Faster-Than-Nyquist Signaling With Index Modulation

In this letter, we propose a novel faster-than-Nyquist (FTN) transmission scheme relying on the time-domain single-carrier index-modulation (IM) concept. In the proposed FTN with IM (FTN-IM) transmitter, a subset of time-domain FTN symbols are activated, where the combination of the activated symbols conveys additional information further to the classic amplitude phase shift keying. Owing to the explicit benefit of sparse FTN-IM signaling, the FTN-specific inter-symbol interference is mitigated in an effective manner, hence attaining a higher spectral efficiency than the conventional FTN counterpart. Furthermore, a low-complexity noise-whitening frequency-domain equalization is developed for our FTN-IM receiver. In order to demonstrate the performance advantage of the proposed FTN-IM scheme over the conventional FTN counterpart, we show that a minimum Euclidean distance of proposed FTN-IM signaling is higher than that of the conventional FTN signaling, while providing their numerical performance comparisons in terms of error rates.

SECTION I. Introduction

Faster-Than-Nyquist (FTN) signaling has the potential of increasing the transmission rate beyond the inter-symbol interference (ISI)-free Nyquist-criterion-based limit [1]–​[3], which is achieved without increasing bandwidth. More specifically, in the conventional time-orthogonal Nyquist-criterion-based transmitter, a minimum symbol interval, which does not cause ISI, is lower-bounded by $T_{0} = 1/(2W)$ for the ideal sinc pulse, where the symbols are strictly band-limited to $W$ [Hz]. By contrast, in the FTN scenarios, the symbol interval $T$ is typically set to lower than $T_{0}$ , i.e., $T=\alpha T_{0}$ , where $\alpha < 1$ is a packing ratio. This implies that FTN signaling allows us to transmit more symbols than those of the traditional Nyquist-criterion-based scheme. However, this merit specific to FTN signaling is achieved at the cost of increasing ISI, hence imposing an increased demodulation complexity at the receiver.

In order to achieve a practically low demodulating complexity, in [4] the hard-decision frequency-domain equalization (FDE), based on the minimum-mean square error (MMSE) criterion, was developed for the uncoded FTN scenario. Also, the soft-decision counterpart of [4] was proposed for a multi-stage serially concatenated turbo FTN architecture in [5]. The complexity of these FDE-aided FTN receivers [4], [5] is typically lower than those of the time-domain equalizers (TDEs) [6]–​[8], especially for a high-ISI (high transmission rate) FTN scenario, having a low-$\alpha $ value. Furthermore, while the FDE-aided receivers of [4] and [5] suffered from the FTN-specific unignorable colored noise effects, the hard-decision and the soft-decision noise-whitening FDEs were proposed in [9] and [10], respectively, in order to combat the limitation associated with the colored noise. More recently, by adjusting the FTN parameters, such as a packing ratio $\alpha $ and a roll-off factor $\beta $ , low-complexity hard-decision symbol-by-symbol FTN detection was enabled in [11]. In [12], a reduced-complexity sphere-decoding algorithm was used for the FTN detection, where the achievable performance was close to that attainable by maximum likelihood (ML) detection.

The recent index modulation (IM) concept allows us to increase a transmission rate, where activation of a subset of indices conveys additional information, further to the classic modulation schemes, such as phase-shift keying (PSK) and quadrature amplitude modulation (QAM). For example, the spatial modulation (SM) [13], [14] is the IM scheme that operates in the spatial domain, which activates one out of multiple transmit antennas in each symbol interval, and the modulated symbol is transmitted from the selected antenna. Moreover, the orthogonal frequency division multiplexing (OFDM) with IM (OFDM-IM) [15]–​[17] is the IM scheme in the frequency-domain, where a subset of subcarriers is activated in each OFDM frame, which conveys information, similar to the SM scheme. Most recently, in [18] the time-domain single-carrier (SC)-IM scheme was proposed, where a subset of time-domain symbols is activated for the additional information transmission. At the receiver, the SC-IM symbols are demodulated, based on the MMSE-based FDE algorithm. Note that the OFDM-IM [17] and the SC-IM schemes [18] tend to achieve a better BER performance than the conventional OFDM and SC counterparts, especially in a low-rate scenario.

Against this background, the novel contribution of this letter is that we propose a novel FDE-aided FTN system exploiting the time-domain IM concept, which is referred to as FTN-IM.¹ In the proposed FTN-IM scheme, a subset of time-domain symbol indices are activated, while other unactivated symbols are set to zero. The combination of the activated indices coveys additional information. Hence, as the explicit benefit of sparse FTN-IM signaling, the FTN-specific ISI effects are reduced, hence achieving a better BER performance than the conventional FTN scheme, while attaining the fundamental benefits of FTN scheme. We investigate a minimum Euclidean distance (MED) of the proposed FTN-IM signals, in order to show that FTN-IM scheme is capable of attaining a higher coding gain than the conventional FTN counterpart. Our simulation results demonstrate that the proposed FDE-aided FTN-IM scheme outperforms the conventional FDE-aided FTN system of [4] and [10] both in the additive white Gaussian noise (AWGN) and the frequency-selective fading channels.

SECTION II. System Model of the Proposed FTN-IM Signaling

A. The FDE-Aided FTN-IM Signaling

In this section, we introduce the system model of our FDE-aided FTN-IM scheme. Here, the AWGN channel is considered for the sake of simplicity. Also, the employment of a root raised cosine (RRC) filter, having an impulse response $a(t)$ and a roll-off factor $\beta $ , is assumed. The schematic of our FDE-aided FTN-IM transceiver is depicted in Fig. 1. At the transmitter, the total $L\cdot B$ information bits are modulated onto the SC-IM symbols $ \mathbf {s}\in \mathbb {C}^{LM}$ , using $\mathcal {P}$ -point amplitude and phase shift keying (APSK) per frame, according to [18]. The symbol vector $ \mathbf {s}$ is composed of $L$ subframes of $ \mathbf {s}= [ \mathbf {s}^{0}, \mathbf {s}^{1},\ldots , \mathbf {s}^{L-1}]^{T}$ , where each subframe contains $M$ symbols $ \mathbf {s}^{l} = [s_{0}^{l}, s_{1}^{l},\ldots , s_{M-1}^{l}]$ . Here, each IM subframe $ \mathbf {s}^{l}$ is modulated by $B=B_{1}+B_{2}$ information bits. More specifically, $K$ out of $M$ symbol indices in the $l$ th subframe are activated based on the first $B_{1}=\lfloor {\log _{2}\binom {M}{K}\rfloor }$ information bits of $ \mathbf {b}_{1}^{l}$ , while $K~\mathcal {P}$ -APSK symbols are modulated by the $B_{2}=K\log _{2}\mathcal {P}$ information bits of $ \mathbf {b}_{2}^{l}$ . Finally, the unactivated $(M-K)$ symbols are set to zeros. Here, in order to maintain the average transmission power to be unity, the activated $K$ symbols in each subframe are scaled by $\sqrt {M/K}$ . Then, a cyclic prefix (CP), which is the first $2\nu $ -length symbols of $ \mathbf {s}$ , is inserted to the end of $ \mathbf {s}$ . The total $(N+2\nu )$ -length CP-inserted symbols are input into the RRC shaping filter $a(t)$ , with the FTN symbol interval of $T=\alpha T_{0}$ . The time-domain FTN-IM signals are represented by\begin{equation} s_{\mathrm {F}}(t) = \sum _{\mathit {n}}s_{\mathit {n}}a(t-nT). \end{equation}

Fig. 1.

System model of the proposed FDE-aided FTN-IM scheme.

At the receiver, the received signals are matched-filtered by $a^{*}(-t)$ , in order to have\begin{equation} y(t)=\sum _{\mathit {n}}s_{\mathit {n}}g(t-nT) + \eta (t), \end{equation} where we have $g(t)=\int a(\tau )a^{\mathrm {*}}(\tau - t)d\tau $ and $\eta (t)=\int n(\tau )a^{\mathrm {*}}(\tau - t)d\tau $ . Moreover, $n(t)$ represents the complex-valued AWGN with zero-mean and noise variance $N_{0}$ . Then, the $k$ th sampled signal is represented by \begin{align} y_{k}=&y(kT) \\=&\sum _{\mathit {n}}s_{\mathit {n}}g(kT-nT) + \eta (kT) . \end{align} Note that due to the non-orthogonal property of $g(t)$ at each FTN sampling instance $kT$ , the noise component $\eta (kT)$ has a non-zero cross-correlation $E[\eta (mT) \eta ^{\mathrm {*}} (nT)]=N_{0} g((m-n)T)$ , hence inducing colored noises.

After removing the CP from the received signal block of (4), the symbols in the $l$ th subframe $\hat { \mathbf {s}}=[\hat { \mathbf {s}}^{0}, \hat { \mathbf {s}}^{1},\ldots , \hat { \mathbf {s}}^{L-1}]^{T}$ are estimated based on the low-complexity noise-whitening MMSE-FDE [9], [10]. Then, the transmitted information bits $( \mathbf {b}_{1}^{l}, \mathbf {b}_{2}^{l})$ in the $l$ th subframe are detected with the aid of ML detection as follows:\begin{equation} (\hat { \mathbf {b}}_{1}^{l}, \hat { \mathbf {b}}_{2}^{l}) = \mathop {\mathrm {arg\,min}} _{( \mathbf {b}_{1}, \mathbf {b}_{2})}\left |{\left |{\hat { \mathbf {s}}^{l} - \mathbf {s}|_{( \mathbf {b}_{1}, \mathbf {b}_{2})}}\right |}\right |^{2}. \end{equation} The detection complexity of (5) is an order of $\mathcal {O}(L\cdot 2^{B})$ , where we have $B=K\log _{2}\mathcal {P} + \lfloor {\log _{2}\binom {M}{K}\rfloor }$ .

Also, note that in the frequency-selective fading channel, the above-mentioned FDE and ML detection are carried out, in a similar manner to in the AWGN and the frequency-flat channels, as shown in [4].

The spectral efficiency $R$ [b/s/Hz] of the proposed FTN-IM signaling is formulated by \begin{equation} R = \underbrace {\frac {N}{N+2\nu }}_{\mathrm {loss\: of\: CP}} \cdot \underbrace {\frac {1}{\alpha (1+\beta )}}_{\mathrm { gain\: of\: FTN}} \cdot \underbrace {\frac {K\log _{2}\mathcal {P} + \lfloor {\log _{2}\binom {M}{K}\rfloor }}{M}}_{\mathrm {gain\: of\: IM}}. \end{equation} The coefficient $N/(N+2\nu )$ represents the overhead imposed by the CP insertion, although it is marginal for the scenario of a sufficiently high block length $N$ . Furthermore, the second and the third terms correspond to the throughput gain achieved by the FTN signaling and that by the IM, respectively.

In the IM schemes, such as the SM, the OFDM-IM and the SC-IM schemes, the achievable detection performance depends on the effects of an inter-channel correlation. In order to combat this limitation, we introduce the concept of symbol interleaving in the proposed FTN-IM scheme, similar to the conventional IM schemes [18], [20]–​[22]. Note that, unlike the conventional SC-IM scheme [18], the FTN-IM scheme has to rely on symbol interleaving, even in the AWGN channels, since FTN-specific ISI is imposed. In the Section IV, we demonstrate that the BER performance of the FTN-IM system improves with the aid of symbol interleaving, both in the AWGN and frequency-selective fading channels.

B. The Computational Complexity of Our FDE-Aided FTN-IM Receiver

In this section, we evaluate the detection complexity, which is imposed on the proposed FDE-aided FTN-IM receiver. As aforementioned, the receiver implements both the MMSE-FDE [9], [10] as well as the ML detection of (5). More specifically, as mentioned in [10], the complexity order of our MMSE-FDE is as low as $\mathcal {O}(N\log N)$ , similar to that operating in the uplink scenario of current cellular standards.

Furthermore, the complexity, imposed by ML detection of (5), is formulated by $\mathcal {O}(L\cdot 2^{B})$ , which becomes high, especially when the system parameters, i.e., $(M, K, \mathcal {P})$ and $L$ , are high. However, as will be shown in Section IV, the proposed FTN-IM scheme having the low system parameters, such as $(M, K, \mathcal {P}) = (4, 1, 4)$ , outperforms the conventional FTN scheme.² In such a scenario, the proposed FTN-IM receiver allows us to attain a high performance with a practically low complexity.

SECTION III. Evaluations of the Minimum Euclidean Distance

In this section, we investigate the MED of the proposed FTN-IM signaling, in order to evaluate its achievable performance limit, while providing the comparison with that of the conventional FTN signaling [3].

The Euclidean distance between two time-domain FTN-IM signals $s_{\mathrm {F_{1}}}(t)$ and $s_{\mathrm {F_{2}}}(t)$ is derived as follows [3]:\begin{align} d^{2}( \mathbf {s}_{i}, \mathbf {s}_{j})=&\int _{-\infty }^{\infty } |s_{\mathrm {F_{1}}}(t)-s_{\mathrm {F_{2}}}(t)|^{2} dt \notag \\=&\sum _{x}\sum _{z}e_{x}e_{z}^{*}g((x-z)T) = \mathbf {e} \mathbf {G} _{\mathrm {d}} \mathbf {e}^{H}, \end{align} where $ \mathbf {s}_{i}, \mathbf {s}_{j} \in \mathbf {s} $ are two different FTN-IM signal vectors, while we have $ \mathbf {e}= \mathbf {s}_{1}- \mathbf {s}_{2}$ . Furthermore, the matrix $ \mathbf {G}_{\mathrm {d}}$ is the Toeplitz matrix, whose $x$ th-row and $z$ th-column element is given by $g((x-z)T)$ . The MED of the proposed FTN-IM and the conventional FTN signals, which is defined by $d_{\mathrm {min}} = \mathop {\mathrm{ min}}\limits _{ \mathbf {s}_{1}- \mathbf {s}_{2} \neq 0} d( \mathbf {s}_{1}, \mathbf {s}_{2})$ , allows us to compare their achievable error-rate limit.

In Fig. 2, the MED comparisons between the proposed FTN-IM and the conventional FTN signaling are provided,³ where the FTN-IM parameters were set to $(M, K, \mathcal {P})=(4, 1, 4)$ , while considering the BPSK modulation for the conventional FTN scheme. The roll-off factor $\beta $ was given by $\beta =0$ , 0.3, and 0.5. Hence, the transmission rate of the two schemes is maintained to be the same. Observe in Fig. 2 that in each $\beta $ scenario, the proposed FTN-IM scheme exhibited a higher MED than the conventional FTN counterpart. More specifically, this advantage of the proposed FTN-IM scheme was explicit especially for the high-$\alpha $ region before the Mazo limit of the conventional FTN scheme. Furthermore, it was found that upon decreasing the roll-off factor $\beta $ , the advantage of the proposed FTN-IM scheme increased. Hence, it is expected that the proposed FTN-IM scheme is capable of attaining a lower error rate limit than the conventional FTN scheme.⁴

Fig. 2.

The effects of the packing ratio $\alpha $ on the MED of the proposed FTN-IM and the conventional FTN signaling. The FTN-IM parameters were set to $(M, K, \mathcal {P})=(4, 1, 4)$ , while considering BPSK modulation for the conventional FTN benchmark.

SECTION IV. Simulation Results

In this section, we provide our simulation results, in order to characterize the performance of the proposed FDE-aided FTN-IM scheme. Here, the conventional FDE-aided FTN scheme [4], [10] and the classic Nyquist-criterion-based scheme with BPSK were considered to be benchmarks, while noting that the performance of the Nyquist-criterion-based scheme corresponds to the no-ISI limit.

Fig. 3 shows the achievable BER performance in the AWGN channel. We considered the block length of $N=512$ , while the FTN parameters were set to $\alpha =0.9, 0.8, 0.7, 0.65$ , $\beta =0.5$ and $\nu =10$ for both the schemes. We employed the FTN-IM parameters of $(M, K, \mathcal {P})=(4, 1, 4)$ , and hence the spectral efficiencies of the FTN-IM and the conventional FTN schemes are the same for each $\alpha $ value. Note that the scenarios of $\alpha =1.0, 0.9, 0.8, 0.7$ and 0.65 corresponded to the spectral efficiencies of $R=0.67, 0.71, 0.80, 0.92$ and 0.99 [b/s/Hz], respectively. Furthermore, in Fig. 3 we also plotted the achievable BER corresponding to the Nyquist-criterion-based system. It was observed in Fig. 3 that the proposed FDE-aided interleaved FTN-IM scheme outperformed the conventional FDE-aided FTN counterpart in each $\alpha $ scenario. Furthermore, it was found that upon decreasing the $\alpha $ value, the performance gain of FTN-IM increased. More specifically, the performance gain of the interleaved FTN-IM system over the conventional FTN system was approximately 2.3 dB, 2.5 dB, 4.0 dB and 13.0 dB for $\alpha = 0.9, 0.8, 0.7$ and 0.65, respectively. This benefit was achieved because the FTN-induced ISI was effectively suppressed, owing to the sparse structure specific to FTN-IM signaling.

Fig. 3.

The achievable BER performance of the proposed FDE-aided FTN-IM and the conventional FDE-aided FTN systems in AWGN channel. The FTN parameters were set to $\beta =0.5$ and $\nu =10$ . Furthermore, the SC-IM parameter was set to $(M, K, \mathcal {P})=(4, 1, 4)$ which results in the same rate to the conventional FTN.

Furthermore, Fig. 4 shows the BER performance for the scenario of $\beta =0.3$ , while other system parameters were the same as those used in Fig. 3. The spectral efficiencies for $\alpha =1.0, 0.9, 0.8$ and 0.7 are $R=0.77, 0.82, 0.93$ and 1.06 [b/s/Hz], respectively. As seen in Fig. 4, the performance advantage of the proposed FTN-IM scheme over the conventional FTN benchmark scheme increased, upon decreasing the $\beta $ value from $\beta =0.5$ to 0.3.

Fig. 4.

The achievable BER performance of the proposed FDE-aided FTN-IM and the conventional FDE-aided FTN schemes in AWGN channel. The system parameters of the proposed FTN-IM scheme were set to $\beta =0.3, \nu =10$ and $(M, K, \mathcal {P})=(4, 1, 4)$ .

Finally, Fig. 5 shows the achievable BER of the proposed and the conventional schemes in the frequency-selective fading channel. The delay spread was set to $L=10$ , and each channel coefficient was randomly generated, according to the complex-valued Gaussian distribution of $\mathcal {CN}(0, 1/L)$ . The system parameters were set to $(\alpha , \beta , \nu ) = (0.8, 0.5, 15), (0.7, 0.5, 15)$ and $(M, K, \mathcal {P})=(4, 1, 4)$ . Observe in Fig. 5 that the performance gains of the proposed scheme were approximately 4.5 dB and 6.5 dB for the scenarios of $\alpha =0.8$ and 0.7, respectively.

Fig. 5.

The achievable BER performances of the proposed FDE-aided FTN-IM system with interleaving in the frequency-selective fading channel having the delay spread $L=10$ . The system parameters of the proposed FTN-IM scheme were set to $(\alpha , \beta , \nu ) = (0.8, 0.5, 15), (0.7, 0.5, 15)$ and $(M, K, \mathcal {P}) = (4, 1, 4)$ .

SECTION V. Conclusion

In this letter, we proposed the novel FTN signaling combined with the time-domain IM concept. The spectral efficiency of our FTN-IM signaling increases, since the FTN-specific ISI effects are effectively mitigated owing to the sparse structure of FTN-IM signaling. We investigated the MED of the proposed FTN-IM signaling, which ensured that the error-rate bound of FTN-IM is lower than that of the conventional FTN counterpart. Our simulation results demonstrated that the proposed FDE-aided FTN-IM scheme is capable of outperforming the conventional FDE-aided FTN scheme.