Complexity Analysis of Highly Improved Hybrid Turbo Codes

Transcription

1 I J C T A, 8(5, 015, pp Iteratioal Sciece Press Complexity Aalysis of Highly Improved Hybrid Turbo Codes M. Jose Ra* ad Sharmii Eoch** Abstract: Moder digital commuicatio systems eed efficiet error correctig codes. Highly Improved Hybrid Turbo Code (HIHTC is a low complex error correctig code with better Bit Error Rate (BER which is comparable to Turbo Covolutioal code (TCC. Rate 1/3 HIHTC shows a BER of 10-5 for E b of 1.7 db which is closer to the E b TCC. I this paper we compare the complexity of TCC, Improved Low Complexity Hybrid Turbo Code (ILCHTC, Low complexity Hybrid Turbo Code (LCHTC with HIHTC. The overall decoder complexity of HIHTC is less tha that of TCC, LCHTC ad ILCHTC. Idex Terms: Zigzag-Hadamard(ZH, A Posteriori Probability (APP, Bit Error Rate (BER, Decoder Complexity (DC, Hybrid Turbo Code (HTC 1. INTRODUCTION A reliable ad efficiet error correctig code is essetial for efficiet commuicatio. Sice reflectio of multiple chaels ad fadig of sigals lead to errors, Turbo covolutio codes is the efficiet error correctig code with better BER. But the decoder complexity of TCC is more. It eeds more iteratio to decode oe bit. Eve though MAP based algorithm is used i logarithmic domai the decoder complexity is more. Decoder complexity remais uchaged, eve after pucturig is used to improve the code rate. HIHTC is the improved versio of ILCHTC which is the combiatio of TCC ad ZH codes. I HIHTC Complexity is very much reduced ad BER is improved ad approaches Shao s limit.. HADAMARD CODES A Hadamard codeword is obtaied from Hadamard matrix. H 1 = [+ 1] X ( = r Hadamard matrix H i [+1, 1] ca be costructed recursively as H H H H H The colums of Hadamard matrix are bi-orthogoal. Each Hadamard Codeword carries r +1 iformatio bits. A ormalized H has ( 1 elemets of 1s ad ( 1 elemets of +1s. Every pair of rows ad every pair of colums differs exactly i places. For a ormalized Hadamard Matrix of order every row or colum has 1s ad 1 s. * Departmet of Electroics ad Commuicatio Egieerig Noorul Islam Ceter for Higher Educatio Kayakumari District, Tamiladu, Idia ** Departmet of Electroics ad Commuicatio Egieerig Noorul Islam Ceter for Higher Educatio Kayakumari District, Tamiladu, Idia

2 434 M. Jose Ra ad Sharmii Eoch Figure 1: ZIGZAG HADAMARD CODE Figure : I ZH code the iformatio bit sequece D is first segmeted ito block d k = [d k (1, d k (, d k r] is represeted by white odes. Figure 3: Upuctured Zigzag Hadamard code Figure 4: Puctured Zigzag Hadamard code

3 Complexity Aalysis of Highly Improved Hybrid Turbo Codes 435 Here the last parity bit of previous segmet is used as the first iput bit to the Hadamard ecoder The coded bits are C k = H[C k (0, C k (1, C k ( r 1] C k (0 = C k 1 ( r 1 ad C k ( 1 = d k ( ad = 1,, r For coveiece the first bit is take as 0. The black ode is the commo bit which is the first bit of a segmet ad last bit of a previous _ segmet. The grey odes are parity bits. The coded bit sequece is 4. ENCODER DESCRIPTION C k = {d k, q k, P k } q k = C k (0 P k Parity bits HIHTC ecoder usig Zigzag Hadamard code ad Recursive Systematic Covolutio (RSC code as show i Fig. d (1 RSc + ZH ZH(1 ( RSc + ZH ZH( (m RSc + ZH ZH(m Iitial data is directly give to ecoder ad the the iterleaver ð(m iterleaves the data ad the data give to RSC ad ZH ecoder. Radom Iterleaver is used i HIHTC. 5. HIHTC DECODER M m ZH ( K d( h, k ZH ( K 1 mod 1 From the received iformatio bits Log Likely hood Ratio (LLR is calculated ad the LLR s are arraged i array form of H K. I soft i Soft out (SISO decodig A Posteriori Probability (APP algorithm is used. Algorithm Process:

4 436 M. Jose Ra ad Sharmii Eoch (i Each row of the row is decoded usig priori LLRS ad is give as the iput to covolutio decoder. The output is take as posteriori LLR. (ii The Dampig Factor (DF is applied to Posteriori LLR for gradual updatig of LLRs Successfully. (iii The each colum of the array is decoded usig A priori LLRs for ZH decoder. (iv The overall decoder is give to the iterative decoder. P( C( i 1/ x L( i log P( C( i 1/ x P( x( i 1 P( C( i 1 log P ( x ( i 1 P ( C ( i 1 C { h }: C( i 1 P( x ( log e P c C { h }: C( i 1 P( P( c x e i = 0, 1, r 1 H(i, is the (i, th etry Hadamard matrix H. For C(i = +1, we have C = h with H(i, = +1 otherwise C = h with H(i, = 1 similarly for C(i = 1. Assumptios shall be made that the coded bits are trasmitted through AWGN chael with variace No The priori LLR Lapr ( i log p( C( i 1 p( C( i 1 h { H : H ( i, i}exp( h x / P( C( i h L( U log h { H : H ( i, i}exp( h x / P( C( i h h { H : H ( i, 1} r( h log h { H : H ( i, 1} r( h Here <... > is ier product. 1 x r( h exp h, Lapr 6. PERFORMANCE ANALYSIS HIHTC, ILHTC, LCHTC ad TCC are simulated with iterleaver legth N = 145 ad RSC code with 15 geerator polyomial G 1, 13 For code Rate 1 RC BER performace of HIHTC, ILCHTC, LCHTC 3 ad TCC are show i Fig 8

5 Complexity Aalysis of Highly Improved Hybrid Turbo Codes 437 Figure 5: Performaces of HIHTC, ILCHTC, LCHTC ad TCC, Rc = 1/3 For Rate 1 3 HIHTC E /N is 1.7 db at BER of which is earest to Shao s limit. For ILCHTC at b o 10 5 BER the E b is 1.7 db which is 0. db higher tha HIHTC ad TCC shows E b of 1.5 db at the BER of 10 5 which is 0. db less tha HIHTC. 7. COMPLEXITY ANALYSIS: The umber of multiplicatio ad additios determie the computatioal complexity of decoders. For a ILCHTC with L = H the trellis legth is N let Q * be the umber of multiplicatios / Iformatio Bit / Iteratio (M/IB/I ad Q + be the umber of Additios / Iformatio Bit / Iteratio (A/IB/I. Whe we use log-map algorithm for a decoder Q * ad Q shall be calculated as follows. + For TCC Q 8 S * P * * t For ILCHTC decoder Q [16 S t * P] For HIHTC decoder Q Q [ L*(8* S 4]/ H * t 16 S t 1 [(5 5 / H ] 1 H Q [ L*(8* S ]/ H * t

6 438 M. Jose Ra ad Sharmii Eoch Q 16 S t 1 [(3 / H ] 1 H Table 1 Computatioal Complexity Aalysis Of Decoders Decoder R Parameter Q* Q+ TCC 1/ M= /3 M= LCTCC 1/ M=. H=3. L= /3 M=. H=3. L= ILCHTCC 1/ M=. H=3. L= /3 M=. H=3. L= HIHTC 1/ M=. H=3. L= 5 60 Table Average Number Of Iteratios Of Decoders 1/3 M=. H=3. L= EB/No EB/No EB/No EB/No EB/No EB/No TCC LCTCC ILCHTCC HIHTC For a TCC decoder of rate 1 3, Q * = 10 ad Q + = 56 ad average umber of iteratios is 4 whe E b =. For the same code rate Q * = 50, = 140 ad average umber of iteratios is 4 for E /N =. The + b o complexity of HIHTC is 50 percet reduced as compared to TCC. The complexity of HIHTC is reduced 0.05 times as compared to ILCHTC. At the same time the BER of HIHTC is better tha ILCHTC. Hece the overall complexity of HIHTC is less tha that of TCC, LCHTC ad ILCHTC. 8. CONCLUSION I this paper a ovel error correctig code called HIHTC ad its complexity is aalyzed. The HIHTC uses the combiatio of covolutio code ad Zigzag Hadamard code. The BER of HIHTC approaches Shao s limit. I HIHTC for iterleaver legth N = 145 BER of 10 5 is achieved at E b = 1.7 db which is 0.1 db more tha TCC ad less tha LCHTC ad ILCHTC. The complexity is 0.5 times reduced as compared to TCC ad 0.05 times as compared to ILCHTC. At the same time BER of HIHTC is 0. db less tha ILCHTC at a referece of 10 5 BER for a rate 1 3 ecoder. Refereces [1] M. Jose Ra ad Dr. Sharmii Eoch, Performace Aalysis of Highly Improved Hybrid Turbo Codes. [] C. Berrou, A Glavieux, ad P. Thitimashima, Near Shao limit error correctig codig ad decodig: turbo-codes, i Proc IEEE It.Cof. Commu., May 1993, pp

7 Complexity Aalysis of Highly Improved Hybrid Turbo Codes 439 [3] S. Beedetto ad G. Motorsi, Uveilig turbo codes: some results o parallel cocateated codig schemes, IEEE Tras. If. Theory, vol. 4, pp , Mar [4] M. C. Valeti ad J. Su, The UMTS turbo code ad a efficiet decoder implemetatio suitable for software defied radios, Iteratioal J.Wireless If. Netw., vol. 8, o. 4, pp , Oct [5] L. Pig, X. Huag, ad N. Phamdo, Zigzag codes ad cocateated zigzag codes, IEEE Tras. If. Theory, vol. 47, o., pp , Feb.001. [6] L. Pig, Turbo-SPC codes, IEEE Tras. Commu., vol. 49, o. 5, pp , May 001. [7] K. Wu ad L. Pig, A improved two-state turbo-spc code for wireless commuicatio systems, IEEE Tras. Commu., vol. 5, o. 8, pp , Aug [8] A. Bhise ad P. D. Vyavahare, Low complexity hybrid turbo codes, i Proc. IEEE Wireless Commu. Netw. Cof., Las Vegas, pp , Mar. 008.