An Extension of the Achievable Rate Region of Schalkwijk’s 1983 Coding Strategy for the Binary Multiplying Channel

H.B. Meeuwissen, J.P.M. Schalkwijk, and ’4.H.A. Bloemen Eindhoven University of Technology, Dept. of Electrical Engineering, Group on Information and Communication Theory,

P.O. Box 513, 5600 MB Eindhoven. the Netherlands.

Abstract A new region R of achievable rate pairs (R1,HL) E R is established for the binary multiplying channel. The new region R has an equal rate point of RI = Ra = 0.63072 bit per transmission.

I. DEFINITIONS This paper is concerned with the binary multiplying chan-

nel (BMC) [l]. The capacity region of the BMC is bounded by the Shannon inner bound region G,, and the Shannon outer hound region Go. These regions are plotted in Fig. 1.

Communication over the BMC by two distant, terminals is inodeled as follows. A message Ot a t terminal t , t = 1 , 2 , is encoded into the channel input sequence X t = ( X t , l , X t , , , ..., Xt,?,.). The common channel outpiit sequence Y I = Ya = Y = (Yl,Yz, ..., Yn) is formed such that

= X I , , ~ X ~ , ~ , X t , , E (0, l}, j = 1 , 2 , ..., n. Note that the first channel input X(, , I is based on the message Ot only, while the k-th channel input X t , k , k = 2 , 3 , ..., n is based on both the local message O t , and the previous channel outputs (YI, Yz, ..., Yk-1). The decoder at terminal t estimates the otlier terminal’s message O3-t from both tjhe channel output sequence Y , and the local message Ot.

A coding strategy for the BMC is described as a progressive subdivision of the [O, 1 ) x [0 ,1) square. Therefore, the proba- bility of each resolution product that occurs in this progressive subdivision of the unit square is equal t o its area.

11. SCHALKWIJK’S 1983 CODING STRATEGY The 1983 coding strategy is composed of alternating so-

called inner and outer bound transmissions. Let P r [i] and P r [o] denote the average code word length of the inner and out,er bound transmissions, respectively. Of course, P r [i] = 1. Let I (O1; Y l O s - t , i ) and I (Ot ; YIOs-t , o) denote the infor- mation rate of an inner and an outer bound transmission from encoder t t o decoder 3 - t , respectively. The achievable rate region of the 1983 coding strategy satisfies R’ = {(RI, Rz) :

products is completed by (i) outer bound transmissions of av- erage code word length 3 P r [o] - L [ loss] , and (ii) three new transmissions of average code word length L [ga in] . In fact, the new coding strategy, see [ 3 ] , is a modification of the 1983 coding strategy that results in both a loss and a gain with respect to its original. Let It [gain] denote the average mu- tual information of the three new transmissions from encoder t t o decoder 3 - t , then the achievable rate region of the new strategy satisfies R = { ( R I , R,) :

1 (3 P r [o] - L [Loss]) I (Ot ; Y l @ - t , o) + It [gain]

3 P r [ ) : I + 3 P r [o] - L [loss] + L [gain]

The new region R has an equal rate point of RI = Rz = 0.63072 bit per ticansmission and includes the region R’. The results of van Overveld [4] prove that all rate pairs (RI, Ra) E R are operationally achievable. The new region R is also plotted in Fig. 1.

1 0.9

0.8 0.7 0.6

0.5 0.4 0.3 0.2 0.1

0 0 0.10.20.30.40.50.60.70.80.9 1

The region R’ has an equal rate point, of RI = Rz = 0.63056 bit, per transmission and includes the region Gi. In the unit square, a message pair ( 0 1 , 6 3 2 ) is always situated in a sub- rectangle after an inner bound transmission and a subsequent outer bound transmission. Thus, the inner and outer bound transmissions can be repeated ad infinitum in all these sub- rectangles.

111. THE NEW CODING STRATEGY The new coding strategy consists of a structure of inner

bound t,ransinissiorrs of average code word length 3 P r [i], such that (i) an efficient resolution product is generated, and (ii) an unlimited number of repetitions of this resolution prod- uct is generated. The subdivision of these efficient resolution

Fig. 1: The new region R of achievable rate pairs.

REFERENCES 111 C. E. Shannon, “Two-way communication channels.” in Proc. . _

4th Berkeley Symp. on Math. Statist. and Prob., vol. 1, 1961,

[2] J. P. M. Schalkwijk, “On an extension of an achievable rate re- gion for the binary multiplying channel,” IEEE Trans. Inform. Theory, vol. IT-29, pp. 445-448, May 1983.

[3] J . P. M. Schalkwijk, H. B. Meeuwissen, and A. 11. A. Bloeinen, “Coding strategies and a new achievable rate region for the binary multiplying channel,” preprint.

[4] W. M. C. J. van Overveld, O n the Gopacity Region for Deter- minist%c Two- Way Channels and Write- Unidirectional Memo- ries. Ph.D. dissertation, Dept. of Electrical Engineering, Eind- hoven Univ. of ‘Technol., Eindhoven, The Netherlands, 1991.

pp. 611-644.

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