Single Parity Check Turbo Product Codes for the DVB-RCT Standard

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1 Single Parity Chek Turbo Produt Codes for the DVB-RCT Standard Angelo Pinelli Martins Samia National Institute of Teleommuniations - Inatel P.O. Box Santa Rita do Sapuaí - MG Brazil angelo.pinelli@inatel.br Dayan Adionel Guimarães National Institute of Teleommuniations - Inatel P.O. Box Santa Rita do Sapuaí - MG Brazil dayan@inatel.br Abstrat This artile suggests some modifiations in onventional single-parity hek turbo produt ode strutures to fulfill the speifiations of the DVB-RCT standard proposed for the return hannel of DVB-T based Digital TV systems. The resultant odes are two and three-dimensional array odes, in whih diagonal parity euations are applied. Simulation results are presented for the AWGN and Rayleigh fading hannels and reveal a good trade-off between performane and omplexity for the suggested hannel oding shemes. Index Terms Array odes, Single Parity Chek Produt Codes, Blok Turbo Codes, DVB-RCT, DVB-T. I. INTRODUCTION Two-way ommuniations are essential to provide real-time interativity between the subsriber and the digital television ontent provider. The so-alled DVB-RCT (Return Channel-Terrestrial) standard [1] defines a wireless interfae that an be shared by a large number of terminals, enabling them to send return signals to a base station, at the same time and through the same antenna that it reeives IP data on the forward link, whih are arried using the DVB-T [2] broadast hannel. A disussion on the utilization of Single Parity Chek Turbo Produt Codes (SPC-TPC) [5] for the reuirements of DVB-RCT is in order. Conventional two and three-dimensional (2D and 3D) SPC are modified through the use of different omponent odes in eah dimension, diagonal parity euations and zero padding, in order to math the speifiations of the DVB-RCT standard, onerning blok sizes and ode rates. The key feature of the hannel-oding sheme suggested here is the low omplexity of the oding and the deoding proesses. The remainder of this paper is organized as follows. Setion II presents the hannel-oding reuirements aording to the DVB-RCT standard, i.e. blok sizes and ode rates assoiated to eah modulation sheme. In Setion III, details about the ode onstrution are presented, in order to fulfill the speifiations listed in Setion II. Setion IV is devoted to the turbo-deoding algorithm, while Setion V presents simulation results for the hannel-oding shemes suggested here, for the AWGN and the Rayleigh fading hannels. Finally, Setion VI onludes the paper, highlighting some open uestions to be addressed in future work. II. CODE REQUIREMENTS Aording to the DVB-RCT, one of the following methods of hannel-oding is used in a given RF hannel: turbo odes or onatenated Reed-Solomon plus onvolutional odes. The turbo ode adopted for the standard uses parallel onatenation of Cirular Reursive Systemati Convolutional (CRSC) odes [3] [9] and its performane is evaluated in [4] for a variety of input blok sizes No matter the hannel-oding method used in the DVB-RCT standard, the data bursts produed after the oding and modulation proess have a fixed length of 144 modulated symbols. Table 1 lists the original sizes of the useful data payloads to be enoded, given the seleted modulation and ode rate [1]. TABLE I USEFUL DATA PAYLOAD OF A BURST (DVB-RCT STANDARD) [1] Modulation Rate = 1/2 Rate = 3/4 QPSK 144 bits 6 bits 16-QAM 288 bits 432 bits 64-QAM 432 bits 648 bits In order to math the ode struture with the reuirements given by Table 1, the number of dimensions of a single parity hek (SPC) produt ode and the length of the omponent odes proposed here are varied. In addition to the onventional multidimensional parity euations, diagonal parity [5] euations are also onsidered as a means to ahieving these reuirements. The resultant struture an be viewed as a form of Array Codes [12]. III. CODE CONSTRUCTION The hannel ode an be onstruted as shown in Figure 1, for the ase where the input blok size is 144 bits. First, the parity is alulated olumn-wise using a (n,k) = (7,6) single parity hek omponent ode, whih will make the original input blok grow by 24 bits. The parity of the rows is then alulated using the same (7,6) omponent ode used in the first dimension. After that, the 49 parity bits orresponding to the enoding in the third dimension are alulated

2 using a (5,4) SPC ode. Diagonal parity is applied to the resulting (7 7 5, 6 6 4) ode by omputing 7 parity bits for eah of the 5 planes, suessively towards the depth of the struture. The final output blok has 280 bits, whih results in an overall ode rate r = Zero-padding the output with 8 symbols brings the ode rate down to ½ in this ase, as reuired. in eah dimension being eual and having an odd number of bits. An independent initiative to ompute the diagonal parity is proposed in [5] and, in this ase, it is enough that the omponent odes have the same length. Beause the reuirements in terms of ode rate and blok length are defined by DVB-RCT, the strutures suggested in this artile were built so as to optimize these parameters, following the guidelines given in [5]. When the reuired ode rate is ¾, a twodimensional produt ode is used, as illustrated in Figure 3, for a 432-bit input blok. Note that the last 9 positions of the array has to be filled with zeros so that the input bits are arranged as a perfet suare and diagonal parity an be applied [5] [8]. Two rows of 16 bits are generated based on the alulation of the diagonal parity bits to the left and to the right of the array formed by the information bits plus vertial and horizontal parity bits. 15 Fig. 1. Blok shape mathed to the speifiations of DVB-RCT for QPSK (R = ½). An example of the proedure used to ompute the diagonal parity on eah plane is shown in Figure 2. The blok formed with the information bits plus the symbols orresponding to the horizontal and vertial parities is first re-arranged to form a new 5 5 array, the olumns of whih orresponding to the left diagonals of the original blok. One way to aomplish this is to do irular shifts on the lines of the struture in Figure 2(a). The first line is not shifted; the seond line is shifted to the left by one bit and the l th line of the original blok is shifted by l 1 bits. The left diagonal parity bits are then alulated by omputing the parity bits along the olumns of the array shown in Figure 2(b) (a) pd pd pd pd pd Fig. 2. Computation of the left diagonal parity for a 2D (5,4) 2 = (5 5, 4 4) produt ode. (a) Array with information bits, horizontal and vertial parity bits; (b) Modifiation in (a) for left diagonal parity alulation. The influene of diagonal parity is analyzed with more details in [8], where the best results were obtained with the restrition that the omponent odes (b) bits left diagonal parity 16 bits right diagonal parity Fig. 3. Blok shape mathed to the speifiations of DVB-RCT for QPSK (R = ¾). The strutures designed for operation with 16-QAM (Quadrature Amplitude Modulation) and 64-QAM follow the same aforementioned priniple. For eah ase, the number of dimensions and the size of the omponent odes had to be adjusted. The odes ere designed so that the resulting ode rate is eual or exeeds the one speified by the DVB-RCT standard. It was found that 3D odes are viable only when the reuired ode rate is R = ½. When R = ¾ it is not possible to use more than two dimensions in the produt ode. As in QPSK mode, diagonal parity is applied to both 3D and 2D strutures, but while in the first ase the diagonal parity an only be applied left or right-wise due to rate restritions, the latter ase implements both left and right diagonal parity. In some onfigurations, both the input and output bloks had to be zero-padded. Figure 4 and 5 show the suggested strutures for ½ and ¾ ode rates, respetively, when using 16-QAM modulation. For the 3D ase, the resulting odeword has = 504 bits, and it has to be appended with 72 null symbols to omply with the R = ½ speifiation.

3 hannels, for the speified blok lengths and rates listed on Table 1, Setion II. Simulation of these shemes stritly in aordane with DVB-RCT is beyond the sope of this work. Similar approahes have been adopted in [4] [7] to evaluate the Cirular Reursive Convolutional Code (CRSC) performane, whih is speified by DVB-RCT as an alternative to the serial onatenation of Reed-Solomon with nonreursive onvolutional odes, as mentioned in Setion II. Fig. 4. Blok shape mathed to the speifiations of DVB-RCT for 16-QAM (R = ½). The reuired input blok for 16-QAM mode and R = ¾ has 432 bits and the 2D produt ode with diagonal parity suggested for this ase is shown in Figure 5. Here the resulting odeword has 528 bits and it is zero-padded with 43 symbols at the output to bring the ode rate down to ¾. Fig. 6. Blok shape mathed to the speifiations of DVB-RCT for 64-QAM (R = ½) bits left diagonal parity bits right diagonal parity 648 Fig. 5. Blok shape mathed to the speifiations of DVB-RCT for 16-QAM (R = ¾). The struture designed for ½ ode rate and 64- QAM modulation is depited in Figure 6. Conerning this 3D ase, it was neessary to pad the input blok with 16 zeros in order for the diagonal parity of the eight suessive planes to be properly alulated from perfet 8 8 suares. For the same 64-QAM modulation sheme, and when the reuired ode rate is ¾, a 2D struture like the one shown in Figure 7 is suggested. This time, 28 bits were appended to the input blok for the diagonal parity omputation. IV. TURBO DECODING A turbo deoding proess was used to deode the hannel-oding shemes desribed in Setion III. For simpliity, suh shemes were evaluated using BPSK (Binary Phase Shift Keying) modulation over AWGN (Additive White Gaussian Noise) and Rayleigh fading bits left diagonal parity 27 bits right diagonal parity Fig. 7. Blok shape mathed to the speifiations of DVB-RCT for 64-QAM (R = ¾). A symbol-by-symbol MAP algorithm was used for turbo-deoding of the SPC produt ode, as in [13, pp ], in whih the extrinsi information is fatored out and passed between deoders in eah dimension, referred to as a SISO (soft-input, soft-output) deoding proess. A simple extension of this algorithm is done here for the ase where diagonal parity is used. In this ase, the diagonal enoding an be onsidered as a form of a new dimension of the ode. The deoding steps are as follows: (i) Calulate the hannel log-likelihood ratios (LLR) for all reeived symbols and set the extrinsi information to zero for all bits in the ode;

4 (ii) Calulate the extrinsi information for all bits and eah dimension. The a priori information for eah bit in a given dimension is the sum of the extrinsi information from the other dimensions in the ode; (iii) The deoding of all dimensions one time is alled iteration. Repeat this proess for as long as reuired. The a posteriori Log-Likelihood Ratio (LLR) of the symbol at the output of the detetor (soft-input of the SISO deoder) an be expressed as [6]: ( = +, ) ( =, ) P x E y g L( xˆ ) = L( x y) = ln P x E y g (1) where g aounts for the fading and is alled hannel gain. Using the Bayes Theorem, from Euation (1) it is obtained: (, ) (, ) ( ) ( ) P y x = + E g P x = + E L( xˆ ) = ln + ln P y x = E g P x = E (2) where the first term on the right of Euation (2), whih will heneforth be designated as L(y), orresponds to the hannel measurement or hannel state information at the detetor output, given the alternate onditions that a 1 or a 0 may have been transmitted. The seond term on the right of Euation (2) will be alled L(x) and is the a priori value of the input symbol, whose value is not known until the first deoding iteration is ompleted. That is why, in general, the a priori values of the input symbols are initially set to zero. Substituting the numerator and the denominator of L( y ) by the euations that desribe the Gaussian distribution, onsidering the expeted values in eah ase, aording to (1), yields: where 2 1 y g E exp 2 σ E L( y) = ln = 4g y (3) 2 N0 1 y g E + exp 2 σ 2 σ = N 0 2 is the average noise power at the mathed filter output and g = 1 if the hannel is purely AWGN. From the LLR at the output of the detetor expressed in Euation (2), it an be shown that, for systemati odes, the log-likelihood ratio at the output of the deoder (soft output), at a given dimension, an be written as [10]: where L ( ) ( ˆ ) = ( ) + ( ) + ( ˆ ) L x L y L x Le x (4) x is the vetor or array that bear the a priori information that feeds the input of the deoder at the th dimension, and Le ( ˆ ) x is the extrinsi information, or additional information due to parity, that is obtained after the deoding proess at the th dimension. The extrinsi information represents the amount of information added to the soft value at the input of the deoder to form the soft output. The extrinsi information for the i th bit (i = 1, 2,, n ) of a reeived odeword is given by [5] [13]: Le ( x ) = ( ) ( ) + ( ) n n L x j L y j ˆi 1 2atanh tanh (5) j=1 2 j i where n is the number of oded bits in the th dimension, L( y j ) is the hannel state information (CSI) at bit position j and L ( x ) is the input LLR of bit j at the th dimension. Euation (5) differs from the one shown in [13] by the term ( ) j n 1, whih is neessary when the logial element 0 is represented by symbol E, as it is the ase of this artile and of the example given in [14, p ]. When the bit 0 is represented by the symbol + E, the term ( ) n 1 is not used [11] [13]. The vetor that holds the a priori values at the input of the deoder, aording to step (ii) of the algorithm desribed above, an be expressed as: D ( ) ( ˆ = i ) L x Le x i= 1 i (6) where D is the number of dimensions of the ode. Note that diagonal parity vetors or arrays are onsidered as additional dimensions in the shemes proposed here. The total LLR at the output of deoder is then alulated through: where LT T D ( ˆ ) = ( ) ( ˆ + i ) L x L y Le x i= 1 (7) ( x ˆ) is a vetor with real numbers whose sign of eah element yields a hard deision about the orresponding transmitted symbol and whose magnitude reveals the reliability of that deision. Pratial aspets reuire the total values of the extrinsi information to be limited to avoid numerial problems during implementation. Aording to what has been suggested in [13], the results of the alulations to find the extrinsi information, at eah dimension, were limited within the range 100 L xˆ An estimation on the T ( )

5 transmitted odeword, ˆx, an be arried out by using the following deision riteria: ( xˆ ) ( xˆ ) 1 LT 0 xˆ = 0 LT < 0 (8) V. SIMULATION RESULTS The performane as a funtion of the number of iterations is shown in Figure 8 for the (10 9 8, 8 8 7) ode depited in Figure 6. Eah deoding yle improves the performane, with dereasing returns, but after a number of iterations euivalent to the number of dimensions of the ode, this improvement, as expeted [13], beomes only marginal. (a) 54-byte input, E b/n 0 = 4 db, 1 iteration. () 54-byte input, E b/n 0 = 4 db, 3 iterations. (b) 54-byte input, E b/n 0 = 4 db, 2 iterations. (d) 54-byte input, E b/n 0 = 4 db, 4 iterations. (e) 54-byte input, E b/n 0 = 4 db, 5 iterations. Fig. 9. Normalized histograms resulting from the measurements of LLR soft values at the output of the turbo deoder for the ode (10 9 8, 8 8 E b/n 0 = 4 db. Fig. 8. BER as a funtion of the number of iterations for the 54- bytes input blok 3D produt odes with diagonal parity. The distribution of LT ( x ˆ), the total LLR out of the turbo deoder after one to five iterations is shown in Figures 9(a) to 9(e), for eually likely input symbols and for E b /N 0 = 4 db. The enter of the horizontal axis of the histograms in Figure 9 is the deision threshold, on the right of whih the deision is favorable to bit 1. Otherwise, bit 0 will be hosen. The soft values of the LLR have greater magnitude the more distant they are from the threshold. It an be seen the similarity between the amplitude distribution after four and five iterations, whih is in agreement with the onvergene observed in Figure 8. This is the reason why the simulation results presented here for BER estimation were obtained using a fixed number of five iterations. Figure 9 also reveals that the soft values of the LLR are onentrated at regular intervals. This behavior is similar to what happens for all odes mentioned in this work, and represents an opportunity for future investigation. A. Performane in AWGN hannel The performane results from the simulation of the 3D produt odes with diagonal parity over AWGN hannel are shown in Figure 10. The number of bytes in the legend of Figure 10 orresponds to the input blok size. Coding gain is approximately 4.8 db for the 54-byte input blok size ode and its distane from apaity ( 0,2 db for R = ½ and BPSK modulation [6]) is around 4.6 db, both results onsidering a BER = This is a uite good result, given the low omplexity of the oding and the deoding proesses and the relatively short odeword length. When the reuired ode rate is ¾, 2D produt odes with diagonal parity are used. The performane for the orresponding odes is shown in Figure 11. If ompared to the 3D ase, the 2D ode that operates with 54 bytes at the input reuires an additional 0.6 db of normalized signal-to-noise ratio to yield the same bit error rate of Still referring to Figures 10 and 11, it is worthy noting that the shorter odes outperforms the longer ones for low signal to noise ratios, a fat that an be attributed to the longer seuenes of zero-padding used in the longer odes. However, the asymptoti gain of the longer odes seems to make the

6 orresponding urves steeper, probably leading to lower BER for higher values of signal-to-noise ratios, when ompared to the shorter odes. The performane of the 3D produt odes without diagonal parity, after 5 deoding iterations, and over the Rayleigh hannel is illustrated in Figure 12. The urves show the simulation results when the hannel state information (CSI) is not available to the deoder. In this ase, the bit error rate seems to be affeted by an error floor after a ertain value of E b /N 0 and this behavior was also observed for odes that implement diagonal parity. The ode that operates with 18 bytes at its input needs a signal-to-noise ratio of about 11 db to yield BER = For the ase when the reeiver is implemented with some sort of mehanism that allows the CSI to be estimated and fed to the deoder input, the performane of the 3D produt odes with diagonal parity is like the one shown in Figure 13. In this ase, there seems to be no error floor, at least for the signalto-noise ratios under onsideration, and the (7 7 5, 6 6 4) ode now reuires only 10 db to operate with BER = This improvement is due to both the diagonal parity and the availability of the CSI. Beause of the error floor, a larger differene on the reuired E b /N 0 should be expeted if the analysis is arried out at a lower BER. Fig. 10. Simulation results for rate ½ 3D produt odes with diagonal parity, after 5 deoding iterations: AWGN hannel. Fig. 12. Simulation results for rate ½ 3D produt odes with diagonal parity, after 5 deoding iterations: Rayleigh hannel, no CSI available for the deoder. Fig. 11. Simulation results for rate ¾ 2D produt odes with diagonal parity, after 5 deoding iterations: AWGN hannel. B. Performane over the Rayleigh hannel Fig. 13. Simulation results for rate ½ 3D produt odes with diagonal parity, after 5 deoding iterations: Rayleigh hannel, with CSI available for the deoder. Aording to Figure 13, oding gains greater than 14 db for the 18-byte input blok size ode an be ahieved (and greater than 11 db for all blok sizes) for bit error rates less than A distane of approximately 6.5 db from apaity ( 1.5 db for R = ½ and BPSK modulation [6]) is obtained for the 18- byte input blok size ode, for a BER eual to The simulation results presented herein for the fading hannel were obtained under the ideal ondition in whih a suffiiently large interleaver is employed, in a manner that the fading that takes plae in a given symbol is unorrelated with the fading that affets the other symbols in the same odeword. Therefore, the

7 simulated fading was generated as independent and identially distributed random variables for eah symbol transmitted through the hannel. Figure 14 shows the performane results for the ase where the reuired ode rate is ¾ and 2D produt odes with diagonal parity are used. An E b /N 0 around 10.5 db is needed for the 54-byte input blok size produt ode to operate with a 10-3 bit error rate. Fig. 14. Simulation results for rate ¾ 2D produt odes with diagonal parity, after 5 deoding iterations: Rayleigh hannel, with CSI available for the deoder. VI. CONCLUDING REMARKS It was found through simulation that, for low signalto-noise ratios, the performane of the produt odes is better the shorter is the blok length. When E b /N 0 inreases, odes that operate with larger bloks tend to perform better. Therefore, it is expeted that there is a rossing point in the urve, from whih longer odes will have better performane. The behavior for low E b /N 0 seems to be aggravated when zeros are inserted at the output. To make matters worse, the suggested strutures that have more null symbols at the output are exatly those with the larger blok length. The inonvenienes brought by zeropadding were partially overome by replaing the null symbols with new parity euations, omputed towards the left and right diagonal of the 2D produt ode and only towards the left diagonal of the several planes that ompose the 3D produt odes. One of the reasons why right diagonal parity has not been applied to the 3D produt odes is beause one of the strutures (with 18 bytes at the input) does not support the additional parity bits while keeping the speified ode rate at ½. A omparison between BER performanes between the produt odes presented in this artile and the turbo ode defined in DVB-RCT standard indiates that the latter has better performane. Although the deoding omplexity of the produt odes is learly low, more investigation is needed for this omplexity/performane trade-off to be exatly uantified. Among the future work that would explore further what was overed so far ould be to investigate the performane of these odes using higher order modulation shemes, speifially 16-QAM and 64- QAM. It would also be worth investigating why the LLR s have suh a onentration of amplitudes at regular intervals, aording to what was seen in Figure 9. Another suggestion for future work is to onfirm and investigate the reasons for the error floor, observed in a Rayleigh hannel when no CSI is available for the deoder. Finally, an investigation using bounds on bit error probabilities would be useful to validate (or not) the onjeture stating that the longer odes onsidered here will outperform the short odes for high values of the signal-to-noise ratio. REFERENCES [1] ETSI EN V1.1.1, Digital Video Broadasting (DVB); Interation hannel for Digital Terrestrial Television (RCT) inorporating Multiple Aess OFDM, European Standard (Teleommuniations series), Deember, [2] ETSI EN V1.4.1, Digital Video Broadasting (DVB); Framing struture, hannel oding and modulation for digital terrestrial television, European Standard (Teleommuniations series), January [3] Berrou, C., Douillard, C. and Jézéuel, M. Multiple parallel onatenation of irular reursive onvolutional (CRSC) odes, Annals of Teleommuniations, Vol. 54, Nº3-4, pp , Marh-April [4] Berrou, C., Jézéuel, M., Douillard, C., Brengarth, N., Toush, J. and Pham, N., The Turbo Code Standard for DVB- RCS. 2 nd Int l Symposium on Turbo Codes & Related Topis: September pp [5] Guimarães, D. A., Turbo Deoding of Single-Parity Chek Produt Codes, Revista Teleomuniações - Inatel, June 2002 (original title in Portuguese: Deodifiação Turbo de Código Produto de Paridade Simples). [6] Guimarães, D. 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