. zi is the AWGN with variance 2 . The capacity of ISI channel in Eq. (3) is [12]
s0 s1
s0
sL s1
s0
sL
s1
s0
(6)
The system diagram is as shown in Fig.1. At the transmitter side, the information vector b is first shaped by a lattice s to generate vector b ' . Then b ' is encoded by the generator matrix G to obtain the shaped lattice point x , which is further transmitted over the ISI channel. It is worth mentioning that the encoding procedures can be altered without changing the power of shaped lattice points. The information vector b can be first encoded by generator matrix G . Then the coded lattice point is shaped by lattice s . This equivalence is interpreted in the following section, as is shown in Eq. (10). At the receiver side, the received lattice point y is first decoded by a BP decoding algorithm based on the factor graph as shown in Fig. 3. Then the decoded information vector bˆ ' is inversely shaped by lattice s . b
b'
shaping mod
x
x = Gb' S
ISI channel
bˆ
mod
bˆ' S
y BP decoding
Figure 1. System of LDLC codes for ISI channels
1 C lim sup I (x; y) n n p ( x x x ... x ) 1, 2, n
(4)
where I (x; y ) is the mutual information between channel input symbols and output symbols. Combining Eq. (1) and (3) can get L
L
yi sl Gi l , j b j zi b j ( sl Gi l , j ) zi (5) l 0
j
j
l 0
where Gi , j is the (i,j)th element of G. Eq. (5) defines a
It can be transformed into the equivalent system below. The receiver is the same with the LDLC codes for ISI channels while in the transmitter side, the generator matrix is C . The information vector is first shaped by a lattice s and then encoded by generator matrix C . We can observe that designing LDLC codes directly in the Euclidean space can naturally match the continuous ISI channels. b
shaping mod
b'
x
x = Cb ' S
new generator matrix C whose (i,j)th element is AWGN channel
L
sl Gi l , j . We can see that LDLC code for ISI channels l 0
with generator matrix G is transformed into lattice code for AWGN channel with generator C. Lattice code (denoted as ' ) defined by C is constructed by concatenating the primary LDLC code with a convolutional lattice code (denoted as * ) defined by S ( viz, C SG ):
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bˆ
mod
bˆ' S
y BP decoding
Figure 2. Equivalent system of Fig. 1
The nominal coding gain of lattice code is defined as [10] 2 c () dmin () / V ()2/ n
(7)
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Theorem 1: Introducing ISI into LDLC code in Eq. (5) provides extra nominal coding gain since the minimum distance is increased. Proof: See Appendix I. According to Theorem 1 we can see that ISI is not regarded as the interference but a useful concatenated convolutional lattice code which can realize the extra nominal coding gain. However, the power of equivalent lattice code ' is increased simultaneously. Shaping operation must be employed to prevent it from being too large so that the potential nominal coding gain is utilized. If the shaping region is well chosen, extra shaping gain can also be obtain compared to the n-cube shaping region. To LDLC codes used for ISI channels, shaping “whiten” the ISI channel. The shaping gain of the power-constrained lattice code is defined as [13]
E ( c ) s ( s ) E ( s )
(8)
generator matrix Cs MC. Lattice point shaped by s is
x' Cb mod s
(9)
Eq. (9) can also be expressed as
x Cb Cs c C(b Mc) ( Cs MC, c
k
)
(10)
The purpose of shaping operation is to find the shaping factor c that minimizes || x' ||2 . This is essentially finding the nearest lattice point of sub-lattice s to the non-shaped lattice point x Cb. The lattice searching algorithms have been studied [14-15]. The purpose of Voronoi shaping in this paper is not to propose a fast lattice searching algorithm but to present an idea to transform the shaping problem for ISI channels into Voronoi shaping problem for AWGN channel. We introduce here a shaping method based on hypercube shaping, which indicates all the shaped lattice
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x' Gb LGk x LGk Gb'
(11)
where L is the side-length of Voronoi region of the hypercube. Without loss of generality, assume that each dimension length of this hypercube is Li . The shaping operation is to find k to ensure that xi' Li / 2 . First decompose the equivalent generator matrix C by using QR decomposition method, (12)
C = TQ Then the encoding can be expressed as
x' = Cb' = RQb' Rb'
(13)
where R lower-triangular matrix and Q orthonormal matrix. Thus we have
b' QT b'
where E (c ) is the average power of lattice codes shaped by c ( Voronoi region is n-cube) and E ( s ) is the average power of lattice codes shaped by s . Extra 1.53 dB shaping gain can be achieved if the Voronoi region of shaping lattice is n-sphere. Voronoi shaping is a shaping method to choose the Voronoi region of a sub-lattice ( s ) of lattice as the shaping region. Voronoi shaping is also called nested shaping. The shaping lattice s is referred as the coarse lattice while is the fine lattice [1]. A good fine lattice (channel codes) combined with a good coarse lattice (quantizer of lattices) achieves the capacity of AWGN channel. In this paper we show that a good fine lattice combined with a good coarse lattice also achieves the capacity of ISI channel by transforming ISI channel system into equivalent AWGN channel system. Suppose sub-lattice s defined by equivalent
'
points are located in a hypercube. Eq. (10) can also be written as
is the (14)
Let b Qb , Eq. (13) can be rewritten as
x' = R(b Lk ) Rb'
(15)
in which b' (b Lk ) . Further there has i 1
' i
x 1 ki (bi L Ri ,i
R
' i ,l l
l 1
b
)
Ri ,i
(16)
Since xi' L / 2 and L Li Ri ,i , then we have i 1
1 ki (bi Li
R l 1
' i ,l l
Ri ,i
b
)
(17)
When we get b ' , the shaped information vector can be obtained as b' QT b' . Further the shaped lattice point is calculated as x' = Cb' . Although the parity check matrix H is sparse, the equivalent generator matrix C is unnecessarily a sparse matrix, and so is the R lower-triangular matrix. As a result, the computational complexity and storage requirements of are O(n2 ) . To reduce the complexity we can keep only the J nonzero values of R that have the largest absolute values, and nullify all the other elements of R . This makes a trade-off between the shaping operation complexity and the shaping gain. A larger number of nonzero elements of R will ensure a better shaping gain but at the cost of higher complexity. This shaping method can be regarded as a generalization of Tomlinson-Harashima precoding scheme for inter-symbol interference (ISI) channels [1617]. The ISI here is the contribution of the xi' components that were already calculated. Therefore the lattice point components will be uniformly distributed.
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III.
BP DECODING ALGORITHM
f l i (
A. Factor Graph Representation The factor graph [18] of LDLC codes for ISI channels is as shown in Fig. 3. There are three kinds of nodes: channel nodes, variable nodes and check nodes. The solid squares represent channel nodes, the solid circles represent variable nodes and the empty squares represent check nodes. Messages are iterated and updated among those nodes. In LDLC code for ISI channels message passing follows a node-by-node strategy. First channel nodes update messages and send them to variable nodes. Then messages are updated at variable nodes and are further sent to check nodes. Instead of making a decision directly, messages from check nodes to variable nodes are updated at variable nodes and then are sent back to channel nodes served as the priori information. This strategy can avoid the excess iteration of initial unreliable channel information and fully utilize the messages generated by the nodes in the factor graph. h3
h1
h1
h3
h2
h2
h2
h1
h2 h3
h3
s1
( x mk ) ). 2Vk 2 V k The decoding algorithm of LDLC codes for ISI channels is described as follows. The variable nodes processing and check nodes processing are the same with that in LDLC codes for AWGN channel. For completion we present all the updating rules here. Channel nodes updating rule: Message from the lth(l =0,...,n-1) channel node to the ith(i =0,...,n-1) variable node is the convolution of initial channel message (the received symbol) and messages from variable nodes in l \ i where l \ i denotes set of all variable nodes connecting to the lth channel node except the ith variable node:
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exp(
(19)
nodes connecting to the ith variable node except the lth channel node:
f i l ( x ) f j i ( x ) f l i ' ( x ) ji
(20)
l ' i \ l
Message from the ith variable node to the jth(j 0,..., k 1) check node is the product of messages from check nodes in i \ j and messages from channel nodes in i where i \ j denotes set of all check nodes connecting to the ith variable node except the jth check node and i denotes set of all channel nodes connecting to the ith variable node:
1
-1
3
Figure 4. Decision of bˆiR when M 4
f i j ( x)
j ' i \ j
(18)
k 1
ck
sl i'
'
)i l \i
where is convolution operation and 2 f ( yl ) ~ N ( yl , ). The derivation of channel nodes updating rule is as seen in Appendix II. Variable nodes updating rules: Message from the ith variable node to the lth channel node is the product of messages from check nodes in i and messages from channel nodes in i \ l where i denotes set of all check nodes connecting to the ith variable node, and i \ l denotes set of all channel
-3
K
2
x
s0
B. BP Decoding Algorithm Messages in LDLC codes for ISI channels are real PDFs, which can be represented by Gaussian mixture distributions [5]
in which ck N (mk ,Vk )
sl i
) f ( yl ) fi' l (
decision region
Figure 3. Factor graph of LDLC codes for ISI channels
fGM ( x) ck N (mk ,Vk )
x
f j' i ( x) fl i ( x)
(21)
li
Check nodes updating rule: First convolve all message from variable nodes in j \ i where j \ i denotes set of all variable nodes connecting to the jth check node except the ith variable node:
f j*i (
x x i'j \i ) fi' j ( ) Hi, j H i' , j
(22)
where H i , j is the (i, j )th element of H. Then f j*i ( x) is periodically extended since we need to search bˆ in the i
constellation M :
f j i ( x) f j*i ( x tM
t ) Hi, j
(23)
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f j i ( x) is the message from the jth check node to
1903
mi j (
the ith variable node. Messages after periodically extended can be approximated as Gaussian mixtures. Decision: After L iterations bˆi can be estimated:
( M 1), f ( M 2) ( M 3), ( M 2) f ( M 4) (24) bˆi ... ... M 3, ( M 4) f ( M 2) f ( M 2). M 1, in which f arg max( fi j ( x
IV.
x i j ) ). Hi, j
CODES CONSTRUCTION
Codes construction is to design an LDLC code for certain ISI channel on the criterion that the BP threshold of decoding algorithm is maximized. The BP threshold of LDLC codes signifies the worst channel parameter (denoted as 2 ) for which BP decoding algorithm of a large dimension LDLC codes can converge. A semianalytical density evolution is used to optimize elements of H as well as to search the BP threshold. A. Single Gaussian Decoder with All-zero Lattice Point Assumption Due to the linearity of lattice codes, error performance of LDLC codes does not depend on the specific transmitted lattice points. For simplicity we can use allzero lattice point as the transmitted codeword when optimizing h. In this case the periodical extension of messages at check nodes is unnecessary so that all passing messages are instinctively single Gaussian, which can be represented by a mean and a variance. By using Claim 1 and Claim 2 in [3] to the BP decoding algorithm in Section III.B we can derive the following simplified decoding algorithm. Channel nodes updating rule: Message from the lth channel node to the ith variable node is single Gaussian fl i ( x) ~ N (ml i ,Vl i ) with
ml i ( yl Vl i ( 2
s
i 'l \ i
l i'
s
i 'l \ i
2 l i'
mi' l ) / sl i and Vi' l ) / sl2i .
Variable nodes updating rules: Message from the ith variable node to the lth channel node is single Gaussian fi l ( x) N (mi l ,Vi l ) with mi l ( m j iV j1 i ml ' iVl '1 i )Vi l and ji
Vi l ( V ji
l ' i \ l
1 j i
V
l i \ l '
1 1 l ' i
) .
Message from the ith variable node to the jth check node is fi j ( x) N (mi j ,Vi j ) with
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j i \ j '
Vi j (
1 1 m j' iV j' ml iVl i )Vi j and i li
V
j ' i \ j
1 j ' i
V li
1 1 l i
) .
Check nodes updating rule: Message from the jth check node to the ith variable node is f j i ( x) ~ N (m j i ,V j i ) with
m j i (
H i' , j mi' j ) / H i , j and
H i2' , jVi' j ) / H i2, j .
i ' j \ i
V j i (
i j \ i '
Decision: Decision rule for single Gaussian decoder is the same with Eq. (17) except that here we have f max( H i , j mi j ). i j
By using this simplified decoding algorithm to received symbols we can statistically get the error probability of large dimension LDLC code whose parity check matrix elements are h under certain channel condition 2 and {s0 ,..., sL }. The cost function can be modeled as
Pe density _ evolution( 2 ; h1 , h2 ,..., hD ; s0 ,..., sL )
(25)
With all-zero lattice point assumption we can optimize h by differential evolution method [11] given {s0 ,..., sL }. The optimization procedures are described as below. Step 1: Set initial channel conditions 2 and {s0 ,..., sL } . Set dimension of LDLC codes (e.g., 104). Step 2: Under certain channel conditions construct an LDLC code via spatial coupling based on a randomly initialized h restricted by convergence condition 1. Generate symbols y from N (0, 2 ) under all-zero lattice assumption. Carry out density evolution in Eq. (17) by using the single Gaussian approximation decoding algorithm. Use differential evolution algorithm [19] to adjust h until certain h that ensures Pe converges is optimized. Step 3: Record 2 under which h has been optimized. Increase 2 by a small step. Return to Step 2 to optimize new h at new 2 . If convergence is satisfied, continue to increase 2 ; otherwise the last recorded 2 is BP threshold *2 . The corresponding h is the optimized elements of H. Differential evolution method can adjust h that maximize 2 while guaranteeing the convergence of decoding algorithm. The maximal 2 is the BP threshold *2 we intend to search. V.
SIMULATION RESULTS
Fig. 5 gives optimized BP threshold measured in SNR (with channel impulse response 1 S ( z) 0.7071 0.7071z ).
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We also give the performances of LDLC codes designed for power-constrained ISI channel at dimension 10,000 by using the node-by-node BP decoding algorithm. The degree of H is 7. The optimized h is {1, 0.8056, 0.3274, 0.1542, 0.1285, 0.0887, 0.0747}. The spectral efficiency is 1 bps/Hz. The performance loss measured in SER (symbol error rate) is about 1.1 dB, which is about 0.3 dB better than LDPC coded modulation for ISI channels in [20]. -1
10
-2
SER
x , x 0
i'
x , x 0
i
(30) Then the nominal coding gain of ' is
c ( ' )
2 2 () d min (' ) s02 d min c () ' 2/ n 2 V ( ) s0 V ()2/ n
(31)
VIII. APPENDIX II CHANNEL NODES UPDATING RULE DERIVATION
[0.7071 0.7071] ISI channel capacity BP threshold SER performance at dimension = 1,000 SER performance at dimension = 10,000
10
2 2 dmin (' ) ' min ( xi''2 ) min ( ( s0 xi )2 ) s02 dmin ( ) ' '
The check equation at channel nodes is as shown in Eq. (3). Consider the lth(l 0,..., n -1) channel node, then Eq. (3) can also be written as
-3
10
sl i xi yl -4
s
i ' l \ i
10
(32)
x
l i ' i'
Then we get the marginal probability density function: -5
10
2.5
3
3.5
4
4.5
5
SNR (dB)
f l i (
Figure 5. BP threshold and the simulation performances
VI.
x sl i
) f ( yl
s
i ' l \ i
x | xi' , y)
l i ' i'
(33)
Since we assume that xi' are independent with each
CONCLUSION
In this paper we propose LDLC codes used for ISI channels, including a node-by-node BP decoding algorithm and a semi-analytical density evolution method to design LDLC codes with all-zero lattice assumption. By introducing ISI into the primary LDLC codes extra nominal coding gain is obtained. Voronoi shaping to the equivalent lattice codes that are generated by concatenating LDLC codes with convolutional lattice codes is described. Further study will focus on the construction of irregular LDLC codes for ISI channels.
other, the probability density functions of linear combination equal to the convolution of probability density functions of its components,
f l i (
x sl i
) f ( yl | xi' , y) f (sl i ' xi' | xi' , y)i l '
(34)
Thus
f l i (
x sl i
) f ( yl ) fi' l (
x sl i'
'
)i l \i
(35)
VII. APPENDIX I PROOF OF THEOREM 1 We will prove that lattice ' has a better nominal coding gain than lattice by introducing ISI into the original LDLC codes, viz, c (' ) c (). Clearly there is
V (' )2/ n (det(C'C))1/ n = det(G'G)1/ n det(S'S)1/ n When L
n and n we have det(S'S)
1/ n
V (' )2/ n s02 V ()2/ n
(26)
s . So 2 0
(27)
It is left to consider the minimum distance of ' . To x' ' , L
xi' s0 xi sl xi l , i 0,..., n 1
(28)
l 1
The minimum distance of ' is 2 dmin (' ) ' min (|| x' ||2 ) ' min ( xi' 2 ) (29) ' ' ' ' x , x 0
x , x 0
i
Notice that since the filtering operations of ISI are linear the shortest vector of generates the shortest vector of ' . Combining Eq. (21) with (22) gives © 2013 ACADEMY PUBLISHER
ACKNOWLEDGMENT This work is supported by National Science Foundation of China (60902048) and Chinese Universities Scientific Fund (BUPT 2013RC0102). REFERENCES [1] Erez U, Zamir R. Achieving 1/2log (1+SNR) on the AWGN channel with lattice encoding and decoding. IEEE Transactions on Information Theory, 2004, 50(10) pp. 2293-2314. [2] Poltyrev G. On coding without restrictions for AWGN channel. IEEE Transactions on Information Theory, 1994, 40(2) pp. 409-417. [3] Sommer N, Feder M, Shalvi O. Low-density lattice codes. IEEE Transactions on Information Theory, 2008, 54(4) pp. 1561-1585. [4] Yoda Y, Feder M. Efficient parametric decoder of lowdensity lattice codes. IEEE International Symposium on Information Theory: June 28-July 3, 2009, Seoul, Korea. New York, NY, USA: IEEE, 2009, 8 pp. 744-748. [5] Kurkoski B, Dauwels J. Message-passing decoding of lattices using Gaussian mixtures. IEEE International Symposium on Information Theory: June 6-11, 2008, Toronto, Canada. New York, NY, USA: IEEE, 2008, 8 pp. 2489-2493.
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[6] Kurkoski B, Yamaguchi K, Kobayashi K. Single-Gaussian messages and noise thresholds for decoding low-density lattice codes. IEEE International Symposium on Information Theory: June 28-July 3, 2009, Seoul, Korea. New York, NY, USA: IEEE, 2009, 8 pp. 734-738. [7] Uchikawa H, Kurkoski B, Dauwels J. Threshold improvement of low-density lattice codes via spatial coupling. IEEE International Conference on Computing, Networking and Communications (ICNC): Jan. 30-Feb. 2, 2012, Maui, HI. New York, NY, USA: IEEE, 2012, 3 pp. 1036-1040. [8] Fischer R. Precoding and signal shaping for digital transmission. New York: John Wiley & Sons, 2002. [9] N. Sommer, M. Feder, O. Shalvi, “Shaping methods for low-density lattice code,” IEEE Information workshop: Oct 11-16, 2009, Taormina, Italy. New York, NY, USA: IEEE, 2009, 12 pp. 238-242. [10] Shalvi O, Sommer N, Feder M. Signal codes: convolutional lattice codes. IEEE Transactions on Information Theory, 2011, 57(8) pp. 2503-5226. [11] Conway J, Sloane N. Sphere packings lattices and groups. New York: Springer-verlag, 1988. [12] Xia X. Modulated coding for Inter-symbol interference channels, New York: McGraw-Hill, 2001 [13] Forney G, Ungerboeck G. Modulation and coding forlinear Gaussian channels. IEEE Transactions on Information Theory, 1998, 44(6) pp. 2384-2415. [14] Conway J, Sloane N. A fast encoding method for lattice codes and quantizers. Transactions on Information Theory, 1983, 29(6) pp. 820-824.
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[15] E. Agrell, T. Eriksson, A. Vardy. Closest point search in lattices. IEEE Transactions on Information Theory, 2002, 48(8): 2201-2214. [16] M. Tomlinson, “New automatic equlizer empoying modulo arithmetic”, Electronics Letters, pp. 138-139, 1971 [17] G. D. Forney and M. V. Eyuboglu, “Combined equalization and coding using precoding”, IEEE Commnun. Magzine, 1991, 29(12): 25-34 [18] F. R. Kschischang, B. J. Frey, H. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Transactions on Information Theory 2001, 47(2), 498-519 [19] Price K, Storn R, Lampinen J et al. Differential evolution—a practical approach to global optimization. New York: Springer, 2005. [20] Franceschini M, Ferarri G, Raheli R et al. LDPC coded modulation, Berlin: Springer, 2009.
Yang Li is a doctoral candidate in the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications. He received a BA degree with major in English Literature and minor in communication engineering from Beijing University of Posts and Telecommunications, and a PhD in Information science from Beijing University of Posts and Telecommunications. Her current research is in coding theory, signal processing in communication system.
