IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 7, JULY 1997 753

Very Low Variable-Rate Convolutional Codes for

Unequal Error Protection in DS–CDMA SystemsAndrew Lientz and John Villasenor

Abstract—Convolutional codes based on Hadamard sequencescan achieve both orthogonality and variable rate. In contrastwith other convolutional coding techniques, these codes incurno processing gain penalty in direct-sequence code-divisionmultiple-access (DS–CDMA) systems. We describe here aclass of such codes called “telescopic protection codes.” Thesecodes enable variable-rate error protection in the manner ofrate-compatible punctured-convolutional (RCPC) codes, butpreserve the Hadamard orthogonality properties necessary tomaintain processing gain that RCPC codes lack.

Index Terms—CDM, orthogonal codes, telescopic protectioncodes, variable-rate error protection.

I. INTRODUCTION

T RADITIONAL DS–CDMA employs modulation of infor-mation sequences by pseudonoise (PN) sequences chosen

for their orthogonality and autocorrelation properties. Providedthat the constraints on orthogonality and autocorrelation aremet, there is flexibility in choosing PN sequences. This opensthe possibility, first proposed by Viterbi in [1], of combiningforward error correction with spectrum spreading, therebyobtaining error protection at no additional cost in processinggain. As noted in [1], to simultaneously perform error codingand spreading requires a low-rate channel coder that producesan output with PN sequence-like orthogonality properties, fol-lowed by modulation by the PN sequence at a rate equivalentto or greater than the channel coder output. Woerneret al.,have advocated the use of biorthogonal codes to accomplishcoding and spreading in DS–CDMA systems [2]. We considerhere the use of Hadamard codes as proposed by Viterbi in[1] to obtain true orthogonality. In addition, Hadamard codesare simply generated, and have the autocorrelation propertiesdesired in a DS–CDMA system.

The combined coding/spreading system uses part or allof the spreading bandwidth to convey redundant informationabout the input information sequence, resulting in significantcoding gains. Consider a desired BER of 0.0057 after decod-ing. A traditional CDMA system with a spreading factor of16 would require a channel SNR of 6.4 dB. If an eight-state,rate-1/16 orthogonal convolutional code is used to performthe spreading, the required SNR for the same BER is 4.02dB. This yields a coding gain of just over 2 dB. In essence,

Paper approved by S. B. Wicker, the Editor for Coding Theory andTechniques of the IEEE Communications Society. Manuscript receivedOctober 10, 1995; revised June 19, 1996, and January 8, 1997. This workwas supported by a contract from Samsung Electronics Company.

The authors are with the Department of Electrical Engineering,University of California, Los Angeles, CA 90095-1594 USA (e-mail:villa@icsl.ucla.edu).

Publisher Item Identifier S 0090-6778(97)05190-8.

a CDMA system acting without the low-rate orthogonal coderis using only a repetition code to spread the sequence, andthe 2 dB improvement represents the performance differencebetween a rate-1/16 orthogonal convolutional coder with softdecisions and a rate-1/16 repetition code. This is similar tothe gains realized by traditional convolutional codes overrepetition codes [3].

We show here that the orthogonal codes proposed byViterbi can be extended to accommodate variable-rate coding.Variable-rate error protection will be advantageous in futurebroad-band CDMA networks carrying information with adiverse set of delay and robustness requirements. Variable-rate convolutional codes are well known in the context ofthe RCPC codes proposed by Hagenauer [4], and one mightexpect that an analogous approach could be used to punc-ture Hadamard sequences. However, puncturing destroys theorthogonality and autocorrelation properties of the Hadamardoutput sequences in the low-rate orthogonal coder. This se-verely limits the utility of punctured codes for spreading in aCDMA system. The fundamental idea presented here is thatone can build a variable rate orthogonal coder in which theoutput alphabet consists only of Hadamard code words of thebase rate, where the base rate is defined as the lowest rateused by the coder. The code then maintains the orthogonalityproperties of base rate codewords for all input rate variations.Because the rate changes are occurring at the input and usedifferent Hadamard matrices as described in the followingsection, we refer to these codes as telescopic protection codes(TPC).

II. CODER DESCRIPTION

To achieve variable-rate error protection usable in a CDMAsystem, the orthogonality of the Hadamard sequences mustbe maintained for all code rates. Telescopic protection codesachieve this by varying the number of input and memory bitsused to select a member from a set of equal-length Hadamardcodewords, thus producing a constant output rate from avariable input rate. This is the opposite of what occurs in anRCPC encoder, where the input sequence is coded at a constantrate, and subsequently is punctured to produce a variable-rateoutput.

Telescopic protection codes achieve variable rates by ex-ploiting the nature of Hadamard sequences which are definedusing the well-known recursion

(1)

0090–6778/97$10.00 1997 IEEE

754 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 7, JULY 1997

Fig. 1. TPC trellis with a transition.

where . We illustrate the coder with a simple example.Consider a system with a base code rate of 1/4 that permitscoding at rates 2/4 and 1/4. The two matrices of interest inthis example are therefore and . For coding at rate 2/4,the telescopic protection encoder reduces to a block coder.Given a set of two input bits , the first bit selects oneof the two codewords (00 or 01) in , and the second bit

then appends either (in the case thatis 0) the outputselected by or (in the case that is 1) the complementof that output. The four possible input combinations (00, 01,10, 11) would then generate the respective outputs (00 00, 0011, 01 01, 01 10), which are, of course, the codewords from

. Coding at rate 1/4 follows the same procedure, with thedifference that a single memory element is used to store theprevious input bit . To code , the pair is formed,and a codeword from is selected using exactly the sameprocedure as described above for rate 2/4.

In general, a base code rate of allows code ratevariations given by where . Regardless ofthe rate, each output word of length is selected using asequence of bits . As the rate is changed, thesource of the bits is altered. For the highest ratecode , the bits are taken directly from theinput, resulting in a block code requiring no memory bits. Forthe codes at rates ( , where is in the interval[0, ], bits – are taken from the input, and bits

– are in memory, and are the finalinput bits that wereused in generating the previous output codeword. Thus, for all

the codes are traditional convolutional codes that canbe decoded using the Viterbi algorithm. For all rates, the basiccoding procedure is identical: bit selects a codeword from

, bit then appends to this either the codeword selectedby or its complement, bit then appends to this eitherthe codeword selected by or its complement, etc. Bits areencoded until a codeword of length (base rate) is chosen.Since this procedure directly implements the recursion of (1),the resulting sequences always will be Hadamard.

The codes can be decoded using the Viterbi algorithm asdescribed in [5]. The code trellis for each rate can be describedby a number of states and a number of branches. Given aTPC with a base rate of , operating at a rate of ,the number of states in the trellis is and the numberof branches is given by . A transition from a lower to ahigher code rate requires that memory bits are discarded inthe encoder. For the code trellis, this implies that a mappingmust occur from a higher to a lower number of states. Froma section rate to a higher rate , the previous state

for state can be found by choosing the statewith thesmallest metric where mod . The trellis in Fig. 1illustrates this for a transition from the base rate 1/16 tothe section rate 2/16. In changing from a higher to a lowerrate, the metric for the higher rate statecan be mapped tostates where . The added complexity for thistransition operation from rate to rate is whichis small in comparison to the complexity of trellis decodingfor incremental rate changes.

One of the advantages of RCPC codes is the embeddedproperty of the coding rates. That is, the output of any codingrate is contained in the output from any lower coding rate.This is useful in networks using ARQ protocols. It is trivial toshow that for TPC codes, all rates have the output of allhigher rates (where embedded in them. For theother coding rates, a method for embedding the codes needsto be investigated.

III. RESULTS AND DISCUSSION

We have implemented the telescopic protection coder fortransmission of a compressed image over an AWGN channel.In this case, a three-level wavelet transform [6] was used onthe 256 256 gray-scale “Lena” image. By applying errorprotection to the subbands of the transform unequally, theimage can be optimally protected. The image in Fig. 2(a) wasprotected using TPC with an average rate of 1/8, and the imagein Fig. 2(b) was protected using TPC with a fixed rate of2/16. Qualitatively, the advantages offered by the variable rateprotection are clearly evident.

A quantitative comparison can be made by plotting the peakSNR (PSNR) for the image versus SNR for an AWGN chan-nel. Simulations for three coding schemes were performed:variable-rate TPC coding with an average rate of 1/8, fixed-rate 2/16 TPC coding, and low rate orthogonal coding at rate1/8. The free distance of the fixed-rate TPC coder is 16, andis approximately equivalent to the average free distance ofthe variable-rate TPC. By contrast, the orthogonal rate-1/8code has a free distance of 12. The curves in Fig. 3 showa significant image PSNR improvement between TPC andthe orthogonal rate-1/8 code due to these distance properties.The curve also shows the advantage of variable-rate errorprotection over equal error protection for TPC codes andlow-rate orthogonal codes. The variable-rate code realizes anincreasing image PSNR gain relative to the fixed-rate codeuntil the channel SNR becomes small ( dB) and maintainslarge gains until low-channel SNR ( dB). At high-channelSNR ( dB), the fixed-rate TPC performs slightly better (

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 7, JULY 1997 755

(a)

(b)

Fig. 2. Images after wavelet coding and transmission with a channel SNR= 7.18 dB. Error correction was performed with the codes described in thispaper at (a) variable-rate 1/8 and (b) fixed-rate 2/16 coding.

dB) than the variable-rate TPC since most of the loss in thevariable-rate coder occurs with rates higher than 2/16. At low-channel SNR ( 2.5 dB), the higher order distance propertiesof the code become significant, and the PSNR’s for all threeschemes converge.

IV. CONCLUSION

We have presented a method for obtaining variable-rateerror protection while still maintaining the orthogonality ofthe output sequence of a DS–CDMA system. As demonstratedin Figs. 2 and 3, TPC codes give significant qualitative andquantitative improvement over traditional codes. For applica-tions such as video and data, these gains will help to enable

Fig. 3. PSNR versus channel SNR for the codes described in this paper at(a) variable-rate 1/8, (b) fixed-rate 2/16, and (c) low-rate orthogonal codingat rate 1/8.

reliable communication without a loss in processing gain overwireless DS–CDMA networks.

An important issue for future study of these codes appliesnot only to TPC, but to low-rate orthogonal codes in general.In the IS-95 CDMA systems, users are made orthogonal byapplying the codewords 1 63 of the Hadamard matrix.This is followed by a PN sequence that randomizes the signaland allows synchronization at the receiver. Using a rate-1/16orthogonal code in this case would reduce the number ofusers from 63 to 4 since they both use the same orthogonalcodewords to spread the information sequence. However, thetraditional convolutional coding that is used in the IS-95system no longer would be necessary, and thus the overallprocessing gain would increase. This results in an increasednumber of users beyond the IS-95 capacity.

ACKNOWLEDGMENT

The authors thank the anonymous reviewers and Prof. G.Pottie of UCLA for providing useful suggestions.

REFERENCES

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[2] B. D. Woerner and W. E. Stark, “Trellis-coded direct-sequence spread-spectrum communications,”IEEE Trans. Commun.,vol. 42, Dec. 1994.

[3] G. D. Forney, Jr., “The Viterbi algorithm,”Proc. IEEE, vol. 61, pp.268–278, Mar. 1973.

[4] J. Hagenauer, “Rate-compatible punctured convolutional codes (RCPCcodes) and their applications,”IEEE Trans. Commun.,vol. 36, pp.389–400, Apr. 1988.

[5] J. Proakis,Digital Communications. New York: McGraw-Hill, 1995.[6] M. Vetterli and C. Herley, “Wavelets and filter banks: Theory and

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