D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, "Transfer function bounds on the performance of turbo codes," Telecommunications and Data Acquisition Progress Rep., vol. 42, no. 122, pp. 44--55, Aug. 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....one iteration. Here, the number of iterations performed is denoted with I. 4. 3 Weight Distribution Aspects The Turbo Code Weight Distribution The weight distributions of the component codes used in a Turbo code are relatively easy to calculate using, for example, transfer function methodology [16]. In order to calculate the distance spectrum of a Turbo code one has to take into account the effect of the interleaver. Calculation of the weight spectrum resulting from a particular interleaver is today an operation that is far too complex for interleaver lengths of practical interest. In [6] ....

.... which is referred to as the input redundancy weight enumerating function (IRWEF) The number (multiplicity) of codewords with weight d, caused by input sequences of weight i, is represented by ta, The multiplicities ta,4, can be enumerated using, for example, the transfer function technique in [16]. The Turbo codewords are combinations of two component code codewords that both result from the same input sequence weight i. The conditional weight enumer ating function (CWEF) Ti (D) ta,iD a, i: 0, 1, K, 4.6) d=0 enumerates the number of codewords of different weights d, ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, and F. Pollara. Transfer function bounds on the perfor- mance of Turbo codes. TDA Progress Report J2-122, pages 44-55, Aug. 1995.

....with Hamming weight and is the pairwise error probability that the ML decoder will prefer a particular codeword of total Hamming weight to the all zero codeword . Since is difficult to obtain for a specific interleaver, an average upper bound constructed with a random interleaver is proposed in [15] and [16] From the results in [15] the average weight distribution is given by (14) where is the probability that an input data sequence with Hamming weight produces a codeword with Hamming weight . Thus, the average union bound is given by (15) Similarly, the average union bound on the ....

....error probability that the ML decoder will prefer a particular codeword of total Hamming weight to the all zero codeword . Since is difficult to obtain for a specific interleaver, an average upper bound constructed with a random interleaver is proposed in [15] and [16] From the results in [15], the average weight distribution is given by (14) where is the probability that an input data sequence with Hamming weight produces a codeword with Hamming weight . Thus, the average union bound is given by (15) Similarly, the average union bound on the probability of bit error (16) In the ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, and F. Pollara, "Transfer function bounds on the performance of turbo codes," Jet Propul. Lab., Pasadena, CA, TDA Progress Rep. 42-122, Aug. 1995.

....square law combining in worst case jamming is (25) 26) The only known way to exactly calculate is via an exhaustive search involving all possible input sequences. One solution is to calculate an average upper bound by computing an average weight function over all possible interleaving schemes [19]. If the average weight function is defined as (27) Fig. 3. Simulation results for 20 bits per hop. where is the probability that an interleaving scheme maps an input weight of to produce a codeword of total weight , and is the number of input frames with weight . An algorithm for calculating ....

....weight function is defined as (27) Fig. 3. Simulation results for 20 bits per hop. where is the probability that an interleaving scheme maps an input weight of to produce a codeword of total weight , and is the number of input frames with weight . An algorithm for calculating was described in [19]. Thus, 28) V. SIMULATION RESULTS For all simulations, the component encoders are rate recursive systematic convolutional encoders with memory four and octal generators . The packet size is 1760 information bits and the number of decoder iterations is five. A helical interleaver [20] is used ....

D. Divsalar, S. Dolinar, F. Pollara, and R. McEliece, "Transfer function bounds on the performance of turbo codes," TDA Progress Rep. 42-122, JPL, Aug. 1995.

....of parallel concatenated convolutional codes (PCCC s) or turbo codes [3] However, approximations and di#cult computational procedures were still needed to derive the bound because, in principle, the complete weight enumerating function (WEF) of a long terminated convolutional code is needed. In [4], a more accurate and systematic approximation to the bound is presented. However, due to a numerical precision problem inherent to the methodology, only block lengths up to about 1000 can be considered. It is also mentioned in [4] that the upper bounds are useful only at signal to noise ratios ....

....(WEF) of a long terminated convolutional code is needed. In [4] a more accurate and systematic approximation to the bound is presented. However, due to a numerical precision problem inherent to the methodology, only block lengths up to about 1000 can be considered. It is also mentioned in [4] that the upper bounds are useful only at signal to noise ratios (SNR s) larger than the channel cuto# rate since, at smaller SNR s, the usual union bound diverges rapidly, especially for large block lengths. Thus, the traditional union bound is primarily useful for approximating the error floor ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, "Transfer function bounds on the performance of turbo codes," The JPL TDA Progress Report vol. 42-122, pp. 44-55, Aug. 1995.

.... of the SNR, calculation of the extrinsic information does (unless a fixed SNR is used, see Item 6) We investigated conventional turbo decoding as well as other decoders on the basis of the following simulation conditions: We used two memory m = 2 RSC constituent codes with generators 5 7 [8, 3]. By periodically puncturing the parity bit, a rate R = 1=2 code is obtained [5, 6] We applied a (non periodic) random interleaver of length L block , where L block is the block length; new permutations were generated for each block. Zero tailing was emulated by transmitting the all zeros ....

....with d k 2 f0; 1g, x s;1p;2p are the coded symbols with x s = 1 Gamma 2 d, and y s;1p;2p is the matched filter output sequence. The index int denotes interleaved sequences. An estimation of the SNR is not required. Upper bounds on the BER for ML decoding were published since 1995 [8, 2, 3]. However, ML sequence decoding (MLSD) by means of a recursive algorithm (i.e. not by an exhaustive search) iteratively solving (1) was only recently invented by Sadowsky [27, 28] Conceptionally, the ML procedure is described as follows [27] First, make complete lists of all N paths for each ....

D. Divsalar, S. Dolinar, F. Pollara, and R.J. McEliece, "Transfer function bounds on the performance of turbo codes," TDA Progress Report 42-122, JPL, Pasadena, California, pp. 4455, Aug. 1995.

....Also, the performance in this region usually improves only slowly with increasing SNR. It can be shown that this error floor depends heavily on the interleaver used (The JPL team show that performance in the error floor region is directly related to the first few terms of the weight distribution [Divsalar et al. 1995] and that the weight spectrum depends on the chosen interleaver [Dolinar Divsalar, 1995] At low to moderate SNR, there is a sharp drop in BER. In this region, a relatively large number of iterations are required for convergence. At very low SNR, increasing iterations do provide a gain in ....

.... shown that this error floor depends heavily on the interleaver used (The JPL team show that performance in the error floor region is directly related to the first few terms of the weight distribution [Divsalar et al. 1995] and that the weight spectrum depends on the chosen interleaver [Dolinar Divsalar, 1995]) At low to moderate SNR, there is a sharp drop in BER. In this region, a relatively large number of iterations are required for convergence. At very low SNR, increasing iterations do provide a gain in performance; however, the overall code performance is poor, and the BER is certainly ....

[Article contains additional citation context not shown here]

DIVSALAR, DARIUSH, DOLINAR, SAM, & POLLARA, FABRIZIO. 1995 (Aug. 15th,). Transfer Function Bounds on the Performance of Turbo Codes. TDA Progress Report 42-122. Jet Propulsion Laboratory, California Institute of Technology.

....of parallel concatenated convolutional codes (PCCC s) or Turbo codes [2] However, approximations and difficult computational procedures were still needed to derive the bound because, in principle, the complete weight enumerating function (WEF) of a long terminated convolutional code is needed. In [3], a more accurate and systematic approximation to the bound is presented. However, due to a numerical precision problem inherent to the methodology, only block lengths up to about 1000 can be considered. It is also mentioned in [3] that the upper bounds are useful only at signal to noise ratios ....

....(WEF) of a long terminated convolutional code is needed. In [3] a more accurate and systematic approximation to the bound is presented. However, due to a numerical precision problem inherent to the methodology, only block lengths up to about 1000 can be considered. It is also mentioned in [3] that the upper bounds are useful only at signal to noise ratios (SNR s) larger than the channel cutoff rate since, at smaller SNR s, the usual union bound diverges rapidly, especially for large block lengths. Thus, the traditional union bound is primarily useful for approximating the error floor ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, "Transfer function bounds on the performance of turbo codes," JPL TDA Progress Rep., vol. 42-122, pp. 44--55, Aug. 1995.

....0 ) Delta fi t (m) 1 C C C A z L j (2.8) A ji is the set of input symbols for which the j th input sequence is i. Decision c d j;t = ae 0 if j (t) 0 1 otherwise b d t = i c d 0;t ; dK Gamma1;t j (2.9) 2. 3 Constituent codes for multi input Turbo Codes In [1, 3, 7] it was shown that Turbo Codes achieve the best results with recursive convolutional codes that have a primitive feedback polynomial. These results can be applied to K input codes such that each input has an individual recursive convolutional code of memory M K , where M is the overall memory ....

....to calculate the bounds with 32 bits is much smaller than the block size of commonly used Turbo Codes. This is the reason why the typical slope and error floor are not visible. The replacement of the systematic part by a convolutional code alone will not give the best performance. According to [7], bit error probability bounds can be calcu29 lated as follows: P b = N X i=1 i N Prob [error event of weight i] N X i=1 N i E dji ( Q r 2dE s N 0 ) 4.1) The conditional expectation E dji f Deltag is over the probability distribution p (d 0 ; d 1 ; d 2 ji) that any ....

D. Divsalar, S. Dolinar, and F. Pollara, "Transfer Function Bounds on the Performance of Turbo Codes," The Telecommunications and Data Acquisition Progress Report 42-122, Jet Propulsion Laboratory, Pasadena, California, Aug. 1995.

.... Gamma) 2 8 Gamma Gamma (2 Gamma) 4 : 26) The only known way to exactly calculate A d is via an exhaustive search involving all possible input sequences. One solution is to calculate an average upper bound by computing an average weight function over all possible interleaving schemes [19]. If the average weight function is defined as A d = k X i=0 k i p(dji) 27) where p(dji) is the probability that an interleaving scheme maps an input weight of i to produce a codeword of total weight d and k i is the number of input frames with weight i. An algorithm for ....

....is defined as A d = k X i=0 k i p(dji) 27) where p(dji) is the probability that an interleaving scheme maps an input weight of i to produce a codeword of total weight d and k i is the number of input frames with weight i. An algorithm for calculating p(dji) was described in [19]. Thus, P word n X d=dmin k X i=1 k i p(dji) D d : 28) V. Simulation Results For all simulations, the component encoders are rate 1 2 recursive systematic convolutional encoders with memory 4 and octal generators (37; 21) The packet size is 1760 information bits and the ....

D. Divsalar, S. Dolinar, F. Pollara, and R. McEliece, "Transfer function bounds on the performance of turbo codes," TDA Progress Report 42-122, JPL, August 1995.

....for an (n; k) block code is Pword n X d=dmin A d P 2 (d) 17) The only way to calculate A d is via an exhaustive search involving all possible input sequences. One solution is to calculate an average upper bound by computing an average weight function over all possible interleaving schemes [4]. We define an average weight function as A d = k X i=1 k i p(dji) 18) where p(dji) is the probability that an interleaving scheme maps an input weight of i to produce a codeword of total weight d and k i is the number of input frames with weight i. Thus, P word n X ....

....i to produce a codeword of total weight d and k i is the number of input frames with weight i. Thus, P word n X d=dmin k X i=1 k i p(dji) P 2 (d) 19) P bit n X d=dmin k X i=1 i k k i p(dji) P 2 (d) 20) An algorithm for calculating p(dji) is given in [4]. Thus, in order to compute this bound, we need only calculate P 2 (d) Over a channel with both full band thermal noise and partial band jamming noise, we can calculate the pairwise error probability by conditioning on the number of jammed symbols. P 2 (d) d X l=0 P (error j E l ) Delta P ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, and F. Pollara, "Transfer function bounds on the performance of turbo codes," Telecom. and Data Aquisition Progress Report 42122, Jet Propulsion Laboratory, August 1995.

....used, the bounds are useful at high enough SNR, where empirical evidence indicates that the performance of the ML decoder will be approached as the number of decoding iterations increases. The word error probability and bit error probability of an (L, N) linear block code can be upper bounded by [3] P e # L X d=d f t d P 2 (d) 2) and P b # L X d=d f b d P 2 (d) 3) respectively, where P 2 (d) denotes the pairwise error probability for codewords with Hamming distance d. Here, t d is the number of codewords with Hamming weight d, and b d is the total weight of the information bits ....

D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, "Transfer function bounds on the performance of turbo codes," The Telecommunications and Data Acquisition Progress Report, vol. 42, no. 122, pp. 44--55, August 1995.

.... Initially greeted with some skepticism, the original results were independently reproduced by several researchers [6] 7] 10] 11] 12] 13] and [30] Subsequently, recent research on Turbo codes has focused on understanding the reasons for their outstanding performance [8] 9] 16] [22] [24] 28] At this point, there are two fundamental questions regarding Turbo codes. First, does the iterative decoding scheme presented in [1] always converge to the optimum solution Second, assuming optimum or near optimum decoding, why do the Turbo codes perform so well In this paper, we ....

....compared to a maximum free 2 distance, rate R =1 2, memory # = 14, i.e. a (2, 1, 14) convolutional code to emphasize the di#erences in performance and structure. Techniques for analyzing the performance of Turbo codes using transfer functions and related methods may be found in [6] 9] and [22]. The paper begins with a detailed examination of the structure of codewords in a Turbo code in Section II. This leads to the calculation of the free distance of a particular Turbo code and an explanation for the error floor in its performance curve. In Section III, the distance spectrum of ....

D. Divsalar, S. Dolinar, F. Pollara and R. J. McEliece, "Transfer function bounds on the performance of turbo codes", TDA Progress Report 42-122, Jet Propulsion Laboratory, Pasadena, California, pp. 44-55, August 15, 1995.

....of HRSC . The even columns are associated to information bits and the odd columns to parity bits. Note also that the PCEs of a convolutional code are not disjoint. The weight distribution of an RSC code (viewed as an (N; K 1 ) block code) is computed using the transfer function method described in [5]. The effect of the trellis termination phase can be neglected, i.e. K1 N r. The number of codewords of weight is denoted N 1 ( 0 : N . Let C 1 be an (N; K 1 ; d 1 ) linear binary block code built from an RSC code. A second (N; K 1 ; d 1 ) block code C 2 = C 1 ) is constructed by ....

D. Divsalar, S. Dolinar, F. Pollara, R.J. McEliece: "Transfer function bounds on the performance of Turbo codes," TDA Progress Report 42-122, August 1995.

....(2 Gamma) 2 8 Gamma Gamma (2 Gamma) 4 (28) The only known way to exactly calculate A d is via an exhaustive search involving all possible input sequences. One solution is to calculate an average upper bound by computing an average weight function over all possible interleaving schemes [9]. If the average weight function is defined as A d = k X i=0 k i p(dji) 29) where p(dji) is the probability that an interleaving scheme maps an input weight of i to produce a codeword of total weight d and k i is the number of input frames with weight i. An algorithm for ....

....is defined as A d = k X i=0 k i p(dji) 29) where p(dji) is the probability that an interleaving scheme maps an input weight of i to produce a codeword of total weight d and k i is the number of input frames with weight i. An algorithm for calculating p(dji) was described in [9]. Thus, P word n X d=dmin k X i=1 k i p(dji) D d (30) 5 Simulation Results For all simulations, the component encoders are rate 1 2 recursive systematic convolutional encoders with memory 4 and octal generators (37; 21) The packet size is 1760 bits and the number of turbo code ....

D. Divsalar, S. Dolinar, F. Pollara, and R. McEliece, "Transfer function bounds on the performance of turbo codes," Telecom. and Data Aquisition Progress Report 42-122, Jet Propulsion Laboratory, August 1995.

....block code is Pword n X d=dmin A d P 2 (d) 3.28) The only known way to exactly calculate A d is via an exhaustive search involving all possible input sequences. One solution is to calculate an average upper bound by computing an average weight function over all possible interleaving schemes [26]. The average weight function in [26] was calculated as A d = k X i=1 0 B B B k i 1 C C C A p(dji) 3.29) where p(dji) is the probability that an interleaving scheme maps an input of weight i to produce a codeword of total weight d and 0 B B B k i 1 C C C A is the number of input ....

....d P 2 (d) 3.28) The only known way to exactly calculate A d is via an exhaustive search involving all possible input sequences. One solution is to calculate an average upper bound by computing an average weight function over all possible interleaving schemes [26] The average weight function in [26] was calculated as A d = k X i=1 0 B B B k i 1 C C C A p(dji) 3.29) where p(dji) is the probability that an interleaving scheme maps an input of weight i to produce a codeword of total weight d and 0 B B B k i 1 C C C A is the number of input frames with weight i. Thus, P word ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, F. Pollara, and R. McEliece, "Transfer Function Bounds on the Performance of Turbo Codes," TDA Progress Report 42-122, JPL, August 1995.

....model, the autocorrelation function is given as ae( e Gammaj2 B j where B is the doppler bandwidth of the fading process. 3 Union Upper Bound The ability to analyze turbo codes in regions of high signal to noise requires lengthy simulations or an analytic bounding technique. In [2] and [5], an average upper bounding technique is developed for turbo codes. It is shown that these bounds are very useful in determining the error floor as well as understanding the impact of constituent encoder choice and block size on performance on the AWGN channel. While these bounds reveal little ....

....of Bound The union upper bound is a popular and effective method of bounding block code performance provided that the weight distribution is known. For turbo codes, deriving this weight distribution for a particular interleaving scheme is very difficult. Therefore, the authors of [2] and [5] have advanced the idea of forming an average weight function, where the average is over all possible interleaving schemes. In this context, it is useful to view the turbo scheme as the concatenation of multiple code fragments . Usually, one of these code fragments is the input frame while the ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, R. J. McEliece, and F. Pollara, "Transfer function bounds on the performance of turbo codes," TDA Progress Report 42-121, JPL, Aug. 1995.

....model, the autocorrelation function is given as ae( e Gammaj2 B j where B is the doppler bandwidth of the fading process. 3 Performance Bounds The ability to evaluate turbo codes in regions of high signalto noise requires lengthy simulations or an analytic bounding technique. In [2] and [5], an average upper bounding technique is developed for turbo codes. It is shown that these bounds are very useful in determining the error floor as well as understanding the impact of constituent encoder choice and block size on performance on the AWGN channel. While these bounds reveal little ....

....Upper Bound The union upper bound is a popular and effective method of bounding block code performance provided that the weight distribution is known. For turbo codes, deriving this weight distribution for a particular interleaving scheme is very difficult. Therefore, the authors of [2] and [5] have advanced the idea of forming an average weight function, where the average is over all possible interleaving schemes. In this context, it is useful to view the turbo scheme as the concatenation of multiple code fragments . One of these code fragments is the input frame while the other ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, R. J. McEliece, and F. Pollara, "Transfer function bounds on the performance of turbo codes," TDA Progress Report 42-121, JPL, Cal Tech, Aug. 1995.

....in the MAP algorithm [9] It should also be noted that for the Rayleigh channel with average energy of 1, EA [a] 0:8862. 3 Performance Bounding The ability to evaluate turbo codes in regions of high signal to noise requires lengthy simulations or an analytic bounding technique. In [10] and [11], an average upper bound is developed for turbo codes. It is shown that this bound is very useful in determining the error floor as well as understanding the impact of constituent encoder choice and block size on performance for the AWGN channel. Here, we apply this bound to the Rayleigh fading ....

....A(d) is the number of codewords with Hammingweight d and P 2 (d) is the probability of incorrectly decoding to a codeword with weight d. For a turbo code with a fixed interleaver, the construction of A(d) requires an exhaustive search. Due to complexity issues involved in this search, 10] and [11] propose an average upper bound constructed by averaging over all possible interleavers. The result of this averaging can be thought of as the traditional union upper bound but with an average weight distribution. As in [11] the average weight distribution can be written as A(d) K X i=1 0 ....

[Article contains additional citation context not shown here]

D. Divsalar, S. Dolinar, R. J. McEliece, and F. Pollara, "Transfer function bounds on the performance of turbo codes," TDA Progress Report 42-121, JPL, Cal Tech, Aug. 1995.

....that N must be an integer multiple of p. 1 We assume, as before, that the convolutional CCs are linear, so that the SCCC is linear as well, and the uniform error property applies. The exact analysis of this scheme can be performed by appropriate modifications of the analysis described in [6] or [13] for PCCCs. It requires the use of a hypertrellis having as hyperstates pairs of states of outer and inner codes. The hyperstates S ij and S lm are joined by a hyperbranch that consists of all pairs of paths with length N p that join states s i and s l of the inner code and states s j and s m of ....

D. Divsalar, S. Dolinar, R. J. McEliece, and F. Pollara, "Transfer Function Bounds on the Performance of Turbo Codes," The Telecommunications and Data Acquisition Progress Report 42-122, April--June 1995, Jet Propulsion Laboratory, Pasadena, California, pp. 44--55, August 15, 1995. http://tda.jpl.nasa.gov/tda/progress report/42-122/122A.pdf

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D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, "Transfer function bounds on the performance of turbo codes," Telecommunications and Data Acquisition Progress Rep., vol. 42, no. 122, pp. 44--55, Aug. 1995.

No context found.

D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece. Transfer function bounds on the performance of turbo codes. The Telecommunications and Data Acquisition Progress Report, 42(122):44--55, August 1995.

No context found.

D. Divsalar, S. Dolinar, F. Pollara, and R.J. McEliece, "Transfer function bounds on the performance of turbo codes", TDA Progress Rep. 42-122, Jet Propulsion Lab., Pasadena, CA, USA, Aug. 15, 1995, pp. 44-55.

No context found.

D. Divsalar, S. Dolinar, R. J. McEliece, and F. Pollara, "Transfer Function Bounds on the Performance of Turbo Codes," TDA Progress Report 42-122, JPL, Caltech, August 1995.

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D. Divsalar, S. Dolinar and F. Pollara, \Transfer Function Bounds on the Performance of Turbo Codes", TDA Progress Report 42-122, August 15, 1995.

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D. Divsalar,S. Dolinar and F. Pollara,\Transfer Function Bounds on the Performance of Turbo Codes",TDA Progress Report 42-122,August 15,1995.

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