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
__________________________________________
Piotr REMLEIN*
Dawid SZŁAPKA*
NEW TAILSYMBOLS RING CONVOLUTIONAL ENCODERS
In this paper a method of using convolutional codes over rings for packet data transmission over additive white gaussian noise (AWGN) channel is proposed. This paper presents a method to create Tailsymbols codes (TS) obtained by the concatenation of feedback convolutional encoder over ring and M-QAM modulator. In the paper it is described how the systematic ring convolutional encoder with feedback can obtain the same starting and ending state. Finally, results of search for systematic feedback ring convolutional encoders for Tailsymbols codes are presented. The search is based on the criterion of maximising Euclidean distance. The best Tailsymbols codes for 16-QAM modulation with different number of memory elements and length of information symbols are tabulated
Keywords: codes over ring, Tailbiting codes
1. INTRODUCTION
Packet data transmission schemes are often used in wireless telecommunication systems. The convolutional codes are used in such systems as an efficient and powerful class of error correcting codes [4]. To be able to use convolutional codes (of rate R=k/n and m memory elements) in the packet transmission we must convert these codes to block codes. There are some methods for this conversion. One of such methods is called Tailbiting. In this method we transmit the convolutionaly encoded data in a block form without known tail. By framing the encoded data we are not adding known bits to the end of the data information stream [1]. The encoder starts and finishes the encoding process in the same state, but this state is not known by the decoder. In the paper [2] it is shown that turbo-codes generally provide the best error rate performance for long blocks (over 150 bits), but for short blocks (under 150 bits) Tailbiting convolutional codes provide the best performance. The motivation for investigating the ring convolutional codes was to explore a natural relation between M-ary modulation and codes over the
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ring of integers modulo M [5]. Up to now in the literature were published the best Tailbiting codes with the greatest minimum Hamming distance [3,6,7]. In case of search for the best convolutional codes for the signals transmitted over the additive white Gaussian noise (AWGN) channel the quality criterion is Euclidean distance [4]. In this article we assumed the Euclidean distance as a parameter to estimate the quality of Tailsymbols codes (TS). To find the best TS codes over ring one can search the full space of the codes. Such method gives the certainty that the found codes are the best. The fault of this method is the exponentially growing complexity with the growing number of memory cells of the encoder and the number of its inputs.
In this paper we are analysed Tailsymbols codes encoded by systematic ring convolutional encoders with feedback. We present results of the search for the best convolutional encoders over ring modulo-4 with code rate R=1/2 used with 16-QAM modulation. We found new TS codes for 16-16-QAM modulation with the best Euclidean distance. These codes were not published yet.
This paper is organised as follows. Section 2 describes the procedure of encoding TS by using the systematic convolutional encoders with a feedback. In Section 3 we present the results of our computer search for TS codes. Finally, Section 4 gives the concluding remarks.
2. TAILSYMBOLS ENCODING METHOD
In this article we generalize the Tailbiting method to convolutional codes over rings of integers modulo-m [2, 3] and we name the resulting codes Tailsymbols convolutional codes over ring (TS). In the proposed method we encode and decode a block of N (M-ary) symbols without a know tail, thus keeping the effective rate of transmission equal to the code rate. This is done by letting the encoder start and end in the same, for the decoder unknown, state. The encoding procedure to achieve this is not difficult if the structure of the encoder is feedforward. In this case the starting state depends on the m last information symbols in the transmited packet, where m is the number of memory cells in the encoder. In case of convolutional encoder with feedback (Fig. 1) the starting state depends on all of the information symbols in the packet. Finding the initial state, wherein the encoder should start the work and after N symbol intervals end encoding in the same state, is complex. One of methods to the find this initial state was proposed in [8] and extended for multilevel codes in [9].
In Fig. 1, we show the realization of the systematic feedback convolutional encoder over ring of integers modulo-M [5, 7] with the code rate R=k/n, n=k+1.
u u u v v v v
Fig. 1. Systematic feedback convolutional encoder over ring of integers modulo-M. At inputs to the encoder, in time t, is given information vector Ut with M-ary
elements ut(i) belonging to the ring ZM={0, 1, 2, ... , M-1}, (ℜ=ZM).
Ut = (ut(1), ut(2), ..., ut(k)) (1)
The convolutional encoder produces a coded sequence of the symbols which belong to the same ring ZM
Vt = (vt(1), vt(2), ..., vt(n)), (2)
where n=k+1.
The coefficients in the encoder (Fig. 1) are taken from the set {0,...,M-1). The memory cells are capable of storing ring elements. Multipliers and adders perform multiplication and addition respectively in the ring of integers modulo-m.
The encoding process described as mapping between the information vector (1) and encoded vector (2) can be determined by:
Vt =Ut G, (3)
where G denotes generator matrix of the encoder [8].
The state of the encoder at time t is determined by the content of memory elements Xt= (xt(1), xt(2), ..., xt(m))T, (4)
where m is the number of encoder memory elements.
In case of using the convolutional encoders with feedback to packet transmission without tail we have to calculate the initial state X0 that will be the same as the
state XN of the encoder after N cycles. This is not quite easy. In the paper to find
this starting state we used the method proposed in [8]. The correct starting state can by calculated using the state space representation. The state of the encoder in time t+1 can be described as:
Xt+1=AXt + BUtT, (5)
where A is the (m x m) state matrix which defines connections between memory elements, B is the (m x k) control matrix which defines connections between encoder inputs and memory elements.
The vector Vt of the the output of the encoder in time t can be described as [8]
where: C is the (n x m) observation matrix which defines connections between encoder outputs and memory elements, D is the (n x k) transition matrix which defines connections between encoder entries and outputs.
In the paper [8] it was shown also that the state, in time t of the systematic convolutional encoder with feedback can be described as the superposition of two vectors Xt[zi] and Xt[zs] which define the ending state of the encoder
Xt= Xt[zi] + Xt[zs] (7)
where Xt[zi] is the vector which defines the encoder state achieved after t cycles
if the encoding process started in state X0 and all inputs symbols are zero, Xt[zs] is
the vector which defines the encoder state achieved after t cycles if the encoding stared in the all zero state (X0=0) and the information sequence is encoded.
From the equations (5) and (7) we can write that: Xt= Xt[zi] + Xt[zs]=
∑
− = τ τ τ − − + 1 t 0 T ) 1 t ( 0 t BU A X A (8)If we assume that the state at time t=N is equal to the initial state X0, we obtain
from (8):
(Im - AN)X0= XN[zs], (9)
This equation can be described for convolutional encoders over ring ℜ=ZM as:
(Im + AN)X0= XN[zs], (10)
where Im is the (m x m) identity matrix.
As we can see form (10), we can calculate correct initial state X0 of the encoder if
the matrix (Im + AN) is invertible.
The matrix A form equation (10) for the systematic convolutional encoder with the feedback is described as [8, 9]: = − 1 1 m m f f f 1 1 0 0 A M O L (11)
Using the obtained above mathematical relations (9) and (10) we can describe the encoding process for TS codes as follows: at first we have to calculate the vector XN[zs] for a given data information packet. Accordingly, the encoder starts in the all
zero state. All of the N·k information symbols are encoded but the output symbols are ignored. After N cycles the encoder will be in the state XN[zs]. In this situation
we can calculate form (10) the correct initial state X0 and the encoder can start the
proper encoding process and a valid codeword results. After N cycles the encoder ends its work in this same state as it started.
Following this description we show the example of TS encoding procedure with feedback systematic convolutional encoder over ring Z4, modulo-4.
EXAMPLE
A packet of four symbols is encoded. The symbols belong to the ring Z4. The
encoder is a systematic convolutional encoder over ring Z4 with feedback, with
code rate R=1/2 and two memory elements m=2. In Fig. 2, we show the structure of this encoder. We encode the information block U=(U0, U1, U2, U3) = (1, 0, 3, 3).
The state matrix is given as A=
3
1
3
0
. Therefore N=4, k=1, and from equation
(9) we can calculate 0 [4zs] 4 2
X
X
3
1
3
0
I
=
−
. From this formula we obtain:] [ 4 0
3
3
1
2
zsX
X
=
. Therefore we have to calculate the stateX
[4zs]. From Fig. 3 wecan see that this state is equal to (3,1)T and the correct state from which we must start the encoding process is equal to
=
=
0
3
1
3
3
3
1
2
X
0 . From Fig. 4 we cansee that, if we start to encode the sequence U from state (3,0)T, than after N=4
cycles we end in the same state and obtain valid codeword V=(13,02,31,30).
Fig. 2. Encoder of the convolutional code
+
+
+
+
=
2 2D
3
D
3
1
D
D
2
3
1
)
D
(
G
Fig. 3. The tree diagram when the zero state response is obtained X4[zs].
Fig. 4. The tree diagram for proper encoding process for Tailsymbols convolutional codes over ring.
3.
SEARCH RESULTS
In this paper we present the results of our computer search for the best Tailsymbols codes over ring modulo-M for transmission over AWGN channel. As the quality criterion we take a minimum Euclidean distance de_min. The codes were
generated by the feedback systematic convolutional encoder over ring. An exhaustive search was used to find TS codes in Table 1. An object of search in this article were Tailsymbols convolutional codes over ring Z4 generated by
concatenation of the systematic encoders with feedback with code rate R=1/2 and 16-QAM modulator. The found encoders have m memory cells, S states and k inputs. N denotes the length of the input symbol sequence of k information bits per symbol. For codes over a ring feedback coefficients f0÷fm are written as a sequence
of decimal numbers and the coefficients in the systematic branches gk0÷gkm. The
description. All TS codes over ring found for 16-QAM are presented in Table 1. We found the best TS codes for encoders with 16, 64 and 256 states. All of these TS codes are the new codes and were not published yet.
Table 1. Tailsymbols codes over ring Z4 with code rate R=1/2 for 16-QAM
modulation.
S 16 TS 64 TS 256 TS
N f, g de_min f, g de_min f, g de_min
4 130,100 12,000 1300,1000 12,000 11100,10000 12,000 5 113,210 14,128 1130,2100 14,128 10132,10000 14,128 6 102,111 14,141 1121,1100 14,828 12330,11000 14,828 7 111,123 16,944 1312,1313 16,970 11331,20110 16,970 8 111,221 16,970 1121,120 18,129 - -
4.
CONCLUSION
In this paper, we generalized the Tailbiting techniques to Tailsymbols encoding over ring of integers modulo-M. We described how the systematic ring convolutional encoder with feedback can have the same starting and ending state. We presented the search results of the best Tailsymbols convolutional codes over ring Z4 for the transmission over AWGN channel. As a criterion of the
optimalisation we took the Euclidean distance.
Tables of the best new Tailsymbols convolutional codes over ring Z4 with rates
R=1/2 for 16-QAM modulation were obtained by computer search. All TS codes shown in Table 1 were not presented in the literature the authors are aware of. The usefulness of the codes one ought to verify by means of computer simulations.
REFERENCES
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[5] J. L. Massey, T. Mittelholzer, „Convolutional codes over rings", in Proc. 4th Joint Swedish-USSR Int. Workshop Information Theory, 1989, pp. 14 -18. [6] H.H. Ma, J.K. Wolf “On Tailbiting Convolutional Codes”. IEEE Trans.
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