Underwater acoustic communication signals suffer from time dispersion due to time-varying multipath propagation in the ocean. This leads to intersymbol interference, which in turn degrades the performance of the communication system. Typically, the channel correlation functions are employed to describe these characteristics. In this paper, a metric called the channel average correlation coefficient (CACC) is proposed from the correlation function to quantify the time-varying characteristics. It has a theoretical negative relationship with communication performance. Comparative analysis involving simulations and experimental data processing highlights the superior effectiveness of CACC over the traditional metric, the channel coherence time.

## 1. Introduction

The ocean serves as a dynamic medium, introducing time-varying characteristics to underwater acoustic (UWA) multipath channels. As a result, acoustic communication signals propagating through the ocean display temporal fluctuations, subsequently affecting the performance of the communication system. Quantifying these time-varying characteristics is crucial for several reasons. First, it enables a more accurate classification of UWA channels based on their time-varying properties. Moreover, it can be exploited to guide the design of channel-adaptive UWA communication systems and predict communication performance.

The study of channel time-varying characteristics can be traced back to the field of radio communications (Bello, 1963). Typically, the channel variations are often characterized using correlation functions of channel impulsive responses (CIRs). The correlation function can then be utilized to calculate the channel coherence time, which is defined as the period during which the CIRs show no correlation. In the field of underwater acoustics, channel correlation functions and coherence time have been extensively utilized to characterize the time-varying properties of diverse UWA channels (Borowski, 2009; van Walree, 2011).

Notably, channel coherence time serves as a quantification metric that provides a more direct measure of the temporal correlation. It was demonstrated that in deep and shallow water, channel coherence time is proportional to the inverse of the first power and 3/2 power of frequency, respectively (Munk and Zachariasen, 1976; Yang, 2006). A comprehensive analysis was conducted on the results of multiple experiments using phase coherent UWA communications with different frequencies in shallow waters (Yang, 2004). Within it, the analysis of the ASCOT01 data (Yang, 2004) at mid frequencies (2–5 kHz) demonstrated a notable correlation between the coherence time and the output signal-to-noise ratio (SNR) of the decision feedback equalizer (DFE) (Stojanovic , 1994). However, it is important to acknowledge that these findings were obtained from communication packets with adequate lengths, enabling accurate calculation of the coherence time. In numerous UWA communication experiments, especially those involving low-frequency applications, determining the coherence time necessitates prolonged channel observation. This poses challenges in real-time communication scenarios.

In coherent UWA communication systems, equalizers are commonly employed to resist the channel fading (Stojanovic , 1994). Based on the observation that the residual error of an equalizer reflects the extent to which the channel affects the communication performance, the mean square error (MSE) between the transmitted symbols and the decision symbols of the linear equalizer (LE) is defined as the residual error in this paper. Subsequently, the relationship between the MSE and the channel correlation function is established, and then a novel metric called channel average correlation coefficient (CACC) is proposed based on the correlation function. The CACC metric shows an inverse relationship with the MSE of the equalizer, allowing it to effectively quantify the impacts of various time-varying characteristics of UWA channels on communication performance. To assess the effectiveness of metrics such as channel coherence time and CACC, the correlations between the metrics and the bit error rate (BER) following the adaptive DFE are analyzed. The results of simulations and data processing from the sea trial indicate that CACC exhibits stronger correlation with the communication BER, both for the quasi-wide-sense stationary uncorrelated scattering (quasi-WSSUS) and non-wide-sense stationary uncorrelated scattering (non-WSSUS) channels (Eggen , 2000; van Walree , 2008). This implies that CACC can quantify the time-varying characteristics of UWA channels and their impacts on communication performance more accurately than channel coherence time.

## 2. Channel model and coherence time

The UWA channel is modeled as a time-varying, discrete time system in this paper, which is described by the complex baseband impulse response. The *m*th tap of the CIR at time *n* is represented by $ g m ( n )$; then the CIR at time *n* can be given by the vector form of $ g \u0303 ( n ) \u225c [ g \u2212 N a ( n ) , \u2026 , g 0 ( n ) , \u2026 , g N c \u2212 1 ( n ) ] T$ (Preisig, 2005). Here, *N _{a}* and

*N*denote the number of acausal and causal taps of the CIR, respectively. Consequently, the total number of taps in the CIR can be determined as $ N = N a + N c$. The communication symbol period is assumed to be

_{c}*T*; then the corresponding channel duration is denoted as $ ( N \u2212 1 ) T$.

*n*

_{0}, and $ n 0 + \Delta n$. It is also referred to as the normalized cosine between the two different snapshots. Setting $ n 0 = 0$ and denoting the total number of snapshots as $ N \u2032$, the available range of time intervals is $ \Delta n \u2208 [ 0 , 1 , \u2026 , N \u2032 \u2212 1 ]$. The channel autocorrelation function is defined as $ R \u0303 ( \Delta n ) = E [ I \u0303 ( n 0 , \Delta n ) ]$ to obtain the channel coherence time. Based on the correlation threshold $ X c = 0.8$ (Yang, 2004), channel coherence time can be expressed as $ T coh = ( min \Delta n { \Delta n | R \u0303 ( \Delta n ) \u2264 X c} ) T$.

## 3. The proposal of CACC

Equalizers are commonly employed in UWA communication systems to mitigate the impact of the intersymbol interference (ISI). Therefore, the magnitude of the residual error following equalization serves as an indicator of the extent of channel variation. In this section, the LE is utilized as a baseline to investigate the relationship between the MSE after LE and the channel autocorrelation function. This analysis ultimately leads to the proposal of CACC. Specifically, the MSE is divided into two parts first, which allows for a deep understanding of contributions of different components to the MSE. Additionally, an error correlation matrix within one component is introduced and its correlation with the channel variations is demonstrated. Finally, CACC is proposed based on the matrix, which is calculated based on the channel autocorrelation function.

### 3.1 The decomposition of the MSE of LE

*N*

_{0}. Let

*L*and

_{a}*L*represent the number of acausal and causal taps of LE, respectively, with the total size of $ L = L a + L c$ for the equalizer. The vector consisted of received symbols is defined as $ u ( n ) \u225c [ u ( n + L a ) , \u2026 , u ( n ) , \u2026 , u ( n \u2212 L c + 1 ) ] T$; then $ u ( n )$ can be obtained through the equation

_{c}*L*rows and $ L + N \u2212 1$ columns, with each row packed with leading and trailing zeros in the appropriate position corresponding to the elements of $ d ( n )$. It can be represented using column vectors $ G ( n ) = [ g \u2212 ( N a + L a ) ( n ) , \u2026 , g \u2212 1 ( n ) , g 0 ( n ) , g 1 ( n ) , \u2026 , g ( N c + L c \u2212 2 ) ( n ) ]$ (Preisig, 2005). The decision symbol of LE is given by $ d \u0302 s ( n ) = h H ( n ) u ( n )$, where the vector $ h ( n )$ represents the equalizer coefficients. Based on the minimum mean square error (MMSE) criterion, the MMSE at time instant

*n*can be determined and denoted as

Based on the structure of $ G ( n )$, the error correlation matrix $ R E G$ is reasonably approximated by a Toeplitz matrix (Preisig, 2005). Under the assumption of a WSSUS channel (Bello, 1963; Vaughan and Andersen, 2003), $ R E G$ is a diagonal matrix with identical diagonal elements. In real-world marine environments, there are channels that exhibit wide-sense stationarity and uncorrelated scattering over a specific observation period (van Walree, 2013). Therefore, it is reasonably assumed that the WSSUS channel is modeled for theoretical analysis. If there exists a certain correlation between the quantitative metric of time-varying characteristics of the channel and the diagonal element of $ R E G$, this metric can be used to quantify the effects of channel time-varying characteristics on UWA communication. Section 3.2 will revolve around this observation to derive the corresponding metric.

### 3.2 The relationship between $ R E G$ and CACC

*i*th diagonal of matrix $ R E G$ are equivalent to the summation of the elements spanning along the

*i*th diagonal of matrix $ R \epsilon , \epsilon ( 1 )$. It can be observed that $ R \epsilon , \epsilon ( 1 )$ is comprised of two components, as shown in Eq. (6). The first component encompasses the channel correlation matrix $ R g \u0303 , g \u0303 ( k )$, which arises from the time-varying properties of the channel. The second component comprises the noise variance

*N*

_{0}induced by ambient noise.

*λ*. In addition, it can be deduced that the trace of $ R g \u0303 , g \u0303 ( k )$ is equivalent to the unnormalized channel autocorrelation function $ R \u0303 ( k )$. Thus, the trace of $ R g \u0303 , g \u0303 ( k )$ can be replaced with $ R \u0303 ( k )$, which is subsequently substituted into the first term of Eq. (6), denoted as

*λ*are constants and $ \u2211 m = 0 \u221e \lambda k R \u0303 ( k + 1 )$ is a variable that is only related to the CIR. As the value of this variable increases, the trace of $ R \epsilon , \epsilon ( 1 )$ decreases, and the diagonal element of $ R E G$ becomes smaller. Consequently, the residual error at the output of the equalizer decreases, indicating that the impacts of channel variation on communication performance is reduced.

*k*, thereby weakening the effectiveness of the corresponding correlation coefficient $ R \u0303 ( k )$. This results in a significant coupling between the metric value and the number of static channels $ N \u2032$, as well as the sampling rate along the temporal axis of the time-varying channel, leading to a decrease in its overall effectiveness. Therefore,

*λ*= 1 is chosen for CACC computation, and the value of CACC is represented as

Drawing upon the research findings from Preisig (2005), the analysis above employs theoretical derivations to establish the correlation between the quantitative metric of channel time-varying characteristics, defined as CACC, and UWA communication performance. Sections 4 and 5 will validate the effectiveness of CACC for quasi-WSSUS channels (van Walree , 2008) and non-WSSUS (Matz, 2005) channels through simulation and processing of field trial data.

## 4. Computer simulations

As described previously, the proposal of CACC is based on the assumption of the WSSUS channel. In this section, numerical simulations are employed to compare the effectiveness of CACC and coherence time as quantitative metrics for assessing the time-varying characteristics of quasi-WSSUS channels.

The channels extracted from the Norway-Oslofjoa (NOF1) experiment based on the Watermark data platform (van Walree , 2017) are utilized for the analysis. The NOF1 channels were measured using linear frequency modulated (LFM) pulses with a bandwidth of 10–18 kHz, and the signals were emitted by a single transducer and subsequently received by a hydrophone. The transceiver devices were both fixed on the seabed, indicating that the channel variations were predominantly influenced by the marine environment. The impulse response of the time-varying channel is shown in Fig. 1(a), with a duration of 32.9 s and a delay coverage of 128 ms. The sound arrivals were primarily concentrated within 10 ms, with some faint paths appearing between 50 and 80 ms. Figure 1(b) gives the delay-Doppler spread function. It can be seen that the first arrival (zero delay) exhibited no frequency dispersion, while the subsequent arrivals showed Doppler spread caused by sea surface interactions. In addition, a distinct streak of delayed line is observed at zero frequency shift, attributable to the presence of stationary scatterers. From Fig. 1(c), it can be observed that the autocorrelation function quickly decreased to around 0.75 and maintained relative stability, primarily due to the presence of stable seabed reflections. As a result of the rapid decrease in the correlation coefficient, the coherence time of this time-varying channel was notably short. The channel autocorrelation function and the instantaneous correlation function displayed a consistent overall trend, supporting the conclusion that the assumption of wide-sense stationarity was reasonably satisfied throughout the observation duration in the NOF1 experiment (van Walree, 2013). The normalized cross-covariance matrix, shown in Fig. 1(d), exhibited a prominent bright diagonal, indicating uncorrelated between different channel taps. Consequently, the channels in the NOF1 experiment can be considered to satisfy quasi-WSSUS.

In the NOF1 experiment, a total of 60 sets of channels with different time-varying characteristics were obtained. These channels were used to conduct computer simulations. In each simulation, *N*′ = 200 000 binary phase shift keying (BPSK) symbols were randomly generated and transmitted through the NOF1 channel, followed by the addition of Gaussian white noise at the receiver. Subsequently, the received symbols were equalized using DFE, resulting in the calculation of the corresponding BER for each NOF1 channel. The parameters of DFE remained consistent throughout the simulations. In each simulation, the impacts of channel variations on communication performance differ due to varying degrees of channel variability, as evidenced by the different BERs after equalization. Hence, the effectiveness of the aforementioned metrics can be evaluated based on their correlations with the BER. A strong correlation indicates exceptional performance in using the metric to quantify the channel time-varying characteristics and evaluate their impacts on communication performance.

A set of 60 quantitative metrics and corresponding BERs were obtained based on the simulation results. The relationships between the metrics and BERs are illustrated in Fig. 2. For the purpose of presentation, we uniformly set the coherence times exceeding the observation time of 32.9 s to be 32.9 s and plotted them in Fig. 2(a). It is evident that the coherence times of the majority of channels were less than 1 s. Within this 1 s duration, a clear negative correlation between the coherence time and BER was observed. However, among the 60 sets of time-varying channels, the coherence times of 13 sets were longer than the channel observation time, indicating that the autocorrelation coefficients of these 13 channels consistently remained above 0.8 within the 32.9 s duration. The above analysis demonstrates a certain correlation between the coherence time and communication BER. Nonetheless, it is not sufficient for conducting an in-depth assessment of the time-varying characteristics of UWA channels within the observation window. From Fig. 2(b), a notable inverse relationship between CACC and BER was observed. This validates the effectiveness of CACC in quantifying the time-varying characteristics of channels, where channels with CACC values closer to 1 exhibit less variations.

To quantitatively assess the effectiveness of channel coherence time and CACC, the Pearson correlation coefficient was employed to examine the linear relationships between these metrics and the BER. The Pearson correlation coefficient is a statistical measure that quantifies the linear correlation between two variables (Hauke and Kossowski, 2011). It ranges between +1 and −1, where +1 indicates a strong positive linear correlation, 0 indicates no linear correlation, and −1 indicates a strong negative linear correlation. The average coefficients of different SNRs between channel coherence time and CACC with the BER are −0.6 and −0.9, respectively. Note that both Pearson correlation coefficients were negative, indicating negative correlations between the metrics and the BER. Furthermore, the absolute values of the correlation coefficients for CACC were greater than that for channel coherence time, maintaining that CACC provides a stronger representation of the time-varying characteristics of quasi-WSSUS channels compared to channel coherence time.

## 5. Experimental validation

In this section, data from the ASIAEX in the South China Sea (SCS) were used to further compare the effectiveness of above metrics in quantifying the channel variations. During the experiment, a sound source operating at a frequency of 400 Hz was deployed on the 120 m isobath, while an L-shaped hydrophone array with both horizontal and vertical apertures was also moored on the isobath with the same depth of the source. The communication range was approximately 19 km. A rich field of strongly nonlinear internal solitary waves (solitons) was observed during the course of the experiment (Ramp , 2004), so the channels are suitable for our analysis due to the strong time-varying characteristics. The data of a single horizontal array element were used. The sound source and hydrophone can be considered as predominantly stationary: thus, the variations in the channel primarily stem from environmental factors such as internal waves in the ocean.

The transmission signals had a center frequency of 400 Hz and a symbol rate of 100 Hz. The received signals were demodulated to complex baseband, interpreted as BPSK symbols with a specific modulation offset. The communication data were organized into packets consisting of 3000 symbols. The received symbols were equalized using the DFE and phase locked loop (PLL) structure (Stojanovic , 1994). The first 1000 symbols were allocated as training sequences, while the subsequent 2000 symbols were used as decision sequences to measure the BER. A total of 111 communication packets, corresponding to 111 sets of time-varying channels, were processed with SNRs ranging from 10 to 12.5 dB.

The channels in the ASIAEX exhibited distinct characteristics, as depicted in Fig. 3. The delay spread was primarily within 500 ms. From Figs. 3(a) and 3(b), it can be observed that even the direct path, corresponding to the delay time of zero, underwent noticeable variations. This indicates significant variations in the ocean environment, which are substantially different from NOF1 channels. Figure 3(c) shows that the channel autocorrelation coefficient continuously decreased over time, without maintaining a steady state. At approximately 17 s, the correlation coefficient dropped below 0.8, indicating that the coherence time of this channel was approximately 17 s. Additionally, Fig. 3(d) reveals a significant correlation among different channel taps, meaning that this channel does not satisfy uncorrelated scattering. Almost all channels in the ASIAEX experiment are non-WSSUS channels.

Based on the experimental results, a total of 111 metrics and BERs were obtained. The relationships between the metrics and BERs are illustrated in Fig. 4. Several channels had a coherence time greater than 30 s, as shown in Fig. 4(a), with a display time of 30 s. The red line in the graph represented the curve fitted based on the original scatterplot, clearly indicating a stronger monotonic correlation between the metric of CACC and BER, as depicted in Fig. 4(b). The Pearson coefficient between channel coherence time and BER showed a weak negative correlation of −0.2. However, the correlation between CACC and BER was stronger, with a coefficient of −0.6. Although the communication performance during actual data processing is inevitably affected by ambient noise and channel multipath characteristics, monotonic correlations between CACC and BER were still observed. These results further emphasize the advantage of using CACC to measure the temporal variations of non-WSSUS channels.

## 6. Conclusion

A novel metric named CACC has been proposed to quantify the time-varying characteristics of UWA channels from the perspective of channel correlations. The extents of channel temporal variations are reflected in the residual error of the equalizers in UWA communications; this specific relationship has been ingeniously utilized to propose the CACC. Specifically, the excess error influenced by channel time-varying characteristics has been decomposed from the residual error of the LE. Equations have been formulated to establish the relationship between the excess error and the channel autocorrelation function. Based on these equations, the CACC, represented by the channel autocorrelation function, has been derived.

Another commonly used metric, the channel coherence time, which utilizes the channel autocorrelation function, has also been introduced as a benchmark for comparison. Two types of time-varying channels, including quasi-WSSUS and non-WSSUS channels, have been considered to compare the effectiveness of the two metrics. The simulation and experiment results reveal a strong negative correlation between CACC and BER. Conversely, the correlation between coherence time and BER is relatively weak. This can be attributed to the unavailability of coherence time for certain channels within the observation time, which limits its effectiveness in assessing the time-varying characteristics of channels. The superior performance of CACC can be attributed to its comprehensive utilization of delay and correlation coefficient information obtained from the channel autocorrelation function.

Based on the analysis above, CACC can be regarded as a novel quantitative metric for assessing the time-varying characteristics of channels and their influence on UWA communication performance. This motivates future work on the prediction of communication performance based on the specific time-varying characteristics of UWA channels. Specifically, under the conditions of determined channel length, SNR, communication architecture, and so on, the relationships between CACC and other time-varying characteristic quantitative metrics with communication BER or post-equalization decision error can be analyzed using extensive datasets. Subsequently, communication performance is expected to be predicted using CACC or in conjunction with other metrics.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 62301551). The authors would like to express gratitude to the sea trial team members for their invaluable contribution of data to this study.

## Author Declarations

### Conflict of Interest

The authors have no conflicts of interest to disclose.

## Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

*Channels, Propagation and Antennas for Mobile Communications: IET Electromagnetic Waves Series 50*