1 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]. Submarine-to-submarine or submarine-to-AUV 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 time-varying 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 hydro-acoustic 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 low-power and long-range 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 hydro-acoustic communication, unlike multi-carrier 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 filter-based receiver. The nonlinear frequency-modulated 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 multi-rate 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 non-orthogonal. 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 cross-correlation between the pair of complete sine cycle at nonzero time shift is high; as a result, its performance degrades.

The M-ary technique is traditionally used in single-carrier as well as multi-carrier communication to increase the data rate. The M-ary chirp system can be implemented by using various techniques; it can be employed by transmitting in different frequency sub-bands, 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, higher-order modulation in different frequency bands cannot be achieved easily. On the other hand, M-ary modulation can be detected using coherent or non-coherent approaches [24]. The authors in [25] employed DPSK and different chirp rates for transmitting M-ary 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. M-ary-based cyclic shift keying spread spectrum using VTRM presented in [26] requires a high spreading sequence for a higher data rate. The M-ary 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 bandwidth-time (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 inter-symbol interference caused by underwater acoustic channels, M-ary Sine Chirp Spread Spectrum (MS-CSS) based on VTRM is proposed in this paper. MS-CSS is a combination of two techniques used in UWA communication. The nonlinear sine upchirp and downchirp is used to represent data bits, and M-ary 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 M-ary technique, whereas the system model of the M-ary sine chirp spread spectrum is given in Sect. 4. The channel estimation using matching pursuit and virtual time-reversal 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.

2 Analysis of linear and nonlinear chirp signal

2.1 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

$$\left\{ \begin{gathered} {\text{Up}}\;{\text{Chirp}}:\quad \, c_{0} (t) = A\cos (2\pi f_{o} t + \pi \mu t^{2} ) - \frac{{T_{LC} }}{2} \le t \le \frac{{T_{LC} }}{2} \hfill \\ {\text{Down}}\;{\text{Chirp}}:\quad \, c_{1} (t) = A\cos (2\pi f_{o} t - \pi \mu t^{2} ) - \frac{{T_{LC} }}{2} \le t \le \frac{{T_{LC} }}{2} \hfill \\ \end{gathered} \right.$$
(1)

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 filter-based receiver is used for detection. The expression for the autocorrelation and cross-correlation in [28] and it can be given as,

$$r_{{c_{0} c_{0} }} (\tau ) = \sqrt \mu \frac{{\sin [\pi \mu t(T_{LC} - \left| \tau \right|)]}}{2\pi \mu \tau }\cos (2\pi f_{o} \tau ) - \frac{{T_{LC} }}{2} \le \tau \le \frac{{T_{LC} }}{2}$$
(2)
$$r_{{c_{0} c1}} (\tau ) = \frac{{\cos (2\pi f_{o} \tau )}}{{\sqrt {T_{LC} B} }}\left[ {C\left( {\frac{\pi }{2}\left( {\sqrt {T_{LC} B} - \left| \tau \right|\sqrt \mu } \right)^{2} } \right) + jS\left( {\frac{\pi }{2}\left( {\sqrt {T_{LC} B} - \left| \tau \right|\sqrt \mu } \right)^{2} } \right)} \right]$$
(3)

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 cross-correlation coefficient of LFM is defined by

$$\left| {\rho_{l} } \right| = \frac{1}{{\sqrt {T_{LC} B} }}\sqrt {\int\limits_{0}^{{\sqrt {T_{LC} B} }} {\cos \left( {\frac{{\pi v^{2} }}{2}{\text{d}}v} \right)^{2} } + \int\limits_{0}^{{\sqrt {T_{LC} B} }} {\sin \left( {\frac{{\pi v^{2} }}{2}{\text{d}}v} \right)^{2} } }$$
(4)
Fig. 1
figure 1

Linear frequency-modulated correlator output

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Figure 2 shows that the cross-correlation 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 quasi-orthogonal [29].

Fig. 2
figure 2

Time bandwidth versus cross-correlation coefficient graph

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2.2 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

$$\left\{ \begin{gathered} {\text{Up:}}\quad c_{s0} (t) = A\cos \left[ {2\pi f_{o} t + \frac{{BT_{S} }}{\eta }\cos \left( {\pi \eta \frac{t}{{T_{s} }}} \right)} \right] - \frac{{T_{S} }}{2} \le t \le \frac{{T_{S} }}{2} \hfill \\ {\text{Down}}:\quad c_{s1} (t) = A\cos \left[ {2\pi f_{o} t - \frac{{BT_{S} }}{\eta }\cos \left( {\pi \eta \frac{t}{{T_{s} }}} \right)} \right] - \frac{{T_{S} }}{2} \le t \le \frac{{T_{S} }}{2} \hfill \\ \end{gathered} \right.$$
(5)

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), half-cycle sine chirp is obtained, whereas if \(\eta\) is set to “2” in Eq. (5), full-cycle 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.

$$\begin{aligned} r_{{c_{s1} c_{s1} }} (\tau ) & = \left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\sum\limits_{n = - \infty }^{n = + \infty } {\left\{ {( - 1)^{n} J_{2n} (k_{{s_{1} }}^{m} )\sin c[(\omega_{L} + 2n\theta )(T_{s} - \left| \tau \right|)]} \right\}} \\ & \quad + \left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\cos (\omega_{L} \tau )J_{o} (k_{{s_{2} }}^{m} ) + 2\left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\cos (\omega_{L} \tau )\sum\limits_{n = 1}^{\infty } {\left\{ {J_{2n} (k_{{s_{2} }}^{m} )\sin c[(2n\theta )(T_{s} - \left| \tau \right|)]} \right\}} \\ \end{aligned}$$
(6)

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

$$r_{{c_{s1} c_{s1} }} (\tau ) \approx \left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\cos (\omega_{L} \tau )J_{o} \left( {\frac{{2BT_{s} }}{\eta }\sin \left( {\frac{\pi \eta }{{2T_{s} }}\tau } \right)} \right)$$
(7)

The following equation gives the cross-correlation coefficient of pair of sine chirp

$$\begin{aligned} \rho_{s} (\tau ) & = \left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\sum\limits_{n = - \infty }^{n = + \infty } {\left\{ {( - 1)^{n} J_{2n} (k_{{s_{1} }}^{u} )\sin c[(\omega_{L} + 2n\theta )(T_{s} - \left| \tau \right|)]} \right\}} + \left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\cos (\omega_{L} \tau )J_{o} (k_{{s_{2} }}^{u} ) \\ & \quad + 2\left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\cos (\omega_{L} \tau )\sum\limits_{n = 1}^{\infty } {\left\{ {J_{2n} (k_{{s_{2} }}^{u} )\sin c[(2n\theta )(T_{s} - \left| \tau \right|)]} \right\}} \\ & \quad - 2\left( {1 - \frac{\left| \tau \right|}{{T_{s} }}} \right)\sin (\omega_{L} \tau )\sum\limits_{n = 0}^{\infty } {\left\{ {J_{2n + 1} (k_{{s_{2} }}^{u} )\sin c[(2n\theta )(T_{s} - \left| \tau \right|)]} \right\}} \\ \end{aligned}$$
(8)

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 cross-correlation coefficient at time shift \(\tau = 0\) can be simplified as in [30] and given by the equation below.

$$\rho_{s} (0) \approx J_{0} \left( {\frac{{2T_{s} B}}{\eta }} \right)$$
(9)

Figure 4 shows that the cross-correlation 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 half-sine period has a cross-correlation 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 cross-correlation 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 half-sine 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 cross-correlation coefficient and bandwidth time product of nonlinear chirps. The cross-correlation between the pair to full and half-period sine chirp at 18sHz and 20sHz in Fig. 3b shows that at time shift (\(\tau\) = 0) the value of cross-correlation 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 auto-correlation increases for both linear and nonlinear chirp signals (Fig. 4).

Fig. 3
figure 3

Auto- and cross-correlation of a full cycle and half cycle sine chirp a auto-correlation of full sine and half sine cycle nonlinear chirp, b cross-correlation of full sine and half sine cycle nonlinear chirp

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Fig. 4
figure 4

Cross-correlation coefficient versus bandwidth time product graph of nonlinear and linear chirp

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3 M-ary chirp spread spectrum

M-ary 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 8-ary 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 half-sine cycle can also be represented using a time–frequency graph.

Fig. 5
figure 5

Graphical representation M-ary chirp spread spectrum

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Fig. 6
figure 6

Time–frequency representation of M-ary sine chirp spread spectrum

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4 System model of MS-CSS

The block diagram of the M-ary 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 time-shifted 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.

Fig. 7
figure 7

Block diagram of 8-ary sine chirp spread spectrum communication system

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5 Channel estimation

The channel estimation is one of the requisite components of VTRM-based 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

$$y(n) = \sum\limits_{n = 0}^{L - 1} {x(t - n)} *h(n) + v(n)$$
(10)

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

$$\underbrace {{\left[ \begin{gathered} y(0) \\ y(1) \\ \vdots \\ y(l - 2) \\ y(l + L - 1) \\ \end{gathered} \right]}}_{Y} = \underbrace {{\left[ {\begin{array}{*{20}c} {x(0)} & 0 & \cdots & 0 \\ {x(1)} & {x(0)} & \cdots & 0 \\ \vdots & \vdots & \cdots & 0 \\ \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} & \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} & \begin{gathered} \ddots \\ \cdots \\ \end{gathered} & \begin{gathered} x(l - 1) \\ x(l) \\ \end{gathered} \\ \end{array} } \right]}}_{X}*\underbrace {{\left[ \begin{gathered} h(0) \\ h(1) \\ \vdots \\ h(L - 2) \\ h(L - 1) \\ \end{gathered} \right]}}_{h} + \underbrace {{\left[ \begin{gathered} v(0) \\ v(1) \\ \vdots \\ v(L + l - 2) \\ v(L + l - 1) \\ \end{gathered} \right]}}_{V}$$
(11)

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 mth iteration, the residual \(\Upsilon_{m - 1}\) for which the \(X\) has the maximal rank-one projection can be expressed as \(x_{m}\) and it is given by the following equation

$$x_{m} = \;\mathop {\arg \max }\limits_{{k \notin I_{m - 1} }} \frac{{\left| {X_{k}^{H} \Upsilon_{m - 1} } \right|}}{{\left\| {X_{k} } \right\|^{2} }}$$
(12)

\(X_{k}\) 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

$$\mathop {h_{m} }\limits^{ \wedge } = \frac{{\left| {X_{{x_{m} }}^{H} \Upsilon_{m - 1} } \right|}}{{\left\| {X_{{x_{m} }} } \right\|^{2} }}$$
(13)

Finally, the residual vector can be calculated as

$$\Upsilon_{m} = \Upsilon_{m - 1} - \mathop {h_{m} }\limits^{ \wedge } X_{{x{}_{m}}}$$
(14)

6 Virtual time-reversal mirror method

The virtual time-reversal mirror method is widely used for equalization in many types of communication channels, including underwater acoustic communication channel [36]. VTRM-based 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 time-varying 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 time-reversed \(h^{^{\prime}} ( - t)\), and then it is convolved with the received signal [42].

Fig. 8
figure 8

VTRM-based system design

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Let \(x_{r} (t)\) be the transmit signal passed through the UWA channel can be given as

$$x_{r} (t) = x(t) \otimes h(t) + n(t)$$
(15)

Then after virtual time reversal, the received signal can be expressed as

$$r(t) = x_{r} (t) \otimes h^{^{\prime}} ( - t)$$
(16)
$$r(t) = \left\{ {x(t) \otimes h(t) + n(t)} \right\} \otimes h^{^{\prime}} ( - t)$$
(17)
$$r(t) = x(t) \otimes \left[ {h(t) \otimes h^{^{\prime}} ( - t)} \right] + n(t) \otimes h^{^{\prime}} ( - t)$$
(18)

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 inter-symbol interference but also improves the SNR.

7 Results and discussion

7.1 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.

Table 1 Simulation parameters
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Fig. 9
figure 9

Underwater acoustic simulation setup for BELL HOP channel

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Fig. 10
figure 10

Underwater acoustic actual and estimated channel impulse response at a distance of 1 km using BELLHOP a actual impulse response, b estimated impulse response using matching pursuit

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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.

Fig. 11
figure 11

BER versus SNR graph of HCS Chirp, FCS Chirp and LFM with SF 9sHz and 8.65, respectively, and Chirp Period 10 ms

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Fig. 12
figure 12

BER versus SNR graph of HCS Chirp, FCS chirp and LFM with SF 4.4sHz and 5.5, respectively, and Chirp Period 10 ms

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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 half-cycle have similar performance but outperform the linear chirp, as illustrated in Fig. 14. The simulation results validate that the MS-CSS 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.

Fig. 13
figure 13

BER versus SNR graph of HCS Chirp, FCS Chirp and LFM with SF 9sHz and 8.5, respectively, and Chirp Period 5 ms

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Fig. 14
figure 14

BER versus SNR graph of HCS Chirp, FCS Chirp and LFM with SF 4.4sHz and 5.5 respectively and Chirp Period 5 m

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7.2 Simulation using watermark reply channel

The watermark replay channel is a time-varying 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.

$$y(t) = \int\limits_{ - \infty }^{\infty } {\mathop h\limits^{ \wedge } (t,\tau )x(t - \tau )d\tau } + n(t)$$
(19)

where \(y(t)\) is the output of the time-varying 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 time-varying 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.

Table 2 Watermark benchmark NOF1 simulation channel parameters as given in [43]
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Fig. 15
figure 15

Time-varying channel impulse response of a NOF1 watermark channel

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Fig. 16
figure 16

BER versus SNR graph proposed system in watermark channel a BER versus SNR graph of HCS Chirp, FCS Chirp and LFM with SF 9sHz and 8.65, respectively, and Chirp Period 10 ms, b BER versus SNR

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Fig. 17
figure 17

BER versus SNR graph proposed system in watermark channel a BER versus SNR graph of HCS Chirp, FCS Chirp and LFM with SF 9sHz and 8.65, respectively, and Chirp Period 5 ms, b BER versus SNR graph of HCS Chirp, FCS chirp and LFM with SF 4.4sHz and 5.5, respectively, and Chirp Period 5 ms

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8 Conclusion and future work

In this paper, MS-CSS-VTRM is proposed. The mentioned technique uses nonlinear Sine chirp, which can perform better at low spreading factor, which plays a significant role in hydro-acoustic communication's overall performance. However, to increase the information rate, M-ary 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 M-ary sine chirp spread spectrum is presented. Then, the concept of VTRM in underwater acoustic communication to mitigate inter-symbol interference is presented. Finally, the MS-CSS 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 real-world 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.