Decision Feedback Equalization-Based Low-Complexity Interference Cancellation and Signal Detection Technique Based for Non-Orthogonal Signaling

Abstract

FTN signalling is an effective communication method that achieves a high spectral efficiency. However, employing a symbol rate faster than the Nyquist rate disrupts the orthogonality between symbols, leading to unavoidable inter-symbol interference (ISI). To mitigate the effects of ISI, interference cancellation and signal detection processes are essential for FTN receivers. Conventional ISI reduction techniques often utilize trellis-based algorithms. However, as the number of states increases due to additional interference symbols, the complexity of these algorithms grows exponentially. To address this challenge, this paper explores a matrix computation-based interference cancellation technique tailored for FTN communication systems, aiming to significantly reduce the complexity of the ISI mitigation process. To execute ISI cancellation and signal detection more precisely, the proposed technique includes iterative interference cancellation and a signal detection process. When six interference symbols are considered, the complexity of the proposed technique is reduced by 97% compared with that of the conventional Viterbi algorithm. Furthermore, in the case of τ = 0.85, the performance of the proposed technique is about 1 dB better than that of the Viterbi algorithm at BER = ${10}^{-4}$.

1. Introduction

To increase the spectral efficiency of a communication system, faster-than-Nyquist (FTN) signalling has received considerable attention. FTN signalling, first introduced by Mazo in the 1970s [1], demonstrated that binary data with an ideal sync waveform could achieve transmission rates up to 25% faster than the Nyquist rate without a reduction in the minimum distance. In recent years, research has focused on adapting FTN signalling for practical communication scenarios, incorporating M-ary data symbols and square-root-raised cosine pulses [2,3]. Furthermore, to adopt an FTN signal in a practical communication system, various studies have been executed [4,5,6,7,8]. Hefinawy and co-workers [4] investigated a spectral shaper to constrain both the input and output power in an FTN signalling system. In [5], the researchers investigated the H-ARQ technique for the creation of a single carrier frequency domain equalization environment for an uplink with FTN signalling. Furthermore, for an FTN system, a phase detection scheme was investigated in [6].

The primary goal of the FTN scheme is to enhance the transmission rate while maintaining the same bandwidth, transmission power, and number of antennas. This is achieved by employing a symbol rate that exceeds the Nyquist rate [2]. In conventional communication systems, the Nyquist rate is typically used to ensure the maximum transmission rate without causing inter-symbol interference (ISI). However, since FTN signalling operates at a rate beyond the Nyquist limit, the occurrence of ISI becomes unavoidable.

1.1. Related Work

In communication systems, inter-symbol interference (ISI) is often caused by multipath fading channels. Traditional methods for mitigating ISI include the maximum likelihood sequence estimation (MLSE) approach [8] and the Viterbi algorithm (VA) [9]. Although MLSE provides the optimal results, its computational demands are exceptionally high. For FTN communication systems, researchers have explored ISI mitigation techniques leveraging the VA [10]. Prlja and co-authors proposed a truncated version of the VA to handle ISI in FTN signalling scenarios [11]. Following this, the Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm was investigated as an alternative for mitigating ISI in FTN systems [12,13,14].

To date, the majority of ISI cancellation approaches for FTN signalling have relied on trellis decoders. Since their introduction in the 1970s [9,11], trellis decoders have undergone continuous enhancements [10,12,13,14,15,16]. However, the computational burden of trellis decoders significantly increases with the number of interference symbols and the modulation order, as the number of states expands exponentially. Methods such as the truncated VA [12] and the reduced BCJR [15] have been introduced to address this issue by reducing the number of states. Unfortunately, this simplification comes at the cost of a reduced performance. Recently, various detection techniques have been researched for FTN signalling [17,18,19]. Ref. [17] proposes a spatially coupled approach for FTN signals to effectively deal with ISI. It integrates detection and code design while reducing complexity using EXIT chart analysis. Since this work focuses on reducing complexity by integrating code design and detection, this work requires complicated signal processing for the transmitter. However, our work does not require additional signal processing for the FTN-based transmitter. Ref. [18] further improves the decoding performance by solving the error floor problem of FTN signals using LDPC-RS concatenation and a decoding scheme. However, since the study carried out in [18] focuses on error floor reduction by channel coding, an active signal detector is required to reduce the ISI. The study carried out in [19] explores frequency shift monitoring of optical filters in FTN-WDM transmission systems, focusing on optical communication applications. Since this research requires frequency doming signal processing, additional fast Fourier transform [FFT] should be inserted to the receiver.

Recently, to reduce the ISI, various interference cancellation techniques have been investigated, such as a transmission technique [20,21] and an interference alignment scheme [22,23]. In particular, in multiple input, multiple output (MIMO) signal processing, interference cancellation is simply executed based on a signal matrix computation [24,25,26]. Because the interference from a MIMO system is generated at one symbol time, the MIMO signal can be easily detected through simple signal processing. Unlike MIMO signal processing, FTN interference spreads across multiple symbols, which are continuously connected with each other. This study explores an ISI mitigation approach that leverages matrix computation. A straightforward method for interference cancellation within this framework involves utilizing an inverse matrix [24]. Additionally, techniques such as decision feedback equalization (DFE) and successive interference cancellation (SIC) can also be applied using matrix computation [27,28,29]. These approaches are well-known signal detection techniques commonly employed in MIMO systems. However, due to the distinct characteristics of MIMO and FTN communication environments, directly applying MIMO detection techniques to FTN signalling is challenging without modifications.

1.2. Contribution and Novelty of This Work

In our work, efficient interference cancellation and signal detection schemes are described for an FTN-based communication system. The proposed interference cancellation technique includes previous interference cancellation and post interference cancellation to reduce continuous interference. The main contribution of this paper is to develop a low-complexity interference cancellation and signal detection technique for FTN-based communication systems. Unlike existing methods that rely on computationally intensive trellis-based algorithms, our approach significantly reduces the computational burden by leveraging matrix-based computation. The main contributions are as follows:

Complexity Reduction: The proposed method achieves a complexity reduction of about 97% compared to the Viterbi algorithm while maintaining equivalent or better performance.

Scalability and Practical Implementation: The simplicity of the matrix-based method ensures both memory efficiency and scalability, making it suitable for practical deployment in modern communication systems.

Novelty: The novelty of this work lies in combining iterative interference cancellation and signal detection within a matrix-based computational framework. This approach provides a simple solution for ISI reduction without the exponential complexity increase experienced in grid-based solutions.

The simulation results indicate that the complexity of the proposed technique with a matrix is reduced by 97% compared with that of a conventional Viterbi algorithm with six interference symbols. Furthermore, in the case of τ = 0.85, the performance of the proposed technique is about 1 dB better than that of the VA at BER = ${10}^{-4}$.

The rest of this paper is organized as follows. Section 2 presents the FTN system model based on a matrix computation. Section 3 describes the interference cancellation and signal detection technique with a matrix computation for an FTN signal. Next, the proposed partial DFE detection techniques are described in Section 4 and Section 5. Simulations are reported in Section 6, followed by some concluding remarks in Section 7.

2. FTN Signal Modelling Based on Matrix Computation

A transmission signal with a baseband pulse g(t) whose bandwidth is 1/(2T) can be written as

(1)$s(t)={\displaystyle \sum _k{a}_k}•g(t-k\tau T),$

where $a_k$ is the k-th data symbol, and τT is the symbol time $(0<\tau <1)$. Therefore, factor τ controls the symbol time in which the pulses come quickly by a factor of 1/τ [3]. Figure 1 shows the signal waveform for the Nyquist rate without ISI and the FTN signal waveform with ISI. In the conventional Nyquist rate where τ is not applied, orthogonality between symbols is maintained, so no interference occurs. However, when τ < 1 is applied, the spacing between symbols becomes narrow. In this case, the symbol transmission rate increases, but ISI occurs due to the overlap between symbols. In this paper, the data symbol $a_k$ is considered as a M-ary quadrature amplitude modulation (QAM) or phase shift keying (PSK) symbol. The pulse g(t) has a narrower band than 1/2τT, and consequently, lSI occurs. In addition, g(t) has a unit energy of ${\displaystyle {\int }_{-\infty }^{\infty }|}g(t){|}^{2}dt=1$. For a practical model, a root-raised cosine pulse is considered for g(t). For the excess bandwidth α, a value of 0.3 is considered, i.e., α = 0.3, which is a common and practical value.

After an additive noise channel and post filtering are applied, the received signal can be written as

(2)$y(n\tau T)={\displaystyle \sum _k{a}_k}•x((n-k)\tau T)+w(n\tau T),$

where $x(t)=g(t)*g{(-t)}^{*}$, and $w(t)$ is additive noise with an $N_0/2$ power spectral density. The signal processing procedure is as follows [14]: data for each $n\tau T$ → transmit filter $g(t)$ → additive noise $w(t)$ → $y(n\tau T)={\displaystyle \sum _k{a}_k}•x((n-k)\tau T)+w(n\tau T)$-matched filter $g{(-t)}^{*}$ → sample at nτT.

Let us consider two previous symbols and two post symbols as interference symbols in FTN signalling. In the detection of the k-th symbol $a_k$, the two previous and post symbols of $a_k$ are interference in the form of $\left[a_{k-2},\hspace{0.17em}{a}_{k-1},\hspace{0.17em}{a}_{k+1},\hspace{0.17em}{a}_{k+2}\right]$. The number of interference symbols is decided based on the value of τ. In this paper, the number of interference symbols is defined by parameter N. In the above case, the value of N is 2, i.e., N = 2. And, the filter coefficients, $\left[x_{-2},\hspace{0.17em}{x}_{-1},\hspace{0.17em}{x}_1,\hspace{0.17em}{x}_2\right]$ are the coefficients for the interference symbols. The values of $x_{-1}$ and $x_1$ are the first interference coefficients, and are the same. In the same way, the values of $x_{-2}$ and $x_2$ are the second interference coefficients, and are also the same. According to the above explanation, the k-th-received signal $y_k$ can be described as

(3)$$\begin{gathered} y_k=\left[x_N{x}_{N-1}\cdots x_1{x}_0{x}_1\cdots x_{N-1}{x}_N\right]\times \\ [a_{k-N}{a}_{k-N+1}\cdots a_{k-1}{a}_k{a}_{k+1}\cdots a_{k+N-1}{a}_{k+N}{]}^T+w_k. \end{gathered}$$

When the sequence length L is 6 (L = 6), the overall received signal can be expressed as

(4)$$\begin{gathered} \mathit{Y}=\mathit{X}\mathit{A}+\mathit{W} \\ =\left[\begin{array}{c}\begin{array}{c}{x}_0\\ x_1\\ x_2\\ 0\\ 0\\ 0\end{array}\begin{array}{c}{x}_1\\ x_0\\ x_1\\ x_2\\ 0\\ 0\end{array}\begin{array}{c}{x}_2\\ x_1\\ x_0\\ x_1\\ x_2\\ 0\end{array}\begin{array}{c}0\\ x_2\\ x_1\\ x_0\\ x_1\\ x_2\end{array}\begin{array}{c}0\\ 0\\ x_2\\ x_1\\ x_0\\ x_1\end{array}\begin{array}{c}0\\ 0\\ 0\\ x_2\\ x_1\\ x_0\end{array}\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6\end{array}\right]+\left[\begin{array}{c}{w}_1\\ w_2\\ w_3\\ w_4\\ w_5\\ w_6\end{array}\right], \end{gathered}$$

where the received signal Y is represented by multiplying the data symbols by the filter coefficients. For a general expression, Equation (4) can be written as follows:

(5)$$\begin{gathered} \mathit{Y}=\mathit{X}\mathit{A}+\mathit{W} \\ =\left[\begin{array}{c}\begin{array}{c}{\mathit{x}}_{\mathbf{1,1}}\\ \mathbf{⋮}\\ {\mathit{x}}_{\mathit{L}\mathbf{,}\mathbf{1}}\end{array}\begin{array}{c}\mathbf{\cdots }\\ \mathbf{\ddots }\\ \mathbf{\cdots }\end{array}\begin{array}{c}{\mathit{x}}_{\mathbf{1}\mathbf{,}\mathit{L}}\\ \mathbf{⋮}\\ {\mathit{x}}_{\mathit{L}\mathbf{,}\mathit{L}}\end{array}\end{array}\right]\left[\begin{array}{c}{\mathit{a}}_{\mathbf{1}}\\ \mathbf{⋮}\\ {\mathit{a}}_{\mathit{L}}\end{array}\right]\mathbf{+}\left[\begin{array}{c}{\mathit{w}}_{\mathbf{1}}\\ \mathbf{⋮}\\ {\mathit{w}}_{\mathit{L}}\end{array}\right]. \end{gathered}$$

3. Matrix Computation-Based Interference Cancellation

Interference cancellation and signal detection can be efficiently achieved using the inverse of the filter coefficient matrix. This approach involves multiplying the received signal vector Y by the inverse of the filter coefficient matrix X. The Moore–Penrose pseudo-inverse, represented as ${(•)}^{†}$, is employed for the matrix inversion. The inverse of the filter coefficient matrix is calculated as follows:

(6)$\mathit{H}={\mathit{X}}^{†}=({\mathit{X}}^H\mathit{X}{)}^{-1}{\mathit{X}}^H,$

where ${(•)}^H$ denotes the conjugate transpose, X is filter coefficient matrix with $L\times L$ size, and the size of H is $L\times L$. The interference cancellation and signal detection are executed as follows:

(7)$\widehat{\mathit{A}}=\mathit{H}\mathit{Y}=\mathit{A}+\mathit{H}\mathit{W},$

where $\widehat{\mathit{A}}$ is the signal detected with L $\times$ 1, Y is the signal received with L $\times$ 1, and W is the additive noise with $L\times 1$.

By utilizing a baseband demodulator and $\widehat{A}$, the transmitted symbols can be easily detected. Nevertheless, this approach may result in significant errors due to the noise amplification that occurs when the inverse matrix H is multiplied.

The DFE cancellation method, which relies on QR decomposition, is commonly applied for signal detection in MIMO systems [27]. This technique can also be applied for FTN signalling by utilizing the filter coefficient matrix. Initially, the QR decomposition of the filter coefficient matrix X is performed as follows:

(8)$\mathit{X}=\mathit{Q}\mathit{R}=\mathbf{Q}\left[\begin{array}{cccc}{r}_{\mathrm{1,1}}& r_{\mathrm{1,2}}& \cdots & r_{1,L}\\ 0& r_{\mathrm{2,2}}& \cdots & r_{2,L}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & r_{L,L}\end{array}\right],$

where R is an upper triangular matrix with $L\times L$, and Q is an orthonormal matrix with $L\times L$ satisfied by $Q^{H}Q=I_L$. By multiplying $Q^H$, an $L\times 1$ output vector can be expressed as

(9)$$\begin{gathered} \mathit{Z}={\mathit{Q}}^H\hspace{0.25em}\mathit{Y}=\mathit{R}\mathit{A}+\eta \\ =\left[\begin{array}{cccc}{r}_{\mathrm{1,1}}& r_{\mathrm{1,2}}& \cdots & r_{1,L}\\ 0& r_{\mathrm{2,2}}& \cdots & r_{2,L}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & r_{L,L}\end{array}\right]\times \left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_L\end{array}\right]+\left[\begin{array}{c}{\eta }_1\\ {\eta }_2\\ ⋮\\ {\eta }_L\end{array}\right], \end{gathered}$$

where $\eta$ represents the noise vector resulting from the transformation of the additive noise W through the orthonormal matrix Q, $\eta ={\mathit{Q}}^H\mathit{W}$, and $\mathit{Z}=[z_1{z}_2\cdots z_L]$. The first symbol is detected as

(10)${\widehat{a}}_L=\zeta [Z_L/r_{L,L}]=\zeta [(r_{L,L}{a}_L+{\eta }_L)/r_{L,L}],$

where $\zeta [•]$ is the mapping function, which maps its inner term to the nearest QAM/PSK symbol by checking the Euclidean distance. After the first symbol detection, the remaining symbols are detected as

(11)$\begin{array}{l}{\widehat{a}}_{L-1}=\zeta [(Z_{L-1}-r_{L-1,L}•{\widehat{a}}_L)/r_{L-1,L-1}]\dots \\ {\widehat{a}}_l=\zeta [(Z_l-{\displaystyle \sum _{i=l-1}^L{r}_{l,i}•{\widehat{a}}_i})/r_{l,l}]\cdots \\ {\widehat{a}}_1=\zeta [(Z_1-{\displaystyle \sum _{i=2}^L{r}_{1,i}•{\widehat{a}}_i})/r_{1,1}].\end{array}$

The two aforementioned schemes can be directly applied to FTN signalling without modifications. However, since the matrix size corresponds to the frame size, L, the memory required for matrix computation can become excessively large.

Figure 2 illustrates the BER performance of the proposed method utilizing the overall matrix computation, as described in Equations (7) and (11). As the value of τ decreases, the BER performance deteriorates. For τ = 0.9, the proposed techniques outperform the VA, with both linear and DFE detections showing comparable performances. In cases where τ = 0.85 and τ = 0.8, similar to MIMO systems, the DFE detection scheme demonstrates a superior performance compared to linear detection [27]. Additionally, at higher Eb/N0 values, the DFE detection provides better performance than the VA. However, as these interference cancellation and symbol detection methods rely on full matrix computation, the memory requirements are substantial, making practical implementation challenging.

4. Partial DFE Methods for FTN Signalling

In this paper the parentheses surrounding superscripts such as (1) and (2) are intentionally used to distinguish them from exponents or powers. To achieve a low complexity and efficient signal detection, this section presents a partial DFE technique tailored for FTN signalling. In this approach, C symbols are detected simultaneously in a single detection step. This process is then repeated M times, where $M=⌈L/C⌉$, and $⌈\Delta ⌉$ represents the smallest integer greater than or equal to $\Delta$. Figure 3 describes the FTN signal model with C = 3, N = 2, L = 9, and M = 3. In Figure 3, three symbols are detected in one detection step using a $3\times 3$ filter matrix. In the first detection step, $[a_1,\hspace{0.17em}{a}_2,\hspace{0.17em}{a}_3]$ are detected by $X_1$ with a $C\times C$ size, and $[a_4,\hspace{0.17em}{a}_5,\hspace{0.17em}{a}_6]$ are then detected by $X_2$ with a $C\times C$ size. Finally, $[a_7,\hspace{0.17em}{a}_8,\hspace{0.17em}{a}_9]$ are detected by $X_3$ with a $C\times C$ size. In Figure 3, the filter matrix X has a repetitive pattern. The partial diagonal matrices are exactly the same as each other, i.e., $\left(X_1=X_2=X_3\right)$. In addition, the partial lower filter matrix $X_1^{(-)}$ with $C\times C$ size has the same elements as $X_2^{(-)}$ with $C\times C$ size, $\left(X_1^{(-)}=X_2^{(-)}\right)$. Furthermore, the partial upper filter matrices are identical $\left(X_2^{(+)}=X_3^{(+)}\right)$. ${\mathit{X}}^{(-)}$ is applied to the previous interference. Therefore, ${\mathit{X}}^{(-)}$ can be used to remove the previous interference. And ${\mathit{X}}^{(+)}$ is applied to the post interference. Therefore, ${\mathit{X}}^{(-)}$ can be used to remove the post interference. Furthermore, each partial filter matrix can be generalized. The diagonal partial filter matrix can be generalized as

(12)$\begin{array}{c}X=\left[\begin{array}{cccc}{X}_{1,1}& X_{1,2}& \cdots & X_{1,C}\\ X_{2,1}& X_{2,2}& \cdots & X_{2,C}\\ ⋮& ⋮& \ddots & ⋮\\ X_{C,1}& X_{C,2}& \cdots & X_{C,C}\end{array}\right]\end{array}$

Equation (12) is similar to Equation (8). But, Equation (8) is a formula included in Section 3 that operates on the entire matrix, and Equation (12) is a formula included in this Section that operates on only a partial filter matrix, so, the sizes of the matrices are different.

The lower and upper partial filter matrices can, respectively, be generalized as

(13)$\begin{array}{c}{X}^{(-)}=\left[\begin{array}{cccc}{X}_{1,1}^{(-)}& X_{1,2}^{(-)}& \cdots & X_{1,C}^{(-)}\\ X_{2,1}^{(-)}& X_{2,2}^{(-)}& \cdots & X_{2,C}^{(-)}\\ ⋮& ⋮& \ddots & ⋮\\ X_{C,1}^{(-)}& X_{C,2}^{(-)}& \cdots & X_{C,C}^{(-)}\end{array}\right]\end{array},$

(14)$\begin{array}{c}{X}^{(+)}=\left[\begin{array}{cccc}{X}_{1,1}^{(+)}& X_{1,2}^{(+)}& \cdots & X_{1,C}^{(+)}\\ X_{2,1}^{(+)}& X_{2,2}^{(+)}& \cdots & X_{2,C}^{(+)}\\ ⋮& ⋮& \ddots & ⋮\\ X_{C,1}^{(+)}& X_{C,2}^{(+)}& \cdots & X_{C,C}^{(+)}\end{array}\right]\end{array}.$

In the first step, a QR decomposition is executed for partial diagonal matrix X, which can be written as

(15)$$\begin{gathered} \mathit{X}=\mathit{Q}\mathit{R} \\ =\mathbf{Q}\times \left[\begin{array}{cccc}{\mathit{r}}_{1,1}& {\mathit{r}}_{1,2}& \cdots & {\mathit{r}}_{1,\mathit{C}}\\ 0& {\mathit{r}}_{2,2}& \cdots & {\mathit{r}}_{2,\mathit{C}}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\mathit{r}}_{\mathit{C},\mathit{C}}\end{array}\right]. \end{gathered}$$

For the detection of C symbols, the first C signals received are selected

(16)$$\begin{gathered} {\mathit{Y}}^{\left(1\right)}=[y_1{y}_2\cdots y_C{]}^T \\ =[y_1^{\left(1\right)}{y}_2^{\left(1\right)}\cdots y_C^{\left(1\right)}{]}^T, \end{gathered}$$

where the superscript of $y_c^{\left(l\right)}$ indicates the detection step, and the subscript indicates the order of symbols detected in each detection step. The DFE is then executed. By multiplying $Q^H$, the $C\times 1$ output vector can be expressed as

(17)$$\begin{gathered} {\mathit{Z}}^{\left(1\right)}={\mathit{Q}}^H\hspace{0.25em}{\mathit{Y}}^{\left(1\right)}=\mathit{R}{\mathit{A}}^{\left(1\right)}+{\psi }^{\left(1\right)} \\ =\left[\begin{array}{cccc}{r}_{\mathrm{1,1}}& r_{\mathrm{1,2}}& \cdots & r_{1,C}\\ 0& r_{\mathrm{2,2}}& \cdots & r_{2,C}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & r_{C,C}\end{array}\right]\times \left[\begin{array}{c}{a}_1^{\left(1\right)}\\ a_2^{\left(1\right)}\\ ⋮\\ a_C^{\left(1\right)}\end{array}\right]+\left[\begin{array}{c}{\psi }_1^{\left(1\right)}\\ {\psi }_2^{\left(1\right)}\\ ⋮\\ {\psi }_C^{\left(1\right)}\end{array}\right]. \end{gathered}$$

In (17), ${\psi }^{(1)}$ is

(18)$$\begin{gathered} {\psi }^{\left(1\right)}={\mathit{Q}}^{\left(1\right)}\left({\mathit{W}}^{\left(1\right)}+{\mathit{P}}^{\left(1\right)}\right) \\ \hspace{0.17em}={\eta }^{\left(1\right)}+{\mathit{Q}}^{\left(1\right)}{\mathit{P}}^{\left(1\right)} \end{gathered}$$

where $P^{(1)}$ is the interference term. In the first detection step, $P^{(1)}$ is demonstrated as

(19)$$\begin{gathered} {\mathit{P}}^{\left(1\right)}={\mathit{X}}^{(+)}{\mathit{A}}^{\left(2\right)} \\ =\left[\begin{array}{cccc}{X}_{\mathrm{1,1}}^{(+)}& X_{\mathrm{1,2}}^{(+)}& \cdots & X_{1,C}^{(+)}\\ X_{\mathrm{2,1}}^{(+)}& X_{\mathrm{2,2}}^{(+)}& \cdots & X_{2,C}^{(+)}\\ ⋮& ⋮& \ddots & ⋮\\ X_{C,1}^{(+)}& X_{C,2}^{(+)}& \cdots & X_{C,C}^{(+)}\end{array}\right]\times \left[\begin{array}{c}{a}_{C+1}\\ a_{C+2}\\ ⋮\\ a_{2C}\end{array}\right]. \end{gathered}$$

The first C symbols are then detected as

(20)$\begin{array}{c}{\widehat{a}}_C^{(1)}=\zeta [Z_C^{(1)}/r_{C,C}]\\ {\widehat{a}}_{C-1}^{(1)}=\zeta [(Z_{C-1}^{(1)}-r_{C-1,C}•{\widehat{a}}_C^{(1)})/r_{C-1,C-1}]\\ ⋮\\ {\widehat{a}}_c^{(1)}=\zeta [(Z_c^{(1)}-{\displaystyle \sum _{i=c-1}^M{r}_{c,i}^{(1)}•{\widehat{a}}_i^{(1)}})/r_{c,c}]\\ ⋮\\ {\widehat{a}}_1^{(1)}=\zeta [(Z_1^{(1)}-{\displaystyle \sum _{i=2}^C{r}_{1,i}^{(1)}•{\widehat{a}}_i^{(1)}})/r_{1,1}].\end{array}$

The first detected symbols can be written as

(21)${\widehat{A}}^{(1)}=[{\widehat{a}}_1^{(1)}{\widehat{a}}_2^{(1)}\cdots {\widehat{a}}_C^{(1)}]$

For detection of the second C symbols, the second received C symbols are selected

(22)$\begin{array}{ll}{Y}^{(2)}& ={\left[y_{C+1}{y}_{C+2}\cdots y_{2C}\right]}^T\\ & ={\left[y_1^{(2)}{y}_2^{(2)}\cdots y_C^{(2)}\right]}^T\end{array}$

Here, $Y^{(2)}$ is written simply as follows:

(23)$$\begin{gathered} {\mathit{Y}}^{\left(2\right)}=[y_1^{\left(2\right)}{y}_2^{\left(2\right)}\cdots y_C^{\left(2\right)}{]}^T \\ =\mathit{X}{\mathit{A}}^{\left(2\right)}+{\mathit{X}}^{(-)}{\mathit{A}}^{\left(1\right)}+{\mathit{X}}^{(+)}{\mathit{A}}^{\left(3\right)}+{\mathit{W}}^{\left(2\right)}. \end{gathered}$$

In the second detection step, the target symbols are $A^{(2)}=[a_1^{(2)},\cdots ,a_C^{(2)}]$, whereas ${\mathit{X}}^{(-)}{\mathit{A}}^{\left(1\right)}$ and ${\mathit{X}}^{(+)}{\mathit{A}}^{\left(3\right)}$ are interference terms. Because ${\widehat{A}}^{(1)}$ was detected in the previous detection step as in (21), ${\mathit{X}}^{(-)}{\mathit{A}}^{\left(1\right)}$ can be removed. In this paper, the interference cancellation process of the previously detected symbols is called previous interference cancellation, which can be depicted as

(24)$\begin{array}{c}{\mathit{Y}}^{(2{)}^{\prime }}={\mathit{Y}}^{\left(2\right)}-\left[{\mathit{X}}^{(-)}{\widehat{\mathit{A}}}^{\left(1\right)}\right]\end{array}$

After the previous interference cancellation in (17), the multiplication of $Q^H$ is executed as

(25)$Z^{(2)}=Q^H Y^{{(2)}^{\prime }}$

As in (20), the second C symbols are then detected, which can be written as ${\widehat{A}}^{(2)}=[{\widehat{a}}_1^{(2)}{\widehat{a}}_2^{(2)}\cdots {\widehat{a}}_C^{(2)}]$. For the detection of the remaining symbols, the above detection process is repeated.

5. Partial DFE Methods with Post Interference Cancellation for FTN Signalling

In the above section, post interference is not considered. However, as shown in Figure 3, the process of detecting $A^{(1)}$ has post interference. Here, $Y^{(1)}$ can be expressed as

(26)${\mathit{Y}}^{\left(1\right)}=\mathit{X}{\mathit{A}}^{\left(1\right)}+{\mathit{X}}^{(+)}{\mathit{A}}^{\left(2\right)}+{\mathit{W}}^{\left(\mathbf{1}\right)}$

where ${\mathit{X}}^{(+)}{\mathit{A}}^{\left(2\right)}$ is the post interference. Unfortunately, in the detection step for $A^{(1)}$, $A^{(2)}$ is not detected. Therefore, in the detection process for $A^{(1)}$, post interference exists. Furthermore, post interference also exists in the detection step for $A^{(2)}$, which can be written as ${\mathit{X}}^{(+)}{\mathit{A}}^{\left(3\right)}$. Because a post interference signal is not detected in each detection step, cancellation of the post interference is impossible. In this section, for the cancellation of a post interference signal, recursive detection is proposed.

First, as described in Section 4, the detection is executed without considering the post interference cancellation. After the first detection process, recursive detection is executed including post interference cancellation. In this scheme, for the post interference cancellation, the first detected signal is used. The detected signal shown in Section 4 can be expressed as

(27)$\widehat{A}=[{\widehat{a}}_1{\widehat{a}}_2\cdots {\widehat{a}}_C{\widehat{a}}_{C+1}\cdots {\widehat{a}}_L]$

In the second recursive detection process, the first C symbols, $[a_1{a}_2\cdots a_C],$ are detected using $Y^{(1)}$. Before the detection of $A^{(1)}$, post interference cancellation is executed as follows:

(28)$\begin{array}{l}{\tilde{\mathit{Y}}}^{\left(1\right)}={\mathit{Y}}^{\left(1\right)}-{\mathit{X}}^{(+)}{\widehat{\mathit{A}}}^{\left(2\right)}\\ =\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_C\end{array}\right]-\left[\begin{array}{cccc}{X}_{\mathrm{1,1}}^{\left(+\right)}& X_{\mathrm{1,2}}^{\left(+\right)}& \cdots & X_{1,C}^{\left(+\right)}\\ X_{\mathrm{2,1}}^{\left(+\right)}& X_{\mathrm{2,2}}^{\left(+\right)}& \cdots & X_{2,C}^{\left(+\right)}\\ ⋮& ⋮& \ddots & ⋮\\ X_{C,1}^{\left(+\right)}& X_{C,2}^{\left(+\right)}& \cdots & X_{C,C}^{\left(+\right)}\end{array}\right]\times \left[\begin{array}{c}{\widehat{a}}_{C+1}\\ {\widehat{a}}_{C+2}\\ ⋮\\ {\widehat{a}}_{2C}\end{array}\right].\end{array}$

As in (17) and (20), the interference cancellation and symbol detection process are then executed. In the interference cancellation and symbol detection process, ${\tilde{Y}}^{(1)}$ is used instead of $Y^{(1)}$. The detection process can be written as

(29)${\tilde{Z}}^{(1)}=Q^H {\tilde{Y}}^{(1)},$

(30)$\begin{array}{l}{\tilde{a}}_C^{(1)}=\zeta [{\tilde{Z}}_C^{(1)}/r_{C,C}]\cdots \\ {\tilde{a}}_{C-1}^{(1)}=\zeta [({\tilde{Z}}_{C-1}^{(1)}-r_{C-1,C}•{\tilde{a}}_C^{(1)})/r_{C-1,C-1}]\cdots \\ {\tilde{a}}_c^{(1)}=\zeta [({\tilde{Z}}_c^{(1)}-{\displaystyle \sum _{i=c-1}^M{r}_{c,i}^{(1)}•{\tilde{a}}_i^{(1)}})/r_{c,c}]\cdots \\ {\tilde{a}}_1^{(1)}=\zeta [({\tilde{Z}}_1^{(1)}-{\displaystyle \sum _{i=2}^C{r}_{1,i}•{\tilde{a}}_i^{(1)}})/r_{1,1}].\end{array}$

After the above process, the detected symbols are acquired as

(31)$\begin{array}{l}{\tilde{A}}^{(1)}=[{\tilde{a}}_1^{(1)}{\tilde{a}}_2^{(1)}\cdots {\tilde{a}}_C^{(1)}]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=[{\tilde{a}}_1{\tilde{a}}_2\cdots {\tilde{a}}_C]\end{array}$

For the second detection step, $Y^{(2)}$ is used, and both the previous interference cancellation and post interference cancellation are executed. In this second step, the previous interference cancellation is similar to (24), except for ${\widehat{A}}^{(1)}$. In this previous interference cancellation, ${\tilde{A}}^{(1)}$ is used instead of ${\widehat{A}}^{(1)}$ as follows:

(32)$\begin{array}{c}{\mathit{Y}}^{(2{)}^{\prime }}={\mathit{Y}}^{\left(2\right)}-\left[{\mathit{X}}^{(-)}{\tilde{\mathit{A}}}^{\left(1\right)}\right]\end{array}.$

Furthermore, the post interference cancellation is executed using ${\widehat{A}}^{(3)}$ as

(33)$\begin{array}{c}{\tilde{\mathit{Y}}}^{\left(2\right)}={\mathit{Y}}^{(2{)}^{\prime }}-\left[{\mathit{X}}^{(+)}{\widehat{\mathit{A}}}^{\left(3\right)}\right]\end{array}.$

The detection processes are then executed as

(34)${\tilde{Z}}^{(2)}=Q^H {\tilde{Y}}^{(2)},$

(35)$\begin{array}{c}{\tilde{a}}_C^{(2)}=\zeta [{\tilde{Z}}_C^{(2)}/r_{C,C}]\\ ⋮\\ {\tilde{a}}_1^{(2)}=\zeta [({\tilde{Z}}_1^{(2)}-{\displaystyle \sum _{i=2}^C{r}_{1,i}•{\tilde{a}}_i^{(2)}})/r_{1,1}].\end{array}$

This detection process is repeated until the last symbol has been detected. In the last M-th detection step, post interference does not exist. The last M-th received signal can be written as

(36)$$\begin{gathered} {\mathit{Y}}^{\left(M\right)}={\left[y_1^{\left(M\right)} y_2^{\left(M\right)} \cdots y_C^{\left(M\right)}\right]}^T \\ =\mathit{X}{\mathit{A}}^{\left(M\right)}+{\mathit{X}}^{(-)}{\mathit{A}}^{(M-1)}+{\mathit{W}}^{\left(M\right)}. \end{gathered}$$

Therefore, only the previous interference cancellation is executed as

(37)$\begin{array}{c}{\tilde{\mathit{Y}}}^{\left(M\right)}={\mathit{Y}}^{\left(M\right)}-\left[{\mathit{X}}^{(-)}{\tilde{\mathit{A}}}^{(M-1)}\right]\end{array}$

Signal detection is then executed as follows:

(38)$\begin{array}{c}{\tilde{a}}_C^{(M)}=\zeta [{\tilde{Z}}_C^{(M)}/r_{C,C}]\\ ⋮\\ {\tilde{a}}_1^{(M)}=\zeta [({\tilde{Z}}_1^{(M)}-{\displaystyle \sum _{i=2}^C{r}_{1,i}•{\tilde{a}}_i^{(M)}})/r_{1,1}].\end{array}$

6. Computational Complexity

This section expresses the computational complexity of the proposed detection technique and a conventional VA. To compare the computational complexity, the number of multiplications is considered. For the multiplication, QPSK modulation is considered. The number of multiplications $K_V$ for the VA can be written as

(39)$\begin{array}{l}{K}_V=\left({\displaystyle \sum _{i=1}^{C-1}{2}^{i-1}}\times 2\times (1+(C+1))+2\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left({\displaystyle \sum _{i=M}^L{2}^M}\times 2\times (1+(M+1)+2)\right)\end{array}$

where C is the number of interference symbols considered, and L is the frame length. In Equation (39), the first set of parentheses indicates the amount of multiplication based on the number of states according to the number of interference symbols considered. At each point, the number of states according to the number of interferences is the same as the exponent of the number of interference symbols. Therefore, the value 2^C in the first set of parentheses becomes the most dominant value as the exponent of the number of interference symbols increases. Also, the second set of parentheses indicates the amount of multiplication that occurs according to the frame length. Therefore, in the Viterbi operation, the number of operations increases exponentially according to the number of interference symbols considered.

The computational complexity of the proposed algorithm without post interference cancellation is as follows:

(40)$K_1=C^3+2\left[M\left(C^2+{\displaystyle \sum _{m=1}^C(m)}\right)+(M-1)•C^2\right]$

where C is the partial matrix size and M indicates the L/C. In Equation (40), the highest exponent C³ is the operation that occurs when QR decomposing occurs in the interference matrix in Equation (15). The C² in the parentheses after that is the number of operations that occur when multiplying the interference matrix by the Q matrix after QR decomposing it as in Equation (17). The operation in the innermost parentheses is the operation that occurs when performing DFE as in Equation (20), and this operation is repeated M times. Since the DFE operation is performed repeatedly for the previous interference and the current interference, a multiplication of 2 is added.

The computational complexity of the proposed algorithm with post interference cancellation is

(41)$K_2=C^3+2\left[2\left(M\left(C^2+{\displaystyle \sum _{m=1}^C(m)}\right)+(M-1)C^2\right)+(M-1)C^2\right].$

Equation (41) considers both previous interference and post interference cancellation. In addition, since previous interference cancellation and the current symbol are re-detected after the interference, Equation (40) is structured in a form that repeats twice.

Table 1 describes the number of multiplications according to the frame length. As shown in Table 1, the detection complexity of the proposed detection technique is much lower than that of a conventional VA.

7. Simulation Results

To assess the effectiveness of the proposed method, the FTN system was evaluated using QPSK modulation. FTN signalling scenarios with τ values of 0.9, 0.85, and 0.8 were examined. For comparison, the performance of standard QPSK without FTN signalling was included as a reference, along with the performance of the application of the Viterbi algorithm (VA) to FTN signalling [9,10]. The simulations, carried out using MATLAB, analyzed the bit error rate (BER) under these conditions.

Figure 4, Figure 5 and Figure 6 present the BER performance of the proposed detection method described in Section 4 for τ values of 0.9, 0.85, and 0.8, respectively. An FTN system with τ = 0.9 theoretically achieves an 11% increase in the transmission rate, while τ = 0.8 theoretically provides a 25% rate improvement. However, this gain comes at the cost of BER performance degradation due to ISI. The figures also evaluate various partial matrix sizes. Increasing the matrix size enhances performance. In Figure 4, for example, the BER performance at C = 6 is approximately 0.7 dB better than at C = 1 for a BER of ${10}^{-4}$. Since the performance at C = 6 is comparable to that at C = 5, either C = 5 or 6 is deemed an optimal matrix size. Similarly, in Figure 5, the performance improvement for C = 6 compared to C = 1 is about 1.5 dB. In Figure 6, the optimal matrix sizes appear to be C = 6 or 7.

Figure 7 illustrates the effect of a change in the matrix size C on the BER performance of the partial DFE technique. The dotted lines are the performances of Eb/N0 = 8 dB, and the solid lines are the performances of Eb/N0 = 10 dB. In all cases, a performance improvement can be found according to the increase in the matrix size. Larger values of C allow for a more accurate matrix representation of the interference symbols, thereby improving the accuracy of rejection. Therefore, a larger C increases the computational burden, but also improves the detection accuracy by reducing residual interference. However, due to the influence of C on error propagation in the recursive detection process, BER improves up to the optimal C value, but stagnates at an excessive C due to noise amplification.

The performance of the partial DFE scheme with the post interference cancellation described in Section 5 is depicted in Figure 8. According to the results of Figure 4, Figure 5 and Figure 6, the matrix size C = 6 is considered for all τ values. A performance improvement in the partial DFE with post interference cancellation was found, regardless of the τ values. In the case of τ = 0.9, the partial DFE with post interference cancellation has only a 0.15 dB performance degradation compared with the general QPSK at BER = ${10}^{-4}$. Furthermore, the proposed partial DFE with post interference cancellation has a better performance than that of the VA for all τ values. Therefore, the proposed technique can be efficiently used for the FTN communication system to improve the detection performance and reduce the complexity.

This paper considers the most basic modulation techniques that can demonstrate the performance of FTN well. We wanted to focus on a performance analysis according to the application of FTN, not the modulation method. Therefore, we considered only the most commonly known modulation methods. For the τ value, various τ values have been studied, but a value up to 0.8 is the most commonly considered and studied value. Mazo, who first announced the concept of FTN, also proved that the performance of the entire system is guaranteed when considering up to t = 0.802 [1]. Therefore, this paper considers only the most commonly used τ values.

8. Conclusions

To reduce the burden of ISI cancellation of FTN signalling, this paper introduced simple interference cancellation and a symbol detection technique for an FTN-based communication system. As a complexity reduction strategy, interference cancellation is investigated based on a matrix computation. The simulation results indicate that the complexity of the proposed technique with a $6\times 6$ matrix computation is reduced by 97% compared with that of the conventional Viterbi algorithm with six interference symbols. Furthermore, in the case of τ = 0.85, the performance of the proposed technique is about 1 dB better than that of the VA at BER = ${10}^{-4}$.

Future Work

• -. Extend the comparative analysis by including additional state-of-the-art algorithms.

• -. Extend the proposed method to more complex and realistic communication environments by including fading channels and non-AWGN noise models.

• -. Investigate the adaptability and robustness of the method under dynamic conditions such as mobility, varying transmission distances, and heterogeneous network settings.

• -. Evaluate the integration of the method into real-world communication systems such as 5G/6G networks, IoT, and satellite communication systems.

We plan to address these aspects in future studies to provide a comprehensive understanding of the applicability and limitations of the proposed method in real-world communication environments. Furthermore, the future studies will provide a more extensive and comprehensive evaluation of the advantages and limitations of the proposed method compared to other advanced techniques.

Author Contributions

Conceptualization, M.-S.B.; methodology, M.-S.B.; software, M.-S.B.; validation, M.-S.B. and H.-K.S.; formal analysis, M.-S.B. and H.-K.S.; investigation, M.-S.B. and H.-K.S.; resources, H.-K.S.; data curation, M.-S.B.; writing—original draft preparation, M.-S.B. and H.-K.S.; writing—review and editing, M.-S.B. and H.-K.S.; visualization, M.-S.B.; supervision, H.-K.S.; project administration, H.-K.S.; funding acquisition, H.-K.S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data will be made available by the authors upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1. Signal comparison between Nyquist rate and FTN.

Figure 2. BER performance of FTN system with overall matrix computation.

Figure 3. FTN signal model with L = 9, C = 3, N = 2 and M = 3.

Figure 4. BER performance of FTN-based system with τ = 0.9.

Figure 5. BER performance of FTN-based system with τ = 0.85.

Figure 6. BER performance of FTN-based system with τ = 0.8.

Figure 7. BER performance according to the values of matrix size C for the proposed scheme.

Figure 8. BER performance of FTN-based system with post interference cancellation.

Table 1

Interference cancellation and signal detection process of partial DFE method for FTN signalling.

Interference CancellationTechnique	L = 600			L = 6000			L = 60,000
Viterbi	C = 4	C = 5	C = 6	C = 4	C = 5	C = 6	C = 4	C = 5	C = 6
	203,488	571,932	1,524,440	2,046,688	5,755,932	15,348,440	20,478,688	57,595,932	153,588,440
PDFE w/o PIC	C = 4	C = 5	C = 6	C = 4	C = 5	C = 6	C = 4	C = 5	C = 6
	8432	13,075	18,744	84,032	130,075	186,144	840,032	1,300,075	1,860,144
PDFE with PIC	C = 4	C = 5	C = 6	C = 4	C = 5	C = 6	C = 4	C = 5	C = 6
	19,968	30,975	44,400	199,968	309,975	444,000	1,999,968	3,099,975	4,440,000

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