TPC Together with Overlapped Time Domain Multiplexing System Based on Turbo Structure

Overlapped time domain multiplexing (OvTDM) is a novel technique for utilizing inter-symbol interference (ISI) to benefit a communication system. We implement the OvTDM technique based on turbo structure and associate a turbo product code (TPC) to construct a novel coded turbo-structure OvTDM system. Two schemes of the iterative receiver and soft input and soft output (SISO) decoding algorithms are presented. Simulation results show the advantage of structures in this paper. In addition, an attractive transmission rate and symbol efficiency of the designed system can also be observed.


I. INTRODUCTION
It is well known that most traditional communication systems are designed based on Nyquist criterion [1], in which intersymbol interference (ISI) should be avoided between consequent symbols. In fact, the communication system without ISI is physically unrealizable. On the other hand, people tend to design a communication system with controlled ISI, such as Fasterthan-Nyquist (FTN) signaling [2] and partial response signaling (PRS) [3]. However, these methods also treat the overlap between symbols as interference and do not really utilize it to collect the extra gain.
Based on ISI to benefit a communication system, overlapped time domain multiplexing (OvTDM) is proposed in [4]- [6].
The idea of OvTDM is to shift a data-weighted and band-limited multiplexing waveform in the time domain to achieve an overlap between different transmitted symbols and a high transmission rate. It can help to form a convolution structure among consequent symbols, so OvTDM can also be regarded as one kind of waveform convolution coding. In the OvTDM system, these overlapped parts are never regarded as ISI but rather as a beneficial encoding constraint relationship. Therefore, OvTDM can show great performance of the transmission rate and the symbol efficiency that is defined as bits per symbol [4] [5].
Maximum likelihood sequence detection (MLSD) [7] and maximum a posteriori (MAP) detection can be utilized to detect OvTDM signals. From the point of view of waveform convolution coding, the detection process can also be called OvTDM decoding.
Most previous studies have focused on the single structure of the OvTDM system. However, another way to improve the OvTDM system is to extend the coding structure [8] [9]. So, we construct a turbo structure inspired by the turbo code [10] for OvTDM. In addition, turbo product code (TPC) is employed as the forward-error-correction (FEC) module together with the turbo structure of OvTDM. In comparison to another popular FEC code, the low-density parity-check code (LDPC) [11] [12], TPC is suitable to be constructed with a shorter code length and requires fewer iterations for decoding [13] [14]. Finally, a coding system with three layers is formed, which contains the FEC code, the turbo structure and OvTDM respectively.
Comparative simulation studies with the coded QAM system show the significant advantage of the coded turbo-strucutre of OvTDM.

A. OvTDM Scheme
In the OvTDM system, we artificially introduce ISI to form an overlap among different symbols. The mapping relationship between original bits and constellation symbols can follow the rule of ordinary modulation methods. Assuming the transmitted signals followed BPSK as x = [x 0 , x 1 , · · · , x L−1 ] with length L and the multiplexing waveform as h(t), t ∈ [0, T s ) with symbol duration T s , then the transmitted signal after overlapping can be expressed as where ∆T = T s /K is the time shift between symbols. In (1), K is the number of overlapped symbols during ∆T , which is named the overlapping coefficient or the constraint length. Notice that, the larger the coefficient K is, the more serious the ISI introduced.

B. Turbo-Structure OvTDM with FEC
Fig .1 shows the transmitter of the turbo-structure OvTDM with FEC. The coded sequence that has passed the FEC encoder and one interleaver is sent to the first OvTDM encoder for the I channel. Meanwhile, the same sequence is sent to pass the other interleaver to form another sequence with a different order, which is encoded by the second OvTDM encoder for the Q channel. The output sequences from both two OvTDM encoders are combined to form a complex sequence.
The decoding process at the receiver is the key to the system design. It is based on the idea of iteration and the extrinsic information transformation [10]. During each iteration, the extrinsic information is exchanged between different decoders. The interleaver and the de-interleaver are employed to match the order of received sequences. Exchanging extrinsic information with low correlation can help to improve performance with the increase of iterations. Together with FEC, two schemes are addressed as follows: Scheme A: After one round of decoding between two OvTDM decoders, the soft information is sent to the FEC decoder.
Then the FEC decoder sends the soft information back to the OvTDM decoder. The model is shown in Fig.2. In this scheme, the FEC decoder needs to be involved in every iteration of the turbo structure.
Scheme B: As shown in Fig.3

A. MAP Algorithm for OvTDM
As discussed in the above sections, OvTDM utilizes the ISI as the encoding constraint. Thus, it can also be represented as a trellis graph [5]. The BCJR algorithm [16] is regarded as an optimal MAP method based on the trellis graph, so it can be modified to calculate the maximum a posteriori probability (APP) for OvTDM encoded bits.
Denoting the input bit at time t as x t and the received sequence at the receiver as r with length N , the log-likelihood-ratio where S t and S t−1 are assumed as states of time t and t − 1 based on the trellis graph. According to properties of OvTDM, p(S t−1 , S t , r) can be further expressed by where α t (S t−1 ) and β t (S t ) can be calculated by forward and backward recursion: Assuming the corresponding output bit at time t after OvTDM encoding as y t , then It is worth noting that there is no input bit in the tail part of OvTDM. So, β L (S L ) can be initialized directly. In the AWGN channel, where σ 2 is the variance of noise.
Let LLR of a prior probability and the extrinsic information be µ t and e t , in the iterative decoder, the extrinsic information can be obtained by which is the output of the OvTDM decoding module.

B. FBBA for TPC
One mainstream method of TPC decoding algorithm is augmented list decoding (ALD) [13] [14]. The key idea of ALD is to form a list including the most likely codewords. Based on ALD, the Fang-Battail-Buda-Algorithm (FBBA) [14] is an efficient SISO algorithm for TPC decoding to achieve near-optimum performance. FBBA is concluded as follows: Step 1: Sort the received symbols d in a decreasing order according to the LLR metric l.
Step 2: Permute the check matrix H according to the permutation pattern from the Step 1. Then, it has to be adjusted by Gauss-Jordan elimination to obtain a systematic one H π that is used to re-code the component codeword to generate a new one c π(0) .
Step 3: A codebook list is obtained through the reversal of certain positions of c π(0) and sorted in an increasing order according to where c π(i) and n are denoted as the ith codeword in the codebook list and the component codeword length.
Step 4: The soft output can be calculated by the first codeword c π(0) and the opposite to the first codeword in jth position in the codebook list.
Following the above process, soft outputs can be calculated. Generally, four iterations are sufficient for the BER performance to converge.

IV. SIMULATION STUDIES
In this section, we need to investigate the performance through some comparative simulation. We choose the Chebyshev waveform with attenuation level 80dB as the multiplexing waveform for OvTDM. Extended BCH(64, 57) is employed as the component codeword to construct a squared TPC. Thus, the code rate is R T P C = (57/64) 2 = 0.7932. The AWGN channel is considered as the transmission channel in the simulation.
As mentioned before, the turbo-structure OvTDM uses the same information sequence for both I and Q channels, so its equivalent coding rate with TPC is R OvT = 1/2 · R T P C · L/N . When the length of the information sequence is large enough,  L/N ≈ 1, so we ignore it in the simulation. BPSK is used as the original modulation for the OvTDM system. Thus, the symbol efficiency of the turbo-structure OvTDM with TPC is η OvT = R OvT · 2K = R T P C · K (bits/symbol). On the other hand, if we select M -ary QAM with TPC for comparison, its symbol efficiency is η QAM = log 2(M ) · R T P C (bits/symbol). In order to do the comparative studies under the same symbol efficiency, we select K = 6 and 64-QAM in Fig.4 as well as K = 8 and 256-QAM in Fig.5. The BER plots in both Fig.4 and Fig.5 show the significant advantage of the coded turbo-structure OvTDM. In Fig.4, the same symbol efficiency is 4.7592 (bits/symbol) and the required E b /N 0 of 64-QAM with TPC to achieve the BER < 10 −5 is 10dB, but the turbo-structure OvTDM with TPC can achieve BER < 10 −5 at 5.8dB. In Fig.5, the same symbol efficiency is 6.3456 (bits/symbol) and the required E b /N 0 of the turbo-structure with TPC using Scheme B to achieve the BER < 10 −5 is 7.4dB less than that of 256-QAM with TPC.
Moreover, Fig.6 illustrates the comparison result of the symbol efficiency among different schemes when BER < 10 −5 . The symbol efficiency of the single structure OvTDM has been shown in [5]. Also, we plot the corresponding symbol efficiency of the Shannon Limit [18]. In Fig.6, the turbo-structure OvTDM with TPC has a obvious improvement, compared with the single structure OvTDM. On the other hand, with the increase of the symbol efficiency, schemes in this paper can achieve it at a lower E b /N 0 than the Shannon Limit that represents traditional communication systems.

V. CONCLUSION
This paper mainly focuses on structures and SISO decoding algorithms of the turbo-structure OvTDM with TPC, which demostrate a significant improvement over the single structure OvTDM. In addition, compared with the coded QAM system of the same symbol efficiency, the BER performance of the turbo-structure OvTDM with TPC is much better. Simulation results shows the advantage of the turbo-structure OvTDM with TPC in the communication scenario requiring high transmission rate at a relatively low E b /N 0 .