Dual-Mode Time-Domain Index Modulation for Nyquist-Criterion and Faster-Than-Nyquist Single-Carrier Transmissions

In this paper, we propose a novel dual-mode time-domain single-carrier (SC) index modulation (DM-SCIM) scheme, where the combination of two constellation modes carries information bits further to modulated symbols, hence increasing the transmission rate. Accordingly, the proposed DM-SCIM scheme is capable of attaining a higher transmission rate than the conventional SC and SCIM schemes, without imposing any additional performance penalty. Furthermore, in order to maintain the detection complexity low enough to be tractable at the handset receiver, a successive detection algorithm, comprising minimum mean-square error (MMSE)-based frequency-domain equalization and log-likelihood ratio detection, is developed for both the conventional SCIM and the proposed DM-SCIM schemes. We also derive the error-rate bound of the proposed DM-SCIM scheme under the assumption of a channel-uncoded scenario. Moreover, in order to further increase bandwidth efficiency, the proposed DM-SCIM scheme is extended to the scenario supporting faster-than-Nyquist signaling, where the symbol interval in the time domain is set to be smaller than that defined by the Nyquist criterion. Our simulation results demonstrate the explicit performance advantage of the proposed schemes over the existing SC schemes. It is also revealed that our DM-SCIM scheme is capable of achieving a significantly lower peak-to-average power ratio than orthogonal frequency division multiplexing (OFDM) and OFDM with index modulation, while benefiting from a high reliability, which is achievable by multipath diversity.

SECTION I. Introduction

The recent concept of index modulation (IM) [2], [3] introduced in the spatial [4]–​[7] and frequency domains [8]–​[10] is a promising approach for next-generation energy-and bandwidth-efficient wireless communications. For example, orthogonal frequency-division multiplexing with IM (OFDM-IM) [8], [9] adopts an IM-based symbol mapping principle in the frequency domain, where a subset of the multicarriers is activated in each OFDM frame, and its activated multicarrier index conveys additional information relative to the classic modulation scheme.¹ More specifically, in [9], it was clarified that the OFDM-IM scheme tends to be more effective than the classic OFDM scheme, especially in a low-rate scenario. Note that both the OFDM and OFDM-IM schemes typically suffer from a high peak-to-average power ratio (PAPR) [9]. Additionally, in [11], Chung proposed the generalized OFDM framework, referred to as orthogonally multiplexed orthogonal amplitude modulation (OMOAM), where the orthogonal multiplexing and grouping methods were used in the frequency domain with pulse amplitude signaling, and hence it is capable of striking a tradeoff between the transmission rate, the error rate, and the PAPR.

In order to expand the OFDM-IM scheme’s beneficial throughput regime relative to the classic OFDM scheme, the dual-mode (DM) OFDM-IM scheme was proposed in [12], [13]. Specifically, while in the original OFDM-IM family, the multicarriers in each subset are composed of zeros and non-zero symbols associated with the same single constellation mode, most recently in the DM OFDM-IM scheme [12], two constellation modes are employed in each multicarrier subset. When comparing the OFDM-IM and DM OFDM-IM schemes having the same index combination, the total number of information bits conveyed by the modulated symbols in the DM OFDM-IM scheme is greater than that for the OFDM-IM scheme. Furthermore, in [13], the DM OFDM-IM scheme is generalized so that the number of subcarriers modulated by the same constellation mode in each multicarrier subset is alterable, hence further increasing the transmission rate. Naturally, a high PAPR limitation is imposed on the DM OFDM-IM scheme, similar to the conventional OFDM and OFDM-IM schemes.

As a means to reduce the PAPR of the above-mentioned OFDM family, single-carrier (SC) symbol transmission with frequency-domain equalization (FDE) [14]–​[16] has been widely employed in uplink scenarios of the current standards [17]. Since the OFDM and FDE-aided SC schemes are close relatives [16], the FDE-aided SC scheme is capable of achieving comparable error-rate performance to that of OFDM while also attaining a significantly lower PAPR than OFDM. Note that, as mentioned in [18], the FDE-aided SC scheme also benefits from multipath diversity gain, which is attainable without relying on channel coding, unlike its OFDM counterpart. Most recently, the SC counterpart of the OFDM-IM scheme was proposed in [19], where the positions of non-zero symbols in the time-domain subframe carry additional information based on the IM principle. However, the performance advantage of the time-domain SC-based IM (SCIM) scheme [19] over the conventional SC scheme is seen particularly in a low-rate regime, in a similar manner to that of OFDM-IM over OFDM [9].

Since its first proposal in the 1970s, the concept of faster-than Nyquist signaling (FTN) [20], [21] has been rediscovered as a means of increasing bandwidth and energy efficiencies. More specifically, in FTN signaling, the symbol interval in the time domain is set to be smaller than that defined by the Nyquist criterion. This allows the transmission rate to be increased without increasing the bandwidth, but this has the detrimental effect of increasing inter-symbol interference (ISI). To deal with this situation, low-complexity time- and frequency-domain equalizers were developed in [22]–​[26], respectively. Most recently, in [27], the IM principle was incorporated into FTN signaling, in order to reduce the effects on ISI, by decreasing the number of transmitted symbols.

Against this background, the novel contributions of this paper are as follows.

• We propose a DM-SCIM scheme, where multiple constellation modes are employed for time-domain SC symbols, and the combination of the constellation modes conveys more information relative to the classic amplitude and phase shift keying (APSK)-modulated symbols. Accordingly, the proposed scheme is capable of increasing the effective transmission rate of the conventional single-mode SCIM scheme [19], without imposing any additional performance loss.

• In order to achieve a low decoding complexity, namely, one tractable at the receiver, a successive detection algorithm comprising minimum mean-square error (MMSE)-based FDE and log-likelihood ratio (LLR) detection is employed for both the conventional SCIM and the proposed DM-SCIM schemes. We also derive the error-rate bound, in order to verify the system model of the proposed scheme. For the sake of further improving the performance of the DM-SCIM scheme, we also introduce a symbol mapping scheme [19], which is capable of reducing the inter-channel correlation between the symbols in the subframe.

• Moreover, our DM-SCIM scheme is amalgamated with the concept of FTN signaling, in order to further enhance the bandwidth and energy efficiencies. By reducing the symbol interval between the time-domain DM-SCIM symbols, a transmission rate is further increased without increasing a bandwidth and a transmit power. It is also ensured that the low-complexity successive detector developed for the DM-SCIM scheme is readily applicable to this FTN counterpart, by introducing a minor modification.

• Our simulation results demonstrate the explicit performance advantage of the proposed schemes over the existing SC scheme. Furthermore, the multipath diversity gain achieved by the DM-SCIM scheme is shown to be beneficial over the OFDM-IM scheme, while the DM-SCIM scheme exhibits a significantly lower PAPR than the OFDM-IM scheme.

The remainder of this paper is organized as follows. Section II presents the system model of our FDE-aided DM-SCIM scheme, while in Section III the analytical error-rate bound is derived. In Section IV, the concept of FTN signaling is incorporated into the proposed DM-SCIM scheme. In Section V, we present the results for the FDE-aided DM-SCIM scheme, and finally this paper is concluded in Section VI.

SECTION II. System Model

In this section, we present the system model of the proposed DM-SCIM scheme. Figs. 1(a) and 1(b) show the frame structures of the conventional SC scheme and the proposed DM-SCIM scheme, respectively. While a single constellation mode having a constant average transmit power is employed in the conventional SC scheme (Fig. 1(a)), in the DM-SCIM scheme, each frame is divided into a train of subframes that contain two constellation modes having different average transmit powers (Fig. 1(b)). In the DM-SCIM scheme, information is conveyed not only by the modulated symbols but also by the combination of constellation modes in each subframe.

FIGURE 1.

Frame structures of the conventional SC and proposed DM-SCIM schemes. (a) Conventional SC. (b) Proposed DM-SCIM.

A. Transmitter Model

Fig. 2 shows the system model of our DM-SCIM transceiver. In the proposed DM-SCIM scheme, $N$ -length data symbols $\mathbf {s} = \left [{s_{1},s_{2},\cdots ,s_{N}}\right ]^{T}$ are transmitted in each frame. These data symbols $\mathbf {s}$ include $L$ subframes, i.e., $$\begin{equation} \mathbf {s} = \left [{\mathbf {s}^{(1)},\mathbf {s}^{(2)},\cdots ,\mathbf {s}^{(L)}}\right ]. \end{equation}$$ Furthermore, each subframe spans $M$ symbol intervals as follows: $$\begin{equation} \mathbf {s}^{(l)}= \left [{s_{1}^{(l)},s_{2}^{(l)},\cdots ,s_{M}^{(l)}}\right ]^{T}. \end{equation}$$

FIGURE 2.

System model of DM-SCIM transmitter and receiver.

The system model of the DM-SCIM transmitter is depicted in the upper part of Fig. 2. First, information bits are serial-to-parallel converted into $L$ bit sequences, where each sequence has $B=B_{1}+B_{2}$ bits. The first $B_{1}$ out of the $B$ information bits are referred to as index bits, which are used for selecting the constellation modes of the $M$ symbols. More specifically, $K$ out of $M$ symbols are selected to be modulated by the $P_{A}$ -point APSK constellation, while the remaining $M-K$ symbols employ the $P_{B}$ -point APSK constellation. The combination of two constellation modes in the $M$ symbols conveys information bits, and hence the index bits have the relationship $$\begin{equation} B_{1} = \left \lfloor{ \log _{2} \binom {M}{K} }\right \rfloor . \end{equation}$$ In the rest of this paper, let us define these two constellation modes as Modes A and B.

Additionally, the $B_{2} = B-B_{1}$ bits are divided into $B_{2,A}$ and $B_{2,B}$ bits; hence, they have the relationship $B_{2}= B_{2,A}+B_{2,B}$ . The first $B_{2,A}$ bits are used for modulating $K~P_{A}$ -point APSK symbols, and the remaining $B_{2,B}$ bits are used for modulating $M-K~P_{B}$ -point APSK symbols. Therefore, we have $$\begin{align} B_{2,A}=&K \log _{2} P_{A}, \\ B_{2,B}=&\left ({ M-K }\right ) \log _{2} P_{B}. \end{align}$$

For example, let us consider the DM-SCIM scenario of $(M,K) = (4, 2)$ . Then, the legitimate combination of two constellation modes is represented by $$\begin{align} \mathbf {s}^{(l)}\in&\left \{{ \left [{ \begin{array}{c} S_{A}^{(1)}\\[0.3pc] S_{A}^{(2)} \\[0.3pc] S_{B}^{(1)} \\[0.3pc] S_{B}^{(2)} \end{array} }\right ], \left [{ \begin{array}{c} S_{B}^{(1)}\\[0.3pc] S_{A}^{(1)} \\[0.3pc] S_{A}^{(2)} \\[0.3pc] S_{B}^{(2)} \end{array} }\right ], \left [{ \begin{array}{c} S_{B}^{(1)}\\[0.3pc] S_{B}^{(2)} \\[0.3pc] S_{A}^{(1)} \\[0.3pc] S_{A}^{(2)} \end{array} }\right ], \left [{ \begin{array}{c} S_{A}^{(1)}\\[0.3pc] S_{B}^{(1)} \\[0.3pc] S_{B}^{(2)} \\[0.3pc] S_{A}^{(2)} \end{array} }\right ] }\right \}\!,\qquad \end{align}$$ where $S_{A}^{(1)}$ and $S_{A}^{(2)}$ are $P_{A}$ -point APSK symbols, and $S_{B}^{(1)}$ and $S_{B}^{(2)}$ are $P_{B}$ -point APSK symbols. Let us assume that $\mathbf {C}_{A} = \left \{{C_{A}(1), \cdots , C_{A}(P_{A}) }\right \}$ and $\mathbf {C}_{B}= \left \{{ C_{B}(1), \cdots , C_{B}(P_{B}) }\right \}$ are the $P_{A}$ - and $P_{B}$ -point APSK constellations, respectively. Then, the relationship $\mathbf {C}_{A} \wedge \mathbf {C}_{B} = \emptyset$ has to be satisfied.

To be more specific, we consider the two APSK constellation sets $P_{A} = 4$ and $P_{B} = 8$ , as shown in Fig. 3. In the figure, $\mathbf {C}_{A}$ is the QPSK constellation, and $\mathbf {C}_{B}$ is the 8PSK constellation. By rotating the phase of $\pi /8$ for the outer 8PSK constellation, the minimum Euclidean distance of the joint constellation $\mathbf {C}_{A} \vee \mathbf {C}_{B}$ is increased. Furthermore, let us consider an example of three constellation mapping models, namely, the Model A of $d_{A} = d_{B}$ , Model B of $d_{A} = d_{AB}$ , and Model C of $d_{B} = d_{AB}$ , where $d_{A}$ and $d_{B}$ are the Euclidean distances between the closest constellation points in $\mathbf {C}_{A}$ and $\mathbf {C}_{B}$ , respectively, while $d_{AB}$ corresponds to the Euclidean distance between the closest points of $\mathbf {C}_{A}$ and $\mathbf {C}_{B}$ , as shown in Fig. 3. The minimum Euclidean distance $d_{\mathrm {min}}$ and the PAPR of each model are listed in Table 1, where the average transmit power of the joint constellation $\mathbf {C}_{A} \vee \mathbf {C}_{B}$ is fixed to unity. As shown, Model B exhibits the highest minimum Euclidean distance, while Model A has the lowest average PAPR value. Hence, the coding gain and PAPR have the tradeoff relationship in our DM-SCIM scheme. Note that although the PAPR of the DM-SCIM scheme is higher than that of conventional SC, i.e., 0 dB, it is significantly lower than those of the OFDM and the OFDM-IM schemes.

TABLE 1 Minimum Euclidean Distances and PAPRs

FIGURE 3.

Constellation design of the proposed DM-SCIM scheme.

Having generated the DM-SCIM frame $\mathbf {s}$ of (1), it is then interleaved for the entire DM-SCIM subframe, in a similar manner to that proposed for the previous SCIM scheme [19]. Finally, a sufficiently long cyclic prefix (CP) is added to the interleaved frame, for the sake of allowing a low-complexity FDE-based MMSE equalization at the receiver, where we assume that the CP length $\nu$ is greater than the delay spread.

The normalized transmission rate of our FDE-aided DM-SCIM scheme is given by $$\begin{align} R \!=\! \frac {N}{N+\nu } \cdot \frac {\left \lfloor{ \log _{2} \binom {M}{K} }\right \rfloor \!+\! K \log _{2} P_{A} \!+\! \left ({ M-K }\right ) \log _{2} P_{B}}{M}.\notag \\ {}\end{align}$$ Since the conventional SCIM scheme [19] includes zero symbols in each subframe, it is a challenging task to increase the transmission rate in comparison to the conventional SC scheme. However, in the proposed DM-SCIM scheme, while exploiting the IM principle, all the symbols are modulated, unlike in the conventional SCIM scheme. This contributes to the effective increase in the transmission rate of the DM-SCIM scheme.

B. Receiver Model

The DM-SCIM receiver model is shown in the lower part of Fig. 2, where the MMSE-FDE is used for estimating the DM-SCIM frame $\mathbf {s}$ . After removing CP symbols, the received signals in the time domain $\mathbf {y}\in \mathbb {C}^{N}$ are represented by $$\begin{equation} \mathbf {y} = \mathbf {H}\mathbf {s} + \mathbf {n}, \end{equation}$$ where $\mathbf {H}$ is the channel matrix and $\mathbf {n}$ comprises the additive white Gaussian noise (AWGN) components, following the complex-valued Gaussian distribution $\mathcal {CN}(0, N_{0})$ , in which $N_{0}$ is the noise variance. Furthermore, $\mathbf {H}$ exhibits a circulant structure whose first column corresponds to the $J$ -length channel impulse responses (CIRs) and $N-J$ zeros, i.e., $\mathbf {h}=[h_{1},\cdots ,h_{J},0,\cdots ,0]$ .

The time-domain received signals are transformed to frequency-domain counterparts by invoking the inverse discrete Fourier transform (IDFT)-based eigenvalue decomposition $\mathbf {H}=\mathbf {Q}^{T}{\Lambda _{\mathbf {f}}}\mathbf {Q}^{*}$ , where ${\Lambda }_{f}=[\lambda _{(1,1)},\cdots ,\lambda _{(N,N)}]$ is a diagonal matrix whose diagonal elements are the IDFT coefficients of the CIRs, and $\mathbf {Q}$ is the IDFT matrix. More specifically, the frequency-domain received signals $\mathbf {y}_{f}$ are given by $$\begin{align} \mathbf {y}_{f}=&\mathbf {Q}^{*}\mathbf {y} \\=&{\Lambda }_{f}\underbrace {\mathbf {Q}^{*}\mathbf {s}}_{\mathbf {s}_{f}}+\underbrace {\mathbf {Q}^{*}\mathbf {n}}_{\mathbf {n}_{f}}. \end{align}$$ In order to estimate the transmitted DM-SCIM symbols $\mathbf {s}$ , the low-complexity diagonal MMSE equalization and the discrete Fourier transform (DFT) are carried out. More specifically, the $i$ th component of the diagonal MMSE weights $\mathbf {W}$ is represented by [14] $$\begin{equation} w_{(i,i)}=\frac {\lambda _{(i,i)}}{\lambda _{(i,i)}+N_{0}}. \end{equation}$$ Thus, the time-domain symbols are attained by $$\begin{equation} \hat {\mathbf {s}} = \mathbf {Q}^{T}\mathbf {W}\mathbf {y}_{f}. \end{equation}$$

Having attained the estimates of the DM-SCIM frame $\hat {\mathbf {s}}$ , the bit sequence of the $l$ th subframe is estimated by the maximum likelihood (ML) criterion as follows: $$\begin{equation} \hat {\mathbf {b}} = \arg \min _{\mathbf {b}}\left \|{ \hat {\mathbf {s}}^{(l)} - \mathbf {s}^{(l)} }\right \|, \end{equation}$$ where $\hat {\mathbf {b}}$ is the estimated bit sequence. The computational complexity of ML detection is $\mathcal {O}\left ({2^{B_{1}}P_{A}^{K}P_{B}^{M-K}/B}\right )$ per bit. Note that the detection complexity of (13) is significantly high if the value of $M$ is large.

C. LLR Detection for DM-SCIM

The successive detection of MMSE-FDE and exhaustive ML search, introduced in Section II-B, may exhibit an excessively high complexity for the large search space of (13). In order to reduce the associated detection cost, we herein introduce the concept of LLR detection, which was originally developed for OFDM-IM [12], [28], into our DM-SCIM.

More specifically, the LLR of the a posteriori probabilities of Modes A and B for the $m$ th symbol in the $l$ th subframe is represented by $$\begin{equation} \gamma ^{(l)}_{m}=\ln \left ({\frac {\sum _{i=1}^{P_{A}}P\left ({ s_{n}^{(l)} = C_{A}\left ({i}\right ) |\hat {s}^{(l)}_{m} }\right )}{\sum _{j=1}^{P_{B}}P\left ({s_{n}^{(l)} = C_{B}\left ({j}\right ) |\hat {s}^{(l)}_{m} }\right )}}\right )\!.\quad \end{equation}$$ Let us consider Bayes’ formula $$\begin{equation} P\left ({ s_{n}^{(l)} = C_{k}\left ({i}\right ) |\hat {s}^{(l)}_{m} }\right ) = \frac {P\left ({ s_{n}^{(l)} = C_{k}\left ({i}\right )}\right )}{P\left ({\hat {s}^{(l)}_{m} }\right )} (k=A,B), \quad \end{equation}$$ and the probabilities $P\left ({ s_{n}^{(l)} = C_{A}\left ({i}\right )}\right )=K/M$ and $P\left ({ s_{n}^{(l)} = C_{B}\left ({i}\right )}\right )=(M-K)/M$ . Then, we arrive at $$\begin{align} \gamma _{n}^{(l)}=&\ln \left ({\frac {K}{M-K}}\right ) \!+\! \ln \left ({\sum _{i=1}^{P_{A}}\exp \left ({\!-\!\frac {1}{N_{0}}\left |{\hat {s}_{n}^{(l)}-C_{A}\left ({i}\right )}\right |^{2}}\right ) }\right ) \notag \\&\qquad -\, \ln \left ({\sum _{j=1}^{P_{B}} \exp \left ({-\frac {1}{N_{0}}\left |{\hat {s}_{n}^{(l)}-C_{B}\left ({i}\right )}\right |^{2} }\right )}\right ). \end{align}$$ From (16), each LLR is calculated with a complexity of $\mathcal {O}(M)$ . Practically, in order to avoid the potential numerical overflow in the calculations of (16), the Jacobian logarithm may be used in a similar manner to [28].

Having attained an LLR value for each symbol in the DM-SCIM subframe $\mathbf {s}$ according to (16), we now detect the combination of modulation modes and then demodulate each symbol. More specifically, the LLRs are sorted in descending order, and modulation modes are detected from the symbols having higher LLR values. Once the combination of modulation modes in a subframe, i.e., the IM bit sequence, is detected in a unique manner, other remaining LLRs will not be used. Finally, each APSK symbol is demodulated, based on the detected modulation modes.

In order to elaborate a little further, using Fig. 4, we compared the achievable bit error probability (BER) performance of our DM-SCIM schemes using the exhaustive ML search of (13) and the LLR detection of (16), where the DM-SCIM parameters were set as $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ and the CIR length was $J=20$ . Observe in Fig. 4 that LLR detection exhibited almost the same BER values as those achievable by exhaustive ML search.

FIGURE 4.

Achievable BER of our DM-SCIM schemes using the exhaustive search of (13) and the LLR detection of (16). DM-SCIM parameters, $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ ; delay spread, $J=20$ .

In a similar to LLR detection derived here for our DM-SCIM scheme, the one that operates in the previous SCIM scheme [19] is readily derivable, which has not been presented before. Further note that the LLR detection of the DM-SCIM and SCIM schemes is enabled by exploiting signals filtered by low-complexity FDE-MMSE, which is different from LLR detection of the DM-OFDM-IM and OFDM-IM schemes.

SECTION III. Analytical Error-Rate Bound

In this section, we derive the analytical upper bound of BER for the proposed scheme, under the assumption of a channel-uncoded scenario. The received signals in the frequency-domain represented in (10) are rewritten as $$\begin{equation} \mathbf {y}_{f} = \mathbf {S}_{f}{\lambda } + \mathbf {n}_{f}, \end{equation}$$ where ${\lambda }=[\lambda _{(1,1)},\cdots ,\lambda _{(N,N)}]^{T}$ . The conditional pairwise error probability (PEP) is given by [29] $$\begin{equation} P(\mathbf {S}_{f}\rightarrow \mathbf {S}_{f}'| {\lambda }) = Q\left ({\sqrt {\frac {{\left \|{ (\mathbf {S}_{f} - \mathbf {S}_{f}') {\lambda } }\right \|^{2}}}{2N_{0}}} }\right )\!, \end{equation}$$ where $Q(\cdot )$ is the $Q$ -function. The unconditional PEP can be obtained by averaging the conditional PEP of (18). Hence, we have $$\begin{equation} P(\mathbf {S}_{f}\rightarrow \mathbf {S}_{f}') = \mathbb {E}\left [{P(\mathbf {S}_{f}\rightarrow \mathbf {S}_{f}'| {\lambda })}\right ]. \end{equation}$$ The performance of the DM-SCIM scheme relying on ML detection can be estimated according to the unconditional PEP. More specifically, the analytical upper bound of BER is calculated as [30] $$\begin{equation} P_{\mathrm {BER}} \le \frac {1}{BL2^{B}} \sum _{\mathbf {S}_{f}} \sum _{\mathbf {S}_{f}\ne \mathbf {S}_{f}'} d(\mathbf {S}_{f},\mathbf {S}_{f}') P(\mathbf {S}_{f} \rightarrow \mathbf {S}_{f}'), ~~\end{equation}$$ where $d(\mathbf {S}_{f},\mathbf {S}_{f}')$ represents the Hamming distance between $\mathbf {S}_{f}$ and $\mathbf {S}_{f}'$ .

SECTION IV. DM-FTN-IM Scheme

In this section, we introduce another scheme that amalgamates our DM-SCIM and FTN signaling, which is referred to as DM-FTN-IM in this paper. Let us define the ISI-free symbol interval limit as $T_{0}=1/(2W)$ , which is determined by the Nyquist criterion, for the case that the symbols are strictly bandlimited to $W$ [Hz].

In the proposed DM-FTN-IM scheme, information bits are modulated onto an $N$ -length time-domain DM-SCIM frame $\mathbf {s}$ , which is the same as that modulated in (1) of our DM-SCIM scheme (Section II-A). After adding a sufficiently long CP (length $\nu$ ) to $\mathbf {s}$ in order to have $\tilde {\mathbf {s}} = [\tilde {s}_{1},\cdots ,\tilde {s}_{N+\nu }]^{T} \in \mathbb {C}^{N+\nu }$ , the DM-FTN-IM transmitter transmits the frame with the symbol interval shortened to $T=\alpha T_{0}$ , where $\alpha <1$ is the symbol packing ratio. Hence, the time-domain transmit signal is represented by $$\begin{equation} \sum _{i=1}^{N+\nu }a(t-i\alpha T_{0}) \tilde {s}_{i}, \end{equation}$$ where $a(t)$ represents the impulse response of a root-raised-cosine (RRC) filter, which is used for pulse shaping in this paper. Note that when we have $\alpha =1$ , the system model of the DM-FTN-IM scheme becomes equivalent to that of the DM-SCIM scheme.

The normalized transmission rate of the DM-FTN-IM scheme is expressed by $$\begin{equation} \frac {1}{\alpha }\frac {1}{1+\beta }R~ \textrm {[bps/Hz]}, \end{equation}$$ where the coefficient $1/\alpha$ represents the effects of a rate increase achievable owing to FTN signaling. Also, the coefficient $1/(1+\beta )$ corresponds to the effects of bandwidth expansion caused by the RRC filter, which one should note in practice is unavoidable even for other transmission schemes, such as the SC, SCIM, DM-SCIM, OFDM, and OFDM-IM schemes.

At the receiver, the received signal is firstly matched-filtered by an RRC shaping filter $a^{*}(-t)$ . Then, the signals associated with the CP are removed from the samples, in order to obtain the block-wise received symbols of [25] $$\begin{equation} \mathbf {y} = \mathbf {H}\mathbf {G}\mathbf {s}+\boldsymbol \eta , \end{equation}$$ where $\mathbf {G} \in \mathbb {R}^{N\times N}$ represents a circulant matrix having the tap coefficient vector $\mathbf {g} = [g(-\xi T),\cdots ,g(0),\cdots ,g(\xi T)]^{T}$ , which corresponds to the ISI effects induced by FTN signaling [25]. Here, $\xi$ is the effective tap length of FTN-induced ISI, while $g(t)=\int a(\tau )a^{*}(\tau -t)d\tau$ . Furthermore, $\boldsymbol \eta \in \mathbb {C}^{N}$ represents colored (correlated) noises, which have the relationship $\mathbb {E}[\boldsymbol \eta (iT)\boldsymbol \eta ^{*}(jT)]=N_{0}g((i-j)T)$ . In order to maintain circulant structures for $\mathbf {H}$ and $\mathbf {G}$ , it is assumed that the CP length is set larger than the resultant tap length, which is caused by the FTN-induced ISI effects, as well as the long CIR dispersive channel.

Furthermore, since both the matrices $\mathbf {H}$ and $\mathbf {G}$ exhibit a circulant structure, we have the relationship $\mathbf {HG}=\mathbf {Q}^{T}{\Lambda }_{f} {\Lambda _{\mathbf {G}}}\mathbf {Q}^{*}$ with the aid of the IDFT operation, where ${\Lambda _{\mathbf {G}}}$ represents the diagonal matrix calculated from the IDFT coefficients of $\mathbf {G}$ . This implies that the received signal model of DM-FTN-IM in (23) is similar to that of DM-SCIM. Hence, at the receiver, the low-complexity successive algorithm of MMSE-based FDE and LLR detection proposed in Section II-C can be readily carried out in the DM-FTN-IM receiver. It should be noted that in order to eliminate the colored noise effects in (23), the low-complexity noise-whitening MMSE detector of [26] has to be employed instead of the conventional one.

SECTION V. Performance Results

In this section, we present our simulation results, in order to characterize the proposed DM-SCIM and DM-FTN-IM schemes, employing the successive MMSE-aided FDE and LLR detection at the receiver. The achievable BER and PAPR performances were investigated, based on Monte Carlo simulations. The basic system parameters are listed in Table 2. We assumed a frequency-selective Rayleigh fading channel having a CIR length of $J=20$ . The block length was set to $N = 128$ and the CP length was $\nu = 20$ .

TABLE 2 Parameters Used in Our Simulations

A. Error-Rate Performance

Fig. 5 shows the BER performances of the DM-SCIM schemes employing different constellations, namely, Models A, B, and C shown in Fig. 3 and Table 1. The DM-SCIM parameters were set as $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ . Hence, we had $B_{1}=1$ bit, $B_{2,A}=2$ bits, and $B_{2,B}=3$ bits, as well as the normalized transmission rate of $R = 2.59$ bps/Hz. As seen from Fig. 5, the DM-SCIM scheme employing Model B achieved the best BER performance among the three. Thus, it was found to be effective to design the two constellations modes based on the minimum Euclidean distance of the joint constellation $\mathbf {C}_{A} \vee \mathbf {C}_{B}$ .

FIGURE 5.

BER performances of the DM-SCIM schemes of Models A, B, and C. DM parameters, $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ ; delay spread, $J=20$ .

Fig. 6 shows the BER performances of the DM-SCIM scheme employing Model B and of the conventional SC scheme. The DM-SCIM parameters were set as $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ , while 8PSK was used for the conventional SC scheme. Hence, the transmission rate of both the schemes was $R = 2.59$ bps/Hz. Furthermore, the delay spread was $J = 1, 10$ , or 20. As seen from Fig. 6, our scheme archived a better BER performance than the conventional SC scheme in the $J=10$ and 20 scenarios. More specifically, the performance advantage of the DM-SCIM scheme over the conventional SC scheme, which was recorded for $\mathrm {BER} = 10^{-5}$ , was approximately 1 dB for the delay spread of $J=20$ . Furthermore, this advantage of the DM-SCIM scheme increased upon increasing the SNR value. Also, observe in Fig. 6 that upon increasing the delay spread, the achievable diversity order increased, owing to the explicit benefit of the multipath diversity gain.

FIGURE 6.

BER performances of the DM-SCIM scheme employing constellation Model B, the conventional SCIM scheme, and the conventional SC scheme employing 8PSK. DM-SCIM parameters, $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ .

Fig. 7 shows the BER performances of the DM-SCIM scheme, the conventional SC scheme, and the conventional SCIM scheme for the case of two receive antennas. The system parameters of the DM-SCIM scheme and the conventional SC scheme were the same as those used for Fig. 6. The parameters used for the conventional SCIM scheme were set as $(M,K,P_{IM})=(2,1,32)$ , where $P_{IM}$ is the constellation size. As seen from Fig. 7, regardless of the number of receive antennas, our scheme tended to outperform the conventional SC scheme. Since the SCIM scheme has zero symbols in each subframe, the constellation size had to be high in order to attain the same transmission rate as that of our DM-SCIM scheme. This imposes a performance penalty on the SCIM scheme, and hence the BER performance of our DM-SCIM scheme was significantly better than that of the SCIM benchmark scheme.

FIGURE 7.

BER performances of the DM-SCIM scheme, the conventional SC scheme using 8PSK, and the conventional SCIM scheme, each employing two receive antennas. DM-SCIM parameters, $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ ; SCIM parameters, $(M,K,P_{IM})=(2,1,32)$ .

Fig. 8 shows the effective SNRs of the DM-SCIM scheme and the conventional SC scheme, which were recorded for the BER value of $10^{-4}$ . In the simulations, the CIR length was varied from $J=1$ to $J=20$ in steps of one. As seen from Fig. 8, our scheme achieved $\mathrm {BER}=10^{-4}$ at a lower SNR than the conventional SC scheme for $J\ge 2$ . Furthermore, the performance gain of our scheme increased with the increase of the CIR length. Note that in the frequency-selective fading channel, the delay spread is typically tens of symbol intervals.

FIGURE 8.

Effective SNRs of the DM-SCIM scheme and SC scheme at a BER value of $10^{-4}$ .

Using Fig. 9, we investigated the achievable BER performance of the DM-FTN-IM scheme, which was compared with the conventional FTN-IM and FTN schemes, employing the proposed successive MMSE-based FDE and LLR detection. The symbol packing ratio was $\alpha =0.7$ or 0.8. Observe in Fig. 9 that the proposed DM-FTN-IM scheme achieved the best performance among the three FTN-related schemes. Also note that both the FTN-IM and DM-FTN-IM schemes plotted here were not tractable in the previous exhaustive ML search [27], due to the excessively high detection complexity. These schemes only became possible to use because of the explicit benefit of the low-complexity successive detection introduced in this paper.

FIGURE 9.

BER performances of the conventional FTN scheme employing 8PSK, the FTN-IM scheme, and the proposed DM-FTN-IM scheme. DM-FTN-IM parameters, $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ ; FTN-IM parameters, $(M,K,P_{IM})=(8,7,8)$ ; symbol packing ratio, $\alpha =0.7$ or 0.8.

B. PAPR Analysis

Finally, using Fig. 10, we compared the complementary cumulative distribution functions (CCDFs) of PAPRs of the DM-SCIM, SCIM, SC, OFDM-IM, and OFDM schemes. Here, 8PSK was used for the SC and OFDM schemes, while the parameters were set as $(M,K,P_{IM})=(8,7,8)$ for the SCIM and OFDM-IM schemes. The parameter values $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ were employed for the DM-SCIM scheme. The transmission rate was 3 bps/Hz when ignoring the effects of the CP. The effects of pulse shaping were also investigated for the SC, SCIM, and DM-SCIM schemes. The number of subcarrier was set to 1024 for the OFDM and OFDM-IM schemes. Observe in Fig. 10 that the SC-based transmission schemes without pulse shaping, such as the DM-SCIM, SCIM, and SC schemes, exhibited significantly lower PAPR profiles than the OFDM-IM and OFDM schemes, as expected. Upon introducing pulse shaping into the SC transmission family, their PAPR profiles increased but remained much lower than those of OFDM-IM and OFDM without pulse shaping.

FIGURE 10.

CCDFs of PAPRs of the DM-SCIM, SCIM, and OFDM-IM schemes, and the SC and OFDM schemes employing 8PSK. SCIM and OFDM-IM parameters, $(M,K,P_{IM})=(8,7,8)$ ; DM-SCIM parameters, $(M, K, P_{A}, P_{B}) = (2, 1, 4, 8)$ ; transmission rate (ignoring the effects of CP), 3 bps/Hz.

SECTION VI. Conclusions

In this paper, we have introduced the concept of DM SCIM in the context of broadband SC systems, in order to increase the transmission rate of the conventional SC and SCIM schemes. A low-complexity successive detection algorithm, comprising MMSE-aided FDE and LLR detection, was developed for the proposed DM-SCIM scheme, as well as the conventional SCIM scheme. Moreover, in order to further enhance the beneficial operational region of our DM-SCIM scheme, the concept of FTN signaling was incorporated. Our simulation results demonstrated that the proposed schemes are capable of the increasing transmission rates of the existing SC and SCIM schemes. It was also verified that these benefits are attained while maintaining significantly lower PAPRs than those of OFDM-IM and OFDM.