Fully Quadrature Subcarrier-Index Shift Keying for Efficient Underwater Acoustic Communications

A novel fully quadrature subcarrier-index shift keying orthogonal frequency division multiplexing (FQSISK-OFDM) modulation scheme for underwater acoustic communications (UWAC) is proposed in this paper. In the FQSISK, the data is conveyed by active and non-active OFDM-subcarriers indices. Moreover, the active odd and even subcarriers are marked with separable predefined dummy variables (DV). Herein, the DV <inline-formula> <tex-math notation="LaTeX">$\pm \delta $ </tex-math></inline-formula> (i.e., it can take any predefined value), and its quadrature <inline-formula> <tex-math notation="LaTeX">$\pm j\delta $ </tex-math></inline-formula> are exploited to distinguish between active even and odd subcarriers. Hence, the possibility of power leakage from active to non-active subcarriers is reduced, which minimizes the effect of severe UWAC inter-carrier-interface (ICI). Moreover, the spectral efficiency of the FQSISK hardly depends on the subcarriers grouping size, which reduces the proposed FQSISK decoder complexity significantly. The simulation results demonstrate the proposed FQSISK superiority, as it achieves higher spectral and energy efficiencies than the plain OFDM and the recently introduced OFDM index modulation (OFDM-IM) techniques.

made them non-practical for many underwater communication scenarios. Hence, UWAC is still considered the backbone of underwater applications [1]- [7], and many modulation techniques were introduced to mitigate the UWAC drawbacks above.
For example, the orthogonal frequency division multiplexing (OFDM) was exploited in [15] to compensate the ISI of UAC. Unfortunately, a long cyclic prefix (CP) should be added as overhead with the OFDM transmitted symbols to cope with the UAC long channel impulse response. Further, owing to the low propagation speed of acoustic waves and the wide-band nature of the UWC, a severe Doppler shift is generated [1], [10], [15]- [17]. Moreover, due to the acoustic band limitations, it becomes necessary to divide the acoustic bandwidth between as many OFDM number subcarriers as possible for generating a long OFDM symbol interval to cope with UAC wide multi-path delay spread [1], [16], [17]. Therefore, a high ICI results from the UWC Doppler shift, which degrades the overall system performance.
OFDM-IM introduced an improvement of the overall system power-saving and in average bit error rate (ABER) performance compared to the conventional OFDM, but its spectral efficiency (SE) was lower than the conventional OFDM spatially with higher-order modulations [22], [23], [25], [26], [29]- [33]. Hence, many research works were introduced to improve the OFDM-IM spectral efficiency. For example, OFDM with in-phase and quadrature IM (OFDM-IM-IQ) was introduced in [37] to improve the SE of conventional OFDM-IM system by applying the IM in both in-phase and quadrature domains. While, in [38], a generalized version of OFDM-IM was proposed to expand the indexing space of OFDM-IM. Moreover, in [25], [39], the dual-mode OFDM-IM and its generalized version were introduced, whereas different modes of modulated symbols were transmitted over both active and non-active subcarriers, which improves their SEs compared to the conventional OFDM-IM. Further, in [29]- [32], the multiple-mode OFDM-IM (MM-OFDM-IM) was proposed, in which all OFDM subcarriers were exploited in the transmission process of the modulated data, and the modulated data itself was generated from different constellations, while the permutations of multiple distinguishable constellations convey the IM bits. In [33], the layered-OFDM-IM (L-OFDM-IM) was proposed to improve the OFDM-IM spectral efficiency by dividing the subcarriers indices into different layers with overlapping subcarriers, while different modulation constellations were used to differentiate between the different layers.
Although, the aforementioned OFDM-IM techniques [22], [23], [25], [26], [29]- [33], [37], [39] can improve UWAC spectral efficiency, but they have three major drawbacks when they applied in UWAC. a) Its huge maximum likelihood (ML) decoder complexity, as its SE depends on the OFDM-subcarrier grouping size (N ), and the computational complexity (CC) of the ML is exponentially proportional to N , where N is the OFDM sub-block size. b) It still suffers from a high peak average power ratio (PAPR) owing to its QAM modulation, i.e., its energy efficiency (EE) is low. c) Owing to the UWAC Doppler shift, there will be a high possibility for power leakage from active to non-active subcarriers, which distorts the OFDM-IM sub-block framing structure and significantly degrades its performance.
Therefore, a novel index shift keying (ISK) modulation termed as fully quadrature subcarrier-index shift keying OFDM (FQSISK-OFDM) for UWAC is proposed in this paper. FQSISK is a class of OFDM-IM but with higher reliability and low decoding complexity, which makes it very suitable to fulfill the UWAC PHY layer design requirements. Considering the above-mentioned concepts, the paper's main contributions are as follows: • A novel FQSISK-OFDM modulation scheme for UWAC is proposed. In the FQSISK, two different flexible and extendable modulation constellations are proposed for even and odd OFDM subcarriers. Hence, it reduces the possibility of power leakage from active to nonactive OFDM-subcarriers, which increases the FQSISK immunity against the severe ICI resulting from the Doppler shift. Thus the overall system performances are improved.
• A mathematical analysis for the FQSISK upper bound average block error rate (ABLER) is driven and discussed in Sec. III. Further, the driven formula is validated by using extensive Monte-Carlo simulation.
The paper remainder sections are arranged as follows: FQSISK system model is discussed in Section II. Section III addresses FQSISK performance analysis. Simulation results are given in Section IV. Finally, conclusion is given in Section V. Figure.1, shows the general structure of the proposed FQSISK-OFDM transmitter. For each OFDM symbol, the incoming data B is divided into G groups of bits, each group G i contains P bits, with B = GP. The groups above are exploited to construct the OFDM sub-blocks with subcarriers number N = N F /G, where N F is the Fast Fourier Transform (FFT) length.

II. PROPOSED FQSISK-OFDM SYSTEM MODEL
Every group of P of bits is exploited to jointly select the active subcarriers pattern (ASP) as well as their predefined dummy variables (DV) values from E and O for active even and odd subcarriers, respectively.
Unlike OFDM-IM [18], [22], and the recently introduced hybrid OFDM-IM with subcarrier number modulation  (OFDM-HNIM) [40], the index shift keying (ISK) concept is exploited in the proposed FQSISK, where all possible combinations of the active and non-active subcarriers are exploited along with the proposed DV constellation to transmit more data. Hence, it achieves higher SE than its OFDM-IM counterparts.
Different DV constellations designs are shown in Table 1, where E / O are the even/odd subcarrier constellations with size of i = i, i ∈ [1,2,3].
Two examples are given here to FQSISK working mechanism. The first example of the FQSISK working mechanism is shown in Table 2. The example uses 2 , N = 3, and P = 4 bits, i.e., with a cardinality of X = 2 P , where X is the cardinality of the set that contains all the possible active subcarriers indices (ASI) with different E and O realizations. However, to jointly select the ASI ( G i ) and the DV values for those ASI (d G i ) for the group of subcarriers G i , it is assumed that the subcarrier with index S i 1 in Table 2 is an even subcarrier. The aforementioned assumption is presumed as in the proposed FQSISK two different DV modulation constellations are used for even and odd subcarriers, i.e., E and O , respectively.
Hence, if the incoming P = [1001], then the 10 th state in Table 2 will be assigned to G i , i.e., X ( G i , d G i ) = [δ jδ 0] ∈ X , with active subcarriers indices of G i = [i 1 , i 2 ] ∈ G i , and d G i = [δ jδ] are the DV assigned for those active subcarriers.
Another example for the proposed FQSISK working mechanism is given in Table 3. The example uses 1 , N = 4, hence, P = 4 bits. Likewise, if the incoming P = [1000], then the 9 th state in Table 3 will be assigned to The examples above show the flexibility of FQSISK, whereas different active and non-active subcarriers realizations (states) are used to encode (convey) the transmitted data. Moreover, the state of non-active subcarriers is used to convey/encode information, as shown in Table 2 and Table 3. Furthermore, the FQSISK can work with any N , i.e., N not restricted to be 2 i as in OFDM-IM [18], which makes the proposed FQSISK more practical than its counterparts. As in real-world scenarios, OFDM subcarriers is generally divided into two parts; the data part and the pilot part. The pilot subcarriers are used for channel estimation, and their number and location are varied according to the channel. Moreover, in an underwater acoustic communication scenario, the channel estimation uses a larger number of pilots compared to free space communication [16], [17]. Therefore, the remaining data subcarriers can be any number (not 2 i ).
After selecting the indices of the active subcarriers and the DV values for all the groups G, the resultant frequency domain (FD) vector is given by: where P t and T are the total FQSISK-OFDM symbol power and the average number of active subcarriers, respectively, and The conventional steps for constructing the OFDM symbol are then applied, as shown in Fig.1. Whereas IFFT with N F points is applied on the FD vector X , then a CP of a length L CP is added to the resultant time domain (TD) vector x to avoid inter-symbol-interference (ISI). However, L CP should be longer than the channel impulse response (CIR). Afterward, the OFDM symbol is converted from parallel to serial and sent over UAC.
The number of transmitted bits P i achieved by any subblock with fixed active subcarrier number T = i out of N is given by [38], [41]: where . , . . , and are the floor operator, binomial coefficient, and the size of odd/even subcarrier constellation (i.e., both even and odd constellations have the same size), respectively.
The number of active subcarriers in FQSISK is varied from the case where no active subcarriers to the case where all N subcarriers are active. Thus, total number of bits P conveyed by N subcarriers in a group G i is given by: In FQSISK size is designed to cope with different UWAC channel conditions, where the size of 1 = 0.5 BPSK, 2 = BPSK, and 3 = 1.5 BPSK in Table. 1, are selected to be used with poor, medium, and good UAC conditions. However, for very short range UWAC with very good channel conditions, E and O can be jointly used regardless of the subcarriers being odd or even, which almost duplicates the SE of the FQSISK as follows.
The FQSISK receiver is the reversal of the transmitter blocks, which contains CP elimination, FFT, and FQSISK detection. However, after applying the CP and FFT steps, the resultant received FD vector is given by: where H ∈ C N F ×N F is diagonal channel matrix with N F channel state information in its diagonal, and C is the complex space. While n, is the frequency domain of the additive white Gaussian noise (AWGN) vector, n(k) ∼ CN (0, N 0 F ). Finally, optimal ML is applied on (6) for every FQSISK sub-block G i to retrieve the transmitted data as followŝ where, Y G i is the frequency domain received signal with N subcarriers, and H G i is N × N diagonal sub-channel matrix associated with the G i .
Moreover, the BPSK log-likelihood ratio (LLR) decoder [22] is used here as a low CC decoder for the FQSISK. However, the LLR [22] is adapted to cope with different used with FQSISK.
In this context, when FQSISK uses 2 , LLR is adapted to individually detect the even and odd subcarriers as follows where a j = While, the adapted LLR decoder for FQSISK with 3 is given as 46978 VOLUME 9, 2021 Please note that, sim-equal is used in (10) and (11) as we drop the constant term ln T N −T ) form their right hand sides, because for any sub-block (G i ) average the number of active subcarriers T = 0.5N .
Moreover, the simple energy detection (ED) with a hard threshold γ = |δ| 2 is used with FQSISK when 1 is used, which is considered the simplest linear version of BPSK LLR [22].

III. FQSISK PERFORMANCE ANALYSIS
In this section, the SE, EE, CC, and error performances of the proposed FQSISK scheme will be discussed.

A. FQSISK SPECTRAL EFFICIENCY
Since the achievable data rate of the proposed FQSISK is dependent on the block based mutual information where H(X G i ) and H(Y G i /X G i ) are the entropy of X G i and the conditional entropy of Y G i given X G i , respectively. Thus, by assuming lossless data transmission over a very good channel or at a very high signal to noise ratio (SNR), H(Y /X ) = 0 in (12). Therefore, (3) and (5) can be used as an expression for the FQSISK achievable rate. With size of ∈ [1, 2, 3] and for even N numbers, (3) and (5) can be approximately rewritten as It is clear from (14) that η FQSISK is not dependent on N , it depends on the size of . As Table 1 shows, seems to be as the classical BPSK. However, there are two major merits of the proposed over the BPSK. a) Two different constellations E and O are used with FQSISK for even and odd subcarriers, respectively. Which mitigates the effect of the power leakage between active and non-active subcarriers. Thus it increases the proposed FQSISK immunity against the ICI effect of UWAC. b) Unlike BPSK, the constellation size of the FQSISK is not restricted to be 2 i any more (i.e., it is flexible), where i is a positive integer number. The size of can be any i ≥ 1 as shown in Table 1.

B. FQSISK ENERGY EFFICIENCY
The concept of energy-saving factor (ESF) was used in [40] to indicate the OFDM-IM schemes energy-saving compared to the conventional OFDM, and it is given by.
Based on ESF, the energy efficiency (EE) ratio (EE r ) of any OFDM-IM scheme compared to EE of the conventional OFDM is given by [40].
Thus, the EE r of the FQSISK is equal to 2 × η FQSISK since on average T 0.5N .
Since, the SE of FQSISK is not dependent of N (III-A), thus at same SE, FQSISK will use lower N and modulation order compared to other OFDM-IM techniques. Therefore, the FQSISK decoder CC is significantly reduced, especially for the ML decoder case. This also reflects on the overall systems energy consumption. In other words, the proposed FQSISK is low computationally complex and more energy efficient than its OFDM-IM counterparts

D. FQSISK ERROR PERFORMANCE
The upper bound error rate for the FQSISK with ML decoder will be given here. However, due to OFDM's framing structure, it is more convenient to use the block-based error rate (i.e., error on the G i sub-block) rather than bit error rate. Thus, the block level pairwise error probability (PEP) is used. However, the Q−function formularization of the conditional block PEP can be written as [43]   where , and T i is the average number of active subcarriers in the group G i . Furthermore, the Q−function in (17) can be approximately written in exponential form as [40], [43] where κ 1 = 1/12, κ 2 = 1/4, τ 1 = 1/2, and τ 2 = 2/3. By using (18), (17) can be written as where (ω) is ω element in the diagonal of H that associated However, the marginal block base PEP can be found by taking the expected value of (19) over H as follows [43] With a very big amount of data bits the equiprobable can be assumed, then the ABLER upper bound is given as [43] where ( G i ,ˆ G i ) is the error in data bits due to erroneously detectingˆ G i instead of G i .
The simulation results were obtained based on the following parameters, the DV is selected from a normalized  Table. 4.
The systems were tested with a UAC generated by BELLHOP ray-tracing simulator using East Mediterranean Sea average sound-speed profile given in [8], whereas the receiver (RX) and transmitter (TX) were placed at 20 and 30 meters (m) depth, respectively, with average water depths of 55m and 300m distance between RX and TX.
However, the channel paths with a maximum power delay ≤ 15 ms (i.e., maximum channel impulse response length = 15 ms) are selected from the generated power delay profile as shown in Fig. 2. Moreover, based on the sound-speed profile [8] the relative velocity between TX and RX is set to υ = 0.1 m/s and the average sound propagation velocity is c = 1515 m/s, hence the Doppler shift α = υ c = 6.6 × 10 −5 . The SE and of the FQSISK with N = 2 compared to the SEs of OFDM, OFDM-IM (with T = N /2), and OFDM-HNIM (Eq. (14) in [40]) at very high signal to noise ratio (SNR) are listed in Table 5.
As Table 5 shows, the SE of FQSISK with 2 and 3 is superior to SE of OFDM-IM with BPSK and QPSK regardless the value of N . Compared to OFDM with BPSK, FQSISK with 2 achieved higher SE, but the same SE is fulfilled when the OFDM and FQSISK used QPSK and 3 , respectively. Moreover, FQSISK and OFDM-HNIM achieved same SE when OFDM-HNIM used N = 4, while OFDM-HNIM achieved a low SE improvement over FQSISK when OFDM-HNIM used N = 8. However, OFDM-HNIM's SE improvement came up with a very high computational complexity (CC) and degradation on its ABLER.
In this context, Fig. 3 shows achievable rates of FQSISK with 3 and N = 2 compared to OFDM, OFDM-IM (with T = 3, N = 4), OFDM-GIM1 (with E = 1, 2, 3, N = 4), and OFDM-HNIM (with N = 4) when QPSK is used. In Fig. 3, the achievable rate is calculated by multiplying the  SE for a given OFDM-IM scheme with ABLER subtracted from one [40]. As Fig. 3 demonstrates, at small and medium values of SNR, the proposed FQSISK achieves better achievable rate than its counterparts, which demonstrates its SE superiority.
Moreover, with N = 4 and BPSK the value of EE r (16) for the OFDM-IM, OFDM-HNIM, and FQSISK with respect to OFDM, are 2, 3, and 3, respectively. Thus, FQSISK archives better EE r than OFDM-IM, while it achieves the same EE r of OFDM-HNIM but with lower N or lower CC.
Further, the peak average power ratio (PAPR) of the proposed FQSISK with 3 compared to conventional OFDM and OFDM-IM (N = 4, T = 3) with QPSK is shown in Fig. 4. The FQSISK achieved an average PAPR reduction of 0.5 dB compared to its counterparts.  1, 2, 3, N = 4). Also, FQSISK outperforms OFDM-HNIM, and OFDM-GIM1 by on average 3.4 and 4.5 dB saving, respectively. Moreover, the ABLER upper bound (21) for FQSISK with 2 is tested and verified against SNR as shown in Fig. 5. Likewise, the ED decoder with |δ|/2 and the LLR for 2 (8) and (9) are tested.  The FQSISK with ED and LLR decoders have on average 0.8 dB degradation compared to its ML decoder. However, the ABLER performance of FQSISK with ED and LLR still outperform their counterparts with ML decoder.
It is worth noting that, ABLER margins between FQSISK and other schemes becomes clear and increases after SNR = 5 dB. When SNR increases, the effect of the FQSISK odd and even subcarriers modulation feature (i.e., using two different separable constellations E and O for even and odd subcarriers, respectively) starts to appear. However, at lower SNR, the effect of this feature is concealed due to noise power.   To show the robustness of the FQSISK against ICI of UWAC, FQSISK transmitted TD signal is exposed to an artificial carrier frequency offset (CFO) [44], i.e., y CFO (u) = y(u).e 2jπuµ N F , where µ is the normalized fractional CFO. Figure. 7 shows the FQSISK ABLER performances compared to OFDM-IM, OFDM-GIM, OFDM-HNIM, and the Enhanced DCT-OFDM system with index modulation (EDCT-OFDM-IM) [45] when a CFO with µ = 0.05 is applied. As the results show, the FQSISK is more robust against ICI than its counterparts, and it outperforms the EDCT-OFDM-IM (8,2,1024) with BPSK, which is considered one of the most robust IM schemes against ICI [45]. Moreover, When subcarrier inter-leaver [23] is used with FQSISK (IFQSISK) its robustness to ICI is improved as shown in the Fig. 7.
The above results confirmed that the FQSISK is able to fulfill UWAC requirements. As, at the same achievable rate, FQSISK can give from 3 to 5 dB power-saving with lower PAPR. Thus, FQSISK has higher EE, lower CC, and more robust against ICI than its counterparts.

V. CONCLUSION
A novel and highly efficient OFDM index modulation scheme with flexible and extendable cardinality was proposed in this paper for UWAC. The proposed FQSISK handles the main drawbacks of multi-carrier OFDM with index modulations in UWAC scenario. Whereas two different separable constellations were used to distinguish between the active even and odd subcarriers. Thus, the possibility of power leakage from active to non-active OFDM-subcarriers is reduced, which minimizes the severe ICI effect of UAC. Moreover, FQSISK spectral efficiency is not dependent on N , which significantly reduced its decoder CC. The simulation result proved that the proposed FQSISK is an efficient multi-carrier modulation scheme that can fulfill the UWAC requirements.