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Superposition Coding to SupportMultiple Streams, Priorities, and Channel Capacities

in the Context of GMSKThomas A. Courtade, Jiadong Wang, and Richard D. Wesel

Department of Electrical EngineeringUniversity of California, Los AngelesLos Angeles, California 90095

Email: {tacourta, wjd, wesel} @ ee.ucla.edu

AbstractIt is well known in information theory that super-position coding is an optimal technique for providing a singletransmission that both provides a high data rate when channelconditions are good and still provides some information whenthe channel conditions are poor (perhaps due to jamming).Typical superposition coding schemes for the additive whiteGaussian noise channel combine signals after the modulator.For modulation schemes such as GMSK, this standard formof (additive) superposition is not acceptable because it doesnot maintain a constant envelope. This paper introduces anew approach in which newly designed nonlinear codes allowsuperposition to take place before the GMSK modulation. Thesecodes are demonstrated to operate very close to the boundary ofthe achievable rate region for the broadcast binary symmetricchannel which would result from GMSK transmission withhard decoding. The results also suggest that similarly optimalperformance might be obtainable in the soft decoding regime.

I. INTRODUCTIONIn tactical networks, the signal-to-noise ratio (SNR) of

the communication channel is frequently unknown (due toa dynamic environment, adversarial jamming, etc.). However,even if the channel quality is poor, it is desirable to maintainsome measure of connectivity so that users have access tocritical information. Moreover, in cases where the channelquality is relatively good, it is also desirable for the receiverto have access to additional, non-critical information. Thus,the challenge is to design a transmission scheme in which areceiver can decode a stream of critical data in all anticipatedchannel conditions, and can also recover an additional streamof non-critical data when the channel quality is relatively good.One way to approach this problem is to consider the

two-receiver1 degraded broadcast channel (DBC) in which atransmitter broadcasts a common transmission to two differentreceivers. In the case of the DBC, it is assumed that the signalreceived at the second receiver is a degraded (i.e., noisier)version of the signal received at the first receiver. Typically,two independent data streams at rates R1 and R2 respectivelyare encoded together and broadcast to the receivers. The

This research was supported by Rockwell Collins through contract#4502769987.1We restrict our discussion to the two-receiver degraded broadcast channel

in this paper, however the results can be extended to more users.

first receiver is able to decode both data streams (for atotal information rate R1 +R2), while the second (degraded)receiver is only able to decode the second data stream at rateR2. By encoding critical information into the second datastream at rate R2 and non-critical information into the first datastream at rate R1, we can ensure that a receiver has access tothe critical information when the channel quality is poor, andthat receivers can recover additional, non-critical informationat rate R1 when the channel quality is sufficiently good.

For many DBCs (in particular the AWGN-DBC), an optimaltechnique for achieving any point in the capacity region iscalled superposition coding. In superposition coding, the twodata streams are independently encoded and modulated, andthe modulated signals are scaled and added together. Receiversdecode the streams through successive decoding. For somemodulation schemes, such as m-ary QAM, this approachis perfectly acceptable. However, many tactical waveformsemploy constant-envelope modulation (e.g., GMSK), and thisadditive superposition of modulated signals is unacceptablebecause it destroys the constant-envelope characteristics of themodulation scheme.

In this paper, we present a coding scheme (based on non-linear codes) which circumvents this problem by performingthe superposition prior to the modulation. Thus, our encodingscheme supports the independent encoding of two distinct datastreams (which are recoverable at different channel qualities),while the modulation and spectral characteristics of the trans-mitted signal remain unchanged.

This paper is organized as follows. In Section II-A, we in-troduce the fundamentals of nonlinear turbo-codes. In SectionII-B, we formally define the DBC and give capacity results forthe special case of the binary-symmetric DBC. A systematictechnique for designing nonlinear turbo-codes is described inSection III. Section IV presents the architecture for our systemand also provides simulation results, demonstrating that ourscheme is nearly optimal. Section V delivers the conclusionsand discusses directions for future work.

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Fig. 1. Block diagram representation of a nonlinear turbo-encoder.

II. BACKGROUND

In this section, we provide the background material requiredto describe our superposition coding approach. In the firstsubsection, we briefly review the concepts behind nonlinearturbo-codes, and in the second subsection we formally definethe degraded broadcast channel.

A. Fundamentals of Nonlinear Turbo-Codes

Nonlinear turbo-codes, similar to their linear counterparts,consist of constituent encoders connected via interleavers. Thekey difference between nonlinear and linear turbo-codes isthe following: In a linear turbo code, the output bits froma consituent encoder are linear combinations of the encoderstate and the input bits. In a nonlinear turbo-code, the outputbits from a constituent encoder correspond to the output of alookup-table addressed by the encoder state and the input bits.See Fig. 1 for a block diagram of a nonlinear turbo-code.The critical advantage afforded by nonlinear turbo-codes is

their capability of producing codewords with ones densitiesother than 50%. As we will demonstrate in the remainder ofthis paper, the ability to design a nonlinear turbo-code witha desired ones density allows one to independently encodedifferent data streams, XOR the codewords together, send themodulated sum over a channel, and then successively decodeto recover one or both of the streams depending on channelquality. As we noted in Section I, this strategy preservesthe constellation of the original modulation scheme, whichis a requirement for constant-envelope modulations such asGMSK.A systematic method for designing nonlinear turbo-codes is

given in Section III.

B. Introduction to the DBC

In this section, we define the Degraded Broadcast Chan-nel (DBC) and introduce the Broadcast Binary SymmetricChannel (BBSC), which will be of particular interest to usin Section IV.Definition 1 (e.g., [1]): A broadcast channel consists of

an input alphabet X and two output alphabets Y1 and Y2 and a

Fig. 2. Channel model for two-user broadcast binary-symmetric channel.

Fig. 3. Encoder scheme for two-user broadcast binary-symmetric channel.

probability transition function p(y1, y2|x). The broadcast chan-nel is said to be stochastically degraded (or just degraded)if there exists a distribution p(y2|y1) such that

p(y2|x) =

y1

p(y1|x)p(y2|y1).

In this paper, we will restrict our attention to memorylessbroadcast channels, i.e. p(yn1 , yn2 |xn) =

ni=1 p(y1i, y2i, xi).

Of particular interest to us is the BBSC, which is suf-ficient to model any binary modulation scheme in whichthe demodulator provides hard decisions to the decoder. Thetwo-user BBSC consists of two binary symmetric componentchannels, one with transition probability and the other withtransition probability , as shown in Fig. 2. Without loss ofgenerality, we assume < . A simple and optimal encodingscheme is an independent-encoding approach in which binarycodewords from independent codebooks are added togetherusing the XOR operation [2][7]. We refer to this scheme assuperposition coding, and the encoder structure is shown inFig. 3.The capacity region of a degraded broadcast channel was

established by Cover [1], Bergmans [8] and Gallager [9].Cover [10] introduced an independent-encoding scheme fortwo-user broadcast channels. This scheme is known to achievethe boundary of the capacity region for the broadcast binary-symmetric channel (BBSC) and is investigated in [2] [3] [4][5]. The class of channels for which independent encoding isoptimal was recently extended in [6], [7].The capacity region for a BBSC is given by

R1 h( ) h()

R2 1 h( ),(1)

606

where h() is the binary entropy function, is the ones densityof X1, X2 has 50% ones density and the operation is definedby

a b = a(1 b) + b(1 a), 0 < a, b < 1. (2)

In order to use a superposition coding scheme with R2 > 0,the codes of the two users cannot both have ones densities of50%. This precludes the exclusive use of linear codes. Usingthe techniques described in Section III, we design a familyof nonlinear turbo codes that can provide a controlled onesdensity. Superposition of one of our nonlinear codes with alinear turbo code produces an overall transmission with thepotential to approach an optimal point the capacity region ofthe BBSC.

III. A DESIGN SCHEME FOR NONLINEAR TURBO CODESA nonlinear turbo code is defined by look-up tables that

map state and input bits to output bits in the trellises ofthe constituent codes. By controlling the number of ones inthe look-up table, any desired ones density can be closelyapproximated. In general, a brute-force search to find thelook-up table yielding the largest effective free distance isimpractical be

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