Serial Concatenated Systematic Convolutional Codes for Space
`Communications
`
`A Letter
`
`Submitted to the
`IEEE Transactions on Communications
`
`6 October 1995
`
`by
`
`Jean Yves Couleaud1 and Steven S. Pietrobon2
`
`Abstract
`
`This letter describes the performance of the serial concatenation of two recursive systematic
`convolutional codes at very low Eb/N0 iteratively decoded with the MAP (maximum a posteriori)
`algorithm.
`
`Index Terms
`
`Error correction, Turbo codes
`
`_________________
`This work was supported in part by the Institute for Telecommunications Research, University
`of South Australia and the Australian Research Council under an Australian Postdoctoral Re-
`search Fellowship.
`1Ecole Nationale Supérieure d’Ingénieurs de Constructions Aéronautiques, 49, Avenue Léon
`Blum, 31056 Toulouse Cedex, France.
`2Satellite Communications Research Centre, University of South Australia, The Levels SA 5095,
`Australia.
`
`ERICSSON EXHIBIT 1037
`Ericsson v. IV
`IPR2014-01149
`
`

`
`1 Introduction
`
`The Galileo space probe on its way to Jupiter is currently limited to using its Low Gain Antenna
`
`(LGA) for transmitting data back to Earth (the 134,400 bits/s High Gain Antenna having failed to
`
`open). It is currently planned to use a serial concatenation of a memory 13, rate 1/4 convolutional
`code with a time varying GF(28) Reed Solomon outer code of depth 8 and length 255. This scheme
`is mostly implemented in software and achieves a Bit Error Ratio (BER) of 10–7 at an Eb/N0 of 0.58
`dB [1] achieving a transmission rate up to 160 bits/s.
`
`The recent discovery of “turbo” codes [2] has allowed schemes to perform up to 0.7 dB from
`
`Shannon capacity. Unfortunately these schemes cannot be used on Galileo since the LGA is
`
`hardwired to a standard memory 6, rate 1/2 non–systematic convolutional encoder with
`
`polynomials g10 + 1718 and g20 + 1338. In this letter, we present a modification of the turbo
`
`coding scheme by serially concatenating a rate 1/2 systematic convolutional code with an outer rate
`
`1/2 systematic convolutional code with interleaving between the two codes [3]. The encoder is
`
`shown in Figure 1. Data are sent trough an AWGN channel. For Galileo, the non–systematic
`
`encoder can be easily converted to systematic form by dividing the input sequence by either
`
`
`
`
`polynomial.
`
`g10 or g20 (we chose g10) using a linear feedback shift register. The outer code has g11 as the divisor
`
`2 Decoding Scheme
`
`A block diagram of the decoding scheme is shown in Figure 2. We chose to decode the outer
`code first and feed the a posteriori information from both the information and parity bits we get
`
`
`
`
`
`
`On the first iteration z0k + x0k. D is the delay of each iteration.
`
`from the MAP decoder [4] to the inner code. After transmitting X0k, Y0k we receive the noisy x0k, y0k.
`
`2.1 Interleaver Design
`
`Since MAP decoding is very sensitive to correlation in the noise, the interleaver we designed
`
`had to spread error bursts amongst the frame. In [3] the influence of the interleaver type for block
`
`interleaving, pseudo–random linear interleaving, and pseudo random interleaving for a
`
`1
`
`

`
`concatenated code was investigated. It was found that pseudo random linear interleaving gives the
`
`best performance, as in turbo codes.
`
`The influence of the interleaver size for pseudo random interleaving was also investigated in
`
`[3]. It was found that for low SNR, interleaving becomes effective for sizes higher than 10,000 bits.
`
`In the next sections the interleaver size has been set to 16,384, almost the same as that used by
`
`Galileo.
`
`2.2 Choice of Polynomials
`
`0ńg20 + 171ń133 as the inner code and g11ńg21 + 3/1,
`
`
`
`Figure 3 shows the performance of using g1
`5/7, 17/15 and 37/21 as the outer code at an Eb/N0=1.0 dB. For low complexity codes around –2.5
`dB Eb/N0 (where the inner code effectively operates in its first iterations), the individual codes
`performance reverses, i.e., the less complex the code the better the performance [3]. The 171/133
`code performs very badly at this low of an Eb/N0, so a more “powerful” less complex outer code
`needs to be used. As can be seen from Figure 3, the least complex two state code performs best.
`
`This also explains why we choose to decode the outer code first, since it is more powerful in
`
`the first few iterations. However, as the number of iterations increases, the difference between
`
`starting with the inner or outer code becomes negligible.
`
`If we choose polynomials other than the 171/133 NASA standard, we can greatly improve the
`performance of our scheme. For the 5/7–5/7 code a BER = 10–7 can be achieved at an Eb/N0 of 0.32
`dB after 20 iterations.
`
`3 Effects of Puncturing
`
`The achieved BERs for the previous codes using 171/133 as the inner code are very far from
`
`the performance of the scheme on Galileo as can be seen from Figure 3. In order to improve our
`
`results, we tried to puncture the outer code so as to “improve” the outer codes performance at the
`
`ƫ performing the
`
`required operating point. Various puncturing matrices were tried with P + ƪ1
`
`10
`11
`
`1
`
`best with an overall code rate of 3/10. Rates lower or higher than this performed worse. Figure 4
`shows that a BER = 10–7 is achieved at an Eb/N0 of 0.57 dB after 15 iterations, similar to the
`
`2
`
`

`
`performance currently achieved by Galileo, but at a higher coding rate (0.3 compared to 0.219 for
`
`Galileo).
`
`4 Conclusions
`
`The coding schemes presented here give very high coding gains, but not as great as that achieved
`
`with parallel concatenated systematic convolutional codes. However, a code that could be used on
`
`Galileo (where parallel concatenated codes cannot be used) was found that is at least as good as
`
`the current code and which operates at a higher coding rate. Note that the analogy with turbo codes
`
`is obvious: in both cases the performance of the scheme is determined by the ability of the
`
`interleaver to randomise error bursts.
`
`References
`
`[1]S. Dolinar and M. Belongie, “Enhanced decoding for the Galileo Low Gain Antenna mission:
`Viterbi redecoding with four decoding stages,” TDA progress Report 42–121, pp. 96–109, 15
`
`May 1995.
`
`[2]C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error correcting coding and
`decoding: Turbo codes,” IEEE Int. Conf. Commun., pp. 1064–1070, Geneva, Switzerland, May
`
`1993.
`
`[3]J. Y. Couleaud, “High gain coding schemes for space communications,” ENSICA Final Year
`Project Report, Univ. of South Australia, Sep. 1995.
`
`[4]L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing
`symbol error rate,” IEEE Trans. Inform. Theory, vol. IT–20, pp. 284–287, Mar. 1974.
`
`3
`
`

`
`
`
`g11, g21
`
`
`
`X1
`k
`
`Y1
`k
`
`INT
`
`Figure 1: Encoder.
`
`
`
`g10, g20
`
`
`
`X0
`k
`
`Y0
`k
`
`DELAY
`
`DELAY
`
`MAP
`
`g10, g20
`
`
`
`–
`
`ȍ
`
`+
`
`+
`
`ȍ
`
`+
`
`DELAY
`
`INT
`
`X^
`
`j
`
`DELAY
`
`DELAY
`
`x0
`k*D
`
`z0
`k*D
`
`y0
`k*D
`
`DELAY
`
`– –
`
`ȍȍ
`
`+ +
`
`x1
`j
`
`y1
`j
`
`MAP
`
`g11, g21
`
`
`
`DELAY
`
`DEINT
`
`Figure 2: Decoder.
`
`Xk
`
`x0
`k
`
`z0
`k
`
`y0
`k
`
`(5)
`
`(1)
`
`(4)
`
`(3)
`
`(2)
`
`1
`
`0.1
`
`0.01
`
`0.001
`
`B.E.R
`
`0.0001
`
`1e-05
`
`1e-06
`
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`Iterations
`Figure 3: Performance of 171/133 based codes (1) Code 3/1–171/133 Eb/N0=0.5 dB, (2) Code
`3/1–171/133 Eb/N0=1.0 dB, (3) Code 5/7–171/133 Eb/N0=1.0 dB,
`(4) Code 17/15–171/133 Eb/N0=1.0 dB, (5) Code 37/21–171/133 Eb/N0=1.0 dB
`
`4
`
`

`
`iteration 1
`iteration 2
`iteration 3
`iteration 4
`iteration 5
`iteration 6
`iteration 7
`iteration 8
`iteration 9
`iteration 10
`iteration 11
`iteration 12
`iteration 13
`iteration 14
`iteration 15
`
`0.35
`
`0.4
`
`0.45
`Eb/No [dB]
`
`0.5
`
`0.55
`
`0.6
`
`1
`
`0.1
`
`0.01
`
`0.001
`
`0.0001
`
`1e-05
`
`1e-06
`
`1e-07
`
`1e-08
`
`1e-09
`0.3
`
`B.E.R
`
`Figure 4: Performance of the 3/1–171/133 code with punctured outer code (R=3/10).
`
`5