Capacity and Outage Probability Analysis of Faster-Than-Nyquist Cooperative NOMA

This letter proposes a novel cooperative non-orthogonal multiple access (C-NOMA) based on precoded faster-than-Nyquist (FTN) signaling. The additional gain in degrees of freedom obtained from precoded FTN signaling is exploited in the context of relaying-induced diversity inherent to C-NOMA. We derive the achievable ergodic rate of the proposed scheme while also deriving a closed-form expression of the proposed scheme’s outage probability, which shows improved fairness to a far user. Our simulation results substantiate the advantages of the proposed scheme over conventional benchmark methods.

SECTION I. Introduction

The next-generation wireless communication system is expected to leap the current generation in terms of peak data rate, spectral efficiency, and connection density [1]. Towards achieving this target, several techniques in multiplexing and multiple access have gained increasing research interest. Non-orthogonal multiple access (NOMA) [2] has been shown to offer a range of advantages over conventional orthogonal multiple access (OMA) schemes. For example, in a typical two-user power-domain NOMA downlink, a base station (BS) relies on superposition coding by allocating high power to a far user and low power to a near user. The multiplexed signals for the two users are transmitted by the BS in the same time and frequency resources. The far user demodulates its signal by exploiting the higher power allocation in the presence of the inter-user interference (IUI), while the near user performs successive interference cancellation (SIC) to remove the signal associated with the far user.

The benefits of NOMA can be categorized as (a) the high spectral efficiency owing to the flexible use of entire time and frequency resources, (b) the high number of simultaneous connections, and (c) the reduced latency by the relaxed requirement for scheduling. However, a NOMA system typically suffers from the problem of fairness between the near and far users. While higher power allocated to the far user improves the performance, it, in turn, reduces the relative power allocated to the near user. In order to alleviate this fairness problem, cooperative NOMA (C-NOMA) was introduced in [3]. The central idea of C-NOMA is exploiting the ability of the near user to decode the data of the far user during SIC, which is unused in the conventional NOMA system. The redundant copy of the far user’s signal is used to improve the achievable rate of the far user owing to the diversity gain without imposing multiple antennas. In [3], the near user acts as a half-duplex relay, while in [4], the near user is a full-duplex relay.

Analogous to the non-orthogonality introduced in the multiple-access domain, i.e., NOMA, a faster-than-Nyquist (FTN) signaling [5], [6], [7] technique constitutes non-orthogonal time-division multiplexing. More specifically, FTN signaling relaxes Nyquist’s orthogonality criterion in the time domain by reducing the time separation between adjacent data-bearing pulses below the threshold necessary for zero intersymbol interference (ISI). For a bandlimited channel, FTN signaling can recover the capacity loss [8] imposed in conventional Nyquist-rate signaling when practical filters (e.g., the root-raised cosine (RRC) filter) are employed. The capacity advantage is gleaned from the excess bandwidth of an RRC filter (captured by its roll-off factor $\beta$ ) and the reduction in symbol packing ratio $\tau (0 < \tau \le 1)$ . By appropriate choice of $\tau$ and $\beta$ under the condition of ${1}/({1+\beta })\leq \tau \leq 1$ , the FTN-induced ISI matrix can become positive definite (thus, full-rank) with sufficiently large eigenvalues [9], which accounts for the available degree of freedom (DoF) in FTN signaling. Since the FTN symbols arrive at the receiver earlier than the Nyquist rate, ISI is unavoidable in the received signal. Diverse FTN transceivers have been investigated in the literature for ISI cancellation. Specifically, pre-equalization [10], [11], precoding [12], [13] have been employed to compensate for ISI at the transmitter. Also, maximum likelihood (ML) sequence estimation [14], its reduced-complexity variant [15], frequency-domain equalization (FDE) [16], [17], iterative detection employing near-capacity channel codes [18] have been studied. To expound a little further, ML-based detection of FTN signal is NP-hard [19], [20], which led to the development of the low-complexity FDE and precoding-based techniques. More recently, non-cooperative FTN-NOMA has been introduced in asynchronous uplink [21], [22], as well as in downlink [23] as an attractive alternative to increase spectral efficiency in the multiuser scenario.

Against the above background, in this letter, we propose a novel FTN-signaling-based power-domain C-NOMA downlink to improve fairness to the far user. Eigendecomposition precoding is employed to pre-equalize FTN-induced-ISI and enable diagonalization at the receiver. Motivated by the idea of cooperative diversity [24], half-duplexed relaying based on the decode-and-forward (DF) strategy is performed by the near user at an FTN rate to improve the capacity of far user. We derive the ergodic rate and show the advantage of the proposed system over benchmark techniques in terms of capacity and outage.

Notations: Boldface uppercase letters are used to denote matrices, boldface lowercase letters for vectors, and lowercase letters with a suffix to denote elements of a vector. $\mathbb {C}^{j\times k}$ and $\mathbb {R}^{j\times k}$ denote the complex and real fields of dimensions ${j} \times {k}$ , respectively. For vectors or matrices, the transpose and Hermitian operations are denoted by $(.)^{T}$ and $(.)^{H}$ , respectively. The probability of an event is denoted by $\mathbb {P}(.)$ . The expectation of a random variable is denoted by $\mathbb {E}[.]$ .

SECTION II. System Model of Proposed Scheme

The system model of C-NOMA considered in this letter is shown in Fig. 1. A BS and two users, $u_{1}$ and $u_{2}$ , are considered to be located in a single cell, each equipped with a single antenna element. Under the assumption of a frequency-flat Rayleigh fading environment, the channel coefficients for the wireless link between the BS and $u_{1}$ , that between the BS and $u_{2}$ , and that between $u_{1}$ and $u_{2}$ are denoted by $h_{1}$ , $h_{2}$ , and $h_{3}$ , respectively. Also, the pairwise distances in the above three links are represented by $d_{1}$ , $d_{2}$ , and $d_{3}$ , respectively.

Fig. 1.

(a) System model of two-user C-NOMA downlink with power-domain multiplexing, (b) time-division transmission in direct and relay phases.

Each channel coefficient is modeled as $h_{j}=\sqrt {d_{j}^{-\zeta }}\Upsilon (j=1, 2, 3)$ , where $\zeta$ denotes the path-loss exponent, and $\Upsilon \sim \mathcal {CN}(0, 1)$ . Without loss of generality, we assume the relationship of $d_{1}>d_{2}$ , and correspondingly $\mathbb {E}[\lvert h_{1}\rvert ^{2}] < \mathbb {E}[\lvert h_{2}\rvert ^{2}]$ . This implies that in our C-NOMA model, $u_{1}$ is a weak (far) user while $u_{2}$ is a strong (near) user. We present a simple two-user system model here for space constraint; this letter can be extended to a multiuser scenario by employing user-pairing and hybrid NOMA-OMA technique [25] such that NOMA can be employed within a pair, and OMA can be used among different pairs.

A. FTN-Signaling-Based C-NOMA Downlink

Information bits destined for $u_{j}, j\in (1,2)$ are modulated onto ${N}$ complex-valued symbols expressed as $\boldsymbol {s}_{j}=[s_{j,0},\,\, s_{j,1},\ldots, s_{j,N-1}]^{T}\in \mathbb {C}^{N}$ , satisfying the power constraint $\mathbb {E}[\boldsymbol {s}_{j}^{H}\boldsymbol {s}_{j}]=NP_{s}$ , where $P_{s}$ denotes the average energy per symbol. Then, the symbol vectors $\boldsymbol {s}_{j}$ are superposition-coded as follows: $$\begin{align*} \boldsymbol {c}=&[c_{0}, c_{1},\ldots, c_{N-1}]^{T}\in \mathbb {C}^{N} \tag{1}\\=&\sum _{j}\sqrt {\theta _{j} P_{s}}\boldsymbol {s}_{j}, \tag{2}\end{align*}$$ where $\theta _{j}$ denotes the fixed power-allocation coefficients, satisfying the relationships of $\theta _{1}>\theta _{2}$ and $\theta _{1}+\theta _{2}=1$ .

1) Precoding for FTN Signaling:

The symbols in (2) are precoded as follows: $$\begin{align*} \boldsymbol {x}=&[x_{0}, x_{1},\ldots, x_{N-1}]^{T} \tag{3}\\=&\boldsymbol{\Xi } \boldsymbol {c},\tag{4}\end{align*}$$ where $\boldsymbol{\Xi }={\mathbf {V\Psi }}$ denotes an FTN-precoding matrix, such that V and ${\boldsymbol{\Psi }}$ are obtained from the eigendecomposition of the FTN-induced ISI matrix, as described in Section II-A3.

2) Transmit Filtering and FTN Signaling:

The precoded multiplexed symbols of (4) are filtered with a ${T}$ -orthogonal RRC filter with impulse response denoted by ${p}$ (${t}$ ) with unit energy $\int _{-\infty }^{\infty } |p(t)|^{2}dt=1$ . The continuous-time multiplexed FTN signal is represented by $$\begin{equation*} x(t)=\sum _{n=0}^{N-1}x_{n} p(t-n\tau T_{0}),\tag{5}\end{equation*}$$ where $T_{0}$ is the symbol interval corresponding to Nyquist-based orthogonal transmission. Transmission is divided into two phases with equal duration, i.e., the first direct transmission phase, followed by the second cooperative relaying phase, as shown in Fig. 1(b).

3) Direct Transmission With IUI:

In the direct transmission phase, both $u_{1}$ and $u_{2}$ directly receive the multiplexed signal from the BS. The received signal at $u_{j}$ is expressed as $$\begin{equation*} y_{j}(t) = h_{j} \sum _{n=0}^{N-1} x_{n} p(t-nT) + n_{j}(t), \tag{6}\end{equation*}$$ where $n_{j}(t)$ denotes the complex-valued white Gaussian random process with a zero mean and variance of $N_{0}$ . After matched filtering with $p^{*}(-t)$ , the signal is given by $$\begin{equation*} y^{\mathrm {MF}}_{j}(t) = h_{j} \sum _{n=0}^{N-1} x_{n} g(t-nT) + \omega _{j}(t),\tag{7}\end{equation*}$$ where $g(t)=\int {p(\varphi)p^{*}(\varphi -t)d\varphi }$ captures the FTN-induced ISI effects, and $\omega _{j}(t)=\int {n_{j}(\varphi)p^{*}(\varphi -t)d\varphi }$ is the correlated noise. By sampling the signal of (7) at an FTN interval of $\tau T_{0}$ , we obtain $$\begin{equation*} \boldsymbol {y}^{\mathrm {MF}}_{j}=h_{j}{\mathbf {G}}\boldsymbol {x}+\boldsymbol {\omega }_{j} \in \mathbb {C}^{N},\tag{8}\end{equation*}$$ where ${\mathbf {G}}=[\boldsymbol {g}_{0},\ldots, \boldsymbol {g}_{N-1}]{\in }\mathbb {R}^{N\times N}$ is the FTN-induced correlation matrix, and we have $\boldsymbol {g}_{n}=[g(-n\tau T_{0}), g(-(n-1)\tau T_{0}),\ldots, g((N-n-1)\tau T_{0})]^{T}$ for $n=0,\ldots, N-1$ . Also, the covariance of $\boldsymbol {\omega }_{j}$ is $\mathbb {E}[\boldsymbol {\omega }_{j}\boldsymbol {\omega }_{j}^{H}]=N_{0}{\mathbf {G}}$ .

Furthermore, exploiting the Toeplitz matrix properties of G [23], we arrive at the eigenvalue decomposition of $$\begin{equation*} {\mathbf {G}}={\mathbf {V\Lambda V}}^{T}, \tag{9}\end{equation*}$$ where ${\boldsymbol{\Lambda }}=\mathrm {\mathrm{ diag}}/(\lambda _{0},\ldots, \lambda _{N-1})$ represents the ${N}$ eigenvalues of G, while V denotes an orthonormal eigenvector matrix. To remove the effects of the FTN-induced ISI captured by G, and to whiten the additive correlated noise $\boldsymbol {\omega }_{j}$ in (8), the following matrix multiplication is performed: $$\begin{align*} \boldsymbol {y}^{\mathrm {noISI}}_{j}=&{\boldsymbol{\Lambda }}^{-\frac {1}{2}}{\mathbf {V}}^{T}\boldsymbol {y}^{\mathrm {MF}}_{j} \tag{10}\\=&h_{j}{\boldsymbol{\Lambda }}^{-\frac {1}{2}}{\mathbf {V}}^{T}{\mathbf {G}}\boldsymbol {x} + {\boldsymbol{\Lambda }}^{-\frac {1}{2}}{\mathbf {V}}^{T}\boldsymbol {\omega }_{j}. \tag{11}\end{align*}$$ By substituting (2), (4), and (9) into (11), we obtain $$\begin{equation*} \boldsymbol {y}^{\mathrm {noISI}}_{j} =h_{j}{\boldsymbol{\Lambda }}^{\frac {1}{2}}{\boldsymbol{\Psi }}\sum _{j}\sqrt {\theta _{j} P_{s}}\boldsymbol {s}_{j} + {\boldsymbol{\Lambda }}^{-\frac {1}{2}}{\mathbf {V}}^{T}\boldsymbol {\omega }_{j}.\tag{12}\end{equation*}$$ By defining ${\boldsymbol{\Psi }}=\mathrm {diag}(\psi _{0},\ldots, \psi _{N-1})={\boldsymbol{\Lambda }}^{-{}\frac {1}{2}}=\mathrm {diag}(1/\sqrt {\lambda _{0}},\ldots, 1/\sqrt {\lambda _{N-1}})$ , we obtain $\boldsymbol {y}^{\mathrm {noISI}}_{j} =h_{j}\sum _{j}\sqrt {\theta _{j} P_{s}}\boldsymbol {s}_{j} + {\boldsymbol{\Lambda }}^{-{}\frac {1}{2}}{\mathbf {V}}^{T}\boldsymbol {\omega }_{j}$ , where ${\boldsymbol{\Lambda }}^{-{}\frac {1}{2}}{\mathbf {V}}^{T}\boldsymbol {\omega }_{j}$ is additive white Gaussian noise with variance $N_{0}$ . Then the channel fading coefficient can be equalized as $\boldsymbol {y}^{\mathrm {noISI,eq}}_{j}={}\frac {h_{j}^{*}}{|h_{j}|^{2}+N_{0}}\boldsymbol {y}^{\mathrm {noISI}}_{j}$ . Thus, symbol-by-symbol detection of $u_{1}$ ’s signal becomes possible at each user by virtue of the relationship $\theta _{1}>\theta _{2}$ as follows: $$\begin{equation*} \hat {s}_{n}^{(1,j)}=\mathop {\mathrm{ arg\,min}}\limits _{s_{\gamma }\in \Gamma }\Big \lVert y^{\mathrm {noISI,eq}}_{j,n}-s_{\gamma }\Big \rVert ^{2}, \tag{13}\end{equation*}$$ where $\hat {\boldsymbol {s}}^{(1,j)}=[\hat {s}_{1}^{(1,j)},\ldots, \hat {s}_{N}^{(1,j)}]^{T}$ , and $\Gamma$ is the symbol constellation. Subsequently, $u_{2}$ performs SIC to obtain the IUI-free received signal and decodes its own signal $\hat {\boldsymbol {s}}^{(2,2)}=[\hat {s}_{1}^{(2,2)},\ldots, \hat {s}_{N}^{(2,2)}]^{T}$ , which is given by $$\begin{equation*} \hat {s}_{n}^{(2,2)}=\mathop {\mathrm{ arg\,min}}\limits _{s_{\gamma }\in \Gamma }\bigg \lVert \left ({y^{\mathrm {noISI,eq}}_{2,n}-\sqrt {\theta _{1} P_{s}}\hat {s}_{n}^{(1,2)} }\right)-s_{\gamma }\bigg \rVert ^{2}. \tag{14}\end{equation*}$$

4) Relay Transmission Without IUI:

In the cooperative relaying phase, $u_{2}$ transmits the redundant signal1 $\hat {\boldsymbol {s}}^{(1,2)}$ defined in (13) to user $u_{1}$ , following similar operations as described earlier in (3)–​(13). The received signal at $u_{1}$ from $u_{2}$ in the cooperative relaying phase is given by $$\begin{equation*} y_{1}^{\mathrm {coop}}(t) = h_{3} \sum _{n=0}^{N-1} \hat {s}_{n}^{(1,2)} p(t-nT) + n_{1}^{\mathrm {coop}}(t). \tag{15}\end{equation*}$$ Note that power division (i.e., scaling by $\theta _{1} < 1$ ) is not imposed in this phase, and hence the entire transmit power budget is invested. Due to the absence of power-domain multiplexing, there is no IUI. Analogous to (12), the signals after matched-filtering, FTN-rate sampling, and ISI-cancellation in the relaying phase are denoted as $\boldsymbol {y}^{\mathrm {noISI,coop}}_{1}{=}[y^{\mathrm {noISI,coop}}_{1,0},\ldots, y^{\mathrm {noISI,coop}}_{1,0}]^{T}{\in }\mathbb {C}^{N\times N}$ . Then, detection is performed as $$\begin{equation*} \hat {s}_{n}^{(1,1),\mathrm {coop}}=\mathop {\mathrm{ arg\,min}}\limits _{s_{\gamma }\in \Gamma }\Big \lVert y^{\mathrm {noISI,coop}}_{1,n}-s_{\gamma }\Big \rVert ^{2}. \tag{16}\end{equation*}$$ By employing selection combining [27], the detector output $\hat {\boldsymbol {s}}^{(1,1),\mathrm {sc}}{=}[\hat {s}_{0}^{(1,1),\mathrm {sc}},\ldots, \hat {s}_{N-1}^{(1,1),\mathrm {sc}}]^{T}{\in }\mathbb {C}^{N\times N}$ of the far user is obtained as: $$\begin{align*} \hat {s}_{n}^{(1,1),\mathrm {sc}}= \begin{cases} \hat {s}_{n}^{(1,1),\mathrm {dir}}, & \mathrm {if} \rho _{1}^{\mathrm {dir}}>\rho _{1}^{\mathrm {coop}} \\ \hat {s}_{n}^{(1,1),\mathrm {coop}}, & \mathrm {otherwise}. \end{cases} \tag{17}\end{align*}$$ Here, $\rho _{1}^{\mathrm {dir}}$ and $\rho _{1}^{\mathrm {coop}}$ denote the received signal-to-interference plus noise ratio (SINR) at $u_{1}$ during the direct transmission and the cooperative relaying phases, respectively.

SECTION III. Performance Analysis

In this section, we characterize the achievable ergodic information rate and outage performance of the proposed scheme.

A. Ergodic Information Rate

The received SINR at the far user $u_{1}$ during the direct transmission phase is given by $$\begin{equation*} \rho _{1}^{\mathrm {dir}}= \frac {|h_{1}|^{2}\theta _{1} P_{s}}{|h_{1}|^{2}\theta _{2} P_{s} + N_{0}}. \tag{18}\end{equation*}$$ The SINR for the near user $u_{2}$ during the direct transmission phase for detecting the signals of $u_{1}$ and $u_{2}$ are given, respectively, by $$\begin{equation*} \rho _{2,1}^{\mathrm {dir}}= \frac {|h_{2}|^{2}\theta _{1} P_{s}}{|h_{2}|^{2}\theta _{2} P_{s} + N_{0}},\quad \rho _{2,2}^{\mathrm {dir}}= \frac {|h_{2}|^{2}\theta _{2} P_{s}}{N_{0}}. \tag{19}\end{equation*}$$ Furthermore, the received SINR at the far user $u_{1}$ during the cooperative relaying phase is given by $$\begin{equation*} \rho _{1}^{\mathrm {coop}}= \frac {|h_{3}|^{2} P_{s}}{N_{0}}. \tag{20}\end{equation*}$$ Capacity for FTN signaling can be derived from the determinant of a covariance matrix, or alternatively, by using folded spectrum employing Szegö’s theorem [8]. For ease of notation, let us express the diagonalized signals $y_{j}^{\mathrm{ noISI,eq}}$ as $y = s+\eta$ , where $\eta$ is the uncorrelated Gaussian noises. Let $\boldsymbol{\Sigma }_{s}$ denote the covariance matrix of input symbols, while $\boldsymbol{\Sigma }_{y}$ and $\boldsymbol{\Sigma }_{\eta }$ denote that of received symbols and noise after diagonalization (in (12)), respectively. Then, the maximum achievable rate is expressed as $$\begin{equation*} R^{\mathrm {FTN}} = \lim _{N\rightarrow \infty }\frac {1}{N\tau T_{0}}I(\boldsymbol {y};\boldsymbol {s}) \tag{21}\end{equation*}$$ subject to the power constraint $P=\mathrm {trace}(\boldsymbol{\Sigma }_{s}\mathbf {G})$ considering the effects of ISI introduced at the transmitter.2 On simplification, we obtain $$\begin{align*} R^{\mathrm {FTN}}=&\lim _{N\rightarrow \infty }\frac {1}{N\tau T_{0}}h_{e}(\boldsymbol {y})-h_{e}\left ({\boldsymbol {y}|\boldsymbol {s}}\right) \tag{22}\\=&\lim _{N\rightarrow \infty }\frac {1}{N\tau T_{0}}h_{e}\left ({\boldsymbol {y})-h_{e}(\boldsymbol {\eta }}\right). \tag{23}\end{align*}$$ where $h_{e}(\cdot)$ denotes differential entropy.

Using the relations $h_{e}(\boldsymbol {y})\leq \log _{2}((\pi e)^{N})\mathrm {det}(\boldsymbol{\Sigma }_{y}))$ and $h_{e}(\boldsymbol {\eta })\leq \log _{2}((\pi e)^{N})\mathrm {det}(\boldsymbol{\Sigma }_{\eta }))$ , we express capacity as $$\begin{align*} R^{\mathrm {FTN}}=&\lim _{N\rightarrow \infty }\frac {1}{N\tau T_{0}}\sum _{i=1}^{N}\log _{2} \mathrm {det}\left ({\mathbf {I}+\frac {\boldsymbol{\Sigma }_{s}\mathbf {G}}{N_{0}}}\right) \tag{24}\\=&\lim _{N\rightarrow \infty }\frac {1}{N\tau T_{0}}\sum _{i=1}^{N}\log _{2}\left ({1+\frac {\lambda _{i} \psi _{i}^{2} P_{s}}{N_{0}}}\right) \tag{25}\end{align*}$$ where $\lambda _{i}$ denotes the ${i}$ th eigenvalue of $\boldsymbol{\Sigma }_{s}\mathbf {G}$ .

Denoting the two-sided bandwidth of shaping pulse $p(t)$ by $2W$ , and further simplifying (25), the maximum achievable rate for $u_{1}$ and $u_{2}$ in the direct transmission phase is then expressed as $$\begin{align*} R_{1}^{\mathrm {dir}}=&\frac {1}{2}\frac {2W}{\tau }\log _{2}\left({1 {+}\frac {|h_{1}|^{2}\theta _{1} P_{s}}{|h_{1}|^{2}\theta _{2} P_{s} {+} N_{0}}}\right) {[\mathrm {bits/s}]} \quad \tag{26}\\ R_{2,2}^{\mathrm {dir}}=&\frac {1}{2}\frac {2W}{\tau }\log _{2}\left({1 {+}\frac {|h_{2}|^{2}\theta _{2} P_{s}}{N_{0}}}\right) {[\mathrm {bits/s}]}. \tag{27}\end{align*}$$ Also, the scaling factor 1/2 in (26) and (27) indicates that the direct transmission phase is one-half of the total transmission duration. To achieve the maximum DoF gain in our proposed FTN signaling, we apply $\tau =1/(1+\beta)$ in (26) and (27), which gives $$\begin{align*} R_{1}^{\mathrm {dir}}=&W(1 {+}\beta)\log _{2}\left({1 {+}\frac {|h_{1}|^{2}\theta _{1} P_{s}}{|h_{1}|^{2}\theta _{2} P_{s} {+} N_{0}}}\right) {[\mathrm {bits/s}]} \quad \tag{28}\\ R_{2,2}^{\mathrm {dir}}=&W(1 {+}\beta)\log _{2}\left({1 {+}\frac {|h_{2}|^{2}\theta _{2} P_{s}}{N_{0}}}\right) {[\mathrm {bits/s}]}. \tag{29}\end{align*}$$ In the cooperative relaying phase, the maximum achievable rate with FTN signaling is expressed as $$\begin{equation*} R_{1}^{\mathrm {coop}} = W(1+\beta)\log _{2}\left({1+\frac {|h_{3}|^{2} P_{s}}{N_{0}}}\right) {[\mathrm {bits/s}]}. \tag{30}\end{equation*}$$ The maximum achievable rate of the far user $u_{1}$ in the two transmission phases with selection combining3 is expressed as $$\begin{align*}&R_{1}^{\mathrm {dir {+}coop}} {=} W(1 {+}\beta) \\&\;\quad {}\times \log _{2}\left[{1{+}\max \left({\frac {|h_{1}|^{2}\theta _{1} P_{s}}{|h_{1}|^{2}\theta _{2} P_{s}{+}N_{0}}, \frac {|h_{3}|^{2} P_{s}}{N_{0}}}\right)}\right] {[\mathrm {bits/s}]}. \tag{31}\end{align*}$$

B. Outage Probability

Letting the target rate at $u_{1}$ be $R_{\mathrm {thres}}$ , then the outage probability in our proposed FTN-based C-NOMA scheme is given by $$\begin{equation*} P^{\mathrm {out}}_{u_{1}}=\mathbb {P}\big [R_{1}^{\mathrm {dir+coop}} < R_{\mathrm {thres}}\big].\tag{32}\end{equation*}$$ By substituting (31) into (32), we have $$\begin{equation*} P^{\mathrm {out}}_{u_{1}}=\mathbb {P}\left[{\max \big (\rho _{1}^{\mathrm {dir}},\rho _{1}^{\mathrm {coop}}\big) < 2^{\frac {R_{\mathrm {thres}}}{W(1+\beta)}}-1}\right]. \tag{33}\end{equation*}$$ Note that (33) is equivalent to $$\begin{align*} P^{\mathrm {out}}_{u_{1}}=&\mathbb {P}\left[{\rho _{1}^{\mathrm {dir}},\rho _{1}^{\mathrm {coop}} < 2^{\frac {R_{\mathrm {thres}}}{W(1+\beta)}}{-}1}\right] \tag{34}\\=&\mathbb {P}\left[{\rho _{1}^{\mathrm {dir}}{ < }2^{\frac {R_{\mathrm {thres}}}{W(1+\beta)}}{-}1}\right]\mathbb {P}\left[{\rho _{1}^{\mathrm {coop}}{ < }2^{\frac {R_{\mathrm {thres}}}{W(1+\beta)}}{-}1}\right].\tag{35}\end{align*}$$ Here, (35) is obtained from (34) by assuming the independence between the direct and relay links. For the Rayleigh fading channel, the probability density function (PDF) of $\rho _{1}^{\mathrm {dir}}$ is expressed as [27] $$\begin{equation*} f_{\mathbb {E}[{\rho }_{1}^{\mathrm {dir}}]}(\rho _{1}^{\mathrm {dir}})=\frac {1}{\mathbb {E}[{\rho }_{1}^{\mathrm {dir}}]}\exp \left({{-}\frac {\rho _{1}^{\mathrm {dir}}}{\mathbb {E}[{\rho }_{1}^{\mathrm {dir}}]}}\right). \tag{36}\end{equation*}$$ Then, we have $$\begin{align*}&\mathbb {P}\left[{\rho _{1}^{\mathrm {dir}} < 2^{\frac {R_{\mathrm {thres}}}{W(1{+}\beta)}}{-}1}\right]=\int _{0}^{2^{\frac {R_{\mathrm {thres}}}{W(1{+}\beta)}}{-}1}f_{\mathbb {E}[{\rho }_{1}^{\mathrm {dir}}]}(\rho _{1}^{\mathrm {dir}})d{\rho _{1}^{\mathrm {dir}}} \\&\;= 1-\exp \left({{-}\frac {2^{\frac {R_{\mathrm {thres}}}{W(1+\beta)}}{-}1}{\mathbb {E}[{\rho }_{1}^{\mathrm {dir}}]}}\right). \tag{37}\end{align*}$$ Similarly, we obtain $$\begin{equation*} \mathbb {P}\left[{\rho _{1}^{\mathrm {coop}} < 2^{\frac {R_{\mathrm {thres}}}{W(1+\beta)}}{-}1}\right]=1-\exp \left({{-}\frac {2^{\frac {R_{\mathrm {thres}}}{W(1+\beta)}}{-}1}{\mathbb {E}[{\rho }_{1}^{\mathrm {coop}}]}}\right). \tag{38}\end{equation*}$$ Using (37) and (38), (35) is rewritten by $$\begin{equation*} P^{\mathrm {out}}_{u_{1}}{=}\left[{1-\exp \left({\frac {1{-}2^{\frac {R_{\mathrm {thres}}}{W(1{+}\beta)}}}{\mathbb {E}[{\rho }_{1}^{\mathrm {dir}}]}}\right)}\right]\left[{1-\exp \left({\frac {1{-}2^{\frac {R_{\mathrm {thres}}}{W(1{+}\beta)}}}{\mathbb {E}[{\rho }_{1}^{\mathrm {coop}}]}}\right)}\right]. \tag{39}\end{equation*}$$ The benefits of our scheme in terms of the outage probability are evident from (39), where the term associated with (38) corresponds to the cooperative diversity. Also, the gain achievable by FTN signaling is represented by $(1+\beta)$ .

Under the assumption of the independent direct and relaying channels, our scheme achieves the diversity order of two.

C. Performance Results

Here, we present the performance results of the capacity and outage probability of the proposed FTN-based C-NOMA scheme. The BS, $u_{1}$ , and $u_{2}$ are assumed to be located at the coordinates (0, 0), (10, 0), and (5, 0), respectively. The path-loss exponent is set to $\zeta =3$ , and thermal noise power of −114 dBm is assumed. The parameters $(\tau,\beta)=(0.8,0.25)$ are employed for FTN signaling. Also, the NOMA power allocation parameters are set to $(\theta _{1}, \theta _{2})=(0.8, 0.2)$ .

Fig. 2 shows the ergodic information rate of $u_{1}$ , employing the proposed FTN C-NOMA scheme, which is compared with five benchmark schemes, i.e., Nyquist-rate C-NOMA, Nyquist-rate NOMA, FTN-NOMA, FTN-OMA, and Nyquist-rate OMA. It is observed that the proposed scheme exhibited the best maximum achievable rate for the far user of all the schemes. This became possible owing to our joint benefits of cooperative diversity and the FTN-induced gain. Also, observe in Fig. 2 that the performance of Nyquist-rate C-NOMA was close to that of the proposed FTN-based C-NOMA because of the cooperative diversity despite the lack of the FTN-signaling gain. Furthermore, since the far-user rate of non-cooperative NOMA at high SNRs is dominated by interference power as compared to the noise power, the rates of FTN-NOMA and Nyquist NOMA are found to saturate at higher SNRs. Conventional FTN-NOMA lacks the advantage of multiple signal copies at the far user and has to rely on direct transmission only, hence resulting in a low achievable rate.

Fig. 2.

Achievable information rates of the far user $u_{1}$ in the proposed FTN C-NOMA scheme and four benchmark schemes.

Finally, Fig. 3 shows the outage probability of the far user $u_{1}$ of the proposed scheme, compared with the three other NOMA schemes, at $R_{\mathrm {thres}}{=}1$ bps/Hz. Observe in Fig. 3, the far user in our proposed FTN C-NOMA has a lower outage probability than the conventional Nyquist-rate C-NOMA. This is achieved as the benefits of the rate improvement of FTN signaling in the C-NOMA context.

Fig. 3.

Outage probabilities of the far user $u_{1}$ in the proposed FTN C-NOMA scheme and NOMA benchmarks.

SECTION IV. Conclusion

In this letter, we proposed the system integrating FTN signaling and C-NOMA with half-duplex DF relaying. While FTN signaling gleans increased spectral efficiency from the combination of a non-ideal shaping filter and reduced symbol interval, C-NOMA extends the concept of cooperative diversity with the aid of the redundant signal for a weak user, which is decoded by the strong user during SIC. Thus, by employing the higher symbol rate achieved by FTN signaling in the context of C-NOMA downlink, we demonstrated the explicit advantage in terms of capacity and outage probability achieved by the proposed scheme over its orthogonal counterparts as well as the conventional NOMA techniques.