Differential Faster-Than-Nyquist Signaling

In this paper, we propose a novel differential faster-than-Nyquist (DFTN) signaling concept that allows us to dispense with any channel estimation at the receiver, while benefiting from a rate boost specific to faster-than-Nyquist (FTN) signaling. More specifically, at the transmitter, differentially modulated phaseshift keying (DPSK) symbols are transmitted with a symbol interval that is smaller than that defined by the Nyquist criterion. The receiver first equalizes the DPSK symbols, which suffer from the effects of FTN-specific inter-symbol interference (ISI), with the aid of frequency-domain equalization (FDE). Then, the equalized symbols are differentially demodulated in a noncoherent manner. This noncoherent detection is enabled, by relying on the fact that FTN-specific ISI is deterministic when the FTN's symbol packing ratio and the roll-off factor of a shaping filter are known in advance of transmissions. Moreover, we derive an analytical bound on the achievable bit error ratio for the proposed DFTN signaling, based on moment-generating functions. This is achieved by introducing an equivalent signal-to-interference-and-noise ratio for the DPSK symbols after FDE. Our simulation results demonstrate that our DFTN receiver is capable of noncoherently demodulating the DFTN symbols when the symbol packing ratio is not substantially low.

SECTION I. Introduction

In the 1970s, the concept of faster-than-Nyquist (FTN) signaling was invented as a means of increasing a transmission rate, which is achieved without expanding bandwidth and power consumption [1]–​[5]. In FTN signaling, a symbol interval $T$ is set to be lower than that defined by the Nyquist criterion $T_{0}$ while satisfying the relationship $T=\alpha T_{0}\,\,(0\le T\le T_{0})$ , where $\alpha$ is the symbol packing ratio. The main limitation of FTN signaling is that the effects of inter-symbol interference (ISI) are imposed on the receiver, even in frequency-flat or additive white Gaussian noise (AWGN) channels. When assuming the employment of an ideal sinc pulse at the FTN transmitter, the FTN scheme is capable of enhancing a transmission rate without decreasing the minimum Euclidean distance in the range of $0.802\le \alpha \le 1$ [2]. A similar trend is seen when a practical root-raised-cosine (RRC) pulse is employed [6].

In order to eliminate the effects of FTN-specific ISI, several efficient equalizers have been developed in the time [6]–​[10] and frequency domains [11]–​[16]. Time-domain equalizers (TDEs) typically suffer from an excessive detection complexity when the block length is practically high. By contrast, cyclic prefix (CP)-assisted frequency-domain receivers [11], [12], [14] achieve a practical low detection complexity even in the highly dispersive frequency-selective fading channel, although they suffer a performance penalty compared to time-domain non-linear detection. Furthermore, in [13], an iterative decision-feedback scheme was incorporated into frequency-domain equalization (FDE) for the sake of further improving the detection performance. In [15], an FDE-assisted FTN receiver suitable for doubly selective fading channels was developed, while in [17], low-complexity symbol-by-symbol detection was proposed under the idealistic assumption of an AWGN channel. Moreover, a linear pre-equalization scheme that compensates for FTN-specific ISI was developed in [18] that attains a performance close to that of the ISI-free Nyquist-criterion-based system when the relationship $\alpha \geq 1/(1+\beta)$ is satisfied, where $\beta$ is the roll-off factor.

Another issue of FTN signaling is channel estimation (CE), while in most of the previous studies, perfect knowledge of channel state information (CSI) was assumed to be available at the receiver. In order to reduce pilot overhead, the use of an FTN pilot (FTNP) sequence was considered while developing the FTNP-based CE schemes in the time [19] and frequency domains [16], [20]. In [19], the joint CE and data detection (DD) scheme was proposed, based on the approximated Gaussian message passing algorithm. Although the scheme of [19] is capable of effectively estimating the CSI for a moderately high $\alpha$ , its complexity becomes prohibitively high upon increasing the tap-length of ISI for the same reason as the TDEs of FTN signaling. In [16], an FTNP-based semi-blind frequency-domain CE and DD was presented, and a design guideline of FTNP symbols was proposed. However, the benefit of reduced pilot overhead is achieved at the cost of additional complexity imposed by the iterative process of the receiver; more specifically, these FTNP-based CE schemes still rely on the use of pilot symbols.

Classically, differential modulation and noncoherent detection were developed in order to allow a receiver to detect symbols without a coherent phase reference [21], and hence avoiding any related pilot overhead. Note that differential modulation typically imposes an error-doubling effect on the noncoherent receiver. Importantly, in general, noncoherent detection is realistic only for an ISI-free frequency-flat channel, and noncoherent detection for an ISI-induced dispersive channel remains an open issue. The only exception is noncoherent detection assisted by interference rejection spreading code in code-division multiple access (CDMA) systems [22]. Since FTN signaling naturally introduces ISI on the received signals even in a frequency-flat channel, no differential modulation and noncoherent detection schemes have been proposed for FTN signaling systems.

Against this background, the novel contributions of this paper are as follows. We first propose a differential FTN (DFTN) signaling concept, in order to enable pilot-free noncoherent detection at the receiver while achieving the fundamental benefits of conventional FTN signaling. More specifically, under the assumption of a frequency-flat Rayleigh fading channel, noncoherent detection becomes realistic in our scheme by exploiting the fact that FTN-specific ISI is deterministic. This is because the channel impulse response associated with the FTN-specific ISI is accurately acquired when the symbol packing ratio $\alpha$ and the roll-off factor of a shaping filter are known at the receiver in advance of transmissions. Hence, low-complexity FDE [11], [23] is carried out in order to eliminate the deterministic ISI, and then the equalized DFTN symbols are detected with the aid of differential noncoherent detection.

Furthermore, we derive an analytical bit error ratio (BER) bound for our channel-uncoded DFTN signaling system. More specifically, in order to exploit the moment-generating-function (MGF)-based BER calculations developed for the conventional Nyquist-criterion-based schemes [24], we introduce a closed-form signal-to-interference-and-noise ratio (SINR) of the equalized DFTN symbols. Our analytical and numerical results demonstrate the above-mentioned fundamental benefits of the proposed DFTN scheme. Moreover, we show that the performance advantage of the DFTN scheme over the conventional coherent FTN scheme becomes clearer in a rapidly time-varying scenario.

The remainder of this paper is organized as follows. In Section II, we introduce the system model of our DFTN scheme. In Section III, the analytical BER bound is derived, and in Section IV, our performance results are presented. Finally, Section V concludes the present paper.

SECTION II. System Model of DFTN

In this section, we provide the system model of our DFTN transmitter and present a description of FDE-assisted noncoherent detection.

A. Transmitter Model

Fig. 1 shows the transceiver structure of our DFTN scheme, where we consider a block transmission of DFTN signaling over the frequency-flat fading channel. At the transmitter, $N$ phase-shift keying (PSK) symbols $\mathbf {x} = [x_{1},\cdots,x_{N}]^{T} \in \mathbb {C}^{N}$ are modulated per block. Then, the $N$ PSK-modulated symbols are differentially encoded in order to obtain differential PSK (DPSK) symbol block $\mathbf {s} = [s_{0},\cdots,s_{N}]^{T} \in \mathbb {C}^{N+1}$ satisfying the following: $$\begin{equation} s_{i} = x_{i} s_{i-1} \quad \mathrm {for}~1\le i\le N, \end{equation}$$ where we consider an initial reference symbol of $s_{0}= 1$ , for simplicity. Furthermore, a $2\nu$ -length CP is added in each block. Here, $2\nu$ is designed to be sufficiently longer than the effective tap length of ISI. The ($N+2\nu +1$ )-length symbols are bandlimited with the aid of an RRC filter $a(t)$ having a roll-off factor of $\beta$ in order to obtain the following time-domain signals: [11] $$\begin{equation} s(t) = \sum _{n} s_{n}a(t-nT), \end{equation}$$ where we have an FTN symbol interval of $T=\alpha T_{0}$ . Hence, the spectral efficiency of our CP-inserted DFTN signaling is formulated by $$\begin{equation} R=\frac {N}{N+2\nu +1}\frac {\log _{2}\mathcal {M}}{\alpha (1+\beta)}. \end{equation}$$ Note that when the block length $N$ is sufficiently longer than the CP length $2\nu$ , the effective overhead imposed by the CP insertion becomes marginal.

FIGURE 1.

Transceiver model of our DFTN scheme.

B. Receiver Model

Under the assumption of quasi-static frequency-flat Rayleigh fading, the received signals, which are matched-filtered by $a^{*}(-t)$ , are expressed as $$\begin{equation} y(t) = h \sum _{n} s_{n}g(t-nT) + \eta (t), \end{equation}$$ where we consider $g(t) = \int a(\tau)a^{*}(\tau -t)d\tau$ and $\eta (t) = \int n(\tau)a^{*}(\tau -t)d\tau$ , and $n(t)$ is the complex-valued AWGN with a zero mean and a noise variance of $N_{0}$ . Moreover, $h$ is a channel coefficient obeying a complex-valued Gaussian distribution having a zero mean and a unit variance. Assume that $h$ remains constant over each block transmission and that the symbol packing ratio $\alpha$ , the roll-off factor $\beta$ , and the noise variance $N_{0}$ are available at the receiver.

The $i$ th received sample is represented by $$\begin{align} y_{i}=&y(iT) \\=&h\sum _{n} s_{n}g((i-n)T) + \eta (iT), \end{align}$$ where the noise components $\eta (iT)\,\,(i=1,\cdots,N+2\nu +1)$ in each block are correlated and we have the relationship of $\mathbb {E}[\eta (iT)\eta (jT)]=N_{0}g((i-j)T)$ [11], while $\mathbb {E}[\cdot]$ represents the expectation operation. After removing the first and last $\nu$ samples from the ($N+2\nu +1$ )-length received block of (6), we arrive at the tractable signal representation of $$\begin{equation} \mathbf {y} = h\mathbf {G}\mathbf {s} + \boldsymbol {\eta } \in \mathbb {C}^{N+1}, \end{equation}$$ where $\mathbf {G}\in \Re ^{(N+1)\times (N+1)}$ is the circulant matrix determined by vector $\mathbf {g} = [g(-\nu T),\cdots,g(0),\cdots,g(\nu T)]^{T}$ , and we have $\boldsymbol {\eta }=[\eta (0),\eta (T),\cdots,\eta (NT)]^{T}$ . Furthermore, the eigenvalue decomposition of $\mathbf {G}$ is efficiently implemented with the aid of a discrete Fourier transform (DFT) as $\mathbf {G}=\mathbf {Q}^{T}\boldsymbol {\Lambda }\mathbf {Q}^{*}$ ,¹ where $\mathbf {Q} \in \mathbb {C}^{(N+1)\times (N+1)}$ represents the normalized DFT matrix whose $k$ th-row and $l$ th-column element is defined by $$\begin{equation} \frac {1}{\sqrt {N+1}}\exp \left [{-2\pi j\frac {(k-1)(l-1)}{N+1}}\right]\!, \end{equation}$$ and $\boldsymbol {\Lambda } = \mathrm {diag}\{\lambda _{0},\cdots,\lambda _{N}\}$ is a diagonal matrix whose diagonal elements correspond to the DFT coefficients of $\mathbf {G}$ [11].

Hence, by carrying out the inverse DFT (IDFT) operation in (7), we have the frequency-domain received signals $$\begin{align} \mathbf {y}_{f}=&\mathbf {Q}^{*}\mathbf {y} \\=&h\boldsymbol {\Lambda }\underbrace {\mathbf {Q^{*}}\mathbf {s}}_{\mathbf {s}_{f}} + \mathbf {Q}^{*}\boldsymbol {\eta } \in \mathbb {C}^{N+1}, \end{align}$$ where $\mathbf {s}_{f} = \mathbf {Q^{*}s}$ represents the frequency-domain DFTN symbols.

Then, since the matrix $\mathbf {G}$ is also available at the receiver,² the estimates of $h\mathbf {s}_{f}$ are calculated with the aid of the noise-whitening minimum mean-square error (MMSE)-based FDE [16] as follows: $$\begin{equation} \mathbf {v}_{f} = \mathbf {W}\mathbf {y}_{f}, \end{equation}$$ where the weights $\mathbf {W} \in \mathbb {C}^{(N+1)\times (N+1)}$ are given by [16] $$\begin{equation} \mathbf {W} = \boldsymbol {\Lambda }^{H}\left ({\boldsymbol {\Lambda }\boldsymbol {\Lambda }^{H}+N_{0}\boldsymbol {\Phi }}\right)^{-1}, \end{equation}$$ and $\boldsymbol {\Phi } = \mathrm {diag}[\Phi _{0},\cdots,\Phi _{N}]$ is also a diagonal matrix, the $i$ th element of which is calculated by $$\begin{equation} \Phi _{i} = \frac {1}{N\!+\!1}\sum _{k=0}^{N}\sum _{l=0}^{N} g((k-l)T)\exp \left ({2\pi j\frac {(k-l)i}{N+1} }\right)\!.\quad \end{equation}$$ Note that since the weight matrix $\mathbf {W}$ exhibits a diagonal structure, its inverse calculation imposes a low complexity, on the order of $N$ . More specifically, the $i$ th diagonal component of the weight matrix $\mathbf {W}$ is formulated by $$\begin{equation} w_{i} = \frac {\lambda _{i}^{*}}{\|\lambda _{i}\|^{2}+N_{0}\Phi _{i}}. \end{equation}$$

Moreover, the estimates of $h\mathbf {s}$ are given by carrying out a DFT in (11) as follows: $$\begin{align} \mathbf {v}=&[v_{0},\cdots,v_{N}]^{T} \\=&\mathbf {Q}^{T}\mathbf {v}_{f} \end{align}$$

Finally, in a similar manner to the conventional DPSK receiver, the transmitted PSK symbols $x_{i}\,\,(i = 1,\cdots,N)$ are estimated with the aid of differential demodulation without relying on any CE as follows: $$\begin{equation} \hat {x}_{i} = v_{i} v^{*}_{i-1} \quad \mathrm {for}~1\le i\le N. \end{equation}$$

In comparison to the conventional Nyquist-criterion-based DPSK detection, our DFTN detection requires the additional calculations needed for the IDFT and DFT operations and the equalization of (11). However, the IDFT and DFT operations are efficiently performed with the aid of the fast Fourier transform, while FDE of (11) requires only $(N+1)$ complex-valued multiplications. These low-complexity calculations have been used in uplink scenarios of the current wireless standards.

SECTION III. Analytical BER Bound

In this section, we derive an analytical BER bound for our channel-uncoded DFTN signaling system employing $\mathcal {M}$ -point DPSK ($\mathcal {M}$ -DPSK) modulation for a quasi-static frequency-flat Rayleigh fading channel.

A. MGF-Based BER Bound for the Conventional DPSK

The average symbol error probability (SEP) of conventional $\mathcal {M}$ -DPSK symbol transmission over an AWGN channel is given by [25] $$\begin{equation} P_{s}(E) = \frac {\sqrt {\kappa }}{2\pi } \int ^{\frac {\pi }{2}}_{-\frac {\pi }{2}}\frac {\mathrm { exp}\bigl \{-\gamma _{s}(1-\sqrt {1-\kappa }\cos \theta)\bigr \}}{1-\sqrt {1-\kappa }\cos \theta }d\theta,\quad \end{equation}$$ where $\kappa = \sin ^{2}(\pi /\mathcal {M})$ , and $\gamma _{s}$ represents the instantaneous signal-to-noise ratio (SNR). The SEP of a specific frequency-flat fading channel is obtained by averaging (18) over a fading probability density function $p_{\gamma _{s}}(\gamma _{s})$ in terms of $\gamma _{s}$ as follows [24]: $$\begin{equation} P_{s}(E) = \frac {\sqrt {\kappa }}{2\pi } \int ^{\frac {\pi }{2}}_{-\frac {\pi }{2}}\frac {M_{\gamma _{s}}\bigl \{-(1-\sqrt {1-\kappa }\cos \theta)\bigr \}} {1-\sqrt {1-\kappa }\cos \theta }d\theta,\quad \end{equation}$$ where $M_{\gamma _{s}}\{u\}=\int _{0}^\infty p_{\gamma _{s}}(\gamma _{s})\exp (u\gamma _{s})d\gamma _{s}$ represents the MGF. For example, we have $\kappa =1$ in the differential binary PSK (DBPSK) scenario, and the average BER of DBPSK of (19) is changed to $$\begin{equation} P_{b_{1}}(E) = \frac {1}{2}M_{\gamma _{s}}\{-1\}. \end{equation}$$ Also, the average BER of the differential quadrature PSK (DQPSK) scheme is given by [24] $$\begin{equation} P_{b_{2}}(E) = \frac {1}{4\pi }\int ^{\pi }_{-\pi }\frac {1-\tau ^{2}}{\epsilon }M_{\gamma _{s}} \left\{{-\left({1+\frac {1}{\sqrt {2}}}\right)\epsilon }\right\} d\theta,\quad \end{equation}$$ where we have $$\begin{align} \tau=&\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}}, \\ \epsilon=&1+2\tau \sin \theta +\tau ^{2}. \end{align}$$

For a Rayleigh fading channel, the MGF with respect to $\gamma _{s}$ is given by $M_{\gamma _{s}}\{u\}=(1-u\gamma _{s})^{-1}$ , and hence (20) and (23) are formulated in closed forms as $$\begin{align} P_{b_{1}}(E)=&\frac {1}{2(1+\gamma _{b})}, \\ P_{b_{2}}(E)=&\frac {1}{2}\left\{{1-\frac {1}{\sqrt {\frac {(1+2\gamma _{b})^{2}}{2\gamma _{b}^{2}} - 1}}}\right\}, \end{align}$$ where $\gamma _{b}$ denotes the average SNR per bit. Similarly, the average BERs for 8- and 16-DPSK schemes are given respectively by [24] $$\begin{align} P_{b_{3}}(E)=&\frac {2}{3}\left\{{f\left({\frac {13\pi }{8}}\right)-f\left({\frac {\pi }{8}}\right)}\right\},\quad \mathrm { for}~\mathcal {M}=8, \\ P_{b_{4}}(E)=&\frac {1}{4}\left\{{f\left({\frac {15\pi }{16}}\right)+f\left({\frac {13\pi }{16}}\right) -f\left({\frac {11\pi }{16}}\right)-f\left({\frac {9\pi }{16}}\right)}\right\} \notag \\&-\,\frac {1}{2}\left\{{f\left({\frac {3\pi }{16}}\right)+f\left({\frac {\pi }{16}}\right)}\right\},\quad \mathrm { for}~\mathcal {M}=16, \end{align}$$ where we have $$\begin{align} f(x)=&-\frac {\sin x}{4\pi }\int ^{\frac {\pi }{2}}_{-\frac {\pi }{2}}\frac {1}{(1-\zeta)\bigl (1+\gamma _{b}(1-\zeta)\log _{2}\mathcal {M}\bigr)}dt,\notag \\ \\ \zeta=&\cos x\cos t. \end{align}$$

B. MGF-Based BER Bound of Our DFTN

In the proposed DFTN signaling, the FTN-specific ISI effects are eliminated with the aid of MMSE-based FDE in order to estimate the DPSK symbols $\mathbf {v}$ . More specifically, the equalized DFTN symbols of (16) are rewritten as $$\begin{align} \mathbf {v}=&h \mathbf {Q}^{T} \mathbf {W} \mathbf {Q} ^{*} \mathbf {y}, \\=&h \boldsymbol {\Gamma }_{s} \mathbf {s}+ \boldsymbol {\Gamma }_{n} \boldsymbol {\eta }, \end{align}$$ where we have $\boldsymbol {\Gamma }_{n} = \mathbf {Q}^{T} \mathbf {W} \mathbf {Q} ^{*}$ and $\boldsymbol {\Gamma }_{s} = \boldsymbol {\Gamma }_{n} \mathbf {G}$ .

From (31), the components of the desired DFTN symbols, ISI, and AWGNs are represented respectively by $$\begin{align*} \mathbf {s}_{d}=&h \boldsymbol {\Gamma }_{d} \mathbf {s}, \\ \mathbf {s}_{I}=&h(\boldsymbol {\Gamma }_{s}- \boldsymbol {\Gamma }_{d}) \mathbf {s}, \\ \boldsymbol {\eta }_{v}=&\boldsymbol {\Gamma }_{n} \boldsymbol {\eta }, \end{align*}$$ where $\boldsymbol {\Gamma }_{d}$ is the diagonal matrix structured using only the diagonal components of $\boldsymbol {\Gamma }_{s}$ . Then, the average power of the DFTN symbols $\mathbf {s}_{d}$ is calculated as $$\begin{align} P_{d}=&\mathrm {tr}\bigl \{\mathbb {E}\bigl [\mathbf {s}_{d} \mathbf {s}_{d}^{H}\bigr]\bigr \}, \notag \\=&\mathrm {tr}\bigl \{\mathbb {E}\bigl [hh^{*} \boldsymbol {\Gamma }_{d} \mathbf {s} \mathbf {s} ^{H} \boldsymbol {\Gamma }_{d}^{H}\bigr]\bigr \}, \notag \\=&\mathrm {tr}\bigl \{ \boldsymbol {\Gamma }_{d} \boldsymbol {\Gamma }_{d}^{H}\bigr \}. \end{align}$$ Note that (32) comes from the fact that $\mathbb {E}[|h|^{2}]=1$ and $\mathbb {E}[\mathbf {s} \mathbf {s} ^{H}]=\mathbf {I}_{N+1}$ , where $\mathbf {I}_{N}$ is the $(N\times N)$ identity matrix. In a similar manner to (32), the average powers of the ISI and AWGN components are given by $$\begin{align} P_{I}=&\mathrm { tr}\bigl \{\mathbb {E}\bigl [\mathbf {s}_{I} \mathbf {s}_{I}^{H}\bigr]\bigr \}, \notag \\=&\mathrm { tr}\bigl \{\mathbb {E}\bigl [hh^{*}(\boldsymbol {\Gamma }_{s}- \boldsymbol {\Gamma }_{d}) \mathbf {s} \mathbf {s} ^{H}(\boldsymbol {\Gamma }_{s}^{H}- \boldsymbol {\Gamma }_{d}^{H})\bigr]\bigr \}, \notag \\=&\mathrm { tr}\bigl \{| \boldsymbol {\Gamma }_{s}|^{2} + | \boldsymbol {\Gamma }_{d}|^{2} - \boldsymbol {\Gamma }_{s} \boldsymbol {\Gamma }_{d}^{H} - \boldsymbol {\Gamma }_{d} \boldsymbol {\Gamma }_{s}^{H}\bigr \}. \\ P_{\eta }=&\mathrm { tr}\bigl \{\mathbb {E}\bigl [\boldsymbol {\eta }_{v} \boldsymbol {\eta }_{v}^{H}\bigr]\bigr \}, \notag \\=&\mathrm { tr}\bigl \{\mathbb {E}\bigl [\boldsymbol {\Gamma }_{n} \boldsymbol {\eta } \boldsymbol {\eta } ^{H} \boldsymbol {\Gamma }_{n}^{H}\bigr]\bigr \}, \notag \\=&\mathrm { tr}\bigl \{ \boldsymbol {\Gamma }_{n}\mathbf {R} \boldsymbol {\Gamma }_{n}^{H}\bigr]\bigr \}, \end{align}$$ where $\mathbf {R}=\mathbb {E}[\boldsymbol {\eta } \boldsymbol {\eta } ^{H}]$ is the cross-correlation matrix of a colored-noise vector $\boldsymbol {\eta }$ whose $i$ th-row and $j$ th-column element is given by $N_{0}g((i-j)T)$ .

From (32)–​(34), the average SINR_b of the equalized DFTN symbols is given by $$\begin{equation} \mathrm { SINR}_{b} = \frac {1}{\log _{2}\mathcal {M}}\times \frac {P_{d}}{P_{I} + P_{\eta }}. \end{equation}$$ Finally, the analytical BER bound of $\mathcal {M}$ -DPSK-modulated DFTN symbol transmission over a frequency-flat Rayleigh fading channel is formulated by substituting (35) into the $\gamma _{b}\text{s}$ of (24)–​(27).

SECTION IV. Performance Results

In this section, we provide our performance results, based on Monte Carlo simulations, as well as the analytical bound derived in Section III, in order to characterize the proposed noncoherently detected DFTN signaling. The basic system parameters employed in our simulations are listed in Table 1. We employed an RRC filter having a symbol packing ratio $\alpha \in [{0.7, 1.0}]$ and a roll-off factor $\beta =0.3$ . The block size and the CP length were set to $N=512$ and $2\nu =20$ for numerical results, while much longer CP length was considered for analytical results, in order to show the achievable bound. Also, a frequency-flat Rayleigh fading channel was considered. Unless otherwise noted, these system parameters were used throughout the analyses and simulations described in this section. The conventional coherent FTN signaling [16] scheme, assuming the use of perfect CSI at the receiver in a quasi-static fading scenario, was considered as a benchmark scheme.

TABLE 1 Basic System Parameters

A. Performance in a Quasi-Static Frequency-Flat Rayleigh Fading Channel

Fig. 2 shows the analytical and numerical BER curves of our proposed DFTN scheme in a quasi-static Rayleigh fading scenario, where the DBPSK- and DQPSK-modulated DFTN schemes were considered in Fig. 2(a) and Fig. 2(b), respectively. The symbol packing ratio was set to $\alpha =0.9$ , 0.8, and 0.7, which corresponded to spectral efficiencies of 0.82, 0.92, and 1.06 bps/Hz for the DBPSK-aided DFTN scheme and 1.64, 1.85, and 2.11 bps/Hz for the DQPSK-modulated DFTN scheme. We also plotted the achievable BER curves of the conventional Nyquist-criterion-based DBPSK and DQPSK schemes (i.e., $\alpha =1.0$ ), whose spectral efficiencies were 0.77 and 1.54 bps/Hz, respectively. As shown in Fig. 2, it was found that the proposed DFTN scheme is capable of correctly demodulating DFTN signals in a noncoherent manner for $\alpha =0.9$ and 0.8, while an error floor exists for $\alpha =0.7$ . More specifically, the proposed DBPSK- and DQPSK-modulated DFTN schemes with a symbol packing ratio of $\alpha =0.9$ achieved the same performance as those of the classic Nyquist-criterion-based DBPSK and DQPSK schemes, implying that the FTN-specific rate boost was attained without imposing any performance penalty for $\alpha =0.9$ . Additionally, the DBPSK- and DQPSK-modulated DFTN schemes with $\alpha =0.8$ , having a 12% rate increase over their Nyquist-criterion-based counterparts as well as the DFTN schemes with $\alpha =0.9$ , exhibited an approximately 3-dB performance penalty. Moreover, in Fig. 2(a) and Fig. 2(b), for the $\alpha$ values of 0.9 and 0.8, the analytical curves matched well with the numerical curves in the SNRs higher than 20 dB, although a slight gap was seen in the low SNR regime. By contrast, for $\alpha =0.7$ , the analytical and numerical curves exhibit an unignorable gap, while both the curves clearly show an error floor.

FIGURE 2.

Analytical and numerical BER curves of the DBPSK- and DQPSK-modulated DFTN schemes. The symbol packing ratio is set to $\alpha =0.9, 0.8$ and 0.7. (a) DBPSK. (b) DQPSK.

Similarly, in Fig. 3 we show the analytical and numerical BER performance of the proposed DFTN scheme, while varying the constellation size as $\mathcal {M}=2, 4, 8$ , and 16, where the symbol packing ratio was fixed to $\alpha =0.8$ . Similar to Fig. 2, the analytical and numerical curves of the DFTN schemes coincided in each scenario, which verified the system model of our DFTN scheme. Observe in Fig. 3 that the performance gap among the DBPSK- and DQPSK-modulated DFTN schemes was 3 dB as expected, while that between the 8-DPSK- and 16-DPSK-modulated DFTN schemes was approximately 5 dB.

FIGURE 3.

Analytical and numerical BER results of $\mathcal {M}$ -DPSK-modulated DFTN schemes, where $\mathcal {M}$ is set to $M=2, 4, 8$ , and 16. The symbol packing ratio is fixed to $\alpha =0.8$ .

Figs. 4(a) and 4(b) depict the analytical SINR_b and BER curves of our DFTN schemes, respectively. In Fig. 4(a) the DBPSK-modulated DFTN scheme, having $\alpha =0.9, 0.8$ , and 0.7 and $\beta =0.5$ , and the DQPSK-modulated DFTN scheme, employing $\alpha =0.83$ , $\beta = 1.0, 0.7, 0.4, 0.3$ , and 0.25, were considered. The associated SINR_b curves of the conventional DBPSK and DQPSK schemes ($\alpha =1.0$ ) were also plotted. As seen from Fig. 4(a), the analytical SINR_b value decreased, upon decreasing the packing ratio $\alpha$ . More specifically, the DBPSK-modulated DFTN scheme was capable of achieving performance close to that of the conventional DBPSK scheme for $\alpha =0.9$ and 0.8. Similarly, the DQPSK-aided DFTN scheme with $\beta =1.0$ and 0.7 exhibited almost the same SINR_b as that of the conventional DQPSK scheme. Naturally, the achievable SINR_bs of the DBPSK- and DQPSK-modulated DFTN schemes deteriorated when the $\alpha$ and $\beta$ values decrease, i.e., the spectral efficiency increased. Furthermore, in Fig. 4(b), the DQPSK-modulated DFTN scheme with $\alpha =0.83$ is considered, where the roll-off factor is $\beta =0.7, 0.3$ , and 0.25. A trend similar to Fig. 4(a) was observed, where the BER performance deteriorated upon decreasing $\beta$ .

FIGURE 4.

Analytical SINR_b and BER comparisons of the DFTN scheme. (a) Analytical SINR_b of DBPSK ($\beta =0.5, \alpha =0.9, 0.8$ , and 0.7) and DQPSK ($\alpha =0.83, \beta =1.0, 0.7, 0.4, 0.3$ , and 0.25). (b) Analytical and numerical BERs of DQPSK-based DFTN ($\alpha =0.83$ , $\beta =0.7, 0.3$ , and 0.25).

In Fig. 5, we compare the BER performance of our DFTN scheme and the conventional coherent FTN counterpart [16], where both employ MMSE-based FDE at the receiver. Observe in Fig. 5 that a 3-dB performance loss was observed in the DFTN scheme in comparison to the coherent FTN scheme for $\alpha =0.9$ . This penalty was due to the well-known noise-doubling effects imposed by differential demodulation. However, in the high-ISI $\alpha =0.8$ scenario, the performance penalty imposed on the proposed DFTN scheme increased to 5 dB.³ Furthermore, for $\alpha =0.7$ , the BER curves of both the DFTN and coherent FTN schemes exhibited an error floor. Note that the use of powerful channel coding scheme and iterative detection in FTN and DFTN systems allows us to attain an ISI-free performance limit even in a low-$\alpha$ scenario [16].

FIGURE 5.

BER comparison between the proposed DFTN scheme and the conventional coherent FTN scheme of [16]. Here, DBPSK modulation is employed and the packing ratio is set to $\alpha = 1.0, 0.9, 0.8$ , and 0.7.

B. Performance in a Time-Varying Frequency-Flat Rayleigh Fading Channel

Next, we investigated the effects of the time-varying channel on the achievable BER performance of the DFTN scheme. In Fig. 6, we consider the DBPSK-modulated DFTN scheme and its coherent counterpart. Here, the received signals of (4) have been modified to $$\begin{equation} y(t) = \sum _{n} h_{n} s_{n}g(t-nT) + \eta (t), \end{equation}$$ where $h_{n}\,\,(n=0,\cdots,N+2\nu +1)$ represent the coefficients of the time-varying channel in each block, which were generated according to $E[h_{n}h_{n+\tau }^{*}]=J_{0}(2\pi F_{d}T\tau)$ . Furthermore, $F_{d}T$ denotes the normalized Doppler frequency, and $J_{0}$ is the zero-order Bessel function of the first kind. We assumed that, in the coherent FTN scheme, the perfect CSI is acquired only for the initial symbol in each block. Moreover, the packing ratio was fixed to $\alpha =0.8$ . As shown in Fig. 6, the conventional coherent FTN scheme exhibited an error floor, due to the introduction of the effects of the time-varying channel, while the BER performance of the DFTN scheme with the normalized Doppler frequency of $F_{d}T = 1.0\times 10^{-6}$ remained unchanged from that of the quasi-static scenario ($F_{d}T =0$ ). Upon increasing $F_{d}T$ , the performance advantage of the DFTN scheme became higher.

FIGURE 6.

Effects of the normalized Doppler frequency $F_{d}T$ on the achievable BER performance of the DBPSK-modulated DFTN and the coherent BPSK-modulated FTN schemes, where the packing ratio is fixed to $\alpha =0.8$ .

Furthermore, in Fig. 7, we plot the achievable BER performance of the DBPSK-modulated DFTN scheme and its coherent FTN counterpart. The block size was varied from $N=64$ to 2048 while maintaining the FTN’s symbol packing ratio at $\alpha =0.8$ and the SNR value at 40 dB. Observe in Fig. 7 that the coherent FTN scheme with the normalized Doppler frequency as low as $F_{d}T=1.0\times 10^{-6}$ achieved almost the same BER performance regardless of the block size $N$ . However, when considering $F_{d}T=1.0\times 10^{-5}$ and $1.0\times 10^{-4}$ , the coherent FTN scheme suffered from a severe performance deterioration. This is because, in the coherent FTN scheme, the gap between the estimated and actual CSI coefficients increased upon increasing the block size. By contrast, the DFTN scheme was found to be robust for all the $F_{d}T$ scenarios. More specifically, the DFTN scheme for $F_{d}T=1.0\times 10^{-5}$ and $1.0\times 10^{-4}$ exhibited a better BER performance than its coherent FTN counterpart when the block size was greater than $N=512$ and 64, respectively.

FIGURE 7.

Achievable BER performance of the DBPSK-modulated DFTN and the conventional BPSK-modulated FTN schemes at an SNR of 40 dB in the time-varying frequency-flat channels. Here, the packing factor is set to $\alpha =0.8$ and the normalized Doppler frequency is given by $F_{d}T = 1.0\times 10^{-6}, 1.0\times 10^{-5}$ , and $1.0\times 10^{-4}$ .

In order to elaborate a little further, the effects of the CP size relative to the block length on the spectral efficiency are shown in Table 2, where the system parameters are the same as those used in Fig. 7. Upon increasing the block size $N$ while maintaining the CP size $2\nu$ , the effective CP overhead was found to be negligible; note that a block size on the order of thousands of symbols is typically employed.

TABLE 2 Effects of CP and Block Size on Spectral Efficiencies of DBPSK-Modulated DFTN Signaling

SECTION V. Conclusions

This paper first proposed the DFTN concept, which allows CE-free noncoherent detection at the receiver while attaining the explicit benefits of FTN signaling. The proposed DFTN receiver has the capability of correctly demodulating ISI-induced DFTN symbols, which is achieved with the aid of low-complexity MMSE-based FDE and differential detection. Moreover, the MGF-based analytical BER bound is derived for our DFTN signaling by introducing the closed-form SINR of the equalized DFTN symbols. Our simulation results demonstrated that the fundamental benefits of FTN signaling are achievable in our DFTN scheme, without relying on any pilot-assisted CE, when the symbol packing ratio is moderately high. Furthermore, it was found that the performance advantage of the DFTN scheme over the conventional coherent FTN counterpart is observable, especially in a time-varying fading scenario.

ACKNOWLEDGMENT

Part of this paper was submitted for presentation in IEEE ICASSP 2018, Calgary, Canada.