Yang Yang, Wenjing Wang, Xiqi Gao＊

National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

I. INTRODUCTION

Multibeam antennas that can form hundreds of spot beams and support the transmissions of independent information streams for multiple users at the same spectrum band have been widely employed by mobile satellite communication (MSC) system to enhance its capacity.However, in spite of the much improved capacity, inter-beam-interference (IBI) and noise still lead to receiving errors [1]. The multibeam MSC channel with full frequency reuse is essentially a multiple-input multiple-output(MIMO) channel. A hybrid space ground precoding method to manage the downlink IBI of multi-beam MSC can be found in [2]. For the uplink, we propose to employ the MIMO receiver [3, 4] to manage its IBI and reduce the receiving errors.

For forward error correction coded MIMO systems, turbo iterative detection and decoding that exchanges soft information between a soft-in soft-out (SISO) detector and an SISO decoder in an iterative fashion is a kind of near-optimum MIMO receiver [5–10]. A low computational complexity turbo iterative detection and decoding scheme called dual-turbo iterative detection and decoding for turbo coded MIMO systems is proposed in [11], and the dual-turbo method for low-density parity-check (LDPC) coded systems is given in[12].

In this work, we improve the dual-turbo iterative detection and decoding scheme via the approximate message passing (AMP)algorithm for the LDPC coded systems and introduce it into the multibeam MSC uplink to manage its uplink IBI. This improvement can further reduce the computational complexity and achieve much lower bit error rate (BER).

The rest of the paper is organized as follows: in section II, multibeam MSC system model is described. The AMP algorithm aided detection method is introduced in section III.The proposed AMP dual-turbo iterative detection and decoding method is given in section IV, along with the computational complexity analysis. The simulation results are shown in Section V. The conclusions are drawn in section VI.

The authors proposed to hybridize the AMP detector with LDPC SISO decoder to improve the traditional dual-turbo iterative detection and decoding scheme.

II. SYSTEM MODEL

We consider a geostationary earth orbit (GEO)MSC uplink withMspot-beams. Suppose there areNusers randomly distributed in the service area transmitting to the satellite at the same time. As shown in figure 1, for each user a block ofKbinary information bits is encoded by an (K,W) LDPC code. We employ vectors bnand cnto denote a codeword before and after coding for usern. TheWcoded bits cnare then fed into a 2Mc-ary QAM modulator with constellation X and output xn. We useto denote itst-th entry, which is transmitted by usernduring symbol periodtSuppose the carriers of all user terminals(UTs) are locked on the same clock. Then,their symbols can be transmitted and received almost synchronously. Let xtdenotes the symbols transmitted by all UTs during symbol period t, whereThen the general baseband channel model during symbol periodtis given by.

where vectordenotes the signals received by all spot-beams, Ztdenotes channel matrix, and ntdenotes the independent and identically distributed (i.i.d.) additive white Gaussian noise (AWGN) with varianceIt worth noting that both xnand xtare used to denote the transmitted symbols. xnis a row vector, and xtis a column vector. Their relationships are given as follows,

The received data ynand ytare similar.

The channel matrix Ztcan be modeled specifically for the multibeam MSC uplink as follows [3],

The first item B is anM×Nmatrix modeling the beam gains, which is generally approximated by the well accepted method as follows [13],

whereJ1andJ3are thefirst kind Bessel functions of order one and three respectively. θmndenotes the off axis angle of usernwith respect to the axis of spot-beamm, and θ3dBis the angle that corresponds to 3dBpower loss of each spot-beam.Gmaxis the maximum beam gain. Suppose the position of all users remain constant during a codeword. Then B reduces to a deterministic real positive matrix. In most case, B is sparse, because the signals transmitted by one user can be received by just a part of the spotbeams, and each spot-beam can only receive the signals coming from the users nearby.

Fig. 1. Multibeam MSC uplink with MIMO receiver.

The second itemis anN×Ndiagonal matrix modeling the shadowing and rain attenuation of each user, which is also constant during a codeword as well.

The third itemis anN×Ndiagonal matrix containing i.i.d. non-zero mean complex entries, which models the multi-passing fading of each user.is modeled according to the Rician channel model as follows,

where K is the Rician K-factor,is a diagonal matrix modeling the line of sight (LoS)signal fading, which is constant during a codeword as well, andis an i.i.d. complex Gaussian diagonal matrix modeling the scattered fading.

III. AMP DETECTION

AMP algorithm [14, 15] is a distributed message passing method, which can be employed as an MIMO detector to estimate the posterior probability distribution of the transmitted signals [16–18]. Hence, it can be used as a detector to estimatefor the multibeam MSC uplink as well [3, 4]. Through estimatingit can turn the unwanted IBI into useful energy to reduce the detection errors.

We employ the Bayes theorem to writeas

Fig. 2. Factor graph of posteriori distribution

Recall that B is sparse, thus Ztis sparse too. Denote the user set whose signals can be received by spot beammas Um, and the spot beam set that can receive the signals transmitted by usernas Bn. Then, for usern, the marginal probability density function (pdf)ytBnis the subset of ytcorresponding to Bn.Moreover, note that the entries of xtare independent of each other. Therefore,in(6) can be reformulated as

whereis the subset of xt, which is similar to. Then we can use a factor graph to illustrate the posteriori distribution in (7) as given in figure 2.

As a matter of fact, the AMP algorithm is a numerically efficient method to approximate each marginal pdfby a set of message passing equations that go from factor nodes to variable nodes(i.e.,m→n) and from variable nodes to factor nodes (i.e.,n→m) as illustrated in figure 2 [14, 15]. The message passing equations are constructed as follows,

where the superscript(r)is used to denote the number of iterations, and the symbol ? means that two functions are identical upon a normalization factor.

Suppose all random variables follow Gaussian distributions, and letdenote the mean and variance offollows,

Thencan be worked out through(9) according to Gaussian approximation method as [19, 20]

wherezmnis the (m,n)-th entry of Zt. Base on (11) we can use (8) to update

Henceforth, we have the method to update the mean and variance ofas

After enough iterations, we can estimate the mean and variance of

Thus, we estimate the mean and varianceof each marginal pdf ofMore details of the AMP detector are given in APPENDIX.

It is different from traditional linear detector that is aimed at estimating xt, the AMP detector is aimed at estimating the mean and variance off(xt|yt). In another word, the AMP detector achieves more posteriori information of the transmitted signals xt.

IV. AMP DUAL-TURBO ITERATIVE DETECTION AND DECODING

As a kind of MIMO receiver, turbo iterative detection and decoding can further turn the unwanted IBI into useful energy and further reduce the receiving errors, which is a kind of near-optimum MIMO receiver [5]. Dual-turbo iterative detection and decoding that exchanges soft extrinsic information and soft inner information efficiently between the linear SISO detector and the SISO decoder is a low complexity turbo iterative detection and decoding method [11, 12]. (The information exchanged between the SISO detector and the SISO decoder is called soft extrinsic information, while the information exchanged in the SISO decoder is called soft inner information.)For LDPC coded systems, the dual-turbo iterative detection and decoding is given in figure 3 (a). We propose to replace the linear SISO detector with the AMP detector as shown in figure 3 (b), i.e., the proposed AMP dual-turbo iterative detection and decoding scheme.

Finding a way to exchange the soft extrinsic information for AMP detector is the key to employ it in the dual-turbo fashion. Take the variable notefor example that is a symbol of xtas given in section II. We have given the SISO detection method to update the mean and variance of its posteriori pdf in section III. Meanwhile, the mean and variance ofcan also be updated through SISO decoding as shown in figure 3 (b). Hence, the mean and variance ofcan be a bridge to help exchanging the soft extrinsic information between the AMP detector and the SISO decoder. The detail method to exchange the soft extrinsic information for LDPC coded systems is given in the following.

The mean and variance ofgenerate the priori log-likelihood ratio (L-value) of(k). Meanwhile, they can also be rebuilt from(k) that will be given later.That gives a complete iteration of the soft extrinsic information. We use the superscriptsprandposto identify the information before and after SISO decoding. There areMcbits mapped into, and we use(k) to denote thek-th one. The definition of the priori L-value of(k) is given by

We choose to compute it using the information ofas follows [7],

where

is the subset of X of which thek-th mapped bit is 0, andis the subset of which thek-th bit is 1.

Recall thatT=W/Mcsymbol periods can transmit a codeword withWbits. We use vectorto denote the priori L-values of the codeword cnfor usern.

Fig. 3. (a) Traditional dual-turbo iterative detection and decoding. (b) AMP dual-turbo iterative detection and decoding.

It is the belief propagation (BP) algorithm is considered to perform SISO decoding and update the L-values for usern.Suppose Kware the bit notes connect to check notew, and Wkare the check notes connect to bit notek.The soft-information for notekis given by

It worth noting that there are three iterations, the AMP iterations, the iterations to exchange soft inner information for SISO decoding, and the iterations to exchange soft extrinsic information for iterative detection and decoding. It is better to perform several AMP iterations before SISO decoding, because SISO decoding may suffer without relatively accurate prior information. It is suggested to perform about 10 AMP iterations before SISO decoding.

The turbo iterative detection and decoding needs to perform anM×Mmatrix inversion,and generateNprecoded signals via matrix and vector multiplication. Its complexity iswhereTcindicates the iterations of SISO decoding (the number of iterations to exchange soft inner information), andTdindicates the iterations to exchange the soft extrinsic information. Although the traditional dual-turbo method is a kind of low complexity turbo iterative detection and decoding scheme, its linear SISO detector still need to perform matrix inversion. The proposed AMP dual-turbo iterative detection and decoding does not need to perform matrix inversion, and thus has much lower computational load. Suppose the BMs are regularly arranged. Then the number of neighbouring BM for each BM are the same, so the sizes of Unand Bmare also the same for eachmandn. The sizes of Unand Bmare established for the majority of multi-beam systems, so we denote it asN0. In real applications, the multibeam MSC uplink channel matrix is sparse andN0is much smaller thanMandNin large multiple-beam systems. Thus, the computational complexity of the proposed AMP dual-turbo iterative detection and decoding is of orderTadenotes the iterations of AMP SISO detection, which is suggested to be set as 10 generally.TdandTcare no more than 10 as well in most case, which are all much smaller thanMandN. Therefore, the computational complexity of the proposed method can, in fact, be considered as of orderwhile the complexity of the traditional dual-turbo method is ofIt is obvi-ous that the proposed method has much lower computing load.

The proposed AMP dual-turbo method has so low computational load, because thatN0is much smaller thanMandN, that is also why we consider to introduce the proposed method into multibeam MSC scenario. By the way, beside MIMO transmissions, the iterative detection and decoding fashion can also be employed in some other LDPC-coded systems such as Bit-Patterned Media Recording [21].

V. SIMULATIONS

Computer simulations are conducted to evaluate the performance of the proposed method.We consider an GEO MSC system as described in section II with parameters listed in Table I [22, 23]. The channel coherent time is supposed to be long enough to estimate channel state information perfectly. The AWGN power measured at each spot beam is calculat-The maximum beam gain of each spot beam isand the received signal-to-noise ratio (SNR) is

Figure 4 shows the BER performance of the proposed method in comparison with the traditional turbo iterative detection and decoding given in [5] and the dual-turbo method givenin [12]. The result shows that the BER of the proposed AMP dual-turbo iterative detection and decoding is still poorer than the traditional turbo and dual-turbo methods without iterations. However, the proposed method becomes better than traditional methods after performing 3 iterations to feed back the soft extrinsic information. Hence, the proposed method has greater iteration gain than the traditional methods, and can make the iterations more effective to achieve lower BER.

Table I. System configuration.

Fig. 4. BER performance of the proposed method in comparison with the traditional method, where the code rate is K/ W= 1/3.

Fig. 5. BER performance of the proposed method in comparison with the traditional method, where the code rate is K/ W= 1/2.

Both of the proposed method and the method given in [12] employ the so-called dual-turbo idea. To highlight the improvement of the proposed method, we further compare them as shown in figure 5. It is obvious that the proposed method achieves much lower BER with 3 iterations.

VI. CONCLUSION

Multibeam antenna that can form hundreds of spot beams is a key element of current mobile satellite communications to enhance the capacity. However, inter-beam-interference may result in detection and decoding errors.Dual-turbo iterative detection and decoding that exchanges soft extrinsic information between an SISO detector and an SISO decoder iteratively is an efficient method to reduce the detection and decoding errors. We proposed to hybridize the AMP detector with LDPC SISO decoder to improve the traditional dual-turbo iterative detection and decoding scheme. The proposed method had lower computational complexity than traditional method, and has lower BER.

APPENDIX

We also useto save the number of messages [19]. Moreover, usingcan help to connect the AMP detector to the SISO decoder, as SISO decoder generatecan be used to direct update

The detail method to passandis given in algorithm 1.

Algorithm 1. AMP detection.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China under Grants 61320106003 and 61401095, and the Civil Aerospace Technologies Research Project under Grant D010109. The Fundamental Research Funds for the Central Universities under Grant YZZ17009.

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