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Mary nonlinear sine chirp spread spectrum for underwater acoustic communication based on virtual timereversal mirror method
EURASIP Journal on Wireless Communications and Networking volume 2021, Article number: 112 (2021)
Abstract
Linear chirp spread spectrum technique is widely used in underwater acoustic communication because of their resilience to high multipath and Doppler shift. Linear frequency modulated signal requires a high spreading factor to nearly reach orthogonality between two pairs of signals. On the other hand, nonlinear chirp spread spectrum signals can provide orthogonality at a low spreading factor. As a result, it improves spectral efficiency and is more insensitive to Doppler spread than the linear counterpart. To achieve a higher data rate, we propose two variants (half cycle sine and full cycle sine) of the Mary nonlinear sine chirp spread spectrum technique based on virtual timereversal mirror (VTRM). The proposed scheme uses different frequency bands to transmit chirp, and VTRM is used to improve the bit error rate due to high multipath. Its superior Doppler sensitivity makes it suitable for underwater acoustic communication. Furthermore, the proposed method uses a simple, lowpower bank of matched filters; thus, it reduces the overall system complexity. Simulations are performed in different underwater acoustic channels to verify the robustness of the proposed scheme.
Introduction
Underwater acoustic communication in the recent past has attracted many researchers because of its application going beyond military uses such as communication between autonomous underwater vehicles (AUVs) for data collection and ocean exploration [1, 2]. Submarinetosubmarine or submarinetoAUV communication can be achieved through underwater acoustic communication. The underwater acoustic channel is quite challenging compared to traditional terrestrial communication because of the low sound speed of nearly 1500 m/s, high multipath environment, inherit timevarying nature of the ocean, and vehicles' movement adds ups the Doppler effect [3], therefore not only physical layer design is difficult but also designing routing protocol for underwater wireless sensor networks (UWSNs) is also quite challenging [4]. To counter these challenges, researchers have raised many waveforms and modulation schemes. The frequency shift keying was used in traditional hydroacoustic communication. Still, to compensate for multipath interference and high Doppler spread, researchers have raised many spread spectrum techniques, which include Direct Sequences Spread Spectrum (DSSS) and Frequency hop spread spectrum(FHSS) [5]. Linearly modulated signal as a carrier for digital communication was first proposed by the author in [6]. Since then, it is widely studied and implemented in wireless systems [7]. Spread spectrum linear frequency modulation (LFM) signal has gain popularity in lowpower and longrange wide area networks (LoRA) [8]. Simultaneously, in UWA communication, this technique is used for decades [9, 10]. The chirp spread spectrum can be categorized into two types: (1) binary orthogonal keying (BOK) and (2) direct modulation (DM). The BOK uses linear upchirp and downchirp to transmit data [11]. In contrast, DM uses chirp signal for spreading purpose along with different modulation schemes [12, 13]. The linear chirp BOK modulation by its nature is resilient to multipath and immune to Doppler shifts [14]; therefore, it is the right candidate for robust hydroacoustic communication, unlike multicarrier modulation, which requires high complexity receiver. The main advantage of the LFM signal is that it is easy to implement and have a straightforward matched filterbased receiver. The nonlinear frequencymodulated signals are widely used in radar [15] and sonar applications [16]. Many nonlinear waveforms are available for use as a carrier for UWA communication, such as exponential, log, quadratic, and trigonometric chirps. The authors in [17] have used quadratic and exponential chirp for multiple access users. The result shows that it can reduce multiple access interference under a specific chirp rate. A multirate hyperbolic nonlinear chirp spread spectrum proposed in [18] shows better performance than the conventional CDMA system. Many nonlinear chirps have high spreading factor and are nonorthogonal. In contrast, sine chirp signals can be used in many communication and related fields, including ultrasonic positioning [19]. The parameter estimation of sine chirp signal is proposed by the researcher in [20] while, to determine time–frequency bandwidth product authors in [21] proposed discrete sinusoidal frequency modulation transform. Sine chirp as a carrier is implemented for conventional terrestrial communication by the authors in [22], 23. The results show that BOK full period sine chirp is orthogonal at low spreading factor. However, its value of crosscorrelation between the pair of complete sine cycle at nonzero time shift is high; as a result, its performance degrades.
The Mary technique is traditionally used in singlecarrier as well as multicarrier communication to increase the data rate. The Mary chirp system can be implemented by using various techniques; it can be employed by transmitting in different frequency subbands, using different start/stop frequency, also known as pulse positioning, and using different chirp waveforms in the same band. However, in underwater acoustic communication, the bandwidth is limited; therefore, higherorder modulation in different frequency bands cannot be achieved easily. On the other hand, Mary modulation can be detected using coherent or noncoherent approaches [24]. The authors in [25] employed DPSK and different chirp rates for transmitting Mary symbols. In contrast, rake receivers were used to improve the overall performance of the system. However, the performance of the system is degraded as multipath increases. Marybased cyclic shift keying spread spectrum using VTRM presented in [26] requires a high spreading sequence for a higher data rate. The Mary linear chirp spread spectrum proposed in [27] used matched filter to detect chirp signals. However, the linear chirp signal's main problem is that it requires high bandwidthtime (BT) to achieve orthogonality between two chirp signals. As a result, it requires a large bandwidth, which is one of the valuable resources for any communication system, especially in UWA communication, because bandwidth is limited. To increases the data rate of the underwater communication system and to solve the problem of high SF, spectral efficiency, and intersymbol interference caused by underwater acoustic channels, Mary Sine Chirp Spread Spectrum (MSCSS) based on VTRM is proposed in this paper. MSCSS is a combination of two techniques used in UWA communication. The nonlinear sine upchirp and downchirp is used to represent data bits, and Mary technique is utilized to achieve a higher bit rate by transmitting at different frequency bands, whereas correlators are used to detect the symbols. In contrast, VTRM implementation helps improve the bit error rate caused by the multipath environment.
The rest of the paper is organized as follows: Sect. 2 introduces the analysis of linear and nonlinear chirp signals. Section 3 presents the chirp spread spectrum concept using the Mary technique, whereas the system model of the Mary sine chirp spread spectrum is given in Sect. 4. The channel estimation using matching pursuit and virtual timereversal mirror methods is provided in Sect. 5 and Sect. 6, respectively. Section 7 validates the proposed scheme's simulation results and Sect. 8 finally concludes the research article.
Analysis of linear and nonlinear chirp signal
Linear chirp signal
In traditional chirp, signal frequency increases linearly (upchirp) and decreases (downchirp) over a given time period. The LFM signal can be expressed as
where \(A\) is the amplitude of the chirp, \(f_{o}\) is the starting frequency, \(T_{LC}\) is the chirp duration, and \(\mu\) is the frequency sweep rate and can be defined as \(\mu = B/T_{LC}\) where \(B\) is the bandwidth and can be defined as \(B = f_{{{\text{final}}}}  f_{{{\text{initial}}}}\). When the bit “0” is to be sent, the function \(c_{0} (t)\) that is upchirp is transmitted, and when bit “1” is to be sent, function \(c_{1} (t)\) downchirp is transmitted. The matched filterbased receiver is used for detection. The expression for the autocorrelation and crosscorrelation in [28] and it can be given as,
where \(C(x)\) and \(S(x)\) are defined as the Fresnel function. Figure 1 shows that the output function of a matched filter has most of its energy existing from \( 1/B \le \tau \le 1/B\) of period, and the peak of the matched filter is amplified up to \(\sqrt {T_{LC} B}\). This pulse compression property of LFM makes it resilient to multipath and Doppler shifts. The crosscorrelation coefficient of LFM is defined by
Figure 2 shows that the crosscorrelation coefficient \(\rho_{l}\) depends upon the product of time period of linear chirp \(T_{LC}\), and the bandwidth \(B\) of the chirp signal. As \(T_{LC} B\) increases, \(\rho_{l}\) decreases, but it never reaches 0, even if the \(T_{LC} B\) is equal to 100. Therefore, linear chirps are not entirely orthogonal, even at a high \(T_{LC} B\) value and can be considered quasiorthogonal [29].
Nonlinear chirp signal analysis
There are many candidates of nonlinear chirps as a carrier, but one of the preferable contenders for UWA communication can be sine chirp because of its orthogonality between the pair of chirp under certain conditions. The sine chirp can be mathematically written as
where \(T_{S}\) is the duration of the sine chirp, and \(\eta\) represents the number of cycles. For example, if \(\eta\) is set to “1” in Eq. (5), halfcycle sine chirp is obtained, whereas if \(\eta\) is set to “2” in Eq. (5), fullcycle sine chirp is obtained. The autocorrelation property of the sine chirp is similar to the linear chirp signal and can be expressed using the equation.
where \(\theta = \frac{\pi \eta }{{2T_{s} }}\), \(k_{{s_{1} }}^{m} = \frac{{2BT_{s} }}{\eta }\cos \left( {\frac{\pi \eta }{{2T_{s} }}\tau } \right)\), \(k_{{s_{2} }}^{m} = \frac{{2BT_{s} }}{\eta }\sin \left( {\frac{\pi \eta }{{2T_{s} }}\tau } \right)\) and \(J_{2n}\) is the first kind Bessel function. In Eq. (6), the first and third terms have a negligible effect on the autocorrelation coefficient. Hence Eq. (6) is written as
The following equation gives the crosscorrelation coefficient of pair of sine chirp
where \(\theta = \frac{\pi \eta }{{2T_{s} }}\), \(k_{{s_{1} }}^{m} = \frac{{2BT_{s} }}{\eta }\cos \left( {\frac{\pi \eta }{{2T_{s} }}\tau } \right)\), \(k_{{s_{2} }}^{m} = \frac{{2BT_{s} }}{\eta }\sin \left( {\frac{\pi \eta }{{2T_{s} }}\tau } \right)\) and \(J_{n}\) is the first kind Bessel function. The crosscorrelation coefficient at time shift \(\tau = 0\) can be simplified as in [30] and given by the equation below.
Figure 4 shows that the crosscorrelation coefficient and bandwidth time product of nonlinear sine chirps (half and full cycle) are periodically zero; hence, they are entirely orthogonal at the specific condition. For example, the halfsine period has a crosscorrelation coefficient zero at various periodic bandwidth–time product points, including 4.4 sHz, 9 sHz, and 20 sHz, while the full sine period has zero crosscorrelation coefficient at bandwidth–time 5.5 sHz, 8.65 sHz, and 18 sHz. In contrast, the LFM signal never reaches zero. Figure 3a shows that the autocorrelation between the pair of one complete sine cycle and the halfsine cycle is similar to the linear chirp. Still, side lobes of both the nonlinear chirps are greater than linear counterparts, which can be reduced using any side lobe reduction scheme. Figure 3b verifies the relationship between the crosscorrelation coefficient and bandwidth time product of nonlinear chirps. The crosscorrelation between the pair to full and halfperiod sine chirp at 18sHz and 20sHz in Fig. 3b shows that at time shift (\(\tau\) = 0) the value of crosscorrelation is zero for both cycles. However, the full period sine chirp has a large value at non zero time shift, resulting in false detection in the UWA dispersive channel. As a result, the overall performance of the system can be affected. Similar results can be obtained for BT's different values; as the value of BT is decreased, the side lobe of autocorrelation increases for both linear and nonlinear chirp signals (Fig. 4).
Mary chirp spread spectrum
Mary chirp spread spectrum can be implemented by transmitting upchirp and downchirp in two different frequency bands, as shown in Fig. 5. The main issue with linear chirp is that it requires high BT products to achieve acceptable performance. As the number of chirps are increased to increase the data rate, the system's overall performance is worsening. The sine chirp is used in this paper, for which the pair of chirp is orthogonal at low SF. In Fig. 6, the time–frequency graph of 8ary sine chirp at four different frequency bands is shown. When the symbol to be transmitted is “000”, the sine chirp in the first frequency band with a positive gradient is transmitted similarly. If “001” is transmitted, the frequency band is the same but having a negative gradient. As a result, there is a 180 phase shift in sine chirp, as shown in Fig. 6. For the next two symbols, the frequency band ranges from \(f_{2}\) to, \(f_{3}\) and upchirp and downchirp is transmitted according to the symbol assign against it. Similarly, the last two symbols are assigned upchirp and downchirp from frequency \(f_{4}\) to \(f_{5}\)_{.} Similarly, the halfsine cycle can also be represented using a time–frequency graph.
System model of MSCSS
The block diagram of the Mary sine chirp spread spectrum is shown in Fig. 7. First, the stream of serial data bits \(x(n)\) is converted from serial to parallel, then chirp is generated by the chirp generator according to the given symbol; for example, if the symbol to be transmitted is “000”, then upchirp \(c_{s0} (t)\) is generated in the frequency band 1 as shown in Fig. 7, then it is pass through the hydrophone for transmitting it through an underwater acoustic channel in the presence of Additive White Gaussian Noise (AWGN). The transmit signal is received by the transducer and passed through the bank of matched filters. The matched filter works on the principle of correlation. The received signal \(y(t)\) is multiplied with a timeshifted version of the initially generated chirp \(c_{s0} (t  \tau )\)(in this case) and integrated over the chirp period \(T_{s}\). The matched signal gives maximum output while all other unmatched output has low values. As a result, the maximum value detector determines the given chirp signal, and the corresponding symbol is mapped against it. Finally, the output data is received by converting the given symbols from parallel to serial.
Channel estimation
The channel estimation is one of the requisite components of VTRMbased systems. Its performance is highly dependent on how accurately the channel is estimated. The sparse nature of the UWA channel results in a considerable delay spread with minimal nonzero values. Compressed sensing algorithms are widely used to solve the problem of sparsity [31]. In many underwater acoustic communication systems, compressed sensing algorithms are used. In [32], a compressed sensing algorithm and equalization are used to evaluate orthogonal signal division multiplexing performance. In contrast, matching pursuit helps resolve the problem of sparse channel estimation. It is widely used due to its robustness and computational efficiency [33]. Matching pursuit along with VTRM used in [26, 34], 35 proves to be very useful. The channel is estimated by sending a training signal through a sparse UWA channel. The training signal received can be given by the following equation
Let \(y(n)\) is the received training signal and \(l\) its length, where the number of channel taps. The above equation can be rewritten as
It can be concluded from the above equation that most of the channel taps are either zero or have minimal values.
The iteration process estimates the channel until the given threshold is achieved. Initially, the received signal \(Y\) is considered as a residual \(\Upsilon_{0}\) and \(\Upsilon_{0}\) is appropriately matched with the column of \(X\), which is denoted by \(x_{1}\). If the threshold is not achieved, the irritation continues. At m^{th} iteration, the residual \(\Upsilon_{m  1}\) for which the \(X\) has the maximal rankone projection can be expressed as \(x_{m}\) and it is given by the following equation
\(X_{k}\) contains the column of \(X\) and the superscript \(H\) is Hermitian Transpose. The value of \(\mathop {h_{m} }\limits^{ \wedge }\) which can be defined as the components h associated with \(X_{k}\) can be computed as
Finally, the residual vector can be calculated as
Virtual timereversal mirror method
The virtual timereversal mirror method is widely used for equalization in many types of communication channels, including underwater acoustic communication channel [36]. VTRMbased equalization had proven to be effective in underwater acoustic single carrier chirp spread communication. The authors in [37] proposed orthogonal chirp carrier, and to mitigate the effect of inters symbol interference VTRM is used. Moreover, VTRM proves to be useful in multiuser chirp spread spectrum communication proposed in [38]. Furthermore, VTRM is also used in timevarying underwater acoustic communication channel [39, 40].
The idea behind the working principle of the VTRM technique is to estimate the channel impulse response by processing the received training signal. The authors in [41] showed that the performance of VTRM is dependent on the use of appropriate training signals and the duration of the time window. The VTRM system design is shown in Fig. 8; first, the estimated channel impulse response \(h^{^{\prime}} (t)\) is timereversed \(h^{^{\prime}} (  t)\), and then it is convolved with the received signal [42].
Let \(x_{r} (t)\) be the transmit signal passed through the UWA channel can be given as
Then after virtual time reversal, the received signal can be expressed as
The above equation \(h(t) \otimes h^{^{\prime}} (  t)\) is VTRM based on equalized channel. It is a concurrence between the real channel impulse response and estimated channel impulse response. If the channel is estimated precisely, the direct path's energy concentration is much greater than the other reflected paths in the matched channel. As a result, it not only reduces intersymbol interference but also improves the SNR.
Results and discussion
Simulation using BELL HOP ray tracing algorithm
The parameters used for the simulation are shown in Table 1. The simulation setup of the UWA channel generated using the BELLHOP ray tracing algorithm is shown in Fig. 9. The actual impulse response of a channel and estimated channel response when the distance between transmitter and receiver is 1 km is shown in Fig. 10.
The simulation results in Fig. 11 depict that when the chirp period is 10 ms and Spread factor is 9sHz for half cycle sine (HCS) chirp and LFM, whereas 8.65sHz for full cycle sine (FCS) chirp without channel estimation and VTRM, the BER of HCS chirp is zero after − 4 dBs. For the case of FCS chirp and LFM, the BER is zero after − 2 dB. The UWA channel exhibits high multipath. Consequently, the system's overall performance is affected, but using VTRM improves the bit error rate. As shown in Fig. 11, using the VTRM method, all three types of Chirps' performance is improved while HCS Chirp outperforms the other two types. To investigate further, the BT was decreased from 9sHz to 4.4sHz for HCS chirp and LFM, while for FCS chirp, it was decreased from 8.65sHz to 5.5sHz as at the mentioned values, nonlinear chirps are orthogonal. The simulation results shown in Fig. 12 indicate that as the SF is decreased, linear chirp performance deteriorates compared to nonlinear chirp.
To increase the bit rate, the chirp period was decreased from 10 to 5 ms. The result shown in Fig. 13 indicates that the performance of the nonlinear sine full cycle is better than the other two types. In contrast, at low SF the full cycle and halfcycle have similar performance but outperform the linear chirp, as illustrated in Fig. 14. The simulation results validate that the MSCSS solves the problem of increasing the information rate at low spreading factor and improving the system's overall performance influenced by the high multipath due to underwater acoustic channel by using the VTRM technique.
Simulation using watermark reply channel
The watermark replay channel is a timevarying UWA channel; it is widely used to evaluate the performance of different types of the modulation scheme of UWA communication. The watermark channel uses the equation below to evaluate the performance of the system.
where \(y(t)\) is the output of the timevarying UWA channel recorded impulse response \(\mathop h\limits^{ \wedge } (t,\tau )\) of the real channel \(\mathop h\limits^{{}} (t,\tau )\), while \(x(t)\) is the input signal and \(n(t)\) is the noise. The watermark offers many different types of channels. The channel used in this simulation is NOF1 (Norway Oslofjord). The details of the simulation parameters are shown in Table 2. It is to be noted that both the transmitter and receiver are static and timevarying Doppler is compensated using resampling. The impulse response of the channel is shown in Fig. 15. The performance of the HCS and LFM chirp was evaluated at BT 9sHz and 4.4sHz, while the performance of FCS at 8.65 sHz and 5.5 sHz. The results in Fig. 16 shows that the performance of both the nonlinear chirps are better than the linear chirp, but NOF1 channel requires higher SNR to achieve satisfactory performance compared to the BELLHOP channel. In addition, VTRM is used to improve the performance of the system. Moreover, the performance of the linear and nonlinear chirp is also evaluated at 5 ms and SF 9 sHz for HCS and LFM and 8.65 sHz for FCS in NOF1 channel. The results show that linear chirp performance is acceptable after 18 dBs at BT = 9sHz and improves after VTRM processing while the performance of HCS and FCS at 5 ms remains superior then LFM and further improves after using VTRM as shown in Fig. 17a. Furthermore, when the BT of the LFM and HCS was decreased to 4.4sHz, and for FCS 5.5sHz, the LFM showed disappointing results, on the other hand performance of both the nonlinear chirp also degraded, however after VTRM processing the nonlinear chirp showed satisfactory results, while HCS was better than FCS as shown in Fig. 17b.
Conclusion and future work
In this paper, MSCSSVTRM is proposed. The mentioned technique uses nonlinear Sine chirp, which can perform better at low spreading factor, which plays a significant role in hydroacoustic communication's overall performance. However, to increase the information rate, Mary technique is presented by transmitting the chirp at different frequency bands. At the same time, VTRM equalizes the UWA channel to minimize the effect of multipath interference. As a result, the overall reliability of the system is improved. Firstly, the paper presented the sine chirp and proved that at low SF it is orthogonal. Secondly, the system model of the Mary sine chirp spread spectrum is presented. Then, the concept of VTRM in underwater acoustic communication to mitigate intersymbol interference is presented. Finally, the MSCSS scheme proposed in this article is compared to traditional linear chirp. The simulation results proved that the proposed scheme is robust to the multipath environment and performed better than the conventional linear chirp system. In future work, we will perform sea trials to analyze the proposed scheme's performance in the realworld underwater acoustic channel. The time variability due to water surface movements, tides, internal waves, and Doppler shift caused by moving devices such as AUVs will also be analyzed.
Availability of data and materials
Not applicable.
Abbreviations
 AUV:

Autonomous underwater vehicle
 LFM:

Linear frequency modulation
 CDMA:

Codedivision multiple access
 FDMA:

Frequencydivision spread spectrum
 VTRM:

Virtual timereversal mirror method
 SF:

Spread factor
 BT:

Bandwidth time
 LoRA:

Lowpower longrange wide area network
 BOK:

Binary orthogonal keying
 DM:

Direct modulation
 MUCSS:

Mary sine chirp spread spectrum
 MP:

Matching pursuit
 HCS:

Half cycle sine
 FCS:

Full cycle sine
 BER:

Bit error rate
 UWA:

Underwater acoustic
 AWGN:

Additive white Gaussian noise
 UWSNs:

Underwater wireless sensor networks
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Funding
This work was funded by the National Natural Science Foundation of China (Grant Nos. 61771152, 61431004, 61601136 and 61601137), National Key R&D Program of China (Grant Nos. 2018YFC0308500 and 2017YFC0305702) and the Natural Science Foundation of Heilongjiang Province of China (Grant No. YQ2019F002).
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All the authors contributed in the preparation of this manuscript, following are the detail of the contributions Conceptualization, S.L. and H.Z.; methodology, S.L. and H.Z.; software, H.Z.; validation, S.L., L.Y. and M.B; formal analysis, S.L., H.Z. and L.Y; investigation, H.Z.,S.L and L.Y; resources, S.L.; data curation, S.L.,S.S. and WR.; writing—original draft preparation, H.Z.; writing—review and editing, S.L.,L.Y. and M.B.; visualization, S.L. and H.Z; supervision, S.L.; project administration, S.L. All authors read and approved the final manuscript.
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Liu, S., Zuberi, H.H., Lou, Y. et al. Mary nonlinear sine chirp spread spectrum for underwater acoustic communication based on virtual timereversal mirror method. J Wireless Com Network 2021, 112 (2021). https://doi.org/10.1186/s13638021019953
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Keywords
 Underwater acoustic communication
 Nonlinear chirp
 Sine chirp
 Virtual timereversal mirror method