Yang Wang，Dan－Feng Zhao，Xi Liao

(College of Information and Communication Engineering，Harbin Engineering University，Harbin 150001，China)

1 Introduction

MIMO techniques can achieve significant capacity gains in the terrestrial wireless communications［1］.The appealing gain is driving the study of satellite MIMO systems.A promising scheme to form a MIMO matrix channelin satellite communications is the dual polarized satellite MIMO system［2］.

The dual polarized(DP)land mobile satellite (LMS)MIMO channelis investigated through experiments in Refs.［3－6］and the capacity gain is verified.Refs.［7－8］proposed two different DP LMS MIMO channelmodelsin which the polarization correlation， LOS shadowing and cross－polar discrimination(XPD)are considered.The performance of different space time codes in dual polarized mobile satellite broadcasting systems is simulated in Refs.［9－11］.In Ref.［12］，the performance of a DP LMS MIMO system with convolutional codes and iterative detection and decoding(IDD)receivers is evaluated. The IDD algorithm can suppress the cross polarization interference and achieve good performance in the DP LMS MIMO channel，but the complexity is high due to the iteration.

In this paper，we propose a novel IDD algorithm for the DP LMS MIMO system with concatenated codes. The decoding results from the outer code are fed back and exploited to reduce the computational complexity and improve the performance.The proposed algorithm is named asan iterative detection and decoding algorithm with outer code decision feedback(IDDODF).The advantages of the algorithm are verified by extensive simulations under the DP LMS channel.As is known，the concatenated code is employed in many systems to achieve good performance.For example，the BCH－LDPC code is used in the DVB－NGH system and combined with MIMO techniques［12］.The proposed algorithm is also applicable to those systems.

2 System Model and Dual Polarized LMS MIMO Channel

We consider a MIMO system comprising a dual circular polarized satellite and a dual circular polarized user terminal(UT).Fig.1 shows a generic transmitter block diagram.The BCH－LDPC concatenated code is used in the system.The output sequence is V－BLAST coded and two subsequences of length k2are obtained. AfterLDPC encoding and mapping ofthe two subsequences，we get a frame.

Fig.1 Transmitter block diagram

The channel is assumed to be invariant within a frame.Thus，the received symbols during a frame can be written as

where H is the 2×2 channel matrix and its elements hijdenote the fading factors of the sub－channels;X is a matrix consisting of transmitting symbols in a frame;V is the zero－mean complex Gaussian noise.

A two－state DP LMS MIMO channel model is complemented according to Ref.［7］.The sub－channel fading factor incorporates both the large－scale and small－scale fading component，as shown in Eq.(1). The XPD， polarization correlation and temporal correlation are also considered in the model.Besides，the very slow variation of the channel is modeled by the transition between two states.

3 IDD-ODF Algorithm

The traditional joint iterative MIMO detection and LDPC decoding algorithm can achieve the near optimal performance and be applied to the BCH－LDPC coded MIMO system.Fig.2 showsthediagram ofthe traditional IDD algorithm.In this paper，the BM algorithm is used for the BCH decoder.

It is assumed that the system employ a(n1，k1) BCH code and a(n2，k2)LDPC code.Let M denote the size of the constellation.Towards generating a frame，the transmitter buffers a sequence of Lk1bits.The bit sequence isBCH encoded and interleaved.The parameters of the block interleaver are selected to be

Fig.2 Traditional IDD receiving scheme

In the traditional IDD algorithm all received symbol vectors in a frame need to be detected and the a posterioriprobability ofthe symbolhave to be recalculated at each iteration.So the numberof operations in the traditional IDD algorithm is large.In order to reduce the computational complexity and get more performance gain，the IDD－ODF algorithm is proposed for the system.The block diagram of the proposed algorithm is given in Fig.3.

Fig.3 IDD-ODF receiving scheme

Unlike the traditional IDD algorithm，the data is deinterleaved and BCH decoded after each LDPC decoding in the IDD－ODF algorithm.The output from the LDPC and BCH decoder is fed back to the MIMO detector simultaneously at each iteration.The MIMO detector uses the BCH decoding results to control the detecting list and reduce the number of symbol vectors which need to be detected again.As a result，the times of matrix inversion can be reduced significantly when the MMSE detection is used.

Furthermore，the hard－decision code words that are decoded successfully can be used to update the soft information of corresponding bits，which makes the initial probability for the BP decoder more reliable and may improve the system performance further.

The IDD－ODF algorithm has the following steps:

Step 1 According to the decoding flags，the symbol duration indices of symbol vectors corresponding to the successfully decode BCH codeword are founded and generate a set U.Then the complementary set is given by S={n|n=1，2，…N}－U，which represents the detecting list and is kept in the MIMO detector.The received signal and the a priori symbol information Pin(i=1，2;n∈S)for the detector are updated.

where xndis the successfully received symbol vector during the symbol period n;Hdrepresents the related parts of H.

During the first iteration，the flags of the LDPC and BCH decoding are set 0.The priori symbol information is initialized using Eq.(2).

where s∈{s1，s2…sM}denotes a modulation symbol.

Step 2 The MIMO detector calculates the soft output using the updated Pin，yn(n∈S).Then Eqs. (3)and(4)are used to update the initial probability of the BP decoder.

Step 3 The signalis LDPC decoded，deinterleaved and BCH decoded.Then the soft information from the LDPC decoder and the BCH harddecision decoding results are fed back to the MIMO detector and the flags of the LDPC and BCH decoding are updated.

Step 4 Steps 1－3 are repeated until the stop condition is fulfilled.

The MIMO detecting and decoding contribute the most to the total complexity.The number of BCH decoding is increased in the IDD－ODF algorithm due to the feedback.Meanwhile，the number of the symbol vectors need to be detected in the MIMO detector is decreased.Table 1 lists the numbers of the MIMO detecting and BCH decoding when processing a frame with IDD and IDD－ODF algorithm.

Table 1 Number of MIMO detecting and BCH decoding

In the table，Nsis the number of transmitting symbol vectors corresponding to a BCH code word and NBis the number of BCH code words per frame;Imis the maximum numberoftheouteriteration;Ndrepresents the reduced number of symbol vectors need to be detected and is given by

where Pland Pbrepresent the probabilities that LDPC and BCH code words are decoded unsuccessfully.

The number of multiplications is

4 Simulation Results and Analysis

According to the previous sections，we establish a DP LMS MIMO system model.The BCH code with rate 2/3 and QPSK modulation are employed.The performance of the proposed algorithm is evaluated and compared with the traditional one under different channel conditions.Table 2 lists the parameters of the DP LMS MIMO channel［7］.Unless otherwise mentioned，the parameters in Table 2 are used in all the simulations. at least when processing a symbol vector with MMSE algorithm［14］and is 2t2－2t+1 at most for the BM algorithm［15］.Ntand t representthe number of transmitting antennas and the error－correction capability of the BCH code respectively.As a result the reduced number of multiplications can be represented as

Table 2 Parameters of the DP LMS MIMO channel

The time series generated by the DP LMS MIMO channel model using the given parameters is illustrated in Fig.4.The difference of the co－polar and cross－polar sub－channel in power level is obvious because of the XPD.The transition between the“GOOD”and the“BAD”state can be observed in the figure which represents the very slow variation of the channel.

Fig.5 shows the BER performance of the proposed and traditional algorithm under different XPDant.When the XPDantis high，the BER performance is better due to the suppression of the cross polarization interference. Besides，as shown in Fig.5，the IDD－ODF algorithm performs better than the IDD algorithm in both cases. The performance gain is about 0.2 dB.

Fig.4 Time series for the DP LMS MIMO channel

Fig.5 BER performance under different XPDant

In this paper，the processing of one symbol vector in the MIMO detector isregarded asone basic operation.The average number of the basic operation processing a frame is shown in Fig.6.The IDD－ODF algorithm can decrease the operation number by several hundred，which means several hundred times of matrix inversion can be reduced when processing a received frame. Specifically， the reduced number of multiplications is about 1.6×104according to Eq.(5)，when XPDantequals to 10 dB.The numberof multiplications is decreased significantly using the proposed algorithm.The advantage willbe more obvious when a more complex MIMO detector is used. Besides，the reduction issimilarunderdifferent channel conditions.

Fig.6 Average operation number under different XPDant

In Figs.7 and 8，we show the BER performance and the average operation number of the system with different LDPC codes.The results in Fig.7 indicate that the performance gain achieved by the proposed algorithm is about 0.4 dB when the rate 2/3 LDPC code is used and the IDD－ODF algorithm can provide more performance gain when using the higher rate LDPC code.Moreover，the reduction of the operation number is also more considerable in the system with higher rate LDPC code，as illustrated in Fig.8.

Fig.7 BER performance when using different LDPC codes

Fig.8 Average operation number when using different LDPC codes

5 Conclusions

In this work，an IDD－ODF algorithm is proposed forthe dualpolarized LMS MIMO system with concatenated codes.The performance of the algorithm is simulated and compared with the traditional IDD algorithm underdifferentchannelconditions.The results show that the proposed IDD－ODF algorithm can reduce the computational complexity significantly and achieve better performance under the dual polarized LMS channel.Besides，the benefit of the proposed algorithm in the BER performance gain and reducing computational complexity becomes more considerable when the higher rate LDPC code is used.

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