Max-log APP Detection for Non-bijective Symbol Constellations
aa r X i v : . [ c s . I T ] F e b Max-log APP Detectionfor Non-bijective Symbol Constellations
Martin Damrath, Peter Adam Hoeher
Abstract —A posteriori probability (APP) and max-log APPdetection is widely used in soft-input soft-output detection. Incontrast to bijective modulation schemes, there are importantdifferences when applying these algorithms to non-bijectivesymbol constellations. In this letter the main differences arehighlighted.
Index Terms —Digital modulation, Demodulation, Detectionalgorithms, Approximation algorithms.
I. I
NTRODUCTION
Commonly used modulation schemes like square quadratureamplitude modulation or phase-shift keying are characterizedby a bijective symbol constellation. Thus, N bits are mappedonto M = 2 N symbols. In contrast, there also exist non-bijective modulation schemes like superposition modulation(SM) [1], which typically produce a symbol alphabet with acardinality less than N . This reduction in symbol cardinal-ity can be exploited to achieve power efficiency, bandwidthefficiency, and/or to reduce the detection complexity. SM isof particular importance, because active signal shaping canbe avoided, which makes it naturally near-capacity achieving.Furthermore the complexity of the optimal detector can bereduced from O (2 N ) down to O ( N ) by exploiting the tree-based non-bijective modulation structure. In the case of non-bijective modulation, redundancy due to channel coding ismandatory. According to the state-of-the-art, extrinsic infor-mation is exchanged between the demodulator (detector) andthe channel decoder. Hence a soft-output detector is mandatoryin iterative receivers. When performing soft-input soft-output(SISO) detection, the a posteriori probability (APP) algorithmprovides the optimal solution. A common simplification ofAPP detection provides the max-log APP detector [2]. How-ever, these detection algorithms historically have been derivedfor bijective symbol constellations. When considering non-bijective modulation schemes, there are important facts tonotice when applying APP and max-log APP detection. Espe-cially for the max-log APP detector, these facts are essentialwith respect to the performance. Even if they are rather simple,to the best knowledge of the authors, they have not beenpublished yet. Furthermore there is no conventional detectionmethod for max-log APP detection of non-bijective modula-tion schemes in the literature. In this concise letter the maindifferences between bijective and non-bijective modulationschemes for implementing the APP detector or max-log APPdetector are highlighted, and a conventional detection methodfor the max-log APP detector for non-bijective modulationschemes is proposed. M. Damrath, P. A. Hoeher are with the Faculty of Engineering, KielUniversity, Kiel, Germany, e-mail: { md,ph } @tf.uni-kiel.de. II. A P
OSTERIORI P ROBABILITY D ETECTION
For SISO detection, the APP detector provides the optimalsolution. Considering a memoryless channel, the extrinsic log-likelihood ratio (LLR) L n of bit b n can be obtained via thewell-known formula L n . = log p ( y | b n = 0) p ( y | b n = 1) (1) = log P b ∼ n P ( b ∼ n ) p ( y | b ∼ n , b n = 0) P b ∼ n P ( b ∼ n ) p ( y | b ∼ n , b n = 1) (2) = log P x ∈X (0) n P ( z ( x )) p ( y | x ) P x ∈X (1) n P ( z ( x )) p ( y | x ) , (3)where y represents the channel observation, b ∼ n denotes thebit set excluding b n , x is a symbol defined over the alphabet X , X ( b ) n stands for the symbol subset of X with b n = b ∈ { , } ,and z ( x ) represents the set of all those b ∼ n that will leadin combination with b n = b to the symbol x ∈ X ( b ) n . Theterms P ( b ∼ n ) in (2) and P ( z ( x )) in (3) represent a prioriinformation, and hence are time-varying in the case of iterativeprocessing. Both (2) and (3) are applicable independent ofwhether the symbol constellation is bijective or non-bijective.Nevertheless, there is a difference in the number of summandsbetween them for non-bijective symbol constellations. Whileeach sum in (2) consists of N − summands, each sum in (3)consists of just M ≤ N − summands.III. M AX - LOG
APP D
ETECTION
The max-log APP detector [2] is a common simplificationof the APP algorithm, which avoids the high complexity ofexponential and logarithmic operations in APP detection. Es-pecially in high-SNR scenarios the detector achieves an APP-like performance. Nevertheless, the complexity still remainsto be O (2 N ) . The max-log APP concept is based on the APPdetector in the logarithmic domain: L n = log P b ∼ n e log P ( b ∼ n )+log p ( y | b ∼ n ,b n =0) P b ∼ n e log P ( b ∼ n )+log p ( y | b ∼ n ,b n =1) (4) = log P x ∈X (0) n e log P ( z ( x ))+log p ( y | x ) P x ∈X (1) n e log P ( z ( x ))+log p ( y | x ) . (5) Motivated by max ∗ ( a, b ) . = log( e a + e b ) (6) . = max ( a, b ) + log (cid:16) e −| a − b | (cid:17) , (7)the max ∗ -operation can be approximated as max ∗ ( a, b ) ≈ max( a, b ) [2], [3]. Thus, (4) and (5) canbe approximated by: L n ≈ max b ∼ n { log P ( b ∼ n ) + log p ( y | b ∼ n , b n = 0) }− max b ∼ n { log P ( b ∼ n ) + log p ( y | b ∼ n , b n = 1) } (8) L n ≈ max x ∈X (0) n { log P ( z ( x )) + log p ( y | x ) }− max x ∈X (1) n { log P ( z ( x )) + log p ( y | x ) } . (9)While (8) and (9) leads to the same results for bijective symbolconstellations, there is an important difference consideringnon-bijective symbol constellations. Due to the fact that thereis at least one symbol consisting of more than one bit set b ∼ n ,the results of (8) and (9) might no longer be equal. In fact, (9)outperforms (8) and provides a reasonable approximation ofthe APP algorithm, as it will be shown in Section IV. Thus,(9) should be applied as the conventional detection methodfor non-bijective modulation schemes. It is also possible totransform (8) into (9). Therefore, all summands correspondingto a bit set b ∼ n , which leads in combination with b n to theidentical symbol x , have to be summed up in (8), beforeapplying the max -operation: L n ≈ max x ∈X (0) n ( log X b ∼ n → x P ( b ∼ n ) ! + log p ( y | x ) ) − max x ∈X (1) n ( log X b ∼ n → x P ( b ∼ n ) ! + log p ( y | x ) ) ≈ max x ∈X (0) n (cid:26) max b ∼ n → x ∗ (log P ( b ∼ n )) + log p ( y | x ) (cid:27) − max x ∈X (1) n (cid:26) max b ∼ n → x ∗ ( P ( b ∼ n )) + log p ( y | x ) (cid:27) . (10)In comparison, (10) is more complex than (8), because the max ∗ -operator has to be applied. However, non-bijectivesymbol constellations often imply a potential for complexityreduction, when exploiting the non-bijectivity in calculation of(9). In detail, the construction of z ( x ) can be visualized as atree diagram as it is done in [1]. Assuming that the complexityof determining log P ( z ( x )) is proportional to the number ofbranches in the tree diagram, the reduction in complexitycompared to bijective computation becomes clear. In the worstcase z ( x ) is bijective and the complexity of both (8) and (9)are the same. The amount of branches in this case is given by N − . In contrast, if z ( x ) is non-bijective, the complexity ofdetermining log P ( z ( x )) can be reduced by reusing severalstates in the tree diagram for different log P ( z ( x )) . Whenapplying direct superposition modulation with equal powerallocation, the amount of branches needed for log P ( z ( x )) canbe reduced from N − (for bijective calculation) to N − N − − − − Re { y } L n APPMax-Log APP (8)Max-Log APP (9)
Fig. 1. Relationship between extrinsic log-likelihood ratio L n and the realpart of the channel observation y ( N = 16 DSM-EPA, SNR = 12 dB). . . . . . . − − − − SNR (dB) B E R APPMax-Log APP (8)Max-Log APP (9)
Fig. 2. Bit error rate performance for DSM-EPA ( N = 16 ). (for non-bijective calculation). Thus (9) implies a significantpotential of complexity reduction compared to (8).Given (8) and (9), true APP detection is obtained by (re-)substituting all max -operations by max ∗ -operations. Hence,the extra computational complexity of APP detection onlydepends on how to implement the correction term log(1 + e −| a − b | ) , cf. (7). Towards this goal, numerous solutions exist,ranging from a simple table look-up [2] to the direct imple-mentation of the complex correction term.IV. N UMERICAL R ESULTS
So far, the derivation is general and not restricted tomodulation schemes. In fact, it can be exploited in manyapplications like multiple-input multiple-output detection ormulti-user detection. However, the numerical results of thisletter are focused on modulation schemes, especially on direct
TABLE IP
ARAMETER SET OF THE IRREGULAR CONVOLUTIONAL CODE USED INTHE SIMULATION RESULTS . T
HE SUBCODES ARE OBTAINED FROM ARECURSIVE SYSTEMATIC CONVOLUTIONAL CODE BY PUNCTURING ANDREPETITIONS , RESPECTIVELY . T
HE CODE POLYNOMIALS AND THEPUNCTURING / REPETITION TABLE IS GIVEN IN [4]. j R j α j .
10 0 . .
15 0 . .
20 0 . .
25 0 . .
30 0 . .
35 0 . .
40 0 . .
45 0 . .
50 0 . .
55 0 . .
60 0 . superposition modulation with equal power allocation (DSM-EPA) [1]. For the following simulations, the additive whiteGaussian noise channel is assumed. Thus, a channel observa-tion can be written as y = x + w , where w ∼ N (0 , σ ) is azero-mean white Gaussian noise process with the variance σ .Consequently, the conditional probability is given as p ( y | x ) = 1 πσ e − | y − x | σ . (11)Fig. 1 shows the relationship between the extrinsic log-likelihood ratio and the real part of the channel observationsfor DSM-EPA with N = 16 layers for the case that no apriori information is available. DSM-EPA is strongly non-bijective and leads to N / N + 1 ≪ N symbols. The APPdetector from Section II is visualized by the black curve andprovides the optimum performance. The max-log APP detectoris realized for the two different implementations discussed inSection III: Equation (8) corresponds to the blue curve and (9)corresponds to the red curve. As it can be seen from Fig. 1,(9) performs a better approximation of the APP detector than(8) in the case of a non-bijective symbol constellation. Thisstatement is still true, if a priori information is available.Concerning bit error rate (BER) simulations, a bit-interleaved coded modulation [5] system with iterative pro-cessing has been implemented. In order to achieve a near-capacity performance (at least for APP detection) withoutactive signal shaping, an irregular convolutional code ac-cording to [4] with a code rate of about R ≈ / hasbeen matched by means of an EXIT chart design to DSM-EPA employing N = 16 layers, cf. Table I. The bandwidthefficiency is R · N = 4 bits/symbol, which can theoreticallybe achieved by -ary DSM-EPA at an SNR of . dB. Theinformation word length has been chosen to
100 000 bits and An irregular convolutional code C of rate R consists of J puncturedconvolutional codes C j of rates R j implemented in parallel, where R = J P j =1 α j R j and J P j =1 α j = 1 . The EXIT chart characteristic of the irregularconvolutional code can be shaped by optimizing the parameter set α j , ≤ j ≤ J . iterations between the detector and the channel decoderhave been performed. BER simulation results are visualized inFig. 2. The top blue curve shown in broken lines is obtained,when all max ∗ -operations in an APP detector matched tothe symbol alphabet of size N are replaced by the max -operation, which corresponds to (8). Although this is usuallyexactly what is done quite often in the case of bijectivemodulation schemes (and referred to as max-log APP detection[2]), it completely fails in the case of non-bijective modulationschemes like DSM: The iterative receiver does not convergein the interesting SNR range. The max-log APP detectoraccording to (9), which performs the max -operation over thesymbol alphabet of cardinality N / N +1 , provides a muchbetter performance. Compared to the APP detector, it degradesonly by about . dB in the area of the turbo cliff, and theiterative receiver converges. Thus, (9) clearly outperforms (8)in the case of non-bijective symbol constellations. The errorfloor shown in Fig. 2 can be reduced or avoided by means ofdoping, which is beyond the scope of this letter, however.V. C ONCLUSION
It is shown that there are important facts to notice whenapplying APP and max-log APP detection for non-bijectivesymbol constellations. In APP detection, there is no need todistinguish between bijectivity or non-bijectivity, but in max-log APP detection, it is important to distinguish between themfor achieving the best performance for non-bijective symbolconstellations. The main differences with respect to non-bijective modulation schemes are highlighted and supportedby numerical results. Starting off from an APP detector andreplacing all max ∗ -operations by max -operations, as donequite frequently, completely fails for non-bijective modulationschemes, thus a conventional detection method for max-log APP detection of non-bijective modulation schemes isproposed. R EFERENCES[1] P. A. Hoeher and T. Wo, “Superposition modulation: Myths and facts,”
IEEE Commun. Mag. , vol. 49, no. 12, pp. 110–116, Dec. 2011.[2] P. Robertson, P. A. Hoeher, and E. Villebrun, “Optimal and sub-optimalmaximum a posteriori algorithms suitable for turbo decoding,”
Eur. Trans.Telecomm. , vol. 8, no. 2, pp. 119–125, Mar./Apr. 1997.[3] J. Erfanian, S. Pasupathy, and P. Gulak, “Reduced complexity symboldetectors with parallel structure for ISI channels,”
IEEE Trans. Commun. ,vol. 42, no. 234, pp. 1661–1671, Feb. 1994.[4] M. Tuechler, “Design of serially concatenated systems depending on theblock length,”
IEEE Trans. Commun. , vol. 52, no. 2, pp. 209–218, Feb2004.[5] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,”