High-Throughput Low-Complexity Successive- Cancellation Polar Decoder Architecture using One s Complement Scheme

Transcription

1 JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE, VOL.5, NO.3, JUNE, 5 ISSN(Print) ISSN(Online) High-Throghpt Low-Complexity Sccessive- Cancellation Polar Decoder Architectre sing One s Complement Scheme Cheolho Kim, Haram Yn, Sabooh Ajaz, and Hanho Lee Abstract This paper presents a high-throghpt lowcomplexity decoder architectre and design techniqe to implement sccessive-cancellation (SC) polar decoding. A novel merged processing element with a one s complement scheme, a main frame with optimal internal word length, and optimized feedback part architectre are proposed. Generally, a polar decoder ses a two s complement scheme in merged processing elements, in which a conversion between two s complement and sign-magnitde reqires an adder. However, the novel merged processing elements do not reqire an adder. Moreover, in order to redce hardware complexity, optimized main frame and feedback part approaches are also presented. A (4, 5) SC polar decoder was designed and implemented sing 4-nm CMOS standard cell technology. Synthesis reslts show that the proposed SC polar decoder can lead to a 3% redction in hardware complexity and a higher clock speed compared to conventional decoders. Index Terms Polar code, sccessive-cancellation, decoder, one's complement, high-throghpt, lowcomplexity I. INTRODUCTION Polar codes, proposed by Arikan in 9 [], have Manscript received Oct. 5, 4; accepted May. 8, 5 Dept. of Information and Commnication Engr. Inha University, Incheon, 4-75, Korea hhlee@inha.ac.kr attracted a lot of attention becase of their excellent capacity-achieving property over a binary-inpt discrete memoryless channel (B-DMC). De to their explicit strctre and low-complexity encoding/decoding scheme, polar codes have emerged as one of the most important in coding theory. To date, mch of the work has addressed several theoretical aspects of polar codes and is aimed at improving the error correction performance of polar codes of moderate lengths [-7]. However, few pblications have reported implementation of polar decoders. Pamk [8] reported an FPGA implementation of a polar decoder based on the belief-propagation (BP) algorithm. Althogh a BP decoder has particlar advantages in parallel design, de to the reqirement for a large nmber of processing elements (PEs), the BP decoder is not attractive for practical applications. Several researchers have viewed the sccessivecancellation (SC) algorithm as a good candidate for hardware design of polar decoders de to its low complexity [-3]. The semi-parallel SC decoder from Lerox et al. [] has a very low processing complexity, while memory complexity remains similar to previos architectres, also from Lerox et al. [9]. However, de to the inherent serial natre of the SC algorithm, these SC decoders have significant disadvantages with respect to both long latency and low throghpt. Since SC decoding has low intrinsic parallelism, look-ahead techniqes [] were proposed to redce decoding latency and increase throghpt of SC decoders, while sing limited extra hardware resorces. Yan and Parhi [3] presented a b-sc-precomptation decoder that

2 48 CHEOLHO KIM et al : HIGH-THROUGHPUT LOW-COMPLEXITY SUCCESSIVE-CANCELLATION POLAR DECODER redces decoding latency withot performance loss. However, these low-latency architectres do not show detailed implementation reslts and bit error rate (BER) performance. The list SC decoder from A. Balatsokas- Stimming et al. [4] shows higher decoding performance than SC decoder [], whereas the list SC decoders has significant disadvantages with respect to both high hardware complexity and low throghpt. In this paper, we propose a high-throghpt lowcomplexity architectre for SC polar decoding. A novel merged processing element (M-PE) with a one s complement scheme, a main frame with optimal internal word length, and an optimized feedback part architectre are proposed. Generally, the polar decoder ses a two s complement scheme in the M-PE, in which a conversion between two s complement and sign-magnitde reqires an adder []. However, or novel M-PEs do not reqire an adder. Moreover, in order to redce hardware complexity, optimized feedback part approaches are also presented. The rest of this paper is organized as follows. Section II describes design isses related to SC polar decoding and provides analysis of a fixed-point BER simlation. Section III presents the proposed SC polar decoder architectre and the novel design techniqes. In Section IV, the reslts and a comparison are presented. Finally, a conclsion is provided in Section V. II. DESIGN ISSUES RELATED TO SC POLAR DECODING. Constrction of Polar Codes By exploiting channel polarization, polar codes approach the symmetric capacity of a channel as the code length, N, increases. Channel polarization creates N independent channels, W, where, as N, the probability of error-free transmission approaches either or.5 []. In other words, as N, each bit's probability of being sccessflly estimated approaches (perfectly reliable) or.5 (completely seless). The ratio of reliable bits approaches the capacity of the channel. Let N = n (n > ), = (,,... N- ) and c = (c, c,... c N- ), where is the inpt bit and c is the corresponding codeword. A codeword of length N can be represented sing a generator matrix G[]. = s, = s, = s, 3 = s,3 4 = s,4 5 = s,5 6 = s,6 7 = s,7 s, s, s, s,3 s,4 s,5 s,6 s,7 c = G () Here G = F Äm, where Äm F denotes the m-th é ù Kronecker power of F =. For example, for n = 3, ë û F 3 = é ù ë û The eqivalent graph representation of illstrated in Fig., where = information-bit vector and x = () Ä3 F is 7 represents the 7 x is the codeword sent over the channel []. We se the same notation for vectors as that of [], namely s 3, s 3, s 3, s 3,3 s 3,4 s 3,5 s 3,6 b a consists of bits a... b of the vector. An (N, k) polar code ses k bits among the N most reliable bits to transmit the information s k bits and force the remaining N - k bits, called frozen bits, to zero. The location of the information and frozen bits is determined sing the method described by Tal and Vardy [5].. Conventional SC Decoding Algorithm s 3,7 Fig.. Polar codes encoder with N = 8 []. The SC decoding algorithm [] works by sccessively estimating the bits ) i, i =,... N-, sing the channel otpt y and the previosly estimated bits ) ) to i -, as shown in Fig.. As demonstrated by Lerox et al. [9], 4, x 4, x 4, x 4,3 x3 4,4 x4 4,5 x5 4,6 x6 4,7 x7 y y y y 3 y 4 y 5 y 6 y 7

3 JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE, VOL.5, NO.3, JUNE, 5 49 () L y () L y ˆ + ˆ ˆ ˆ () L y () L y3 () L y4 () L y5 () L y6 () L y7 a b ˆ3 ˆ3 û 3 f ( a, b) it can be carried ot withot the se of mltiplication or division by expressing probabilities as log-likelihood ratios (LLRs), denoted L(i, j) from the node at row i and stage j, and by resorting to min sm approximation. With this new notation, LLR vales for the received bit L(i, j) can be denoted as L(i, ). Hence, the decision rle for becomes ) ì, if L( i,) > i = í î, otherwise It is noted that the LLRs with f and g fnctions can be generated by recrsively applying Eqs. (4) and (5), respectively. (3) f(a, b) = sign(a)sign(b)min( a, b ) (4) g(a, b, ) ) = ( ) ) - s a + b (5) s Fnction f can be compted as soon as a and b are available. On the other hand, the comptation of g reqires knowledge of a b ˆ 4 ˆ5 û s g a, b, ˆ ) ( s ) s. Notice that L( i, j) Fig.. Decoding procedre of polar codes with N = 8-bit. û û 5 ) s is the modle- sm of partial previosly decoded bits, which can be compted by sing the constrction of a polar code. The need for partial sm comptations cases strong data dependencies in the SC decoding algorithm. This constrains, the dependencies in which the decoded bit, can be compted in the Fig.. For instance, in Fig., s 3, 3 is estimated by propagating ) in the polar code û û 4 û û 6 û ûi û 4 û û 6 û û 5 û 3 û 7 CC Stage 3 Stage f f f f f f f g f f f g Stage f g f g f g f g Ot ) ) ) ) 3 ) 4 ) 5 ) 6 ) 7 Fig. 3. Schedling for the SC decoding algorithm with N = 8- bit. encoder: ) Å ) Å ) Å ) 3. This partial sm of ) 3 then sed to compte L(, 3). Fig. 3 shows the schedling of the decoding algorithm. At each clock cycle (CC), LLRs are evalated by compting fnction f fnction or g fnction. It is assmed here that those fnctions are calclated as soon as the reqired data is yˆ N - available. Once the channel information is available on the left hand side of the decoder, the bits ) i are sccessively estimated by pdating the appropriate nodes of the graph, from left to right. When the bits ) i are estimated, all partial sms involving ) i are pdated, allowing ftre evalations of g fnction to be carried ot. Therefore, the decoded bits can only be compted in a sccessive manner. 3. Analysis of Internal Word Length In this section, the effects of internal word length parameters in terms of error-correction performance of SC polar codes are described. The optimal internal word length for a SC decoder can be decided. Fig. 4 shows the BER performance comparison of varios SC polar codes in which the code length from 9 to sing q = 5 qantization bits in LLR and internal word length w bits are sed. As show in Fig. 4, BER performance shows that the fixed-point operations with an optimm internal word length have decoding performance close to a floating-point operation. That is, the optimm internal word length W for (4, 5) size and (48, 4) size was decided to 4 bits and 5 bits, respectively, since the BER degradation with respect to the floating point and to the hardware complexity is very small. However, w = 3 bits for (4, 5) and w = 4 g g g g is

4 43 CHEOLHO KIM et al : HIGH-THROUGHPUT LOW-COMPLEXITY SUCCESSIVE-CANCELLATION POLAR DECODER - Bit Error Rate (5, 56) w = (5, 56) w = 3 (5, 56) w = 4 (5, 56) floating-point (4, 5) w = 3 (4, 5) w = 4 (4, 5) w = 5 (4, 5) floating-point (48, 4) w = 4 (48, 4) w = 5 (48, 4) w = 6 (48, 4) floating-point Eb/No (db) Fig. 4. BER performance of varios SC polar codes. ˆ + + ˆ ˆ ˆ û 4 3 ˆ4 ˆ4 ˆ ˆ + + ˆ ˆ ˆ ˆ4 ˆ4 û 4 ˆ ˆ û û6 ˆ i- ˆ i ˆ 5 ˆ6 û û 6 Fig. 6. Proposed N = 8-bit SC polar decoder architectre with optimal internal word length. ˆ i- ˆ i. Main Frame Architectre with Optimm Internal Word Length Fig. 5. Proposed SC polar decoder architectre. bits for (48, 4) show the error floor de to overflow. In order to avoid the overflow in the decoding operation, M-PEs always calclate Eqs. (4) and (5) with an optimm internal word length of W bits. III. PROPOSED SC POLAR DECODER ARCHITECTURE As shown in Fig. 6, the proposed SC polar decoder architectre consists of two major nits: main frame and feedback part. The main frame consists of M-PEs, a decision nit and frozen bit memory. The feedback part, which is defined with a power of recrsively, comptes the partial sms reqired by the M-PEs to calclate Eq. (5). In order to redce hardware complexity and avoid overflow, the optimm internal word length can be fond to design the efficient main frame architectre. The main frame architectre employs tree-based architectre. In order to redce hardware complexity of main-frame architectre, the internal word length of M-PE in each stage can be determined by exploiting the fact that the optimm internal word length in sbseqent M-PE is increased by one bit per stage.. Merged Processing Element with an One's Complement Scheme In this section, we discss the detailed hardware architectres of the proposed M-PE architectre sing one's complement scheme, which is the main arithmetic component of the decoder. It contains the arithmetic logic that carries ot the likelihood estimations sing Eqs. (4) and (5). The hardware complexity of the M-PE is significantly higher than that of other blocks, which reslts in very high hardware complexity for the SC decoder. Ths, minimizing the hardware complexity of the M-PE is critical to redcing the hardware complexity

5 JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE, VOL.5, NO.3, JUNE, 5 43 ˆ = s ˆ = s (a) Fig. 7. (a) Conventional strctre of the merged processing element (M-PE) [], (b) Conventional strctre of the Type I PE []. of the SC decoder. The conventional M-PE [] consists of the TYPE PE, two's complement to sign-magnitde (TtoS) and sign-magnitde to two's complement (StoT), as depicted in Fig. 7. The comptation of Eq. (4) reqires an XOR gate and a mltiplexer (MUX), while B n is already available from the TYPE PE. The TYPE PE is in charge of calclating the two possible g otpts Eq. (5) in parallel and consists of an adder-sbtractor, as shown in Fig. 7(b). For the fll adder, the sm and carry-ot bit are represented by S and C ot. The difference and borrowot prodced by the fll sbtractor are denoted by D and B ot. The TtoS block performs the conversion from two s complement to sign-magnitde representation. The StoT block performs the reverse conversion. The conventional architectre of the TtoS block [] is illstrated in Fig. 8(a). In this work, we propose one's complement to signmagnitde (OtoS) and sign-magnitde to one's complement (StoO) blocks for the merged processing element that employs the one's complement scheme, as shown in Fig. 9. The StoO block is similar to the OtoS block in Fig. 8(b). We see that the proposed OtoS block is mch simpler than the conventional TtoS block. This leads to two benefits. First, since all the half adders are (b) Fig. 8. (a) Conventional strctre of TtoS block [], (b) Proposed strctre of OtoS block. ˆ = s ˆ s = Fig. 9. Proposed strctre of the merged processing element (M-PE). removed, the hardware complexity of the M-PE is redced compared to the conventional TtoS. Second, the critical path of TtoS is only T inv + T MUX, which is mch shorter than that of the conventional TtoS. The conventional two s complement M-PE architectre of Zhang et al. [] calclates Eqs. (4) and (5) by converting two s complement to sign-magnitde representation. On the other hand, the proposed one s

6 43 CHEOLHO KIM et al : HIGH-THROUGHPUT LOW-COMPLEXITY SUCCESSIVE-CANCELLATION POLAR DECODER Table. Calclation of transformed f fnction and g fnction based on one's complement format Case tf(a, b) Case tg(a, b, =) tg(a, b, =) a, b a, b sign(a) sign(b) f a b a <, b > sign(a) sign(b) a > b a <, b < f - a >, b < a <, b < g g - g + g complement M-PE architectre is not reqired to add the carry in the conversion from one s complement to signmagnitde representation. In this section, we describe how inpt LLR vales in the sign-magnitde conversion operation can be calclated, leading s to pdate Eqs. (4, 5) sing the proposed OtoS. Here, Eqs. (4, 5) sing two s complement are denoted as f = f(a, b), g = g(a, b, ) ) and g = g(a, b, =), where a and b are LLR vales and ) s is the partial sm of the decoded bit. Table shows the transformed eqations from Eqs. (4, 5) for the one s complement scheme, which are sed for BER simlation for the proposed one s complement scheme. These transforms depend on the sign of a and b. Therefore, there are for possible examples, as follows: ) a is positive and b is positive in tf(a, b), (a, b ) When both a and b are positive, their sign-magnitde forms are the same as the two s complement forms. As a reslt, the proposed OtoS and StoO in the M-PE will not be sed in this case. ) a is negative and b is negative in tf(a, b), (a <, b < ) When both a and b are negative, tf > ; ths, the magnitde of tf is jst the absolte minimm vale of a and b, whereas the sign of tf is always positive. Since a and b are negative, the sign-magnitde needs to sbtract in the proposed OtoS. However, since the reslt of tf is always positive, StoO is not sed in this case. As a reslt, the transformed f fnction is defined as follows: tf(a, b) = f(a, b) - when a <, b < (6) 3) a is positive and b is negative in tg(a, b, ) ) (a >, b < ) When a is positive and b is negative, a has the same vale of the sign-magnitde and two s complement form. Ths, only the conversion of b is reqired to perform the transformed g fnction. In this case, since b is negative for the one s complement, its magnitde vale is calclated by sbtracting ; that is, b -. As a reslt, the transformed g fnction can be derived as follows: a = a, b = - ( b - ) (7) tg = a + b = a - b + (8) = g(a, b, ) ) + when a, b < 4) a is negative and b is positive in tg(a, b, ) ) (a <, b > ) When a is negative and b is positive, only the conversion of a is reqired to perform the transformed g fnction, as follows: a = - ( a - ), b = b (9) tg = - a + b = a - + b () = g(a, b, = ) - when a <, b > Smmarizing the above for cases, we conclde that instead of sing two s complement TtoS and StoT, the sign-magnitde based addition or sbtraction can still be carried ot with slight modification of Eqs. (4, 5), which is one s complement OtoS and StoO. 3. Feedback Part Every comptation of Eq. (5) reqires a specific inpt corresponding to the sm of a sbset of the previosly estimated bits. Zhang and Parhi [] proposed recrsive constrction of a partial sm nit called the feedback part. In general, for an N-bit length decoder, since the architectre of the feedback part is defined recrsively for powers of, the general parallel pipelined architectre can be constrcted with the recrsive relationship in Fig. (a). Here, modle U n can be constrcted based on modle U n- and (n - ) extra XORpass elements. However, we note that for U n, the nmber of corresponding registers significantly increases with a complexity of n and the feedback part has very high hardware complexity, which is impractical for polar codes with length over. One possible approach to redce the hardware complexity is to employ a RAM instead of flip-flops. However, storing the partial sms in

7 JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE, VOL.5, NO.3, JUNE, ˆ i- - Bit Error Rate -3 (a) (b) ˆ i ˆ i- Fig.. (a) Conventional strctre of feedback part [9], (b) Optimized strctre of the feedback part. ˆ i -4-5 's comp. 's comp. 's comp. with opt Eb/No (db) Fig.. BER performance for a (4, 5) SC polar decoder sing a two s complement scheme, a one s complement scheme, and a one s complement scheme with an optimal internal word length. control signal c n for the feedback part. The control signal c n is generated by m n, which is the MUX control signal for the n-stage in the main frame architectre. The initial state of FSM is S and if FSM arrives into inpt of m n, otpt c n of FSM is ; c n = means the partial sm vales are stored. If k = and S k assmes m n =, otpt c n of FSM is ntil the next m n is set to ; c n = means the partial sm vales are transferred. Until state S N/4, there is no change in c n. If k = {N/, N/+} and S k assmes m n =, S and S states are repeated. IV. RESULTS AND COMPARISON. BER Performance Fig.. Finite state machine (FSM) to generate the control signal cn for the feedback part. a RAM wold lead to scattered memory accesses reqiring mltiple clock cycles. To avoid lowering the throghpt of the decoder, we instead store them in D-FF. In order to redce the complexity of the feedback part, the D-FF and MUX with feedback are sed instead of a shift-register, as shown in Fig. (b). Since data lifetime is n- - clock cycles, where n = log N, the additional control signal c n can be determined accordingly. Fig. shows the finite state machine (FSM) to generate the Fig. shows the BER performance comparison for a (4, 5) SC polar decoder sing a two s complement scheme, a one s complement scheme with the same internal word length, and the proposed one s complement scheme with an optimm internal word length. The qantization bits q denotes the nmber of bit sed for the qantization of the channel LLRs and represents the maximm channel symbol magnitde. The qantization bit of channel LLR q = 5 bits was chosen for BER performance comparison, becase the BER performance of fixed-point operation with q = 5 bits is sfficient to approach the BER performance of floatingpoint operation []. Therefore, the qantized LLR inpts of polar decoder were limited to the dynamic range [-6, 5]. The reslt shows that there is almost

8 434 CHEOLHO KIM et al : HIGH-THROUGHPUT LOW-COMPLEXITY SUCCESSIVE-CANCELLATION POLAR DECODER Table. Synthesis reslts of different (4, 5) SC polar decoders Design [] [3] Proposed Architectre Line-based.5 db degradation of -5 BER between the proposed one s complement and the two s complement scheme, as shown in Fig., becase the one s complement scheme offers one less qantization level than the two s complement. However, we can notice that the proposed one s complement scheme with an optimm internal word length provides almost similar BER performance to the one s complement scheme with the same internal word length in each stage.. Implementation Reslts Treebased Treebased Scheme Two s Two s One s CMOS Tech. 65 nm 45 nm 4 nm Total gate cont (XOR) 85,748 5,9 3,474 Critical path - T comp+t AND+ T OR+T XOR 4T MUX+ 6T XOR Clock speed (MHz) 5 75, Decoding latency (Cycles),8 (4,6 ns) 767 (,5 ns),3 (,3 ns) Throghpt (Mbps) 46,, Gate cont is calclated based on the area information in [] and [3], nit gate area in TSMC 65 nm CMOS library and FreePDK 45 nm library. The proposed polar decoder was modeled in the Verilog HDL and simlated to verify its fnctionality. After complete verification of the design fnctionality, it was then synthesized sing appropriate time and area constraints. Both synthesis and layot steps were carried ot sing the SYNOPSYS design tool and 4-nm CMOS technology. The estimated total nmber of XOR gates is 3,474 from the synthesized reslts, and the clock speed is GHz for the proposed polar decoder. In addition, the proposed decoder also occpies only.4 mm of core area sing TSMC 4-nm CMOS technology. Table lists the implementation reslts of the reported polar (4, 5) SC decoders. The estimated total nmber of XOR gates is 3,474 from the synthesized reslts. By applying the one s complement scheme, the critical path is redced and the clock speed improves compared to the two s complement scheme. The critical path is 4T MUX + 6T XOR and the clock speed is GHz for the proposed polar decoder. In addition, the proposed decoder occpies only.4 mm of core area sing TSMC 4-nm CMOS technology. From Table, it can be seen that the proposed SC polar decoder can achieve a 3% redction in gate cont as well as low latency and Gbps throghpt. V. CONCLUSIONS In this paper, a novel merged processing element with a one s complement scheme, a main frame with an optimm internal word length, and an optimized feedback part for an SC polar decoder are proposed. Based on these techniqes, a high-throghpt lowcomplexity SC polar decoder architectre was presented. The estimated total nmber of XOR gates is 3,474 from the synthesized reslts, and the throghpt is Gbps for the proposed polar decoder. In addition, the proposed decoder reqires.4 mm of core area sing 4-nm CMOS technology. Analysis shows that the proposed architectre has significant advantages with respect to both throghpt and hardware complexity. ACKNOWLEDGMENTS This research was spported by Basic Science Research Program throgh the NRF fnded by the Ministry of Science, ICT and ftre Planning (3RAAA6868) and the MSIP, Korea, nder the ITRC spport program (NIPA-4-H3-4-4) spervised by the NIPA. REFERENCES [] E. Arikan, Channel polarization: A method for constrcting capacity-achieving codes for symmetric binary-inpt memoryless channels, IEEE Trans. Inf. Theory, vol. 55, no. 7, pp , Jl. 9. [] S. B. Korada, E. Sasogl, and R. Urbanke, Polar codes: Characterization of exponent, bonds, and constrctions, IEEE Trans. Inf. Theory, vol. 56, no., pp , Dec.. [3] A. Alamdar-Yazdi and F. R. Kschischang, A simplified sccessive-cancellation decoder for polar codes, IEEE Commn. Lett., vol. 5, no., pp , Dec.. [4] R. Mori and T. Tanaka, Performance of polar codes with the constrction sing density evoltion, IEEE

9 JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE, VOL.5, NO.3, JUNE, Commn. Lett., vol. 3, no. 7, pp. 59 5, Jl. 9. [5] I. Tal and A. Vardy, List decoding of polar codes, in Proc. IEEE Int. Symp. Inform. Theory (ISIT), pp. 5, May.. [6] K. Ni and K. Chen, Stack decoding of polar codes, Elect. Lett., vol. 48, no., pp , Jn.. [7] I. Tal and A. Vardy, How to constrct polar codes," IEEE Trans. Inf. Theory, vol. 59, pp , Oct. 3. [8] A. Pamk, An FPGA implementation architectre for decoding of polar codes, in Proc. 8th Int. Symp. on Wireless Commn. Syst.(ICWCS), pp , Nov.. [9] C. Lerox, I. Tal, A. Vardy, and W. J. Gross, Hardware architectres for sccessive cancellation decoding of polar codes, in Proc. IEEE ICASSP, pp , May.. [] C. Lerox, A. J. Raymond, G. Sarkis, and W. J. Gross, A semi-parallel sccessive-cancellation decoder for polar codes, IEEE Trans. Signal Processing, vol. 6, no., pp , Jan. 3. [] C. Zhang, B. Yan, and K. K. Parhi, Redcedlatency SC polar decoder architectres, in Proc. Int. Conf. Commn., pp , Jn.. [] C. Zhang and K. K. Parhi, Low-latency seqential and overlapped architectres for sccessive cancellation polar decoder, IEEE Trans. Signal Processing, vol. 6, no., pp , May 3. [3] B. Yan and K. K. Parhi, Low-latency Sccessive- Cancellation Polar Decoder Architectres Using -Bit Decoding, IEEE Trans. Circits and Systems I, pp.- 4, Oct. 3. [4] A. Balatsokas-Stimming, A. J. Raymod, W. J. Gross, and A. Brg, Hardware Architectre for List Sccessive Cancellation Decoding of Polar Codes, IEEE Trans. On Circits and Systems-II, Express Briefs, Vol. 6, No. 8, pp , Ag. 4 Cheolho Kim received the B.S and M.S degrees, both in Information & Commnication Engineering from Inha University, Incheon, Korea, in and 4, respectively. His research interests VLSI and SOC architectre design for digital signal processing and commnication systems. Haram Yn received the B.S degree in Information & Commnication Engineering in 4, from Inha University, Incheon, Korea, where he is crrently working toward the M.S degree. His research interests VLSI and SOC architectre design for digital signal processing and commnication systems. Sabooh Ajaz received the B.S degree in electronic engineering from NED University of Engineering and Technology Karachi, in 6 and the M.S degree from University of Wollongong, Wollongong, Astralia, in. Since, he is crrently prsing the Ph.D. degree in Information & Commnication Engineering from Inha University, Incheon, Korea. His research interests VLSI and SOC architectre design for digital signal processing and commnication systems. Hanho Lee received Ph.D. and M.S. degrees, both in Electrical & Compter Engineering, from the University of Minnesota, Minneapolis, in and 996, respectively. In 999, he was a Member of Technical Staff- at Lcent Technologies, Bell Labs, Holmdel, New Jersey. From April to Agst, he was a Member of Technical Staff at the Lcent Technologies (Bell Labs Innovations), Allentown. From Agst to Agst 4, he was an Assistant Professor at the Department of Electrical and Compter Engineering, University of Connectict, USA. Since Agst 4, he has been with the Department of Information and Commnication Engineering, Inha University, where he is crrently Professor. He was a visiting researcher at Electronics and Telecommnications Research Institte (ETRI), Korea, in 5. From Agst to Agst, he was a visiting scholar at Bell Labs, Alcatel-Lcent, Mrray Hill, New Jersey, USA. His research interest incldes VLSI architectre design for digital signal processing, forward error correction architectres, cryptographic systems, and commnications.