SVD-Precoded Faster-Than-Nyquist Signaling With Optimal and Truncated Power Allocation

This study proposes a precoded faster-than-Nyquist (FTN) signaling scheme based on singular-value decomposition (SVD) with optimal power allocation. An information-theoretic analysis is conducted on the conventional and proposed SVD-precoded FTN signaling architectures. The associated information rate bound is derived in a closed-form by extending the classic Shannon capacity to support the SVD-precoded FTN signaling schemes. Our analytical performance results demonstrate that the proposed scheme outperforms its conventional Nyquist-criterion-based counterpart and the conventional SVD-precoded FTN signaling scheme, which does not use power allocation. Several important implementation issues, such as inter-block interference, the use of a frequency-selective (dispersive) channel, truncation, and sampling rate, are also considered. Finally, a suboptimal power allocation scheme that overcomes the truncation issue is presented.

SECTION I. Introduction

The concept of faster-than-Nyquist (FTN) signaling, first introduced in the 1970s [1]–​[4], relies on the transmission of non-orthogonal pulses in the time domain. In FTN signaling, pulses have a symbol interval $T = \tau T_{0} \: (0 < \tau \leq 1)$ , which is shorter than that defined by the Nyquist criterion $T_{0}$ , which guarantees inter-symbol interference (ISI)-free time-orthogonal pulse transmission. Here, $\tau$ is the symbol’s packing ratio. FTN signaling has the potential to increase the transmission rate, without consuming extra bandwidth or power, compared to that achieved using its classic Nyquist-criterion-based counterparts. The minimum Euclidean distance (MED) within an FTN signaling block is the same as that of its ISI-free Nyquist-criterion-based signaling counterpart as long as the symbol’s packing ratio $\tau$ is sufficiently high [2], [5]. For example, the MED of FTN signaling remains unchanged in the range of $0.802\le \tau \le 1$ for sinc pulses [2].

Information-theoretic analyses of FTN signaling have been carried out [6]–​[14]. Rusek and Anderson [6] derived the capacity of FTN signaling, which was found to be higher than that of Nyquist-criterion-based signaling. They attributed this result to the use of the unused excess bandwidth of non-sinc shaping filters, such as the root-raised-cosine (RRC) filter. Another study [7] showed that binary FTN signaling converges to the constrained capacity with decreasing $\tau$ value [6]. In [8], the information rate of cyclostationary FTN signaling was analyzed for the linear time-invariant channel, and presented the water-filling power allocation, to maximize the associated information rate. Note that the power allocation of [8] exhibits a capacity advantage only when using a rectangular shaping pulse (a non-band-limited pulse). There was not any substantial performance gain when employing the RRC shaping pulse. The capacity advantages of FTN signaling over its Nyquist-criterion-based counterpart have been investigated in the context of spectrum-sharing systems [10], multiple-access channels [11], multiple-input multiple-output (MIMO) systems [12], and broadcasting systems [14]. Furthermore, one study [13] calculated the information rate of FTN signaling, precoded using the square root of an FTN-induced ISI matrix, and other studies [6], [7], [10]–​[12], [14] considered unprecoded FTN signaling. The spectral efficiency of the multi-carrier FTN signaling scheme that packs information symbols both in the time and frequency domains was also analyzed [15], [16].

In unprecoded FTN signaling, the effects of FTN-induced ISI have to be eliminated at the receiver, where detection complexity may be excessive for a low-$\tau$ (high-FTN-rate) scenario. In order to overcome this limitation, several computationally efficient detection algorithms have been developed for unprecoded FTN signalling [5], [17]–​[28]. Time-domain FTN equalizers [5], [17]–​[20] can effectively detect FTN signals, even in severe ISI (low-$\tau$ ) scenarios; however, detection complexity exponentially increases with increasing effective tap length of FTN-induced ISI. Linear frequency-domain FTN equalizers [21]–​[25], [27], [28], based on cyclic prefix insertion per block, have a very low detection complexity, even in highly frequency-selective (dispersive) fading channels. Note that the detection performance of linear frequency-domain FTN equalizers is typically lower than that of their non-linear time-domain counterparts.

In contrast to the above-mentioned unprecoded FTN signaling, which imposes unavoidable detection complexity, especially for eliminating FTN-induced ISI, recent precoded FTN signaling systems do not require the receiver to carry out such equalization [13], [29]–​[37]. The fundamental concept of precoded FTN signaling is that FTN-induced ISI is determined by the symbol’s packing ratio $\tau$ and the roll-off factor $\beta$ of a shaping filter, which are known in advance of transmissions. Hence, the ISI effects can be eliminated with the aid of precoding at the transmitter. Some studies [31]–​[33] have applied Tomlinson-Harashima precoding for the pre-equalization of FTN signaling at the FTN transmitter, eliminating FTN-induced ISI at the receiver. Moreover, in [13], the square root of an FTN-induced ISI matrix is utilized for pre-equalizing ISI. In [34], the precoding scheme, which eliminates both of FTN-specific ISI and multi-user interference, was proposed in a downlink scenario. In other studies [29], [30], the MED of FTN signaling was increased with the aid of precoding; however, unlike other precoded FTN signaling schemes, this scheme imposes FTN-induced ISI at the receiver.

Motivated by a singular-value decomposition (SVD)-precoded spatial multiplexing MIMO scheme [38], studies [35], [37] have proposed SVD-precoded FTN signaling schemes for pre-equalizing the ISI of an FTN signaling block. Gattami and Ringh [35] applied SVD-based precoding to FTN signaling to diagonalize an FTN-induced ISI matrix. This allows the exploitation of equivalent independent parallel substreams, similar to the SVD-based precoding scheme developed for spatial multiplexing MIMO. In [36], Kim derived the capacity of the SVD-precoded FTN signaling scheme that does not rely on power allocation, which is consistent with the formula derived by Rusek and Anderson [6]. Note that no power allocation scheme has been considered in the above SVD-aided FTN signaling schemes [35]–​[37]. This is because the energy constraint of spatial multiplexing MIMO is not directly applicable to FTN signaling. The above-mentioned SVD-precoded FTN signaling schemes without power allocation [35], [37] suffer from performance degradation, which is caused by the existence of significantly low singular values in low-$\tau$ (high-FTN-rate) scenarios. To overcome this limitation, an adaptive bit loading scheme has been integrated into the SVD-precoded FTN signaling scheme [37], where appropriate information bits are allocated onto symbols in a block to increase the MED of SVD-precoded FTN signaling. Although SVD-precoded FTN signaling schemes [35], [37] have the potential to outperform the conventional FTN signaling and Nyquist-criterion-based schemes, there are several open issues. Note that in the conventional SVD-precoded spatial multiplexing MIMO scheme, power allocation to parallel independent substreams enhances the information rate. However, the same algorithm is unapplicable to the existing SVD-based FTN signaling scheme, due to the difference in the system models. Hence, no information-theoretic analysis has been carried out for the SVD-precoded FTN signaling schemes with power allocation, while the SVD-precoded FTN signaling with power allocation may outperform its conventional equally power-distributed counterpart [36], in an analogous manner to the relationship between the conventional SVD-precoded spatial multiplexing MIMO system versus its unprecoded counterpart [38].

Against this background, the present study makes the following novel contributions:¹

• Power allocation is applied to the SVD-precoded FTN signaling scheme to increase the information rate, making it significantly higher than those of conventional FTN signaling schemes and the Nyquist-criterion-based scheme. Optimal power allocation is derived for the SVD-precoded FTN signaling scheme by maximizing the associated information rate with the aid of the Lagrangian multiplier method. Note that in the derivation, an FTN-specific energy constraint is imposed, which is different from that considered in the conventional SVD-precoded MIMO scheme.

• The classic Shannon capacity is extended to support the proposed and conventional SVD-precoded FTN signaling schemes in a closed-form. The system model of multiple independent parallel substreams, free from FTN-induced ISI, is considered.

• The analytical performance results, based on the derived capacity, demonstrate that the proposed SVD-precoded FTN signaling scheme with power allocation has a significantly higher information rate bound than those of the conventional SVD-precoded FTN signaling scheme and the classic Nyquist-criterion-based scheme.

• Several important practical implementation issues, such as inter-block interference (IBI), the issues of a frequency-selective fading channel, a sampling rate, spectral broadening associated with FTN signaling, and truncation are discussed. More specifically, to solve the truncation issue, a suboptimal power allocation scheme is proposed. Moreover, the power spectral density (PSD) of the proposed truncated scheme is numerically investigated.

• The bit error ratio (BER) performance of the proposed SVD-precoded FTN signaling with suboptimal truncated power allocation is presented in the context of a multi-stage serially-concatenated turbo coding architecture.

• Based on our simulation results of the PSD and BER, our proposed SVD-precoded FTN signaling with truncated power allocation is shown to achieve a higher information rate than the conventional FTN signaling and the classic Nyquist-criterion-based schemes. This is achieved without sacrificing any substantial spectrum broadening.

The rest of this paper is organized as follows. In Section II, the general system model of the linearly precoded FTN signaling scheme is introduced and the conventional SVD-precoded FTN signaling scheme, as well as its capacity derivation is reviewed. In Section III, a novel SVD-precoded FTN signaling scheme with power allocation is proposed, and its closed-form capacity and optimal power allocation are derived. In Section IV, the analytical performance results of the proposed scheme are presented. In Section V, the practical implementation issues of the proposed scheme are considered. In Section VI, a truncated power allocation scheme is applied to the proposed scheme to facilitate practical implementation. In Section VII, we demonstrate the BER performance of the proposed FTN signaling scheme for the three-stage serially-concatenated turbo-coded architecture. Finally, conclusions are given in Section VIII.

Notation: We use upper- and lower-case bold-faced letters to represent matrices and vectors, respectively. $(\cdot)^{T}$ , $(\cdot)^{*}$ , and $(\cdot)^{H}$ denote the transpose, conjugate, and the conjugate transpose of $(\cdot)$ , respectively. The determinant and trace operations of a matrix $\mathbf {X}$ are represented by $|\mathbf {X}|_{\mathrm{ det}}$ and ${\mathrm{ trace}}\{\mathbf {X}\}$ , respectively. A diagonal matrix whose diagonal elements are given by a vector of $\mathbf {x}$ is denoted by ${\mathrm{ diag}}\{ \mathbf {x}\}$ . $\mathbb {E}[\cdot]$ represents the expectation operation. $\mathcal {CN}(m, v)$ denotes the Gaussian distribution having a mean of $m$ and a variance of $v$ . $I(\mathbf {x}; \mathbf {y})$ is mutual information between a vector of $\mathbf {x}$ and $\mathbf {y}$ . A differential entropy is denoted by $h_{\textrm {e}}(\cdot)$ .

SECTION II. System Model

This section introduces the general system model of a linearly precoded FTN signaling scheme and reviews the conventional SVD-precoded FTN signaling scheme [35], [37].

A. General System Model of Precoded FTN Signaling

It is assumed that the transmission of $N$ -length complex-valued Gaussian symbols $\mathbf {s}=[s_{0}, s_{1}, \cdots , s_{N-1}]^{T} \in \mathbb {C}^{N}$ having an average symbol power of $E_{s}= \mathbb {E}[|s_{i}|^{2}] = 1 \,\,(i=0,\cdots ,N-1)$ .² The Gaussian symbols are precoded using a linear precoding matrix $\mathbf {F} \in \mathbb {C}^{N\times N}$ , and hence the precoded symbols $\mathbf {x} = [x_{0},\cdots ,x_{N-1}]^{T} \in \mathbb {C}^{N}$ are given by: $$\begin{equation*} \mathbf {x} = \mathbf {Fs}.\tag{1}\end{equation*}$$ Then, the precoded symbols are band-limited with the aid of an RRC filter $h(t)$ with the roll-off factor $\beta$ . Note that a sinc filter corresponds to the RRC filter, having the roll-off factor of $\beta =0$ . Finally, the precoded FTN signals are transmitted with an FTN symbol interval of $T=\tau T_{0}$ as follows: $$\begin{equation*} x(t) = {\sqrt {\tau T_{0}}}\sum _{n} x_{n}h(t-nT), \tag{2}\end{equation*}$$ where the coefficient ${\sqrt {\tau T_{0}}}$ normalizes the power of an FTN signaling block so that it remains equal to that of its Nyquist-criterion-based counterpart which has the same block interval as and fewer information bits than those of the FTN signaling scheme [6], [7]. Furthermore, note that the total transmit energy per block is given by $E_{N} = \mathbb {E}\left [{\int _{-\infty }^{\infty }|x(t)|^{2}dt}\right]$ , while the precoding matrix $\mathbf {F}$ is designed for maintaining the average energy consumption $E_{N}$ to be constant.

Assuming an additive white Gaussian noise (AWGN) channel, the received FTN signals are passed through a matched filter $h^{*}(-t)$ , and are represented by: $$\begin{equation*} y(t) = \sqrt {\tau T_{0}}\sum _{n} x_{n}g(t-nT) + \eta (t),\tag{3}\end{equation*}$$ where we have $g(t) = \int h(\xi)h^{*}(\xi -t)d\xi$ and $\eta (t) = \int n(\xi)h^{*}(\xi -t)d\xi$ , while $n(t)$ is a complex-valued random variable that obeys the Gaussian distribution of $\mathcal {CN}(0, {N_{0}})$ .

Ignoring the effects of IBI for simplicity, the $i$ th received sample $y_{i}=y(iT) \,\,(i=0,\cdots ,N-1)$ can be expressed by $$\begin{equation*} y_{i} = \sqrt {\tau T_{0}}\sum _{n} x_{n}g((i-n)T) + \eta (iT), \tag{4}\end{equation*}$$ where $\eta (iT) \,\,(i=0,\cdots ,N-1)$ represent the colored noise, with a correlation of $\mathbb {E}[{\eta (lT)\eta ^{*}(mT)}]={N_{0}}g((l-m)T)$ . Furthermore, the received sample block $\mathbf {y}=[y_{0}, y_{1}, \cdots , y_{N-1}]^{T}$ is represented by $$\begin{align*} \mathbf {y}=&\sqrt {\tau T_{0}} \mathbf {G}\mathbf {x} + \boldsymbol {\eta }\in \mathbb {C}^{N}\tag{5}\\=&\sqrt {\tau T_{0}} \mathbf {G}\mathbf {Fs} + \boldsymbol {\eta }, \tag{6}\end{align*}$$ where $\mathbf {G}\in \mathbb {R}^{N\times N}$ is the FTN-induced ISI matrix, with a Toeplitz structure, the first column of which is given by $[g(0), g(T), g(2T), \cdots , g((N-1)T)]$ , and $\boldsymbol {\eta }= [\eta (0),\cdots ,\eta ((N-1)T)]^{T}$ . Moreover, $\mathbf {G}$ is a positive-definite matrix [13], [36].

B. Conventional SVD-Precoded FTN Signaling

In previous studies [35], [37], the precoding matrix $\mathbf {F}$ was calculated based on the SVD. More specifically, the matrix $\mathbf {G}$ in (6), representing FTN-induced ISI, is factorized into: $\mathbf {G}= \mathbf {U} \boldsymbol{\Lambda } \mathbf {V} ^{T}$ , where $\mathbf {U}\in \mathbb {R}^{N\times N}$ and $\mathbf {V}\in \mathbb {R}^{N\times N}$ are orthogonal matrices, and $\boldsymbol{\Lambda }\in \mathbb {R}^{N\times N}$ is a diagonal matrix, composed of the descending-order singular values $[\lambda _{0}, \cdots , \lambda _{N-1}]$ of $\mathbf {G}$ . Note that since the matrix $\mathbf {G}$ is real and symmetric, the relationship $\mathbf {U}= \mathbf {V}$ holds [35], which corresponds to the eigenvalue decomposition of $\mathbf {G}$ . Therefore, the received samples of (6) can be rewritten by $$\begin{equation*} \mathbf {y}= \sqrt {\tau T_{0}} \mathbf {V} \boldsymbol{\Lambda } \mathbf {V}^{T}\mathbf {Fs} + \boldsymbol {\eta }. \tag{7}\end{equation*}$$ Then, by setting the precoding matrix $\mathbf {F}$ to the orthogonal matrix $\sqrt {P} \mathbf {V}$ and noting the relationship $\mathbf {V} \mathbf {V} ^{T}=\mathbf {I}$ , where ${P}$ is the power factor and $\mathbf {I}$ is the identity matrix, (7) can be further simplified to [35], [37]: $$\begin{equation*} \mathbf {y} = \sqrt {P \tau T_{0}} \mathbf {V} \boldsymbol{\Lambda } \mathbf {s}+ \boldsymbol {\eta }. \tag{8}\end{equation*}$$

Furthermore, multiplying the weight matrix $\mathbf {V}^{T}$ by $\mathbf {y}$ in (8) yields the following diagonalized signal representation: $$\begin{align*} \mathbf {y}_{\textrm {d}}=&\mathbf {V}^{T} \mathbf {y}\in \mathbb {C}^{N} \tag{9}\\=&\sqrt {P \tau T_{0}} \boldsymbol{\Lambda } \mathbf {s} + \boldsymbol {\eta }_{v}, \tag{10}\end{align*}$$ where $\boldsymbol {\eta }_{v}= \mathbf {V}^{T} \boldsymbol {\eta }$ . Note that $\mathbf {y}_{\textrm {d}}$ in (10) is modeled as $N$ independent parallel substreams. The related noise components $\boldsymbol {\eta }_{v}= \mathbf {V}^{T} \boldsymbol {\eta }$ in (10) are no longer correlated. The $i$ th component of $\boldsymbol {\eta }_{v}$ has variance ${N_{0}}\lambda _{i}$ , as follows [35]: $$\begin{align*} \mathbb {E}[ \boldsymbol {\eta }_{v} \boldsymbol {\eta }_{v}^{H}]=&\mathbf {V}^{T}\mathbb {E}[ \boldsymbol {\eta } \boldsymbol {\eta } ^{H}] \mathbf {V}\tag{11}\\=&N_{0} \mathbf {V}^{T} \mathbf {G} \mathbf {V} \tag{12}\\=&N_{0} \boldsymbol{\Lambda },\tag{13}\end{align*}$$ where $\mathbb {E}[ \boldsymbol {\eta } \boldsymbol {\eta } ^{H}] = {N_{0}} \mathbf {G}$ . Hence, the conventional SVD-precoded FTN signaling scheme allows the FTN symbols of (10) to be demodulated based on low-complexity symbol-by-symbol maximum-likelihood detection [35], [37]. It should be noted that SVD is carried out offline before transmission since $\mathbf {G}$ is determined uniquely by $\tau$ and $\beta$ .

C. Capacity of Conventional SVD-Precoded FTN Signaling Without Power Allocation

In [36], the capacity of the SVD-precoded FTN signaling scheme, which does not rely on power allocation unlike our proposed scheme, was provided, although its detail derivation is absent in [36]. In this section, we reformulate the mutual information in [36] with the aid of the independent parallel substreams attainable by SVD-based precoding of (10).

The mutual information between $\mathbf {y}_{\textrm {d}}$ and $\mathbf {s}$ , denoted as $I(\mathbf {s}; \mathbf {y}_{\textrm {d}})$ , is given by $$\begin{align*} I(\mathbf {s}; \mathbf {y}_{\textrm {d}})=&h_{\textrm {e}}(\mathbf {y}_{\textrm {d}}) - h_{\textrm {e}}(\mathbf {y}_{\textrm {d}}| \mathbf {s}) \tag{14}\\=&h_{\textrm {e}}(\mathbf {y}_{\textrm {d}}) - h_{\textrm {e}}(\boldsymbol {\eta }_{v}), \tag{15}\end{align*}$$ where $h_{\textrm {e}}(\cdot)$ denotes a differential entropy. The differential entropy of the SVD-based independent parallel substreams $\mathbf {y}_{\textrm {d}}$ is upper-bounded by [40]: $$\begin{equation*} h_{\textrm {e}}(\mathbf {y}_{\textrm {d}}) \leq \log _{2}\left ({(\pi e)^{N}\left |{\mathbf {C}_{ \mathbf {y}_{\textrm {d}}}}\right |_{\mathrm{ det}}}\right), \tag{16}\end{equation*}$$ where, $\mathbf {C}_{ \mathbf {y}_{\textrm {d}}}$ represents the covariance matrix of $\mathbf {y}_{\textrm {d}}$ , which is given by $$\begin{align*} \mathbf {C}_{ \mathbf {y}_{\textrm {d}}}=&\mathbb {E}[ \mathbf {y}_{\textrm {d}} \mathbf {y}_{\textrm {d}}^{H}] \tag{17}\\=&\mathbb {E}\left [{(\sqrt {P \tau T_{0}} \boldsymbol{\Lambda } \mathbf {s} + \boldsymbol {\eta }_{v})(\sqrt {P \tau T_{0}} \boldsymbol{\Lambda } \mathbf {s} + \boldsymbol {\eta }_{v})^{H}}\right] \qquad \tag{18}\\=&\mathbb {E}\left [{P \tau T_{0} \boldsymbol{\Lambda } \mathbf {s} \mathbf {s}^{H} \boldsymbol{\Lambda }}\right] + \mathbb {E}\left [{ \boldsymbol {\eta }_{v} \boldsymbol {\eta }_{v}^{H}}\right] \tag{19}\\=&\tau PT_{0} \boldsymbol{\Lambda }^{2} + {N_{0}} \boldsymbol{\Lambda },\tag{20}\end{align*}$$ where we have $\mathbb {E}[ \mathbf {s} \boldsymbol {\eta } _{v}^{H}]=\mathbb {E}[ \boldsymbol {\eta }_{v} \mathbf {s}^{H}]=0$ and $\mathbb {E}[ \mathbf {s} \mathbf {s} ^{H}] = \mathbf {I}$ . Furthermore, the differential entropy of the complex-valued white Gaussian noise vector $\boldsymbol {\eta }_{v}$ is represented as $$\begin{equation*} h_{\textrm {e}}(\boldsymbol {\eta }_{v}) = \log _{2}\left ({(\pi e)^{N}\left |{{N_{0}} \boldsymbol{\Lambda }}\right |_{\mathrm{ det}}}\right). \tag{21}\end{equation*}$$ Thus, the mutual information $I(\mathbf {s}; \mathbf {y}_{\textrm {d}})$ is maximized as follows: $$\begin{align*} I(\mathbf {s}; \mathbf {y}_{\textrm {d}})\leq&\log _{2}\left({\frac {\left |{\tau PT_{0} \boldsymbol{\Lambda }^{2} + {N_{0}} \boldsymbol{\Lambda }}\right |_{\mathrm{ det}}}{\left |{{N_{0}} \boldsymbol{\Lambda }}\right |_{\mathrm{ det}}}}\right) \tag{22}\\=&\log _{2}\left ({\left |{\mathbf {I}_{N} + \frac {\tau PT_{0}}{{N_{0}}} \boldsymbol{\Lambda }}\right |_{\mathrm{ det}} }\right) \tag{23}\\=&\sum _{i = 0}^{N-1}\log _{2}\left({1 + \frac {\tau \lambda _{i}PT_{0}}{{N_{0}}}}\right). \tag{24}\end{align*}$$ Finally, similar to [6], [13], (24) is normalized by the symbol interval of FTN signaling $N\tau T_{0}$ , in order to arrive at the capacity of the conventional SVD-precoded FTN signaling scheme as follows: $$\begin{equation*} C_{\textrm {F, conv}} = \lim _{N\to \infty } \frac {1}{N\tau T_{0}}\sum _{i = 0}^{N-1}\log _{2}\left({1 + \frac {\tau \lambda _{i}PT_{0}}{{N_{0}}}}\right). \tag{25}\end{equation*}$$ Note again that the capacity (25) is derived for the SVD-precoded FTN signaling scheme that does not rely on power allocation [36], and hence is not directly applicable to the proposed scheme with power allocation.

Furthermore, since the singular values of the FTN-ISI matrix are approximated by [36] $$\begin{equation*} \lambda _{i} \approx (\tau T_{0})^{-1}H_{\mathrm{ fo}}(f_{i}) \quad (i = 0, \cdots , N-1),\tag{26}\end{equation*}$$ where we have $f_{i}=i/(N\tau T)$ , and $H_{\mathrm{ fo}}(f_{i})$ denotes the folded pulse spectrum.³ Hence, the capacity of (25) is modified to $$\begin{equation*} C_{\textrm {F, conv}} = \int _{-1/(2\tau T_{0})}^{1/(2\tau T_{0})}\log _{2}\left({1 + \frac {P}{{N_{0}}}H_{\mathrm{ fo}}(f)}\right)df. \tag{27}\end{equation*}$$ The folded pulse spectrum $H_{\mathrm{ fo}}(f)$ coincides with the original spectrum, specifically when the sinc shaping filter is considered. This implies that the capacity of (27) has no performance gain over the classic Shannon capacity for Nyquist-criterion-based signaling in the AWGN channel. This holds true for the information-theoretic analysis in the previous studies [6], [8], [10], [14]. By contrast, when employing the RRC shaping filter, a capacity gain is achievable, owing to FTN signaling with the packing ratio of $\tau \geq 1/(1+\beta)$ , as mentioned in [6].

However, in this section, uniform power allocation is assumed for each independent symbol substream. Hence, there is a room for enhancing the achievable information rate with the aid of the power optimization of SVD-precoded FTN signaling.

SECTION III. Proposed SVD-Precoded FTN Signaling With Optimal Power Allocation

Optimal power allocation for the conventional SVD-precoded FTN signaling scheme is proposed in this section.

A. System Model of Proposed SVD-Precoded FTN Signaling With Power Allocation

Fig. 1 shows a schematic diagram of the proposed transceiver structure for SVD-precoded FTN signaling with power allocation. In the proposed scheme, the precoding matrix $\mathbf {F}$ is the product of an orthogonal matrix $\mathbf {V}$ and a linear power allocation matrix $\mathbf {P}$ : $$\begin{equation*} \mathbf {F}=\mathbf {VP},\tag{28}\end{equation*}$$ where $\mathbf {P}$ is a diagonal matrix, represented as $\mathbf {P}=\textrm {diag}\left \{{\sqrt {p_{0}},\cdots ,\sqrt {p_{N-1}}}\right \}\in \mathbb {R}^{N\times N}$ . Recall that in the conventional SVD-precoded FTN signaling scheme without power allocation, the precoding matrix is given by $\mathbf {F}=\sqrt {P}\mathbf {V}$ .

Fig. 1.

Schematic of proposed SVD-precoded FTN signaling transceiver structure.

Hence, from (7), the received samples of the proposed SVD-precoded FTN signaling scheme with power allocation are represented by $$\begin{equation*} \mathbf {y}= \sqrt {\tau T_{0}} \mathbf {V} \boldsymbol{\Lambda } \mathbf {P} \mathbf {s}+ \boldsymbol {\eta }. \tag{29}\end{equation*}$$ Furthermore, similar to (10), multiplying the matrix $\mathbf {V}^{T}$ by $\mathbf {y}$ in (29) yields: $$\begin{equation*} \mathbf {y}_{\textrm {d}} = \sqrt {\tau T_{0}} \boldsymbol{\Lambda }\mathbf {P} \mathbf {s}+ \boldsymbol {\eta }_{v}. \tag{30}\end{equation*}$$ Since $\boldsymbol{\Lambda }\mathbf {P}$ is diagonal, the received sample model of (30) is regarded as $N$ independent parallel substreams as follows: $$\begin{equation*} y_{\textrm {d},i} = \sqrt {\tau T_{0}}\lambda _{i}\sqrt {p_{i}}s_{i} + \eta _{v,i} \quad (i=0,\cdots ,N-1), \tag{31}\end{equation*}$$ where $y_{\textrm {d},i}$ and $\eta _{v,i}$ are the $i$ th elements of $\mathbf {y}_{\textrm {d}}$ and $\boldsymbol {\eta }_{v}$ , respectively. Hence, the equivalent channel coefficient and the equivalent noise variance of the $i$ th substream are given by $\sqrt {\tau T_{0}}\lambda _{i}\sqrt {p_{i}}$ and $\lambda _{i}{N_{0}}$ , respectively. In the rest of this section, the elements of the power allocation matrix $p_{i} \,\,(i=0,\cdots , N-1)$ are optimized to maximize the associated capacity metric.

Note that when $\mathbf {P}=\sqrt {P}\cdot \textbf {I}$ , the proposed scheme corresponds to the conventional SVD-precoded FTN signaling scheme without power allocation.

B. Mutual Information of Proposed SVD-Precoded FTN Signaling Scheme With Power Allocation

In a similar manner to the capacity derivation of the conventional SVD-precoded FTN signaling in Section II-C, mutual information of the proposed $N$ -parallel SVD-precoded FTN signaling with power allocation is given by $$\begin{equation*} I(\mathbf {s}; \mathbf {y}_{\textrm {d}}) \leq \sum _{i = 0}^{N-1}\log _{2}\left({1 + \frac {\tau \lambda _{i}p_{i}T_{0}}{{N_{0}}}}\right), \tag{32}\end{equation*}$$ where the coefficient $P$ of (24) is replaced by the power allocation factor $p_{i}$ in the $i$ th substream.

Note that there are two main differences between the system models of the conventional SVD-precoded spatial multiplexing MIMO scheme [38] and the proposed SVD-precoded FTN signaling scheme, i.e., those in the energy constraints and the variances of the noise vector. Hence, in Section III-C, a power allocation solution specific to the proposed scheme is derived while taking into account these differences.

C. Optimal Power Allocation for Proposed Scheme

The optimal power allocation factors $p_{i} \,\,(i=0,\cdots ,N-1)$ are determined here to maximize the mutual information (32) of the proposed SVD-precoded FTN signaling, in order to derive the associated capacity.

The transmit energy $E_{N}$ per block for the conventional SVD-precoded FTN signaling scheme in Section II-B is calculated as follows: $$\begin{align*} E_{N}=&\mathbb {E}\left [{\int _{-\infty }^{\infty }|x(t)|^{2}dt}\right] \tag{33}\\=&\tau T_{0}\mathbb {E}\left [{\int _{-\infty }^{\infty }\sum _{l}\sum _{m}x_{l}x_{m}^{*}h(t-lT)h^{*}(t-mT)dt}\right] \\ \tag{34}\\=&\tau T_{0}\mathbb {E}\left [{\sum _{l}\sum _{m}x_{l}x_{m}^{*}g((l-m)T)}\right]\tag{35}\\=&\tau T_{0}\mathbb {E}\left [{ \mathbf {x}^{H} \mathbf {G} \mathbf {x} }\right] \tag{36}\\=&\tau PT_{0}\mathbb {E}\left [{ \mathbf {s}^{H} \boldsymbol{\Lambda } \mathbf {s} }\right]\tag{37}\\=&\tau PT_{0} \sum _{i=0}^{N-1}\lambda _{i}\mathbb {E}\left [{|s_{i}|^{2}}\right]\tag{38}\\=&{\tau NPT_{0}}, \tag{39}\end{align*}$$ where we have $\sum _{i=0}^{N-1}\lambda _{i} = {\mathrm{ trace}}\{ \mathbf {G}\} = N$ , since each diagonal element of $\mathbf {G}$ is unity.

Then, in the same manner as (36), the transmit energy $\tilde {E}_{N}$ per block for the proposed scheme is given by: $$\begin{align*} \tilde {E}_{N}=&\tau T_{0}\mathbb {E}\left [{ \mathbf {s}^{H} \mathbf {P} \mathbf {V} ^{T} \mathbf {G} \mathbf {V} \mathbf {P} \mathbf {s} }\right]\tag{40}\\=&\tau T_{0}\mathbb {E}\left [{ \mathbf {s}^{H} \mathbf {P} \boldsymbol{\Lambda } \mathbf {P} \mathbf {s} }\right]\tag{41}\\=&\tau T_{0}\sum _{i=0}^{N-1}p_{i}\lambda _{i}\mathbb {E}\left [{|s_{i}|^{2}}\right]\tag{42}\\=&{\tau T_{0}\sum _{i=0}^{N-1}p_{i}\lambda _{i}}. \tag{43}\end{align*}$$

In order to make the transmission energy of the proposed scheme be the same as that of the conventional SVD-precoded FTN signaling scheme, the following energy constraint is imposed: $$\begin{align*} \tilde {E}_{N}=&E_{N} \tag{44}\\\Leftrightarrow&\sum _{i=0}^{N-1}\lambda _{i}p_{i} = {NP}. \tag{45}\end{align*}$$

Based on the mutual information of (32) and the energy constraint of (45) for the proposed scheme, the power allocation factors $p_{i} \,\,(i=0,\cdots ,N-1)$ are optimized with the aid of the Lagrange multiplier method. More specifically, in order to maximize the mutual information of (32), let us consider the following Lagrange function: $$\begin{equation*} J = \sum _{i=0}^{N-1}\log _{2}\left ({1+\frac {\tau \lambda _{i}p_{i}T_{0}}{{N_{0}}}}\right) \!-\! \alpha \left ({\sum _{i=0}^{N-1}\lambda _{i}p_{i} - NP}\right), \quad ~~\tag{46}\end{equation*}$$ where $\alpha$ is the Lagrange multiplier and the second term of (46) represents the energy constraint. Furthermore, in order to obtain the power allocation factors $p_{i}$ that maximize $J$ of (46), the following problem is solved: $$\begin{equation*} \frac {\partial J}{\partial p_{i}}=0, \quad \textrm {subject to} ~ p_{i}\geq 0. \tag{47}\end{equation*}$$ Then, we have $$\begin{equation*} p_{i} = \frac {1}{\tau \lambda _{i}T_{0}}\left ({\frac {\tau T_{0}}{(\ln 2)\alpha } - {N_{0}}}\right). \tag{48}\end{equation*}$$ Substituting (48) into (45) yields: $$\begin{equation*} \alpha = \frac {1}{(\ln 2)\left ({P + \frac {{N_{0}}}{\tau T_{0}}}\right)}. \tag{49}\end{equation*}$$ Moreover, substituting (49) into (48) yields the optimal power allocation factors: $$\begin{equation*} p_{i} = \frac {P}{\lambda _{i}}. \tag{50}\end{equation*}$$ Note that the constraint $p_{i}\ge 0$ in (47) is always satisfied in (50). Thus, the maximum mutual information is given by $$\begin{align*} I(\mathbf {s}; \mathbf {y}_{\mathrm{ d}})_{\mathrm{ max}}=&\sum _{i=0}^{N-1}\log _{2}\left ({1 + \frac {\tau PT_{0}}{{N_{0}}}}\right) \tag{51}\\=&N\log _{2}\left ({1 + \frac {\tau PT_{0}}{{N_{0}}}}\right) \:\: {\mathrm{ [bits/block]}}. \tag{52}\end{align*}$$ Finally, the capacity of the SVD-precoded FTN signaling with optimal power allocation is formulated as follows: $$\begin{align*} C_{\mathrm{ F, opt}}=&\lim _{N\to \infty }\frac {1}{N} I(\mathbf {s}; \mathbf {y}_{\mathrm{ d}})_{\mathrm{ max}} \tag{53}\\=&\log _{2}\left ({1 + \frac {\tau PT_{0}}{{N_{0}}}}\right) \:\: {\mathrm{ [bits/sample]}}. \tag{54}\end{align*}$$ It is worth noting that the capacity metric (54) does not depend on the shaping filter $h(t)$ as well as on the singular values $\lambda _{i}$ . Furthermore, assuming the use of the strictly band-limited RRC pulse in the frequency range of $[-(1+\beta)W, (1+\beta)W]$ , we arrive at the capacity formula⁴ of [13], [41] $$\begin{equation*} C_{\text {F}_{W}, {\mathrm{ opt}}} = \frac {2W}{\tau }\log _{2}\left ({1 + \frac {\tau P}{{2N_{0}W}}}\right) \:\: {\mathrm{ [bits/sec]}}, \tag{55}\end{equation*}$$ where we have ${1/(\tau T_{0}) = } 2W/\tau$ FTN samples per second and $T_{0} = 1/(2W)$ . Note that for $\tau =1$ , the capacity of (55) corresponds to the classic Nyquist-criterion-based capacity in the complex-valued AWGN channel.

In order to provide further insights, an information rate of (55) for the limit of $\tau \rightarrow 0$ is given in a closed form, by applying l’Hôpital’s rule to (55) in a similar manner to the derivation of the classic Nyquist-criterion-based capacity in the AWGN channel for $W\to \infty$ [40], as follows [41]: $$\begin{align*} \lim _{\tau \to 0} C_{\text {F}_{W}, {\mathrm{ opt}}}=&\lim _{\tau \to 0} \frac {\frac {\partial }{\partial \tau }\left ({2W\log _{2}\left ({1+\frac {\tau P}{2N_{0}W}}\right)}\right)}{\frac {\partial }{\partial \tau }\tau } \tag{56}\\=&\lim _{\tau \to 0} \frac {2W\frac {P}{2N_{0}W}}{\log _{e}2\left ({1 + \frac {\tau P}{2N_{0}W}}\right)} \tag{57}\\=&\frac {1}{\log _{e}2}\times \frac {P}{N_{0}} \:\:\: {\mathrm{ [bits/sec]}}. \tag{58}\end{align*}$$ This indicates that an infinite information rate is unachievable, even if $\tau$ approaches zero.

An important note is that our extended Shannon capacity (55) of the proposed SVD-precoded FTN signaling with optimal power allocation outperforms that of the conventional unprecoded FTN signaling counterpart [6], as well as that of the conventional SVD-precoded FTN signaling counterpart [36] dispensing with any power allocation. This is not surprising since the capacity of SVD-precoded spatial multiplexing with optimal power allocation typically exceeds that of the unprecoded spatial-multiplexing counterpart, as well as that of the SVD-precoded spatial-multiplexing counterpart without any power allocation.

SECTION IV. Analytical Performance Results

This section presents the analytical performance results to characterize the conventional and proposed SVD-precoded FTN signaling schemes and compare them with the conventional Nyquist-criterion-based scheme. In the following analysis, we assume, for the sake of simplicity, that the symbol interval, satisfying the Nyquist criterion, is $T_{0}=1$ [sec] and thus we have $2W=1$ [Hz]. Moreover, the value of $P$ was fixed to 1 [W].

Fig. 2 compares the capacities of the conventional and proposed SVD-precoded FTN signaling schemes, where the symbol’s packing ratio was varied from $\tau =1.0$ to 0.1 in steps of 0.1. More specifically, Figs. 2(a) and 2(b) show the results of the conventional schemes employing a sinc filter and an RRC filter with the roll-off factor of $\beta = 1.0$ , respectively, while Fig. 2(c) shows the result of the proposed scheme. As shown in Fig. 2(a), the conventional scheme employing a sinc filter does not have any substantial performance advantage over the Nyquist-criterion-based counterpart ($\tau =1.0$ ) in terms of capacity. As shown in Fig. 2(b), the conventional scheme employing an RRC filter having $\beta = 1.0$ , which converged at $\tau \simeq 0.5$ , has a nearly two-hold capacity advantage over the Nyquist-criterion-based scheme. This performance advantage, due to the use of a non-sinc shaping filter in the conventional scheme, is consistent with the results shown in the information-theoretic analysis of its unprecoded counterpart [6]. Fig. 2(c) shows that the proposed scheme has substantial performance advantages over the conventional SVD-based FTN signaling scheme and the Nyquist-criterion-based scheme. More specifically, the performance advantages monotonically increased with decreasing symbol’s packing ratio of $\tau$ . These advantages are due to the optimal power allocation scheme derived in this paper.

Fig. 2.

Capacities $C_{\text {F}_{W}, {\mathrm{ opt}}}$ , and $C_{\mathrm{ F, conv}}$ of the proposed and conventional SVD-precoded FTN signaling schemes, where symbol’s packing ratio was varied from $\tau =1.0$ to 0.1 in steps of 0.1. (a) Conventional scheme with the sinc shaping filter, (b) conventional scheme with the RRC shaping filter having $\beta = 1.0$ , and (c) proposed scheme.

Fig. 3 shows the capacity gain of the proposed SVD-precoded FTN signaling scheme over the conventional Nyquist-criterion-based scheme ($\tau =1$ ), which was defined as $C_{\text {F}_{W}, {\mathrm{ opt}}} / C_{\mathrm{ N}}$ , where $C_{\mathrm{ N}}$ denotes the classic Shannon capacity for the Nyquist-criterion-based signaling in the AWGN channel. Here, the average signal-to-noise ratio (SNR) was set to 100 dB and the symbol’s packing ratio was varied from $\tau =0.1$ to 0.9 in steps of 0.1. For comparison, the normalized capacity gains of the conventional SVD-precoded FTN signaling scheme, i.e., $C_{\mathrm{ F, conv}} / C_{\mathrm{ N}}$ when $\beta = 0.0$ and 1.0, were also plotted. The capacity gains of the proposed scheme were 3.16, 4.65, and 9.00 for $\tau = 0.3$ , 0.2, and 0.1, respectively. Note that the proposed scheme’s performance gain did not reach the inverse of the symbol’s packing ratio $1/\tau$ . However, the results show that in order to achieve a capacity gain using FTN signaling, both SVD-based precoding and power allocation are required.

Fig. 3.

Capacity gains of proposed scheme over Nyquist-criterion-based counterpart ($\tau =1$ ) for SNR of 100 dB. The symbol’s packing ratio was varied from $\tau =0.1$ to 0.9 in steps of 0.1.

Fig. 4 shows the singular values $\lambda _{i}$ calculated from the FTN-induced ISI matrix $\mathbf {G}$ , where the symbol’s packing ratio was varied in the range of $0.1\leq \tau \leq 0.9$ with the step of 0.1, while the roll-off factors $\beta =0.0$ and 0.22 were employed in Figs. 4(a) and 4(b), respectively. Note that the number of singular values was equivalent to the block size of $N=10000$ . As shown in Fig. 4, very low singular values were obtained, especially when $\tau < 1/(1+\beta)$ . When the $\tau$ value was further decreased, the number of the low singular values increased. Importantly, such singular values may be too small to be tractable in practice, as discussed in the next section.

Fig. 4.

Singular values of FTN-induced ISI matrix $\mathbf {G}$ , where we employed $\beta =0.0, 0.22$ , and $N=10000$ . The symbol’s packing ratio was varied from $\tau =0.1$ to 0.9 in steps of 0.1.

SECTION V. Practical Considerations

The main focus of this study is the information-theoretic analysis of the proposed SVD-precoded FTN signaling scheme. The analytical results indicate that the proposed scheme has a higher achievable limit than those of the conventional Nyquist-criterion-based scheme and the conventional SVD-precoded FTN signaling scheme.

However, to achieve performance close to the theoretical capacity derived in Section III-C, powerful near-capacity channel coding schemes, such as turbo [42] and low-density parity-check codes [43], as well as an iterative detection algorithm have to be included in the modem. The results presented in this study are based on several idealistic assumptions, and hence there are several issues that have to be considered for the practical implementation of the proposed scheme, as described in the rest of this section.

A. Inter-Block Interference

IBI between FTN signaling blocks is typically imposed when multiple FTN blocks are successively transmitted. However, its detrimental effects were ignored in the information-theoretic analysis conducted here because the aim was to determine the achievable upper bound. Nonetheless, IBI may be efficiently cancelled with the aid of the iterative soft-interference cancellation technique [22], [25], since the effects of FTN-induced IBI are also known at the receiver in advance.

B. Frequency-Selective Channel

In this study, an AWGN channel was assumed for simplicity. The proposed scheme is readily applicable to a frequency-flat fading channel since the calculations of offline SVD are still possible in such a scenario. However, for a frequency-selective fading channel, the SVD of a matrix, including both the effects of the FTN-induced ISI and the dispersive channel, changes depending on the channel conditions. Hence, the highly complex real-time calculations of SVD are imposed on the transmitter; these calculations have to be updated during each channel’s coherence time.

C. Higher Sampling Rate

In the FTN signaling scheme, the symbol interval is $\tau$ times lower than that of the Nyquist-criterion-based counterpart, assuming that a sample in (4) is acquired for each symbol. This implies that in the FTN signaling scheme, the sampling rate is $1/\tau$ times higher than that of the Nyquist-criterion-based counterpart. This may impose an increased receiver cost, especially for the high-rate (low-$\tau$ ) FTN scenario.

D. Spectrum Broadening

The precoding of the FTN signaling may have the possibility of broadening the associated spectrum of $x(t)$ , as mentioned in [44]. In Section VI, we numerically evaluate the effects of spectrum broadening on the proposed SVD-precoded FTN signaling scheme with power allocation.

E. Truncation Issue

Depending on the symbol’s packing ratio $\tau$ employed, the singular values $\lambda _{i}$ may become too low to be tractable in practical modems. Fig. 5 shows the singular values of $\mathbf {G}$ for $(\tau , \beta) = (0.1, 0.0)$ and (0.1, 0.5), where 89% of singular values are below 10⁻¹³ for $\beta =0.0$ , while 84% of singular values are below 10⁻¹⁰ for $\beta =0.5$ . Further decreasing $\tau$ increases the ratio of such low singular values. Importantly, the significantly low singular values have to be carefully treated so that calculations associated with floating point numbers are accurate.⁵ Note that in general, a matrix having the form of $\mathbf {G}$ with sinc shaping filter is known as a prolate matrix [45], [46], which typically contains significantly low singular values when the matrix size is large, as shown in Fig. 5(a). Since it is a challenging task to precisely compute such low singular values, the calculations of the Moore-Penrose inverse of a prolate matrix, which requires the inverse of a singular value, is a numerically ill-posed problem [46]. More specifically, in the proposed scheme, it is needed to compute $1/\lambda _{i}$ for all the substreams, in order to achieve a maximum information rate, and hence the optimal power allocation strategy derived in Section III also suffers from the ill-posed problem.⁶ In order to overcome this limitation, the proposed scheme is modified to support suboptimal truncated power allocation in Section VI.

Fig. 5.

Singular values of the FTN-induced ISI matrix $\mathbf {G}$ : (a) $(\tau , \beta) = (0.1, 0.0)$ and (b) (0.1, 0.5).

SECTION VI. Proposed SVD-Precoded FTN Signaling With Truncated Power Allocation

This section presents truncated power allocation for the proposed scheme, which approximates the optimal power allocation derived in Section III-C. Furthermore, the PSD of truncated SVD-precoded FTN signaling is investigated.

A. Truncated Power Allocation

A thresholding value $t_{h} \in \mathbb {R}$ is introduced for activating or deactivating $N$ independent parallel substreams in (31),

according to: $$\begin{equation*} p_{i} = \begin{cases} \frac {kP}{\lambda _{i}} & (\lambda _{i} \geq t_{h}) \\ 0 & (\lambda _{i} < t_{h}) ,\end{cases} \tag{59}\end{equation*}$$ where $k \in \mathbb {R}$ is a scaling factor used for maintaining the average transmit energy per block. As shown in (59), the ill-conditioned subchannels, satisfying the condition $\lambda _{i} < t_{h}$ , are not used for transmission. With the number of activated substreams denoted as $M ~(\le N)$ , the average transmit energy per block (43) in the proposed suboptimal truncated scheme is given by: $$\begin{align*} \tilde {E}_{N}=&\tau T_{0} \sum _{i=0}^{M-1} \lambda _{i}p_{i} \tag{60}\\=&\tau kPMT_{0}, \tag{61}\end{align*}$$ where it is assumed that $\lambda _{i} \,\,(i=0,\cdots ,N-1)$ are sorted in descending order. Furthermore, according to the energy constraint in (44), $k=N/M$ . Hence, substituting (59) into (32), the information rate of the proposed SVD-precoded FTN signaling with truncated power allocation, employing the RRC shaping filter, is given as follows: $$\begin{align*} C_{\text {F}_{W}, {\mathrm{ sub}}}=&\frac {1}{N\tau T_{0}}\sum _{i=0}^{N-1}\log _{2}\left ({1 + \frac {\tau \lambda _{i} p_{i}T_{0}}{{N_{0}}}}\right) \tag{62}\\=&\frac {2WM}{\tau N}\log _{2}\left ({1 + \frac {\tau (N/M)P}{{2N_{0}W}}}\right) \: {\mathrm{ [bits/sec]}}, \qquad \quad \tag{63}\end{align*}$$ where in a similar manner to (53) and (55), the information rate (62) is normalized by the number of symbols per block $N$ as well as the number of symbols per second $1/(\tau T_{0})$ . With this truncation operation, it is possible to avoid processing the $(N-M)$ substreams, which correspond to singular values lower than $t_{h}$ . However, the main drawback of this truncated scheme is that the associated information rate decreases upon increasing the number of deactivated substreams.

Fig. 6 shows the information rate of the proposed scheme with truncated power allocation, where we have $\tau =0.1$ , $\beta =0.22$ , and $N=10000$ . The threshold $t_{h}$ was set to 10⁻¹ and $10^{a}$ , where the exponent $a$ was varied from −10 to −17 with the step of −1. Observed in Fig. 6 that the information rate decreased, upon increasing the threshold $t_{h}$ , as expected. More specifically, the information rate curve of $t_{h}=10^{-17}$ was comparable to that of the proposed scheme without truncation, which corresponded to the maximum achievable bound. However, upon increasing $t_{h}$ beyond 10⁻¹⁶, the unignorable rate loss was found. The proposed scheme with the threshold of $t_{h}=0.1$ exhibited the information rate similar to that of the conventional SVD-precoded FTN signaling scheme without power allocation.

Fig. 6.

Information rates of the proposed scheme with truncated power allocation, where the system parameters of $\tau =0.1$ , $\beta =0.22$ , and $N=10000$ were used. Moreover, we employed the threshold values of $t_{h}=10^{-17}, 10^{-16}, 10^{-15}, 10^{-14}, 10^{-13}, 10^{-12}, 10^{-11}, 10^{-10}$ , and 10⁻¹.

Fig. 7 shows the information rate of the proposed scheme with truncated power allocation, where $t_{h}=10^{-14}$ , 10⁻¹³, and 10⁻¹⁰ in Figs. 7(a), 7(b), and 7(c), respectively. The symbol’s packing ratio was varied from $\tau =0.9$ to 0.1 with the step of 0.1 as well as given by $\tau =0.05$ and 0.01. The other parameters were the same as those used in Fig. 6. Observe in Fig. 7 that for $t_{h}=10^{-14}$ and 10⁻¹³, the information rate of the proposed scheme exhibited an explicit performance gain over the benchmark schemes, which increased, upon decreasing the $\tau$ value. Even for $t_{h}=10^{-10}$ , the moderate performance gain was found in the proposed scheme over the conventional FTN signaling in the entire range of $\tau$ .

Fig. 7.

Information rate of the proposed scheme with truncated power allocation for $t_{h}=10^{-14}$ , 10⁻¹³, and 10⁻¹⁰, where $\beta =0.22$ and $N=10000$ were employed. The symbol’s packing ratio was varied from $\tau =0.9$ to 0.1 with the step of 0.1 as well as given by $\tau =0.05$ and 0.01.

B. The Effects on PSD

In this section, we investigated the PSDs of the proposed SVD-precoded FTN signaling scheme with truncated power allocation. The PSDs were calculated by averaging the discrete Fourier transform of the samples of $x(t)$ with the aid of the periodogram technique. In our simulations, the transmit energy almost matched with its theoretical value $\tilde {E}_{N} = \mathbb {E}\left [{\int _{-\infty }^{\infty }|x(t)|^{2}dt}\right]$ , so that the small singular values were accurately calculated. Moreover, we considered the samples of transmitted time-domain signals $x(t)$ with the step of $0.01 \times T_{0}$ [sec] within the time interval of $[-100T_{0}, NT_{0}+100T_{0}]$ , such that the side-lobe level of the transmitted signals is maintained to be sufficiently low.

Figs. 8(a) and 8(b) show the PSDs of the conventional SVD-precoded FTN signaling scheme without power allocation and the proposed scheme with truncated power allocation while plotting the associated curve of the Nyquist-criterion-based signaling scheme. A sinc shaping filter was employed in all the schemes. The symbol’s packing ratio and the block length were set to $\tau =0.8, 0.2, 0.1, 0.05, 0.01$ and $N=10000$ , respectively. Moreover, the threshold of the proposed scheme was given by $t_{h} = 10^{-12}$ for $\tau = 0.8$ , and $t_{h} = 10^{-10}$ for other $\tau$ values. The transmitted FTN signals were not normalized by $\sqrt {\tau T_{0}}$ , so that the energy per block was the same as that of Nyquist-criterion-based signaling. As shown in Fig. 8(a), the PSD of the conventional SVD-precoded FTN signaling without power allocation sharply dropped at the frequency of $W=0.5$ [Hz], regardless of the symbol’s packing ratio, hence dispensing with spectrum broadening. Note that upon decreasing $\tau$ , the spectral side-lobe level at the frequencies higher than 0.5 [Hz] increased, while maintaining the width of the spectrum main lobe. Similar to 8(a), in Fig. 8(b) it was found that the spectral main lobe of the proposed scheme with truncated power allocation was not broadened in our simulations with specific FTN parameters.

Fig. 8.

PSDs of (a) the conventional SVD-precoded FTN signaling scheme without power allocation and (b) the proposed scheme with truncated power allocation. A sinc shaping filter was employed in each scheme. In the proposed scheme, the threshold was set to $t_{h} = 10^{-12}$ for $\tau =0.8$ , and $t_{h} = 10^{-10}$ for $\tau =0.2, 0.1, 0.05$ , and 0.01.

Next, Figs. 9(a) and 9(b) show the PSDs of the conventional SVD-precoded FTN signaling scheme without power allocation and the proposed scheme with truncated power allocation. The RRC shaping filter with the roll-off factor of $\beta =0.22$ was used in each scheme. The symbol’s packing ratio was given by $\tau =0.9, 0.8, 0.2, 0.1, 0.05$ , and 0.01, while in the proposed scheme, the threshold was set to $t_{h} = 10^{-14}$ for $\tau =0.9$ and 0.8, as well as $t_{h} = 10^{-10}$ for $\tau = 0.2, 0.1, 0.05$ , and 0.01. In a similar manner to the sinc-filter scenario of Fig. 8, in Figs. 9(a) and 9(b), both the PSDs of the conventional SVD-precoded FTN scheme without power allocation and the proposed scheme with truncated power allocation dropped at the frequency of $(1+\beta)W=0.61$ [Hz], and then upon increasing the frequency, the PSD continued to decrease. More specifically, in Fig. 9(b), the gap in the spectral side-lobe level of the proposed scheme with the RRC shaping filter tended to be higher than that of the sinc shaping filter. Note, however, that the PSD of the proposed scheme for the frequencies higher than 0.61 [Hz] remained sufficiently low, and hence any substantial spectrum broadening is not imposed.

Fig. 9.

PSDs of (a) the conventional SVD-precoded FTN signaling scheme without power allocation and (b) the proposed scheme with truncated power allocation. The RRC shaping filter with $\beta =0.22$ was employed in each scheme. In the proposed scheme, the threshold was set to $t_{h} = 10^{-14}$ for $\tau =0.9$ and 0.8, and $t_{h} = 10^{-10}$ for $\tau =0.2, 0.1, 0.05$ , and 0.01.

Moreover, in Fig. 10, we investigated the effect of the threshold value on the PSDs of the proposed scheme with the RRC filter having $\beta =0.22$ . The packing ratio was fixed to $\tau =0.01$ , and the threshold $t_{h}$ was varied from 50 to 10⁻¹⁰. As shown in Fig. 10, the PSD of the proposed scheme changed depending on the threshold value. More specifically, the PSD was similar to that of the classic Nyquist-criterion-based scheme with a sinc filter for the threshold value as low as $t_{h}\leq 10^{-1}$ . According to Figs. 9 and 10, if we choose the appropriate values of $\tau$ and $t_{h}$ , the proposed scheme is not imposed by any noticeable spectrum broadening, while achieving a higher information rate than the conventional FTN and Nyquist-criterion-based signaling schemes, as shown in Fig. 7(c).

Fig. 10.

PSDs of the proposed SVD-precoded FTN signaling with truncated power allocation and the Nyquist-criterion-based signaling schemes with the RRC shaping filter, where we employed the roll-off factor of $\beta =0.22$ and the packing ratio of $\tau =0.01$ . The threshold $t_{h}$ was varied from 50 to 10⁻¹⁰.

SECTION VII. BER Analysis

In this section, in order to characterize the error-rate performance of our SVD-precoded FTN signaling with truncated power allocation, we carried out Monte Carlo simulations, where we considered the three-stage serially-concatenated turbo-coded architecture.

A. Three-Stage-Concatenated Turbo Coded Architecture

Fig. 11 portrays the three-stage-concatenated turbo coded structure for our SVD-precoded FTN signaling scheme with truncated power allocation, similar to [22]. At the transmitter, information bits are channel-encoded by the half-rate recursive systematic coding (RSC) encoder, and then the RSC-encoded bits are interleaved by the outer interleaver $\Pi _{1}$ . Furthermore, the interleaved bits are encoded by the unity-rate coding (URC) encoder [47]. The URC-encoded bits are interleaved by the inner interleaver $\Pi _{2}$ . Moreover, the interleaved bits are modulated by the proposed SVD-precoded FTN signaling with truncated power allocation of Section VI-A.

Fig. 11.

The transceiver architecture of the proposed three-stage-concatenated turbo-coded SVD-precoded FTN signaling scheme with truncated power allocation.

When activating the proposed truncated power allocation principle, the number of active symbols per block may be lower than the block length $N$ . Hence, to maintain the transmission rate to be the same as that of the conventional FTN signaling scheme, an adaptive multi-level modulation scheme is introduced into each substream in a block. More specifically, one of the several modulation schemes, such as binary phase-shift keying (BPSK), quadrature PSK (QPSK) or $L$ -ary quadrature amplitude modulation (QAM), was assigned to each of the $M$ active substreams, to maintain a target rate regardless of the truncation. Here, we have $L=2^{b}>4$ , and $b$ is even number. In this paper, such an encoding scheme is referred to as bit loading.

At the receiver, iterative detection is carried out after the diagonalization by the matrix $\mathbf {V}^{T}$ at the received samples. At the log-likelihood ratio (LLR) calculation block of Fig. 11, the extrinsic LLRs of the received symbols $y_{\mathrm{ d, i}}$ are calculated from the a priori LLR of $y_{\mathrm{ d, i}}$ which is input from the URC decoder. The URC decoder receives the extrinsic information from the surrounding two blocks, and then it outputs the extrinsic information to the two other soft decoders. The number of iterations between the RSC and the URC decoder and that between the URC decoder and the LLR calculator are denoted as $I_{\mathrm{ out}}$ and $I_{\mathrm{ in}}$ , respectively. After the specified number of inner and outer iterations, the RSC decoder outputs the estimated bit sequences.

B. BER Results

In this section, we show the achievable BER performance of the proposed scheme employing the three-stage-concatenated turbo architecture. Here, the half-rate RSC(2,1,2) code with the constraint length of two and the octal generator polynomials of (3,2) was employed, and the interleaver length was set to 200000. The numbers of inner and outer iterations were set to $I_{\mathrm{ in}}=2$ and $I_{\mathrm{ out}}=40$ , respectively.

In Fig. 12, we plotted the BER curves of our SVD-precoded FTN signaling with truncated power allocation, where the RRC shaping filter having the roll-off factor of $\beta =0.22$ was employed.⁷ The threshold for the proposed scheme was fixed to $t_{h}=10^{-10}$ . For comparison, we also plotted the BER curves of the conventional Nyquist-criterion-based scheme. Here, we employed the symbol’s packing ratio of $\tau = 0.5, 0.25, 0.125$ and 0.1, whose spectral efficiencies corresponded to 0.82, 1.64, 3.28 and 4.10 [bps/Hz], respectively. Here, the spectral efficiency $R$ was calculated as follows [21], [22]: $$\begin{equation*} R = \frac {1}{2}\cdot \frac {1}{N\tau T_{0}}\cdot \frac {1}{2(1+\beta)W}\cdot \sum _{i=0}^{N-1}b_{i} \:\: {\mathrm{ [bps/Hz]}}, \tag{64}\end{equation*}$$ where $b_{i}$ denotes the bits assigned to the $i$ th substream. For the Nyquist-criterion-based scheme, we employed the QPSK, 16-QAM, 256-QAM, and 1024-QAM schemes, to keep the transmission rate same as each scenario of the proposed scheme. In Fig. 12, we also plotted the maximum achievable limits of the SVD-precoded FTN signaling with truncated power allocation and the Nyquist-criterion-based signaling, which were calculated from the associated capacity-versus-SNR curves [48]. As shown in Fig. 12, both the BER curves of the proposed and the Nyquist-criterion-based scheme were comparable for the lower transmission rate of 0.82 [bps/Hz], while the proposed scheme achieved the better performance for the higher transmission rates of 1.64, 3.28 and 4.10 [bps/Hz]. More specifically, the performance gains of the proposed scheme over the conventional Nyquist-criterion-based scheme were 0.6 dB, 1.9 dB and 2.5 dB for the transmission rates of 1.64, 3.28 and 4.10 [bps/Hz], respectively. Furthermore, for the transmission rate of 4.10 [bps/Hz], the performance gap from the achievable performance limit was 2.2 dB for the proposed scheme, while that of the conventional Nyquist-criterion-based scheme was 2.6 dB. Note that these performance benefits were attained without any substantial sacrifice in terms of spectrum broadening, as shown in Fig. 9(b).

Fig. 12.

BER performance of the proposed SVD-precoded FTN signaling with truncated power allocation, where we considered the three-stage-concatenated turbo-coded architecture. We set $\tau = 0.5, 0.25, 0.125, 0.1$ and $\beta =0.22$ , and the threshold of $t_{h}=10^{-10}$ . For comparison, we plotted the BER curves of the Nyquist-criterion-based signaling using the QPSK, 16-QAM, 256-QAM and 1024-QAM modulation schemes.

Fig. 13 shows the achievable BER performance of the proposed scheme for the target transmission rates of 1.64, 3.28 and 4.10 [bps/Hz], where the RRC shaping filter with $\beta =0.22$ was used. The symbol’s packing ratio was given by $\tau = 0.5, 0.25, 0.125, 0.1$ , and 0.01, while the threshold was fixed to $t_{h} = 10^{-10}$ . Note that bit loading was carried out, so that each fixed target rate was satisfied. For comparison, we also plotted the BER curves of the Nyquist-criterion-based scheme with 16-QAM, 256-QAM and 1024-QAM in Fig. 13(a), 13(b) and 13(c), respectively. The dashed lines represent the maximum achievable limits of the SVD-precoded FTN signaling with truncation and the Nyquist-criterion-based signaling. As shown in Fig. 13, the proposed scheme outperformed the conventional Nyquist-criterion-based transmission in all the target-rate scenarios. While the BER performance of the proposed scheme remained almost unchanged for $\tau \ge 0.1$ , an explicit performance gain was observed for $\tau =0.01$ . This benefit was achieved because a low symbol’s packing ratio allows us to use low modulation orders.

Fig. 13.

BER performance of the proposed three-stage-concatenated turbo-coded SVD-precoded FTN signaling scheme with truncated power allocation. Here, we employed the parameters of $\beta =0.22$ , $\tau =0.5, 0.25, 0.125, 0.1$ and 0.01, as well as $t_{h} = 10^{-10}$ . The target transmission rate was given by (a) 1.64 [bps/Hz], (b) 3.28 [bps/Hz], and (c) 4.10 [bps/Hz].

In order to elaborate a little further, Fig. 14 shows the achievable BER performance of the proposed scheme with the RRC shaping filter having the roll-off factors of $\beta =0.0, 0.22$ and 0.5, where the symbol’s packing ratio was maintained to be $\tau =0.1$ . In Figs. 14(a) and 14(b), the target transmission rate was set to 2.0 and 5.0 [bps/Hz], respectively.⁸ The dashed lines exhibit the maximum achievable limits for $\beta =0.0, 0.22$ , and 0.5. As shown in Figs. 14(a) and 14(b), in both the target-rate scenarios, the BER performance of the proposed scheme deteriorated, upon increasing the value of the roll-off factor $\beta$ . This is because increasing the roll-off factor extends the bandwidth, hence leading to the increased modulation order for attaining the target transmission rate. Furthermore, regardless of the roll-off factor, the gap between the BER curve and the theoretical achievable limit remained unchanged, which was 1.3 dB in Fig. 14(a) and 2.5 dB in Fig. 14(b), respectively.

Fig. 14.

BER performance of the proposed three-stage-concatenated turbo-coded SVD-precoded FTN signaling with truncated power allocation. Here, we employed the parameters of $\tau =0.1$ , $\beta =0.0, 0.22$ and 0.5, as well as $t_{h} = 10^{-10}$ . The target transmission rate was set to (a) 2.0 [bps/Hz] and (b) 5.0 [bps/Hz].

SECTION VIII. Conclusion

This study proposed an SVD-precoded FTN signaling scheme with optimal power allocation. To optimize power allocation, the closed-form capacity of the proposed SVD-precoded FTN signaling scheme and that of the conventional counterpart without power allocation were derived. The information-theoretic analysis demonstrated that the proposed scheme outperforms the conventional SVD-precoded and unprecoded FTN signaling schemes, as well as the Nyquist-criterion-based scheme, in terms of capacity. Several implementation issues were considered for the proposed scheme’s practical implementation. To eliminate the effects of significantly low singular values, we extended the proposed scheme to that supporting truncated power allocation. Our theoretical and numerical performance results demonstrate the achievable gains of the proposed SVD-precoded FTN signaling scheme over the conventional FTN signaling and the classic Nyquist-criterion-based signaling schemes.