Improvement of communication quality using compressed sensing in underwater acoustic communication system with orthogonal signal division multiplexing

Underwater acoustic (UWA) communication is a promising technology that achieves medium- and long-range wireless data transmission in marine activities, such as marine research, oceanography, commercial operations, offshore oil extraction, and defense.¹⁾ However, a UWA channel is an extremely complex communication environment characterized by large delay and Doppler spreads.²⁾ Specifically, an acoustic signal emitted by a transmitter (Tx) propagates with numerous reflections toward the receiver (Rx). Hence, the impulse response of a UWA channel has long but sparse reverberation tails. In other words, a channel's impulse response consists of few active taps and numerous zero taps. Furthermore, signals traveling along multiple paths have different Doppler shift in a dynamic environment (Doppler spread). Since the speed of sound underwater is much lower than that of radio waves, the delay and Doppler spreads of a UWA channel become extremely large compared to those of a radio channel.

To provide reliable communication in a UWA channel, transmission methods using single-carrier or multi-carrier modulation (orthogonal frequency division multiplexing, OFDM) techniques have been investigated. In a single-carrier system, the Tx emits a time series of symbols and the Rx equalizes the delay and Doppler spreads by combining a decision feedback equalizer and digital phase-locked loop.^1–11) In an OFDM system, the Tx emits multiple symbols (symbol blocks) simultaneously on the different frequencies (subcarriers) (some subcarriers can be inactivated to avoid interference between subcarriers due to Doppler spread), and the Rx linearly equalizes the delay and Doppler spreads in the frequency domain.^12–22) Furthermore, the combination of transmission method and array signal processing have also been investigated to boost the performance of a communication system.^23–37)

As an alternative, the authors have proposed the use of orthogonal signal division multiplexing (OSDM) for UWA communication. Unlike single-carrier or multi-carrier modulation, in an OSDM system the Tx emits symbol blocks so that each symbol block appears periodically in the time-frequency lattice (some frequencies can be inactivated, as is the case in OFDM), and the Rx linearly equalizes the delay and Doppler spreads. In our previous studies, we have clarified that OSDM can become a viable alternative that achieves reliable communication in UWA channels with large delay and Doppler spreads.^37–47) However, OSDM does not fully exploit the characteristics of UWA channels. Although UWA channels are naturally sparse, a current Rx in OSDM employs a least square (LS) channel estimator, which estimates channel impulse response in the presence of noise components remaining in many taps rather than the desired zero taps, resulting in poor performance. If the channel impulse response is estimated and the message is equalized considering the sparsity of the UWA channel, the communication quality can be expected to improve.

Compressed sensing that solves simultaneous equations by exploiting the sparse nature of sparse signals has a potential to improve OSDM communication quality.⁴⁸⁾ Specifically, compressed sensing solves the sparsity of a linear system by combining L1 regularization (promoting a sparse solution) and L2 regularization (LS) with certain gain. Although the use of compressed sensing has been found to improve the performance of single-carrier and multi-carrier systems,^49–51) it has not been utilized in OSDM. Furthermore, the optimal gain of compressed sensing that determines the performance of the channel estimator has not been clarified yet, to our best knowledge.

Hence, in this paper we apply compressed sensing to OSDM, clarify the optimal gain of the channel estimator using compressed sensing, and evaluate communication quality through simulations and experiments. Section 2 overviews OSDM and the channel estimator using compressed sensing. Section 3 investigates the optimal gain of the channel estimator. Improvement of communication quality using compressed sensing is also demonstrated in simulations. Section 4 demonstrates the improvement of communication quality using compressed sensing in experiments. Section 5 presents our conclusion.

In this section, we overview OSDM and the channel estimator using compressed sensing. Figure 1 shows a block diagram of OSDM in the Tx and Rx, and Table I lists the parameters used in OSDM, where b and 1/T represent the number of bits per symbol and signal bandwidth, respectively.

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Table I.  Parameters used for design of OSDM signal.

Parameters		Value
Message length	M	127
Number of vectors	N	11
Maximum Doppler shift	Q	2
Guard interval	L	127
Effective data rate (bps)	$\tfrac{bM(N-1-4Q)}{T(MN+L)}$	1600

The Tx calculates the transmitting signal as follows:

• 1.  
  Define a vector of size 1 × MN

  $$\begin{eqnarray}&&{\boldsymbol{d}}=\left({{\boldsymbol{m}}}_{0},{{\boldsymbol{m}}}_{1},\,\ldots \,,\,{{\boldsymbol{m}}}_{N-1}\right),\end{eqnarray} \tag{ 1 }$$

  where m_n(n = 0, 1, ..., N−1) is a vector of size 1 × M. Among m₀ is a pilot signal for channel measurement (known to the Rx), m_2Q+1, m_2Q+2, ..., m_N−2Q−1 is a transmission message, and m₁, m₂, ..., m_2Q, m_N−2Q, m_N−2Q+1, ..., m_N−1 is a zero vector whose components are all 0. The structure of vector d is shown in Fig. 2.
• 2.  
  The Tx applies a spreading matrix to d as

  $$\begin{eqnarray}&&{\boldsymbol{x}}={\boldsymbol{d}}\left({{\boldsymbol{F}}}_{N}\otimes {{\boldsymbol{I}}}_{M}\right),\end{eqnarray} \tag{ 2 }$$

  where x, F_N, I_M, and ⨂ represent a transmission signal block of size 1 × MN, an inverse discrete Fourier transform matrix of size N × N, a unit matrix of size M × M, and a Kronecker product, respectively.
• 3.  
  The Tx inserts cyclic prefix of length L to x, converts the baseband signal x to a passband signal with a carrier frequency of f_c, and transmits it to the channel.

The transmitted signal is affected by both the delay and Doppler spreads of the UWA channel. The received signal at the Rx can be represented by using a basis expansion model as⁵²⁾

$$\begin{eqnarray}&&{\boldsymbol{y}}={\boldsymbol{x}}\displaystyle \sum _{q=-Q}^{Q}{{\boldsymbol{H}}}_{q}{{\boldsymbol{\Lambda }}}_{q}+{\boldsymbol{n}},\end{eqnarray} \tag{ 3 }$$

where H_q and Λ_q (q = −Q, −Q + 1, ..., Q) are matrices of size MN × MN representing the effects of delay spread and Doppler shift, respectively, and

$$\begin{eqnarray}{{\boldsymbol{H}}}_{q}=\left(\begin{array}{cccc}{h}_{0,q} & {h}_{1,q} & \cdots & {h}_{MN-1,q}\\ {h}_{MN-1,q} & {h}_{0,q} & \cdots & {h}_{MN-2,q}\\ \vdots & \vdots & \ddots & \vdots \\ {h}_{1,q} & {h}_{2,q} & \cdots & {h}_{0,q}\end{array}\right),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{\boldsymbol{\Lambda }}}_{q}={\rm{diag}}\left({W}_{MN}^{0},{W}_{MN}^{q},\,\ldots \,,\,{W}_{MN}^{(MN-1)q}\right),\end{eqnarray} \tag{ 5 }$$

where ${W}_{MN}^{k}$ = exp(2$\pi \sqrt{-1}$k/MN). Furthermore, n represents an additive noise component. Note that h_l,q represents the channel impulse response at the time of l and the Doppler shift of q, and h_l,q = 0 if l ≥ M.

The Rx calculates the received signal as follows:

• 1.  
  The Rx applies a dispreading matrix to y as

  $$\begin{eqnarray}&&\left({{\boldsymbol{z}}}_{0},{{\boldsymbol{z}}}_{1},\,\ldots \,,\,{{\boldsymbol{z}}}_{N-1}\right)={\boldsymbol{y}}\left({{\boldsymbol{F}}}_{N}^{* }\otimes {{\boldsymbol{I}}}_{M}\right)+({{\boldsymbol{n}}}_{0},{{\boldsymbol{n}}}_{1},\,\ldots \,,\,{{\boldsymbol{n}}}_{N-1}),\end{eqnarray} \tag{ 6 }$$

  where z_n (n = 0, 1, ..., N−1), F_N*, and ${{\boldsymbol{n}}}_{n}$ are a vector of size 1 × M, a discrete Fourier transform matrix of size N × N, and part of n, respectively. In this case, there exist relationships between z_N−Q, z_{N−Q+1, ...,} z_N−1, z_0, z_{1, ...,} z_Q and m₀, and z_Q+1, z_Q+2, ..., z_N−Q−1 and m_2Q+1, m_2Q+2, ..., m_N−2Q−1 as

  $$\begin{eqnarray}&&\begin{array}{l}\left({{\boldsymbol{z}}}_{N-Q,}{{\boldsymbol{z}}}_{N-Q+1,\ldots ,}{{\boldsymbol{z}}}_{N-1,}\,{{\boldsymbol{z}}}_{0,}{{\boldsymbol{z}}}_{1,\ldots ,}\,{{\boldsymbol{z}}}_{Q}\right)\\ \,=\,{{\boldsymbol{m}}}_{0}\left({{\boldsymbol{H}}}_{0,-Q},{{\boldsymbol{H}}}_{0,-Q+1},\ldots ,\,{{\boldsymbol{H}}}_{0,Q}\right)\\ \,+\,\left({{\boldsymbol{n}}}_{N-Q,}\,{{\boldsymbol{n}}}_{N-Q+1,\ldots ,}\,{{\boldsymbol{n}}}_{N-1,}\,{{\boldsymbol{n}}}_{0,}\,{{\boldsymbol{n}}}_{1,\ldots ,}\,{{\boldsymbol{n}}}_{Q}\right),\end{array}\end{eqnarray} \tag{ 7 }$$

  $$\begin{eqnarray}&&\begin{array}{l}\left({{\boldsymbol{z}}}_{Q+1},{{\boldsymbol{z}}}_{Q+2},\,\ldots \,,\,{{\boldsymbol{z}}}_{N-Q-1}\right)\\ \,=\,({{\boldsymbol{m}}}_{2Q+1,}{{\boldsymbol{m}}}_{2Q+2},\,\ldots \,,\,{{\boldsymbol{m}}}_{N-2Q-1})\\ \left(\begin{array}{cccccccc}{{\boldsymbol{H}}}_{2Q+1,-Q} & {{\boldsymbol{H}}}_{2Q+1,-Q+1} & \cdots & {{\boldsymbol{H}}}_{2Q+1,Q} & 0 & 0 & \cdots & 0\\ 0 & {{\boldsymbol{H}}}_{2Q+2,-Q} & \cdots & {{\boldsymbol{H}}}_{2Q+2,Q-1} & {{\boldsymbol{H}}}_{2Q+2,Q} & 0 & \cdots & 0\\ \, & \, & \ddots & \, & \, & \, & \ddots & \,\\ 0 & 0 & \cdots & 0 & {{\boldsymbol{H}}}_{N-2Q-1,-Q} & {{\boldsymbol{H}}}_{N-2Q-1,-Q+1} & \cdots & {{\boldsymbol{H}}}_{N-2Q-1,Q}\end{array}\right)\\ \,+\,({{\boldsymbol{n}}}_{2Q+1},\,{{\boldsymbol{n}}}_{2Q+2},\,\ldots \,,{{\boldsymbol{n}}}_{N-2Q-1}),\end{array}\end{eqnarray} \tag{ 8 }$$

  where H_n,q represents the channel matrix expressed as

  $$\begin{eqnarray}{{\boldsymbol{H}}}_{n,q}=\left(\begin{array}{cccc}{h}_{0,q} & {W}_{MN}^{q}{h}_{1,q} & \cdots & {W}_{MN}^{(M-1)q}{h}_{M-1,q}\\ {W}_{N}^{-n}{h}_{M-1,q} & {W}_{MN}^{q}{h}_{0,q} & \cdots & {W}_{MN}^{(M-1)q}{h}_{M-2,q}\\ \vdots & \vdots & \ddots & \vdots \\ {W}_{N}^{-n}{h}_{1,q} & {W}_{N}^{-n}{W}_{MN}^{q}{h}_{2,q} & \cdots & {W}_{MN}^{(M-1)q}{h}_{0,q}\end{array}\right),\end{eqnarray} \tag{ 9 }$$

  where h_0,q, h_1,q, ..., h_M−1,q represent the channel impulse response at a Doppler scale of q.
• 2.  
  The Rx obtains H_0,q by solving Eq. (7), calculates H_n,q, and obtains the message m_2Q+2, ..., m_N−2Q−1 by solving (8).

Normal OSDM estimates channel impulse response (h_0,q, h_1,q, ..., h_M−1,q) in an LS sense. If we ignore the effect of noise in the UWA channel, Eq. (7) can be written considering the fact that the matrix H_0,q is a circulant matrix

$$\begin{eqnarray}&&\begin{array}{l}{{\boldsymbol{z}}}_{q{\rm{mod}}N}={{\boldsymbol{h}}}_{0,q}{{\boldsymbol{M}}}_{0}\\ \,=\,\left({h}_{0,q},{h}_{1,q},\,\ldots \,,\,{h}_{M-1,q}\right)\,\\ \,\times \,\left(\begin{array}{cccc}{m}_{0,0} & {m}_{0,1} & \cdots & {m}_{0,M-1}\\ {m}_{0,M-1} & {m}_{0,0} & \cdots & {m}_{0,M-2}\\ \vdots & \vdots & \ddots & \vdots \\ {m}_{0,1} & {m}_{0,2} & \cdots & {m}_{0,0}\end{array}\right),\end{array}\end{eqnarray} \tag{ 10 }$$

where q = −Q, −Q + 1, ..., Q and m_0,k (k = 0, 1, ..., M−1) is the kth element of m₀. Furthermore, "q mod N" is the remainder after division of q by N, and ${{\boldsymbol{z}}}_{q{\rm{mod}}N}$ corresponds to the pilot signal m₀ that is affected by the delay spread of the UWA channel with Doppler scale of q and additive noise. Hence, the normal OSDM solves

$$\begin{eqnarray}&&{{\boldsymbol{z}}}_{q{\rm{mod}}N}\,={{\boldsymbol{h}}}_{0,q}{{\boldsymbol{M}}}_{0}+{{\boldsymbol{n}}}_{q{\rm{mod}}N},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\Rightarrow \,\mathop{\min }\limits_{{{\boldsymbol{h}}}_{0,q}}| | {{\boldsymbol{z}}}_{q{\rm{mod}}N}-{{\boldsymbol{h}}}_{0,q}{{\boldsymbol{M}}}_{0}| {| }_{2}^{2},\end{eqnarray} \tag{ 12 }$$

where ∣∣ · ∣∣_k and ${{\boldsymbol{n}}}_{q{\rm{mod}}N}$ represent the L-k norm and noise added to ${{\boldsymbol{z}}}_{q{\rm{mod}}N},$ respectively. In this case, the estimated impulse response ${\tilde{{\boldsymbol{h}}}}_{0,q}$ contains M active taps. This means that the noise component of ${\tilde{{\boldsymbol{h}}}}_{0,q}$ is also regarded as the channel impulse response, which can lead to poor performance in the equalization process.

To address this problem, we propose OSDM using compressed sensing that estimates (h_0,q, h_1,q, ..., h_M−1,q) considering the channel sparsity. Specifically, the proposed channel estimator solves Eq. (10) by solving the equivalent optimization problem of Eq. (14) obtained from the method of the Lagrange multiplier⁵³⁾

$$\begin{eqnarray}&&{{\boldsymbol{z}}}_{q{\rm{mod}}N}\,={{\boldsymbol{h}}}_{0,q}{{\boldsymbol{M}}}_{0}+{{\boldsymbol{n}}}_{q{\rm{mod}}N},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\Rightarrow \,\mathop{\min }\limits_{{{\boldsymbol{h}}}_{0,q}}\left(\,\displaystyle \frac{1}{2}| | {{\boldsymbol{z}}}_{q{\rm{mod}}N}-{{\boldsymbol{h}}}_{0,q}{{\boldsymbol{M}}}_{0}| {| }_{2}^{2}+\tau | | {{\boldsymbol{h}}}_{0,q}| {| }_{1}\right),\end{eqnarray} \tag{ 14 }$$

where τ is a parameter that determines the channel sparsity. In other words, large τ leads to ${\tilde{{\boldsymbol{h}}}}_{0,q}$ of a single tap, while small τ converges LS. Hence, the optimization of τ is necessary to improve OSDM communication quality. In this paper, we use a method called Reconstruction by Separable Approximation (SpaRSA)⁵⁴⁾ to solve Eq. (14). Note that SpaRSA decomposes a problem function into two functions—the convex and differentiable, and the convex and not necessarily differentiable—and optimizes and updates the function iteratively using the proximal gradient method. In Section 3, we clarify the optimal τ and evaluate the performance of the proposed OSDM (using compressed sensing) in simulations.

In this section we perform a simulation of UWA communication that calculates the propagation of a signal in shallow water based on ray-theory. Through this simulation, we investigate the optimal gain of the channel estimator τ. Furthermore, we show that OSDM using compressed sensing outperforms normal OSDM.

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3.1. Simulation environments

Figure 3 shows the simulation environment. As shown in the figure, we consider mobile UWA communication in shallow water. The reflection coefficients of the sea surface and sea bottom are −1 and 0.178, respectively. The Tx and Rx are set at depths of 2 m and 12 m, respectively. The Tx moves at 2 m s⁻¹, while the Rx is fixed at a specific position.

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We first calculate the channel impulse response between Tx and Rx based on ray-theory. Specifically, we consider one direct path and six reflected paths, and we calculate the gain, delay, and Doppler shift of each path by changing the Tx–Rx distance from 100 to 500 m. Figure 4 shows the calculated channel impulse response. Note that the Tx and Rx directivity is considered using an existing UWA transducer (OST-2120, OKI SEATEC).

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We calculate the transmitted signal x using the parameters shown in Table II. We then calculate the received signal y using x and the gain, delay, and Doppler shift of each path. We assume white Gaussian noise of variance ${\sigma }^{2}$ as the noise observed in the UWA channel. Finally, the signal-to-noise ratio (SNR) and bit-error rate (BER) are calculated by changing the parameter of the sparse channel estimator τ.

Table II.  Parameters of OSDM used in simulation and experiment.

Parameters	Value
Carrier frequency (kHz)	32
Bandwidth (kHz)	2.4
Modulation	16 QAM
Effective data rate (kbps)	1.6

3.2. Results and discussion

Figures 5–8 show the simulation results. Figure 5 shows the relationship between σ and optimal τ that achieves the smallest BER. From the simulation, we found that the optimal τ depends mainly on σ (in other words, the optimal τ does not change when σ is the same, even if SNR changes). Hence, OSDM using compressed sensing would be available if the Rx has a function that determines τ by measuring σ from the received signal.

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We next focus on the performance of OSDM using compressed sensing when the parameter of the sparse channel estimator τ is determined from measured σ. Figures 6(a) and 6(b) show the estimated channel impulse response using compressed sensing and LS, respectively. Note that there exist differences between the calculated channel impulse response (Fig. 4) and measured response (Fig. 6). This is because the measured channel impulse response is the discretized channel impulse response at a specific period [time: T, frequency: 1/(MNT)]. As shown in Fig. 6, the estimated channel impulse response by compressed sensing has only a few active taps as well as the ideal one. On the other hand, the estimated channel impulse response by the LS estimator has numerous active taps. This means that compressed sensing would improve the communication quality (BER). Actually, the BER performance of OSDM using compressed sensing outperforms that of the normal OSDM. Figure 7(a) shows the relationship between Tx–Rx distance and SNR. As shown in the figure, SNR gradually decreases as the Tx–Rx distance increases. Note that the SNR sometimes changes drastically (about 3–5 dB) due to signal interference. Figure 7(b) shows the relationship between Tx–Rx distance and BER. As shown in the figure, BER gradually increases as the Tx–Rx distance increases. However, the BER of the proposed OSDM is smaller than that of the normal OSDM. Specifically, there were 1548 error-free data blocks in the proposed OSDM using compressed sensing and 466 in normal OSDM (total number of transmitted data blocks: 4500). Thus, we found that OSDM using compressed sensing outperforms normal OSDM in simulations.

In this section, we perform an experiment on UWA communication. Through this experiment and simulations, we confirm that OSDM using compressed sensing outperforms normal OSDM.

4.1. Experimental environment

4.2. Results and discussion

Figure 8 shows the channel impulse response obtained by the Rx. As shown in Fig. 8, the estimated channel impulse response by compressed sensing has only a few active taps as well as the ideal one. On the other hand, the estimated channel impulse response by the LS estimator has numerous active taps. This means that compressed sensing would improve the communication quality (BER).

Figure 9(a) shows the relationship between Tx–Rx distance and SNR. As shown in the figure, SNR gradually decreases as the Tx–Rx distance increases. Note that the SNR sometimes changes drastically (about 3–5 dB) due to the signal interference. These results are similar to those of the simulation. And Fig. 9(b) shows the relationship between the Tx–Rx distance and BER. Note that there are differences between the experimental results [Fig. 9(b)] and simulation results [Fig. 7(b)], especially when the Tx–Rx distance is 50–150 (m) and 400–500 (m). Specifically, the BER of the experimental results is larger than those of the simulation. One possible reason why is that there were differences between the experimental environment and simulation conditions. In the simulation, the Tx moves with a constant speed of 2 m s⁻¹. On the other hand, in the experiment, the Tx makes a round trip between the barge (Tx–Rx distance: 50 m) and outer area of the bay (Tx–Rx distance: 500 m). When the Tx reached the turning point, the speed of the Tx changed dynamically (deacceleration, change direction, and acceleration). Such dynamic movement of the Tx makes for a large Doppler spread and results in an increase of BER. As shown in Fig. 9, the BER performance of OSDM using compressed sensing outperforms that of the normal OSDM as well as simulation results. Specifically, there were 855 error-free data blocks in the proposed OSDM using compressed sensing and 473 in normal OSDM (total number of transmitted data blocks: 1692).

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Consequently, we found that OSDM using compressed sensing can provide reliable communication in coastal areas.

In this paper, we applied compressed sensing to OSDM, clarified the optimal gain of the channel estimator using compressed sensing, and evaluated communication quality through simulations and experiments. We overviewed OSDM and the channel estimator using compressed sensing. We investigated the optimal gain of the channel estimator. As a result, we found that the optimal τ depends mainly on σ. We demonstrated the improvement of communication quality using compressed sensing in simulations. We also demonstrated improvement of communication quality using compressed sensing in experiments. Consequently, we found that OSDM using compressed sensing can provide reliable communication in coastal areas.

This work was supported by JSPS KAKENHI Grant No. JP19H02351. The authors would like to thank Ms. S. Nishihara, Messrs. H. Iwaya, Y. Umezawa, S. Endo, T. Sato, and R. Chinone of the University of Tsukuba, and Messrs. T. Takekoshi, K. Katsumata, and Y. Takahashi of the Oki Seatec Co. for their assistance during the sea trial.