Product Accumulate Codes on Fading Channels

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1 Product Accumulate Codes on Fading Channels Krishna R. Narayanan, Jing Li and Costas Georghiades Det of Electrical Engineering Texas A&M University, College Station, TX Abstract Product accumulate codes are a secial case of differentially coded low density arity check LDPC) codes. This work analyzes the noncoherent erformance of PA codes and general differentially-coded LDPC codes on flat Rayleigh fading channels using extrinsic information transfer charts, and rooses a convergence-constraint method to design good LDPC ensembles matched to differential coding. Introduction Product accumulate PA) codes are a serially concatenated codes whose inner code is a differential code and outer code is 2 arallel branches of single arity check codes or a secial tye of low density arity check LDPC) codes with only degree- and 2 variable nodes) []. This work investigates the erformance of noncoherently detected PA codes on fading channels, and extend it to the general case where the outer code can be any LDPC code i.e. differentially coded LDPC codes). The motivation is two-fold. First, revious work on PA codes has established them as a class of low-comlexity, caacity-aroaching good codes on additive white Gaussian noise AWGN) channels []. Second, PA codes are inherently differentially coded which ermits simle noncoherent) differential detection. The channel model we consider is flat Rayleigh fading channels with antiodal signaling. The received signal is given by r k = α k e jθ k s k + n k, where s k, n k, α k and θ k are the transmitted signals, the i.i.d. comlex AWGN noise with zero mean and ower sectrum density N 0 /2 in each dimension, the Rayleigh fading amlitude with df A α k ) = 2α k ex α 2 k ) for α k > 0, and the channel hase with a uniform distribution over [0, 2π), resectively. Further, the fading amlitudes and hases are correlated with auto-correlation R k = 2 J 02kπf d T s ), where f d T s is the normalized Doler sread, and J 0 ) is the 0 th order Bessel function of the first kind. To reserve the simlicity of PA codes, instead of using comlex multile-symbol differential detection, we consider ilot symbol assisted modulation PASM) and a simle iterative differential detection and decoding IDDD) receiver which has been shown to erform stably at different Doler rates [2]. In each decoding iteration, an estimation of the fading amlitude and hase is first conducted using a Wiener filter, followed by the coherent decoding of the inner differential code / + D) and the outer code. Soft decision feedback is also used to assist the channel estimation. Detailed discussion on the receiver strategy can be found in [2]. We use extrinsic information transfer EXIT) charts [4] to discuss a few interesting issues concerning noncoherent differential coding. First, we show that the oular ractice of inserting ilot symbols to eriodically terminate the differential trellis could cause additional erformance loss and/or high error floors due to a trellis segmentation effect. Hence, a better way of inserting ilot symbols should be to searate them from the trellis structure. Second, in studying the convergence roerty of the iterative rocess, we show that while the erformance/convergence behavior of the outer code of a high-rate) PA code matches well with that of the differential code, a conventional LDPC code does not. Both analysis and simulations confirm that conventional LDPC codes erform better without a differential code than with one. To further insight into what outer) codes match well with differential coding, we roose and discuss a convergenceconstraint method that uses density evolution on EXIT charts to otimize the degree rofiles of LDPC codes for use with inner differential coding. Unlike the conventional threshold-constraint method that targets at the best asymtotic threshold, the convergence-constraint method controls the convergence behavior of the iterative interaction between the inner differential code and the outer LDPC code. We show that the roosed method is efficient and that the resulting otimal code is more that db better than the PA code. We exect the method to be useful in designing good LDPC ensembles matched with other inner decoder, demodulator and receiver. 2 EXIT Chart Analysis In EXIT charts, the exchange of extrinsic information is visualized as a decoding/detection trajectory, which allows the rediction of the convergence and other erformance behavior of the iterative rocess [3] [4]. We use mutual information between the code bit and the corresonding loglikelihood ratio LLR) to deict the characteristics and relations of the comonent decoders. X-axis denotes the mutual information to the inner code a rior) or from the outer code extrinsic), denoted as I a,i /I e,o, and Y-axis the mutual information from the inner code or to the outer code, denoted as /I a,o. [Pilot Insertion:] It is well-known that either insufficient or excessive ilot symbols could cause erformance degra-

2 dation. The former is due to oor channel estimation, and the latter is attributed to the fact that the erformance gain obtained in channel tracking is not enough to comensate for the energy/rate loss caused by ilot symbols. However, little attention has been aid to the fact that imroerly inserted ilots could cause an intrinsic loss in caacity in a differential code. Fig. A) shows the oular ractice of inserting ilots in a differential code. By eriodically terminating the trellis, ilots here assume a dual role of channel estimation and / D) decoding. Unfortunately, this is in fact not a good strategy since segmenting the trellis into small chunks causes a significant amount of short error events an inverse effect of sectrum thinning), and consequently a loss in caacity. This segmentation effect is best illustrated using Fig. 2, where EXIT curves for the differential decoder with 0%, 4%, 0% and 20% ilots terminating the trellis are lotted for two different SNR values. We assume that the four curves in each family have the same energy er transmitted symbol, and that erfect channel information is known to the receiver irresect of the number of ilot symbols). Hence, the difference of the curves in each family is only due to the difference in ilot sacing. At the left end of the curves, we see that the curves with more ilot symbols are slightly better. This is because when there is little feedback information from the outer code, ilot symbols are the major contribution to a riori information. However, at the right end, when there is sufficient information rovided by the outer code, ilot symbols are no longer an imortant source of a riori information. Rather, their negative imact of segmenting the trellis and shortening the average) error events becomes dominant, causing a considerable erformance loss. The erformance degradation is more severe when more ilot symbols are inserted and when the code is oerating at a lower SNR level. This suggests that the oular ractice of terminating the trellis is not a good strategy and that a better way of inserting ilots may be to searate them the trellis as shown in Fig. B). This is confirmed by the simulation result in Fig. 3 where more than 3 db loss in erformance is observed due to trellis segmentation for a rate /2, code length 64K PA code solid lines assume erfect channel information and dashed line uses noncoherent detection). It is interesting to see that if we overlook the imact of the different strategies of ilot insertion, we might get the surrising result that noncoherent detection dashed line) erforms noticeably better than coherent detection rightmost solid line)! [Codes Matched to Differential Coding:] As mentioned before, the outer code of PA codes is a secial tye of LDPC code. Given PA codes erform well esecially at high rates), one tend to ask how a general LDPC code will erform with differential coding. This is an interesting question, since it directs to the solution of how to erform noncoherent detection with LDPC codes. Before we answer the question, we first note two imortant facts about EXIT analysis. First, in order for iterative decoding to converge successfully, the outer EXIT curve should be strictly below the inner EXIT curve, leaving an oen assage between the curves. Second, the area under the EXIT curve, A = 0 I edi a, has shown to be closely related to the caacity of the code we use caacity to loosely denote the information rate). When the a riori information is coming from the erasure channel and when the decoder is an otimal decoder, the area is exactly the caacity of the code [4]. For other channels, this may not be exact, but is nevertheless a good aroximation as verified by emirical results. The imlication of the above two facts is that, in order to fully achieve the caacity rovided by the inner code, the outer code needs to have an EXIT curve closely matched in shae and in osition to that of the inner code. Unfortunately, this not is the case of a conventional LDPC code outer code) and a differential code inner code). In Fig. 4, we lot a set of three outer EXIT curves corresonding to a regular LDPC code, an irregular LDPC code and the outer code of a PA code, and a set of two inner EXIT curves corresonding to a differential code on correlated Rayleigh channel) and the lain Rayleigh fading channel. The regular LDPC code in the lot is 3,2)-regular, and the irregular one is otimized with variable node degree rofile ρx) = x 20 and check node degree rofile γx) = 0x + 78x x x 29, which has a threshold of 726 about db away from the AWGN caacity) [5]. We observe that while the outer code of high-rate) PA codes shows a good match with an inner differential code, a conventional LDPC code regular or irregular) will either intersect with the differential code decoder failure) or leave a huge area between them a waste in code caacity). The observation that LDPC codes match better with a lain channel than with a differential code indicates that, unless secifically designed, LDPC codes should not be used with a differential encoder or more generally with any recursive inner code/modulation). Put another way, an LDPC code that is otimal in the conventional sense i.e. BPSK modulated on memoryless channels) is not otimal when combined with an inner recursive code/modulation. However, not using differential coding tyically requires more ilot symbols in order to track the channel well. Hence, it is exected that on fast) fading channels where only limited bandwidth exansion is allowed, conventional LDPC codes do not erform well with noncoherent detection whether or not a differential code is used). On the other hand, high-rate) PA codes are able to make use of the intrinsic differential code for noncoherent detection, and are thus a better choice. As a verification of the above EXIT analysis, Fig. 5 lots the erformance curves of noncoherently detected PA codes and LDPC codes on Rayleigh channels. LDPC codes are evaluated either with or without a differential code and their degree rofiles are the same as secified in Fig. 4. First, we see that the differentially-coded irregular LDPC code is more than.7 db worse than its BPSK-coded eer at of 0 4.

3 This confirms that conventional) LDPC codes suffer a erformance loss when used with a differential code. Second, while the erformance ga between BPSK-coded irregular LDPC codes and PA codes is accetable about db) with 4% of ilot symbols, it becomes drastically large when ilot symbols are reduced in half, since 2% of ilot symbols are insufficient for non-differentially coded LDPC codes to track the channel. The observations are in good agreement with the EXIT analysis. 3 Code Design Matched to the Receiver The above analysis leads to a more interesting roblem: what LDPC ensembles are good for differential coding and how to otimize them? Below we rooses a convergenceconstraint method that uses density evolution on EXIT charts to otimize outer) LDPC degree rofiles matched to an inner receiver. The roosed method focuses on the interaction or the convergence behavior) between the inner and outer code during the iterative rocess, and is a useful extension of the conventional method for designing LDPC ensembles with good thresholds call it threshold-constraint method) [6]. In order to design an outer code whose EXIT curve will match closely with the given inner EXIT curve, a nature and simle thinking is to samle the inner EXIT curve and design an EXIT curve that matches with these samle oints or the control oints ). Mathematically, if we choose a set of M control oints in the EXIT chart, denoted as v i, w i ), i =, 2,, M, and if we use T o ) to denote the inutoutut mutual information transfer function of the resulting LDPC code exact exression of T o will be defined later in 4)), the otimization roblem can be formulated as max λi = ρj = { ρj /j R= λi /i T ow k ) v k, k =, 2,, M }, where R denotes the code rate, λ i and ρ i denote the fraction of edges in the biartite grah that are connected to variable nodes and check nodes of degree i. Collectively, we use λx) = λ i x i and ρx) = ρ i x i to describe the degree rofiles from the edge ersective, and similarly, λ x) = λ i xi and ρ x) = ρ i xi from the node ersective [6], where λ i = λi/i, and λj/j ρ i = ρi/i ρj /j. The following functions are also useful for the discussion Ix) φx) = = e z x)2 4x 2πx log + e z )dz,) { z x) 2 4πx tanh z 2 e 4x dz, x>0,, x=0. The code design rocess is a dual constraint otimization rocess that rogressively otimizes λx) and ρx) based on the other. Below we discuss only the otimization of λx) for a given ρx). The otimization of ρx) can be derived similarly. 2) Under the assumtion that the messages assed along all edges are i.i.d. and Gaussian distributed, the average messages variable nodes receive from their neighbors are mixed Gaussian distributed. From l ) th iteration to l th local iteration in the LDPC decoder), the mean of the messages associated with the variable node, m v, evolves as m v l) = i λ i φ m 0 +i ) j ρ j φ m l ) v ) j ), where m 0 denotes the mean of the initial messages received from the channel or the inner code). Let us denote h i m 0, r) = φ m 0 + i ) ρ j φ r) j )), j the evolution of the message mean associated with variable nodes can then be described as r l = hm 0, r l ) = i λ ih i m 0, r l ). The conventional threshold-constraint density evolution forces the resulting code to converge to the zero-error state for a given m 0 by setting r > hm 0, r) for all r 0, φm 0 )] [6]. This has imlicitly use a control oint v, w) =, Im 0 )), i.e., the resulting EXIT curve will stay strictly below oint, Im 0 )). In general, a control oint v, w) can choose any value from 0 to, and the above condition is relaxed to r > hm 0, r) for all r r, φm 0 )], where r is the critical value that ensures T o w) v. Formally, the roblem is stated as: given a check node degree rofile ρx) and a control oint v, w), where 0 v, w, λ i /i, 3) max i λi= subject to: i) i λ i =, ii) i i λ i hi m 0, r) r ) <0, r r, φm 0 )], where m 0 = I w) and r satisfies T o w) = λ i I i ρ j φ r ) j )) v. 4) i j For a set of M control oints, v, w ), v 2, w 2 ),, v M, w M ), we can combine the constraints associated with each individual control oint and erform a joint otimization on all of them, which will result in an EXIT curve whose shae and osition are closely match to the control oints. Note that the above constraint ii) is a nonlinear function of λ i s, and that the comutation of r from 4) requires the knowledge of λx), which is yet to be otimized. To get around with this, one ossible aroach is to consider an aroximation of λx) in 4) to comute r. Secifically, we consider only the two lowest degree variable nodes λ i and λ i2, and aroximate the degree rofile as λx) λ i x i + λ i2 x i2. In a conventional LDPC ensemble, i = 2, i.e., degree- nodes are not allowed, since the outbound messages from

4 these nodes do not imrove in the message-assing decoding. However, when an LDPC code is used together with a differential code or other inner code and/or modulation with memory), weight- nodes in the outer LDPC decoder will get extrinsic information from the inner code as the iteration rogresses and their estimates will imrove accordingly. In this case, the first and the second nonzero λ i s are λ and λ 2. An analytical bound on λ is difficult, but emirical results show that λ R is a reasonable assumtion. This is because, otherwise there are at least two degree- variable nodes, say the th and q th node, connecting to the same check, which creates a very vulnerable link. As shown in Fig. 6, when the four bits denoted by solid circles fli altogether, another valid codeword results and the decoder is unable to detect. In other words, for any finite length construction, the minimum distance of this LDPC ensemble is at the most) 4, which is not desirable. Using the aroximation λx) = R) + Rx in 4), we are able to comute a lower bound of) r to be used in constraint ii). Code design is thus solvable using linear rogramming. Exeriments show that the otimized EXIT curve has a shae as desired, but the osition is slightly lower, i.e. code rate is slightly essimistic. This can be comensated by re-setting the control oints slightly higher than we actually want them to be. [Otimization Results:] We observe that the LDPC ensemble otimal for differential coding always contains degree- and degree-2 variable nodes. For high rate codes above 5, these nodes are dominant, or in some cases the only tyes of variable nodes; for medium rates around, there are also a good ortion of high-degree variable nodes. Hence, it is fair to say that the degree rofile of the outer code of high-rate PA codes is near-)otimal for differential coding. The otimization result of the target rate is shown in Fig. 7. The resulting LDPC ensemble has rate R = 037 and degree rofile λx) = x x x x x x x x 4 and ρx) = x 5. We see that it matches closely with the noncoherent receiver oerating at 0.25 db. Accounting for the rate of the outer code, we see that the resulting differentially-coded LDPC ensemble requires log 0 037) = db asymtotically) in order for the iterative differential detection and decoding to converge successfully. Comared to a rate 0 PA code which requires.26 0 log 0 ) = db, the otimized LDPC ensemble is about.04 db better asymtotically. Simulation results show a good agreement with the analytical result Fig. 8), and the 64K long code erforms about 5 db away from the analytical threshold at of 0 4. The exact code rate is deendent on the otimization result, but we know of the target code rate which is in the vicinity of the final code rate. 4 Conclusion The major conclusions and contributions of this aer are as follows: First, we show that the oular ractice of inserting ilots to eriodically terminate the trellis incurs an intrinsic loss in code caacity and is likely to cause severe erformance loss to the overall code erformance. A better way of inserting ilot symbols is suggested which is to searate ilots from trellis. Second, we investigate the erformance of conventional LDPC codes using noncoherent detection. We show that conventional LDPC codes suffer a erformance loss when used with an inner differential code, yet without the differential code, more ilot symbols are needed to track the channel. Hence, it is fair to say that noncoherently detected LDPC codes do not erform as desirably as the coherent case. Finally, we roose a convergenceconstraint method to design good LDPC ensembles matched with differential coding and in general any receiver). We observe that the LDPC ensemble otimal for differential coding always contains degree- and 2 variable nodes, and that for high code rates, these nodes are dominant. The resulting otimal LDPC code shows a.04 db gain over the existing PA code. It is worth mentioning that otimal differentially-coded LDPC codes are in fact otimal) irregular reeat accumulate IRA) codes [7], but the roosed otimization rocedure has a far-reaching imlication and alication since it has exlicitly taken into account the roerty and the imerfectness of the receiver. References [] J. Li, K. R. Narayanan and C. N. Georghiades, Product accumulate codes: a class of caacity-aroaching, low comlexity codes, submitted to IEEE Trans. Inform. Theory, 200. [2] J. Li, K. R. Narayanan and C. N. Georghiades, On the Performance and Analysis of Product Accumulate Codes over Rayleigh Fading Channels with Coherent and Noncoherent Detection, in reartion, [3] S. ten Brink, Convergence behavior of iteratively decoded arallel concatenated codes, IEEE Trans. on Commun., vol. 49, No. 0, , Oct [4] A. Ashikhmin, G. Kramer, and S. ten Brink, Extrinsic information transfer functions: a model and two roerties, Proc. Conf. Inform. Sciences and Systems, Princeton University, March [5] htt://lthcwww.efl.ch/research/ [6] S-Y. Chung, T. J. Richardson, and R. L. Urbanke, Analysis of sum-roduct decoding of low-density arity-check codes using a Gaussian aroximation, IEEE Trans. Inform. Theory, vol. 47, No. 2, , Feb [7] H. Jin, A. Khandekar and R. J. McEliece, Irregular reeat-accumulate codes, Proc. 2nd Intl. Sym. on Turbo Codes and Related Toics,. -5, France, 2000.

5 0 K=K, R=3/4, fdts=0.0, 0 iter A) 0 2 B) Figure : Different strategies of ilot insertion in a differential code. Effect of ilots segmeting the trellis, Es/No=4.75, db out code of PA codes R=5 Es/No=4.75 db, 0,4,0,20% ilots Es/No= db 0,4,0,20% ilots PA, 2% PA, 4% irr LDPC, 4% reg LDPC, 4% irr LDPC, 2% irr LDPC, 4%, dif dec Eb/No db) Figure 5: Noncoherently detected PA codes and LDPC codes on fast Rayleigh fading channels. checks bits checks bits outer code of PA codes R= I / I a,i e,o Figure 2: The effect of ilot symbols segmenting the trellis K,32K),fdTs=0.0,0% 0%, ideal 0%, ideal, ilots searated 0%, ideal, ilots term trellis 0%, IDDD, ilots searted q LDPC differential code Figure 6: Defect in code structure when λ > R. Code design fdts= db, 0% ilots.26 db, 0% ilots 0 4 erformance ga due to different ilot insertion strategies Eb/No db) Figure 3: Performance of PA codes with different ilot insertion strategies Rayleigh channel irr LDPC fdts=0.0, Eb/No=5.32, R=3/4 differential code PA outer) reg LDPC differential code /+D) fading channel outer code of PA code LDPC code reg) LDPC code irr) I / I a,i e,o Figure 4: EXIT curves of LDPC codes and PA codes R=037, otimized LDPC R=000, outer code of PA codes I a,i / I e,o Figure 7: EXIT chart of a rate LDPC ensemble otimized using convergence-evolution for differential coding K,32K),otimized LDPC,fdTs= % ideal 0% ideal 0% noncoherent Eb/No db) 5 iter 0 iter 5 iter Figure 8: Simulations of otimized LDPC code with differential coding and iterative differential detection and decoding.

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