1. Introduction
Multipath channels involve modifying a transmitted signal into several samples of this signal. So, when quitting the intersymbol interference (ISI) channel, the receiver does not notice only the transmitted signal but an overlap of these distinct samples or replies. Turbo equalization is a digital reception method that processes data altered by multipath communication channels. Turbo equalization merges the work of an equalizer and a channel decoder in conformity with the turbo concept [
1]. Descriptively, this digital reception technique consists of the equalization–interleaving–decoding–processing sequence. First of all, equalization creates an estimate of the sent data. Then, the adjusted decoder information is given to the equalization module. From one iteration to another, the equalization and decoding procedure will put together their information to achieve one single-path channel communication [
2].
It has been established that instead of separately achieving channel equalization and decoding processes, the channel’s ISI effects can be decreased by the execution of both processes iteratively [
3]. This method is developed on turbo equalization [
4]. This approach was designed for systems that use convolutional coding and binary phase-shift keying (BPSK) modulation for communication on dispersive channels. The method executes both channel equalization and decoding iteratively and has been proved to efficiently minimize the consequences of the ISI channel, as demonstrated in [
5].
During the iterations, the SISO (soft-in soft-out) equalizer and decoder exchange information, improving symbol estimation [
6,
7,
8,
9]. In
Section 4.1, this paper presents turbo-equalization based on the Log-MAP criterion [
10,
11,
12].
The current work is a continuation of the previous papers in which a performance comparison was made between the zero forcing (ZF) or MMSE equalization preceded the decoding process using different codes [
13,
14], intending to evaluate the performance of the equalization that takes place inside the decoding process as in the case of turbo equalization. In this respect, the contributions of this article can be summarized as follows:
We proposed a system, respecting the classic turbo equalization scheme, to fix the errors introduced by ISI over an AWGN channel. However, for transmission, we utilized LDPC coding. At reception, we used a system consisting of a Log-MAP equalizer and min-sum LDPC decoding, which differs from the existing systems;
Then, the functional analysis of the proposed system was realized, depending on the number of iterations within the iterative process of equalization and decoding or the number of iterations within the LDPC decoder. The proposed system demonstrated the effectiveness of the equalizer in terms of BER vs. SNR;
Then, taking three impulse responses’ h functions as mentioned in
Section 4.3 and
Section 5, the performances in terms of BER vs. SNR of the proposed system were compared. These performances differed depending on the h function that was used;
The performances in terms of BER vs. SNR of the proposed system were also correlated in comparison with other decoding and equalization systems described in the literature [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13].
The process of LDPC decoding is iterative. Each attempt to decode the input message notifies the next until a valid codeword is obtained. LDPC codes have been adopted by several standards, such as WiFi, WiMax, WiGig, DVB-S2, or 10GbaseT [
15]. We should also include the enhanced mobile broadband (eMBB), ultra-reliable and low-latency communications (URLLC), and massive machine-type communications (mMTC) applications of 5G according to [
15].
2. Related Work
Firstly, a summary of existing SISO equalization algorithms derived from classical equalization methods according to the maximum a posteriori (MAP) or minimum mean-square error (MMSE) criterion is presented. In this respect, we referred to [
4,
5,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27].
Using soft decisions provided by the decoder at the entry of an equalizer was demonstrated to considerably reduce the BER at the reception [
16,
17]. From this point of view, different SISO equalization techniques were built in the literature to improve the information from the decoding levels within the iterative decoding approach.
The idea of using a priori information from the channel decoder to improve equalization was proposed in [
4]. The decoder and the equalizer improve each other the estimate of the transmitted data in an iterative process similar to turbo codes. This first version of turbo equalization uses equalization based on the maximum-likelihood sequence detection (MLSD) criterion using the soft-output Viterbi algorithm (SOVA). The soft-output Viterbi equalizer used in [
4] represented the first soft-in soft-out equalizer within the iterative decoding method at the reception. The soft-output Viterbi equalizer consists of trellis construction of the channel being easily decoded by applying the Viterbi algorithm. Even so, it did not succeed in decreasing the BER. Then, an adaptive turbo-equalization structure was proposed in [
18]. The equalizer created on the MMSE criterion, including a priori information, was provided in [
19], employing a matrix presentation of the channel. In [
20], the principle for a turbo equalization technique whose coefficients are computed in the frequency domain was adopted.
Further, enhancements to the soft-output Viterbi equalizer were performed by the MAP algorithm to succeed in reducing the BER [
5]. In contrast to the soft-output Viterbi equalizer, the better efficiency of the MAP equalizer comes from the higher computational complexity of the MAP equalization algorithm. Max-Log MAP, described in [
21], presents a lower computational complexity in contrast to MAP equalizer but also a lower performance in terms of BER. In this sense, another method (delayed-decision-feedback sequence estimator) employed to decrease the difficulty of the MAP equalizer consisted of reducing the number of states of the trellis.
The soft-output sequential algorithm, a trellis-based equalizer presented in [
7], used the sequential estimation method within the turbo equalization process. In [
22], a soft-output Viterbi algorithm built on soft-output decision-feedback sequence estimation was detailed.
The minimum mean-square error linear equalizer (MMSE LE) was applied in [
23] to decrease computation complexity in the case of an equalizer using a priori information. Also, the minimum mean-square error decision-feedback equalizers (MMSE DFE), including or not a priori information in the finite impulse filters, supplying hard or soft decisions, were presented in [
24,
25].
The iterative techniques created on the MAP or MMSE principle, including a priori information, facilitate the removal of interferences almost totally and obtain the full benefit from the diversity of the channel. These techniques can also be used for other sources of interference. Code division multiple access (CDMA) systems need to regard the interference caused by multiple users. In this respect, in [
26], the iterative handling for CDMA is described. In addition, the principles of turbo equalization can be used in the issue of linear precoding or precoding multiantenna systems [
27].
On the other hand, we referred to other papers [
28,
29,
30,
31] where forward-error-correction decoders (such as spatially coupled low-density parity-check codes, protograph codes, repeat-accumulate codes, spatially-coupled codes, or Bose–Chaudhuri–Hocquenghem codes) were employed in connection with the Bahl–Cock–Jelinek–Raviv (BCJR) detector or minimum mean-square error linear equalizer (MMSE LE) to mitigate the impact of ISI.
In [
28], the authors integrated spatially coupled low-density parity-check codes in turbo equalization. Better said, the authors proved the performances of the spatially coupled low-density parity-check codes over the additive white Gaussian noise and inter-symbol interference channels in obtaining the target BER.
In [
29], the authors depicted the configurations of the LDPC codes through one-dimensional (OD) and two-dimensional (TD) ISI channels, which are usually adopted to describe magnetic recording systems. In this sense, LDPC codes have proved their high quality for employment in magnetic recording systems. They also briefly presented the forwards in studying LDPC versions, for example, protograph codes, repeat-accumulate (RA) codes, or spatially-coupled (SC) codes, through the mentioned ISI channels. Based on this work, the authors have created some operative combined detection and decoding procedures for LDPC codes over OD-ISI channels.
A scalable FPGA-founded architecture for a Bahl–Cock–Jelinek–Raviv (BCJR) equalizer to cancel the ISI was created in [
30]. The authors adopted a turbo BCJR equalizer in connection to a binary forward-error-correction decoder to interchange soft information as extrinsic Log-likelihood ratios in each turbo iteration.
The scope of [
31] was to prove the performance of interference linear equalizer based on the minimum mean-square error in tandem with Bose–Chaudhuri–Hocquenghem (BCH) codes. The authors compared, in terms of BER, the performances of linear turbo equalization adopting both convolutional and BCH codes.
5. Simulation Results and Discussion
The simulation results were attained using Matlab and are presented in the following. As a rule, all graphics were plotted for an SNR that varies between 0–10 dB, but where the graphics display values for SNR lower than 10 dB, it means that in those cases, no more errors were found over the respective values.
In the iterative process of equalization and decoding, there are iterations carried out in both the ensemble process (equalization–decoding) and within the LDPC decoder. It should be mentioned that in this work, five iterations were performed within the iterative process of equalization and decoding. During one iteration of the iterative receiver, the LDPC decoder performs either 10 or 20 iterations, depending on the specific situation being analyzed. This process is illustrated in
Figure 5.
We arranged all the simulation scenarios in
Table 1 to make it easier for the reader to follow.
To notice the role of the Log-MAP equalizer, two scenarios were considered. In the first scenario, iteration 0 (it0) and iteration 1 (it1) were considered in the iterative process of equalization and decoding. In iteration 0 (it0), the Log-MAP equalizer does not receive any a priori information from the min-sum LDPC decoder, thus realizing the a posteriori information only based on the data observed from the channel output [
5]. At each of the two iterations of the iterative receiver, the LDPC decoder makes ten iterations. In the second scenario, there is only one iteration (test) in the iterative process of equalization and decoding. The number of iterations in the decoder is 20, which is the sum of the cumulative iterations of the first scenario (10 by 10).
In it0, the a posteriori information from the equalizer is deinterlaced and then provided to the decoder. In the case of iteration 0 (it0) the Log-MAP equalizer does not receive any a priori information from the decoder. On the other hand, iteration 1 (it1) involves the whole iterative process of equalization and decoding. The a posteriori information from the equalizer is deinterlaced and then provided to the decoder. The Log-MAP equalizer receives the a priori information from the decoder after it has been interlaced. So, the second scenario (test) is like it0, only that the decoder makes more iterations.
The analysis was conducted by considering h1 = [0.18 0.85 0.32]. In this sense,
Figure 6a demonstrates that BER vs. SNR improves when the LDPC decoder performs ten iterations at each of the two iterations of the iterative receiver.
The role of the Log-MAP equalizer is better seen in
Figure 6b. In this case, two scenarios were considered. The first scenario makes two iterations (it0 and it1) within the iterative process of equalization and decoding, as the LDPC decoder makes 20 iterations at each of the two iterations of the iterative receiver. The second one makes a single iteration (test) within the iterative receiver, as the number of iterations in the decoder (40) is the sum of the cumulative iterations of the first scenario (20 by 20). In this case, it was considered the same h1 = [0.18 0.85 0.32].
In
Figure 6b it can be seen that at an SNR of 5 dB, the BER related to the test is located at 10
−4 while the BER related to it1 decreases, being situated between 10
−4 and 10
−5.
Next, taking the three impulse responses h1 = [0.18 0.85 0.32], h2 = [0.302 0.725 0.456], and h3 = [0.407 0.815 0.407] as mentioned in
Section 4.3, the performances in terms of BER vs. SNR of the proposed system were compared, these differing on the h function used, as it could be observed in
Figure 7a–c.
At each iteration in the iterative process of equalization and decoding, 10,000 sequences were generated, and 10 iterations took place in the LDPC decoder.
At the same time, it is observed that these performances, using the same type of channel with ISI, also differed depending on h, the best performance being obtained in the case of h1. For instance, when the signal-to-noise ratio (SNR) is at 5 dB, the bit error rate (BER) ranges between 10
−4 and 10
−5, as shown in
Figure 7a.
On the other hand, for a channel with h2 at the SNR of 5 dB, the BER is approximately 10
−1,
Figure 7b. Similarly, in the case of an ISI channel with h3, the BER is positioned between 10
−1 and 10
−2 when the SNR is at 5 dB,
Figure 7c.
The previous scenario was reconsidered in the case that the LDPC decoder achieves 20 iterations. This is reflected in
Figure 8a–c. If the decoder makes 20 iterations inside itself (within the decoding process), the performance increases, the most visible being for the channel characterized by h1. After the SNR of 4 dB, the proposed system no longer finds any errors.
Table 2 summarizes the BER vs. SNR performances of the proposed system while considering h1, h2, and h3 in two scenarios: when the LDPC decoder executes 10 or 20 iterations at each iteration of the iterative process of equalization and decoding.
To analyze the performance of the proposed system, it was compared with two other equalization and decoding systems, for the same conditions of the transmission channel. The first system consists of turbo equalization that follows the classic scheme, according to the theory mentioned in
Section 3. The recursive systematic convolutional (RSC) encoder was used in this study using the octal generating polynomials G0 = 7 and G1 = 5, at a rate R = 1/2 and constraint K = 3.
The second system consists of the equalization performed separately before the decoding process using MMSE equalizer and LDPC decoder at a rate of ½, similar to [
13].
For example, for a channel with h1, at the SNR of 5 dB, in the case of our proposed system, as was specified earlier, the BER is situated between 10−4 and 10−5 if ten iterations are made in the LDPC decoder and for 20 iterations within the LDPC decoder at the SNR of 5 dB for iterations 1–5 no more errors were found.
In the case of turbo equalization at the same SNR of 5 dB, for a channel with h1, the BER is 10
−3, as can be observed in
Figure 9a. But, for a channel with h2 or h3, the BER is situated between 10
−2 and 10
−3, as can be noticed in
Figure 9b,c.
In
Figure 10a for h1 in the case of the separate equalization performed before the decoding process, using MMSE and LDPC decoder, at the SNR of 5 dB, the BER is located between 10
−5 and 10
−6 (if the decoder makes 40 iterations). Constant performance was shown over 40 iterations with a BER between 10
−5 and 10
−6, as demonstrated in
Figure 10b.
Repeating the previous scenario,
Figure 11a and
Figure 12a show that the BER deteriorates considerably for h2 and h3 at the SNR of 5 dB, being within 10
0 and 10
−1. The graphics in
Figure 11b and
Figure 12b demonstrate a relatively constant level of BER over 40 iterations.
Table 3 summarizes the BER vs. SNR performances of turbo equalization and separate MMSE equalization with LDPC decoding.
Table 4 summarizes the BER vs. SNR performances of the proposed system, turbo equalization, and separate MMSE equalization with LDPC decoding.
It can be noticed that the proposed system’s results for h1 are comparable to separate equalization with MMSE and better than turbo equalization, as can be deduced from
Table 1 and
Table 2. On the other hand, MMSE with LDPC yields poor results, ranging from 10
0 to 10
−1 for h2 and h3.
The system model proposed in this study (LDPC decoder performs 20 iterations) outperforms turbo equalization for an SNR higher than 4 dB for the channel with h1.
For a channel with h2 and h3, for an SNR higher than 7 dB, the system proposed by us (when LDPC decoder performs 20 iterations) is more efficient than turbo equalization and MMSE with LDPC decoding.
If we deviate from the standard and do not puncture the first 2Zc bits and transmit 440 bits in the case of the proposed system, for the three h, the results are presented as follows in
Figure 13a–c. The coding rate is approximately ½, not being adjusted by puncturing. We considered the situation in which the LDPC decoder performs 20 iterations during one iteration of the iterative process of equalization and decoding.
Table 5 highlights the performances of the system proposed in this study (the LDPC decoder makes 20 iterations at one iteration of the iterative equalization and decoding process) with puncturing and without puncturing the first 2Zc bits, for the SNR of 5 dB.
The results are better than in the case of puncturing the first 2Zc bits according to standard.
The equalizers and decoders do not always match equally well to each other to obtain satisfactory results. But in this paper, it was found a combination between the Log-MAP equalizer and the min-sum LDPC decoder that offers the presented performances. In addition, it was made a comparison of the proposed scheme with other established models in the specialized literature, such as turbo equalization or MMSE equalization combined with LDPC decoding.
The impulse responses were not chosen randomly; they can be found in the specialized literature and are referenced in the text.
6. Conclusions
In this study, we proposed a system that respects the classic turbo-equalization scheme. We used an LDPC coder for transmission and a Log-MAP equalizer for reception, connected with a min-sum LDPC decoder.
We performed the performance analysis of the proposed system, BER vs. SNR, considering three different h functions. It has been experimentally demonstrated that increasing the number of iterations performed by the LDPC decoder from 10 to 20 in the iterative process of equalization and decoding leads to improved results (as can be observed in
Table 2,
Section 5).
Then, the proposed system was compared with turbo equalization and separate equalization with MMSE and LDPC decoding in terms of BER vs. SNR, considering the three different h functions.
Based on the analyzed results, the equalization performance depends on both the impulse response of the channel and the chosen decoding and equalization method; some decoding and equalization systems offered better results, depending on h function (as can be observed in
Table 4,
Section 5).
Not puncturing the first 2Zc bits, the proposed system achieved better BER vs. SNR performance (as can be observed in
Table 5,
Section 5).
In conclusion,
Table 6 compares the performances in terms of BER vs. SNR of the proposed approach to others published in the literature, demonstrating its performances.
The comparison with other papers in the literature is not the most eloquent because the simulation parameters and the methods differ from those proposed in this paper.
In the future, channel estimation, using the least-square (LS) channel estimator, can be integrated into the iterative process of equalization and decoding, and thus BER performance can be improved during iterations.
We should also consider performing an EXIT chart analysis as a future perspective.
In this paper, we have limited ourselves to LDPC codes and turbo codes, as they are currently considered to be among the most relevant [
52,
53].