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Application of LDPC codes for Deep Space Communication under Solar Scintillation Condition Liuguo Yin Jianhua Lu Qi Li TsingHua University TsingHua University TsingHua University School of Aerospace Department of Electronic Engineering Department of Electronic Engineering Haidian District, Beijing, 100084 Haidian District, Beijing, 100084 Haidian District, Beijing, 100084 CHINA CHINA CHINA
[email protected] [email protected] [email protected] Abstract: Solar scintillation effects can be significant for deep-space telecommunication links during superior solar conjunction. In this paper, we propose to apply low-density parity-check (LDPC) codes in the Rician Channel characterized by ionospheric scintillation to improve the BER performance. Simulation results reveal that the coding gain with LDPC codes at 8.4 GHz (X-band) and 32 GHz (Ka-band) are better than convolutional codes, with more than 3.2 db and 2.5 db respectively, and the performance degradation of convolutional codes is deeply affected by channel fading, while LDPC codes represents an excellent performance of channel fading resistance. Key–Words: solar scintillation, rician channel, LDPC codes
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Sun
Introduction
During the human exploration of space, the communications link is the only way that enables the Earth to send telecommand to and receive telemetry information from the probe, and directly dictates the success or failure of exploration mission. However, the communication link in deep space environment faces a severe challenge, which, first and foremost, is the very-long distance: for geosynchronous satellite, the distance is 36,000 km, for lunar probe, it reaches 380,000 km, for Mars probe, it can achieve as long as 384,000,000 km at most. The very-long distance transmission can degrade the signals to serious degree. Meanwhile, the signal propagation is under harsh channel conditions characterized by ionospheric scintillation, multipath propagation and interference, which will affect the communication link furthermore. Severe ionospheric scintillation can even make the telecommunication link unavailable. The Earth-Probe link encounters ionospheric scintillation during superior solar conjunction (see Fig. 1), when the Sun lies between the Earth and probe. The Earth-Probe distance is at or near maximum and the received signal is at its weakest level. In addition, the intervening charged particles of the solar corona in the signal path produce significant amplitude scintillation, phase scintillation, and spectral broadening effects, which increase as the Sun-EarthProbe (SEP) angle ( in Fig. 1) decreases [4]. Several solutions are provided to deal with the situation, such as the implementation of channel coding,
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R0
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Figure 1: Relevant solar conjunction geometry.
the increase of antenna aperture and the reduction of project operations, while each of them has its limitation respectively. First, the channel coding method in deep space communication at present is convolutional code, which has insufficient correcting ability in severe condition. Second, the increase of antenna aperture will increase the construction cost as well as the operation complexity. Third, the reduction of project operations, such as invoking command moratoriums, downscaling tracking schedule and lowering data rates, will result in the unreliability of the whole system. In this paper, we proposed to apply low-density parity-check (LDPC) codes instead of convolutioanl codes to obtain high error correctability. Noted for its near Shannon limit performance [2], [3], LDPC codes have been extensively applied in many communication and digital storage systems with high reliability requirements, and these years the applications have
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αsc ( t )
w(t ) = s(t )e θ j
n (t )
tillation effects. The signal fluctuation is measured by the scintillation index S4 [4]: √ var{|αsc (t)|2 } (2) S4 = {E|αsc (t)|2 }2
(t )
Modulated/ coded
The average signal power received via the scat2 tered paths is proportional to: σnsc ≡ E|nscint |2 and the total received signal power is presented by: S = 2 )·P , where P ≡ E|w(t)|2 = E{s(t)2 } (KS +2σnsc S S is the transmitted signal power (see Fig. 2). Assume that the amplitude scintillation causes no loss in the long-term average received signal power. Then the specular and scattered signal power components must 2 = 1. meet the constraint KS + 2σnsc The envelop of the scintillation coefficient RSC = |αsc (t)| has a character of Rician distribution, i.e.,
Input signal
Figure 2: Complex baseband channel simulation model.
been expanded to aerospace field, such as Chang’e 2, Chinese unmanned lunar probe launched in November 2010. As ionospheric scintillation can be statistically modeled as Rician or Rayleigh signal fading with finite signal decoration time and limited coherent bandwidth of transmission channel [1], we optimize the LDPC codes to get better performance in Rician channel. The simulation results reveal that with the application of LDPC codes, the coding gain at 8.4 GHz (X-band) and 32 GHz (Ka-band) are better than convolutional codes, with more than 3.2 db and 2.5 db respectively under solar scintillation condition, and the performance of LDPC codes is much less sensitive to the channel fading than convolutional codes. It is extremely useful to improve communication performance for deep space telemetry. Section II addresses the scintillation modeling of the communication channel. Section III describes the optimization studies of LDPC codes over Rician channel. Section IV presents the simulation and performance analysis of the LDPC codes, finally conclusion is in section V.
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2γ }, R ≽ 0 2 σnsc (3)
2 is the Rician factor, defined as where γ ≡ KS /2σnsc the ratio of received specular power to random power. The relationship of scintillation index S4 and the Rician factor γ can be expressed as [5]
γ=
(1 − S42 )1/2 1 − (1 − S42 )1/2
(4)
2 Then the parameters KS and σnsc are related to the scintillation index S4 by
KS = (1 − S42 )1/2
(5)
1 − (1 − S42 )1/2 (6) 2 Notice that S4 is between 0 and 1. When S4 → 0 , it translates to an ideal non-fading channel, with 0 < S4 < 1 , it is Rician fading, and in the limit S4 → 1 (no line-of-sight received path), it reduces to the Rayleigh fading. The received signal of fading channel in Fig. 2 can be presented by
System Model for Solar Scintillation
2 σnsc =
A simulation model of scintillation communication system is depicted in Fig. 2. Time-varying scintillation coefficient αsc (t) is statistically modeled as a non-zero mean, complex Gaussian random process [8]: √ (1) αsc (t) = KS · ejϕ + nscint (t)
y(t) = αsc (t) · w(t) + n(t) = αsc (t) · s(t)ejθ(t) + n(t)
where KS is deterministic signal power (specular signal component); ϕ is a constant phase shift randomly and uniformly distributed over the interval of [−π, π] and nscint (t) is a (filtered) complex Gaussian process generated from scattered signal propagation over random paths (non line-of-sight) due to ionospheric scin-
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√
R − R2 prsc (R) = 2 · e−γ · e 2σnsc · I0 {R σnsc
(7)
Where θ(t) is derived from channel estimation, and independent of the concerned signal to noise ratio (SNR); n(t) obeys Gaussian random distribution with zero mean and variance σ 2 ; αsc is normalized Rician 192
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2 } = 1 and probability denfading factor with E{αsc sity function (pdf) is given by (3); for binary phase √ √ shift keying (BPSK) modulation, s ∈ {− Es , Es } maps the code symbol x, while Es is the symbol energy [11]. Assume the Rician fading factor αsc is given. Then the conditional pdf of the matched filter output is [10] (y−sαsc 1 p(y|s, αsc ) = √ e− 2σ2 2 2πσ
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(8)
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C = max{I(S; Y |ASC )}
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Figure 3: Shannon channel capacity limit.
ps (s)
ps (s)
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)2
The capacity of the channel is defined by
= max{Ep(s,y,αsc ) {log ∑
Shannon capacity boundary BPSK AWGN BPSK Rician BPSK Rayleigh
p(y|s, αsc ) }} z ps (z)· (y|z, αsc )
(9) the information bit can be transmitted over the channel with an arbitrary small frequency of errors. Let R = C and combine with
where Ep (x) is the mathematical expectation of variable x with probability distribution of p. The pdf p(s, y, αsc ) can be written as (10) due to the independency between s and αsc . p(s, y, αsc ) = p(y|s, αsc ) · p(s) · p(αsc )
{
(10)
Eb 1 Es }R=C = · { }R=C N0 R N0
(13)
the numerical solution of Shannon limit in Rician channel can be calculated. Fig. 3 shows the shannon limits in Rician, Rayleigh and AWGN channel with channel state information of BPSK signal [6]. The rician factor γ is 10 dB. The shannon limits of Rician and AWGN channel is very close to the shannon channel capacity boundary with low bit rate.
If p(s) satisfies the equal-probability distribution, the I(S; Y |ASC ) reaches the maximum. ∫ ∞∫ ∞ 1 ∑ p(αsc ) · p(y|s, αsc ) C= 2 √ −∞ 0 s=± Es
p(y|s, αsc ) ) · dy · dαsc √ z=± Es p(y|z, αsc ) 2 ∫ ∞∫ ∞ √ = p(αsc ) · p(y|s = ES , αsc ) 3 The Proposed Communication −∞ 0 √ System with LDPC Codes (11) 2p(y|z = Es , αsc ) √ √ · log2 ( ) The simulation approach is based on the link model p(y|z = Es , αsc ) + p(y|z = − Es , αsc ) as shown in Fig. 4, with the assumption of an ideal · dy · dαsc receiver, that is, the assumption of perfect carrier and ∫ ∞∫ ∞ √ subcarrier tracking as well as symbol synchronization. =1− p(αsc ) · p(y|s = Es , αsc ) The degradation caused by amplitude scintillation is 0 −∞ √ simulated for the case of BPSK modulation with LDp(y|z = − Es , αsc ) √ ) · dy · dαsc · log2 (1 + PC codes as well as convolutional codes for comparp(y|z = Es , αsc ) ison. The effects due to scintillation are introduced by the relationship between scintillation index S4 and The capacity C can be written as (12) through (8) and 2 through (5) and (6). Rician parameters KS and σnsc (11). The LDPC codes used in the simulation are de∫ ∞∫ ∞ √ fined by column weight distribution polynomial λ(x) C =1− p(αsc ) · p(y|s = Es , αsc ) and row weight distribution polynomial ρ(x) through 0 −∞ (12) √ 2 (14) and (15), with code length n of 2032 and code · log2 (1 + e−2yαsc Es /σ ) · dy · dαsc rate r of 1/2. According to Shannon’s second theorem, if the inforλ(x) = 0.001126 + 0.283784 · x mation transmission rate R is not larger than the chan(14) nel capacity C , there exists a coding system such that + 0.214527 · x2 + 0.500563 · x6 · log2 ( 1 ∑
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LDPC Encoder
R=4R0(S4=0.37) BER Before coding
BPSK Modulator
Convolutional Encoder (7,1/2)
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Figure 5: Telecommunication link performance at Xband with R = 4R0 .
Figure 4: Channel simulation model due to solar scintillation.
S4=0.23
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ρ(x) = 0.006754 · x + 0.993243 · x 4
The decoding of received data is based on iterative belief-propagation (BP) algorithm, which is particularly effective for LDPC codes. For comparison, the communication performance is simulated with no channel coding as well as (7, 1/2) convolutional codes at both 8.4 GHz (X-band) and 32 GHz (Ka-band). The decoder for convolutional codes uses Viterbi algorithm.
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Figure 6: Telecommunication link performance at Xband with R = 5R0 .
Numerical Simulations
The simulation BER performance at X-band is shown in Fig. 5, Fig. 6, Fig. 7 and Fig. 8, where the SunProbe distance (R in Fig. 1) is varied from 4 to 8 solar radii (R0 in Fig. 1) and the corresponding scintillation index S4 decreases from 0.37 to 0.099 [7]. The simulations show that at BER = 10−5 , which is usual requirement for deep-space telecommunication link, the link performance of LDPC codes is better than convolutional codes with coding gain more than 3.2 dB at least, and with the increase of scintillation index S4 , the performance of convolutional codes degrades seriously, while LDPC codes represents an excellent performance of fading resistance. Fig. 9) displays the link BER performance results at Ka-band, where the probe is assumed to be at perihelion with the Sun-Probe distance R = 4R0 and the scintillation index S4 = 0.064, which is much smaller than X-band at the same Sun-Probe distance
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(S4 = 0.37). The link performance of LDPC codes is better than convolutional codes with coding gain about 2.5 db at BER = 10−5 .
5 Conclusion LDPC codes can give very good performance for communication under solar scintillation condition. In addition, their complexity can be reduced to the admissible level by using semi-random structure. Therefore, LDPC codes are very powerful and useful when they are applied to telemetry information transmissions in deep space communications. Acknowledgements: This work was supported in part by National Natural Science Foundation of China (No. 61021001) and Program for New Century Excel-
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lent Talents in University (NCET).
S4=0.162
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[1] Kullstam P. A. and Keskinen M. , “Ionospheric Scintillation Effects on UHF Satellite Communications,” MILCOM, 2000. [2] R. G. Gallager, Low-Density Parity Check Codes. Cambridge, MA: MIT Press, 1963. [3] D. J. MacKay and R. M. Neal, “Near Shannon limit performance of low density parity check codes,” IEE Electronics Letters, vol. 32, no. 18, pp. 1645- 1646, Aug. 1996. [4] E. J. Fremou and H. F. Bates, “Worldwide behavior of average VHF-UHF scintillation,” Radio Sci.,vol. 6, pp. 863-869, Oct. 1971. [5] P. Shaft, “On the Relationship between Scintillation Index and Rician Fading,” IEEE Trans., Comm., pp.731-732, May 1974. [6] Shaoqian Li, Principle and Application of LDPC codes. Chengdu, China: University of Electronic Science and Technology of China Press, May 2006. [7] Y.Feria, M.Belongie, T.McPheeters and H.Tan, “Solar Scintillation Effects on Communication Links at Ka-band and X-band,” The Telecommunications and Data Acquisition Progress Report 42-129,JPL Pasadena,California,May 15,1997. [8] Z. Ye and E. Satorius, “Channel Modeling and Simulation for Mobile User Objective System (MUOS)-Part I: Flat Scitillation and Fading,” ICC’03, pp. 3503- 3510, Nov. 2003. [9] D. Morabito, “Solar Corona Amplitude Scintillation Modeling and Comparison to Measurements at X-Band and Ka-Band,” JPN Progress Report 42-153, May.15 2003. [10] Xu Hua, Xu Cheng-qi and Xiaochuan Zheng, “Optimization of Irregular LDPC codes on Rician Channel,” Proc. Wireless Communications, Networking and Mobile Computing, 2005. [11] Lin Jiaru and Wu Weiling, “Performance of Irregular LDPC codes on Rician-Rading Channel” Acta Electronic Sinica, vol. 33, no. 1, Jan. 2006.
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Figure 7: Telecommunication link performance at Xband with R = 6R0 . S 0
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Figure 8: Telecommunication link performance at Xband with R = 8R0 . R=4R0(S4=0.064) Ka−band
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Figure 9: Telecommunication link performance at Kaband with R = 4R0 .
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