# Self-Interference Channel Characterization in Underwater Acoustic In-Band
Full-Duplex Communications Using OFDM

Mohammad Towliat, Zheng Guo, Leonard J. Cimini, Xiang-Gen Xia and Aijun Song
This work was supported primarily by the Engineering Research Centers Program
of the National Science Foundation under NSF Cooperative Agreement No.
(CNS-1704097, CNS-1704076). Any opinions, findings and conclusions or
recommendations expressed in this material are those of the authors and do not
necessarily reflect those of the National Science Foundation. M. Towliat, L.
J. Cimini and X.-Gen Xia are with the Department of Electrical and Computer
Engineering, University of Delaware, Newark, DE, USA (e-mail: {mtowliat,
cimini, xianggen}@udel.edu). A. Song, and Z. Guo are with the Department of
Electrical and Computer Engineering, University of Alabama, Tuscaloosa, AL,
USA (e-mail<EMAIL_ADDRESS>song@eng.ua.edu}).

###### Abstract

Due to the limited available bandwidth and dynamic channel, data rates are
extremely limited in underwater acoustic (UWA) communications. Addressing this
concern, in-band full-duplex (IBFD) has the potential to double the efficiency
in a given bandwidth. In an IBFD scheme, transmission and reception are
performed simultaneously in the same frequency band. However, in UWA-IBFD,
because of reflections from the surface and bottom and the inhomogeneity of
the water, a significant part of the transmitted signal returns back to the
IBFD receiver. This signal contaminates the desired signal from the remote end
and is known as the self-interference (SI). With an estimate of the self-
interference channel impulse response (SCIR), a receiver can estimate and
eliminate the SI. A better understanding of the statistical characteristics of
the SCIR is necessary for an accurate SI cancellation. In this article, we use
an orthogonal frequency division multiplexing (OFDM) signal to characterize
the SCIR in a lake water experiment. To verify the results, SCIR estimation is
performed using estimators in both the frequency and time domains. We show
that, in our experiment, regardless of the depth of hydrophone, the direct
path of SCIR is strong, stable and easily tracked; however, the reflection
paths are weaker and rapidly time-varying making SI cancellation challenging.
Among the reflections, the first bounce from the water surface is the
prevalent path with a short coherence time around $70$ ms.

###### Index Terms:

In-band full-duplex system, OFDM, Multipath self-interference, Underwater
acoustic communication

##  I Introduction

Underwater acoustic (UWA) communication is an interesting research topic
because of its wide applications such as monitoring the effect of human
activities on marine life and forecasting disasters. However, due to limited
available bandwidth [1], UWA communication systems inherit a low data rate
which cannot satisfy the rising demands in this field. Using in-band full-
duplex (IBFD) transmission in UWA systems provides promise for improving the
data rate, without increasing the bandwidth. Generally, the main motivation
for IBFD is to provide a higher spectrum efficiency by transmitting and
receiving simultaneously in the same frequency band. Unfortunately, the
transmitted signal in this scenario will corrupt the received signal, causing
self-interference (SI) [2]. The key challenge for IBFD schemes is the accurate
cancellation of the SI.

The feasibility of radio frequency-IBFD (RF-IBFD) has been confirmed by
employing advanced SI cancellation methods [3, 4]. Even though the idea of
UWA-IBFD is similar to that of RF-IBFD, the SI cancellation task in UWA-IBFD
is much more challenging. Consider a node in an UWA-IBFD network transmitting
and receiving signals by using a transducer and a hydrophone, respectively. In
addition to the direct path between the transducer and the hydrophone, the sea
surface and sea bottom reflect a considerable portion of the transmitted
signal toward the hydrophone and cause strong SI. In addition, the
inhomogeneity and fluctuations of the water lead to time-variations in the SI
signal [5]. Accordingly, eliminating the strong and time-varying SI is the
bottleneck in UWA-IBFD, especially in shallow water which naturally has more
reflections.

Despite the significant number of works on RF-IBFD, few works have studied
UWA-IBFD. In [6], a time-reversal transmission has been introduced to
eliminate the SI. A multi-user UWA-IBFD using frequency division and code
division multiple access (CDMA) are presented in [7] and [8], respectively. In
[9] a joint analog and digital adaptive SI cancellation is investigated. The
authors in [10] present a fully digital SI cancellation by using recursive
least-squares (RLS) adaptive filters. In [11] a two-stage method is proposed,
in which the relatively stable part of the SI is eliminated first, and the
more rapidly time-varying residual part is separately treated at the second
stage.

From the previous works, it is clear that by better understanding the self-
interference channel impulse response (SCIR), one can better cancel the SI
[12]. In this regard, our main concern in this paper is to study the
statistical characteristics of the SCIR in UWA-IBFD. We employ orthogonal
frequency division multiplexing (OFDM) [13] to estimate the SCIR in a lake
water. For verifying the achieved results, estimations are performed in both
the frequency domain (FD) and the time domain (TD). Using the estimated SCIR,
statistical characteristics of the channel are obtained; including the power
delay profile (PDP), the autocorrelation function (ACF) and the coherence time
(COT). According to our measurements, regardless of the hydrophone’s depth,
the direct path is relatively stable and contributes to the principal part of
the SI; on the other hand, the reflected paths are rapidly time-varying. Among
all reflections, the first bounce from the water surface is the dominant path
with a short COT. In the lake experiment, the COT of this path is determined
to be around $70$ ms, which is even shorter than the OFDM block length. This
means that tracking such a severe time-varying channel cannot be performed by
using the UWA OFDM channel estimation methods proposed in [14, 15, 16], which
assume that the channel coefficients remain constant over at least one OFDM
block.

The rest of this paper is organized as follows. In Section II, we describe the
OFDM system model and the methods we use to estimate PDP, ACF and COT from the
experimental data. In Section III, the experimental steps and results are
presented and discussed. Finally, Section VI summarizes the paper.

##  II SCIR estimation using OFDM

In this section, we first present the OFDM system model and the signal
structure that is used in the experiment. Then, we explain the FD and TD
methods for estimating SCIR and calculating PDP, ACF and COT.

###  II-A System Model

Due to the fact that the carrier frequencies and bandwidths are generally
small in UWA communications, there exist analog-to-digital (A/D) converters
that can sample the received signal much faster than the Nyquist rate. Thus,
in our formulations, we simply consider a discrete-time system model. Fig. 1
illustrates the OFDM-IBFD system that we use in this work. As shown, $s[n]$ is
the SI signal at time instance $n$. Assuming that the underwater environment
behaves linearly, it is reasonable to model $s[n]$ as the convolution of the
transmitted signal $x[n]$ and the SCIR, which could be time-varying in general
[17]. Let us denote the time-varying SCIR by $h[m,n]$, where $m$ is the delay
time and $n$ is the geotime. As a consequence, the SI signal can be
represented as [18]

$s[n]=\sum\limits_{m=0}^{{M}-1}{h[m,n]x[n-m]},$ (1)

where ${M}$ is the SCIR length. In the absence of the remote end signal, the
received signal is given as

$y[n]=s[n]+w[n],$ (2)

where $w[n]$ is the additive ambient noise. With an estimate of the SCIR,
$\hat{h}[m,n]$, the receiver can estimate the SI as

$\displaystyle\hat{s}[n]=\sum\limits_{m=0}^{{M}-1}{\hat{h}[m,n]x[n-m]},$

and the subtract it from the received signal, such that

$\begin{array}[]{*{20}{l}}{e[n]=y[n]-\hat{s}[n]}\\\ {{\mkern 1.0mu}{\mkern
1.0mu}{\mkern 1.0mu}{\mkern 1.0mu}{\mkern 1.0mu}{\mkern 1.0mu}{\mkern
1.0mu}{\mkern 1.0mu}{\mkern 1.0mu}{\mkern 1.0mu}{\mkern
1.0mu}\,\,\,\,\,\,=\sum\limits_{m=0}^{{M}-1}{(h[m,n]-\hat{h}[m,n])x[n-m]}+w[n]},\end{array}$
(3)

Figure 1: UWA-IBFD system using OFDM for SCIR estimation, where $x[n]$ is the
transmitted signal, $y[n]$ is the received signal, ${\hat{s}}[n]$ is the
anticipated SI signal, and $e[n]$ denotes the residual.

where ${e}[n]$ denotes the residual signal after interference cancellation. In
UWA-IBFD, the ultimate goal is to cancel the SI down to the level of the
ambient noise to make sure that its effect is eliminated. According to (3),
the closer $\hat{h}[m,n]$ is to $h[m,n]$, the smaller the residual that
remains. Since in UWA-IBFD the power of the SI is very high, sometimes even a
relatively small deviation between $\hat{h}[m,n]$ and $h[m,n]$ can result in a
large residual, significantly degrading the performance of UWA-IBFD. Based on
this discussion, it is clear that the problem of accurate SI cancellation can
be narrowed down to precise tracking of the SCIR, which in turn is based on a
better understanding of the SCIR’s statistical characteristics.

In order to statistically characterize the SCIR, the first step is to estimate
the channel during a long period of geotime. In the experiment, we use an OFDM
signal for channel estimation. There is a wide range of sophisticated UWA OFDM
channel estimators [14, 15, 16]. Since the focus of this article is channel
characterization, we simply employ the well-known FD and TD methods to
estimate the SCIR; then, we use this estimate to extract the relevant
statistical characteristics.

###  II-B OFDM Signal Structure

According to Fig. 1, consecutive symbol vectors
${{\bf{D}}_{i}}={[{D_{i}}[0],\ldots{D_{i}}[K-1]]^{T}}\in{{\mathbb{C}}^{K\times
1}}$, for $i=0,\ldots N-1$ are the inputs of the OFDM system in the frequency
domain, where $K$ is the number of subchannels and $N$ is the number of
transmitted OFDM blocks. The corresponding vectors in the time domain are
${{\bf{d}}_{i}}={[{d_{i}}[0],\ldots{d_{i}}[K-1]]^{T}}\in{{\mathbb{C}}^{K\times
1}}$, which are the IDFTs of ${{\bf{D}}_{i}}$. For each $i$, a cyclic prefix
(CP) with length $\upsilon$ is appended to ${{\bf{d}}_{i}}$ resulting in the
OFDM block
${{\bf{b}}_{i}}={[{b_{i}}[0],\ldots{b_{i}}[{L_{{\rm{blk}}}}-1]]^{T}}\in{{\mathbb{C}}^{{L_{{\rm{blk}}}}\times
1}}$, where ${L_{{\rm{blk}}}}=K+\upsilon$ denotes the OFDM block length. We
consider that the CP is the repetition of the end of the block appended to the
head of the block, such that
${{\bf{b}}_{i}}={[\left.{{d_{i}}[K-\upsilon],\ldots{d_{i}}[K-1]}\,\,\right|{\bf{d}}_{i}^{T}]^{T}}$.
For the purpose of studying the channel characteristics, we let $\upsilon=K$;
thus, the CP is an exact copy of the data part of the block. Finally, the
output of the OFDM transmitter, $x[n]$, is generated by serializing the
parallel vectors ${{\bf{b}}_{i}}$, such that

$x[n]={b_{i}}[l];\,\,\,n=i{L_{{\rm{blk}}}}+l,$ (4)

where $i=0,\ldots N-1$ and $l=0,\ldots{L_{{\rm{blk}}}}-1$. Since the main goal
of this article is to provide the statistical characteristics of the SCIR, we
simply assume that the entire OFDM signal is dedicated to channel estimation.

###  II-C Frequency Domain SCIR Estimation

As shown in Fig. 1, for FD-SCIR estimation, the serial signal $y[n]$ is
converted to parallel OFDM received blocks
${{{\bf{\hat{b}}}}_{i}}={[{{\hat{b}}_{i}}[0],\ldots{{\hat{b}}_{i}}[{L_{{\rm{blk}}}}-1]]^{T}}$,
such that

${{\hat{b}}_{i}}[l]=y[n];\,\,\,n=i{L_{{\rm{blk}}}}+l.$ (5)

Then, the CP is removed from ${{{\bf{\hat{b}}}}_{i}}$ giving
${{{\bf{\hat{d}}}}_{i}}={[{{\hat{d}}_{i}}[0],\ldots{{\hat{d}}_{i}}[K-1]]^{T}}$.
Assuming that the channel remains constant (denoted by
$h[m,i{L_{{\rm{blk}}}}]$) over the $i$th OFDM block, the DFT of
${{{\bf{\hat{d}}}}_{i}}$,
${{{\bf{\hat{D}}}}_{i}}={[{{{\hat{D}}_{i}}[0],\ldots{{\hat{D}}_{i}}[K-1]}]^{T}}$,
is the element-wise multiplication of ${{{\bf{D}}}_{i}}$ by the channel
frequency response [19], so that

${{\hat{D}}_{i}}[k]=H[k,i{L_{{\rm{blk}}}}]\,{D_{i}}[k]+{W_{i}}[k],$ (6)

where

$\displaystyle
H[k,i{L_{{\rm{blk}}}}]=\sum\limits_{m=0}^{{M}-1}{h[m,i{L_{{\rm{blk}}}}]{\mkern
1.0mu}\,{e^{-j2\pi km/K}}}$

is the $k$th DFT component of $h[m,i{L_{{\rm{blk}}}}]$ and ${W_{i}}[k]$
denotes the corresponding noise. According to (6), the estimation of
$H[k,i{L_{{\rm{blk}}}}]$, for $k=0,\ldots K-1$, can be achieved by the
element-wise division of ${{\hat{D}}_{i}}[k]$ to ${D_{i}}[k]$ as

$\hat{H}[k,i{L_{{\rm{blk}}}}]=\frac{{{{\hat{D}}_{i}}[k]}}{{{D_{i}}[k]}}.$ (7)

Eq. (7) presents the estimated frequency components of the SCIR. Thus, the
SCIR is obtained by taking the IDFT of $\hat{H}[k,i{L_{{\rm{blk}}}}]$,

${{\hat{h}}_{{\rm{FD}}}}[m,i{L_{{\rm{blk}}}}]=\sum\limits_{k=0}^{K-1}{\hat{H}[k,i{L_{{\rm{blk}}}}]\,{e^{j2\pi
km/K}}},$ (8)

where ${{\hat{h}}_{{\rm{FD}}}}[m,i{L_{{\rm{blk}}}}]$ denotes the estimation of
the SCIR using the FD method, which is separately performed for each received
OFDM block.

###  II-D Time Domain SCIR Estimation

In the previous section, we explained SCIR estimation in the frequency domain.
One can also use TD methods on the OFDM signal, to verify the FD estimation
results. As shown in Fig. 1, the received signal $y[n]$ is directly used for
this purpose. We use the least squares (LS)111In order to reduce the noise
effect, usually regularized LS estimations, such as the ridge method, are used
for UWA channel estimation. However, since the noise is small compared to the
SI signal, we consider the primary LS method. method, in which the sliding
observation window at time instance $n$ is defined as
${\bf{y}}[n]={[y[n],\ldots
y[n-{L_{{\rm{win}}}}+1]]^{T}}\in{{\mathbb{C}}^{{L_{{\rm{win}}}}\times 1}}$,
where ${L_{{\rm{win}}}}$ denotes the window size. Assuming that the channel
remains constant over each observation window, according to (2), the observed
signal vector can be presented as

${\bf{y}}[n]={{\bf{\bar{X}}}}[n]{\bf{h}}[n]+{\bf{w}}[n],$ (9)

where $\mathbf{h}[n]={{[h[0,n],\ldots
h[{M}-1,n]]}^{T}}\in{{\mathbb{C}}^{{M}\times 1}}$ is the channel vector and
${{\bf{\bar{X}}}}[n]\in{{\mathbb{C}}^{{L_{{\rm{win}}}}\times{M}}}$ is the
transmitted signal matrix defined as

$\begin{array}[]{l}{{\bf{\bar{X}}}}[n]=\\\
\left[{\begin{array}[]{*{20}{c}}{x[n]}&{x[n-1]}&\cdots&{x[n-{M}+1]}\\\
{x[n-1]}&{x[n-2]}&\cdots&{x[n-{M}]}\\\ \vdots&{}\hfil&\ddots&\vdots\\\
{x[n-{L_{{\rm{win}}}}+1]}&{}\hfil&\cdots&{x[n-{L_{{\rm{win}}}}-{M}+2]}\end{array}}\right].\end{array}$
(10)

Assuming that ${L_{{\rm{win}}}}\geq{M}$, the LS estimation of the SCIR in the
time domain is

${{{{\bf{\hat{h}}}}_{{\rm{TD}}}}}[n]={({{{\bf{\bar{X}}}}^{H}}[n]{{\bf{\bar{X}}}}[n])^{-1}}{{{\bf{\bar{X}}}}^{H}}[n]{\bf{y}}[n],$
(11)

where
${{{\bf{\hat{h}}}}_{{\rm{TD}}}}[n]={[{{\hat{h}}_{{\rm{TD}}}}[0,n],\ldots{{\hat{h}}_{{\rm{TD}}}}[{M}-1,n]]^{T}}$
is the estimated channel vector using the TD method. To guarantee that, in
(11), the singularity of ${{{\bf{\bar{X}}}}^{H}}[n]{{\bf{\bar{X}}}}[n]$ is
very unlikely, we take ${L_{{\rm{win}}}}=2{M}$. In contrast to the FD method,
in the TD method, ${\bf{y}}[n]$ is a sliding observation window and (11)
provides an estimation for each received sample.

###  II-E Power Delay Profile Estimation

One of the most relevant characters of a channel is the power delay profile
(PDP) defined as
${{p[m]}}={{\rm{E}}}\\{{{{\left|{{{h}}[m,n]}\right|}^{2}}}\\}$, where
${{\rm{E}}}\left\\{.\right\\}$ denotes the expected value of the variable
[17]. By transmitting $N$ OFDM blocks (each with length $L_{{\rm blk}}$), the
FD method provides $N$ estimations of the SCIR; however, the TD method
provides $\mathcal{N}=N{L_{{\rm{blk}}}}$ estimations by using the sliding
observation window. According to the notations we have used in (8) and (11),
the estimated PDPs, by using the FD and TD methods, are respectively given as

$\begin{array}[]{l}{{{{\hat{p}}}}_{{\rm{FD}}}}[m]=\frac{1}{N}\sum\limits_{i=0}^{N-1}{{{\left|{{{{{\hat{h}}}}_{{\rm{FD}}}}[m,i{L_{{\rm{blk}}}}]}\right|}^{2}}},\\\
\\\
{{{{\hat{p}}}}_{{\rm{TD}}}}[m]=\frac{1}{\mathcal{N}}\sum\limits_{n=0}^{\mathcal{N}-1}{{{\left|{{{{{\hat{h}}}}_{{\rm{TD}}}}[m,n]}\right|}^{2}}}.\end{array}$
(12)

###  II-F Autocorrelation Function and Coherence Time Estimation

The other important statistical characteristics for the SCIR are the
autocorrelation function (ACF) and the coherence time (COT). The ACF of the
$m$th path of SCIR is defined as $q[m,{\eta}]={{\rm{E}}}\left\\{{h[m,n]{\mkern
1.0mu}{h^{*}}[m,n-{\eta}]{\mkern 1.0mu}}\right\\}$. Since $q[m,{\eta}]$ is
generally a complex value with the maximum amplitude ${q[m,0]}$, usually the
normalized ACF, ${\bar{q}[m,\eta]}=\left|{q[m,\eta]}\right|/q[m,0]$, is used
to study the channel variation. The COT, $\eta_{{\rm{CO}}}$, is also an
interval for which $\bar{q}[m,\frac{{{\eta_{{\rm{CO}}}}}}{2}]>\mu$. In the
literature, different values for $\mu$ is considered; however, we take
$\mu=0.8$ [20].

Given the estimated SCIRs in (8) and (11), the estimations of the ACF are
given as

$\begin{array}[]{l}{{{{\hat{q}}}}_{{\rm{FD}}}}[m,\eta{L_{{\rm{blk}}}}]=\frac{1}{N}\sum\limits_{i=0}^{N-1}{{{{{\hat{h}}}}_{{\rm{FD}}}}[m,i{L_{{\rm{blk}}}}]\,{{\hat{h}}}_{{\rm{FD}}}^{*}[m,(i-\eta){L_{{\rm{blk}}}}]},\\\
\\\
{{{{\hat{q}}}}_{{\rm{TD}}}}[m,\eta]=\frac{1}{\mathcal{N}}\sum\limits_{n=0}^{\mathcal{N}-1}{{{{{\hat{h}}}}_{{\rm{TD}}}}[m,n]\,{{\hat{h}}}_{{\rm{TD}}}^{*}[m,n-\eta]}.\end{array}$
(13)

From (13), it is obvious that, for the FD estimator, the ACF resolution is
${L_{{\rm{blk}}}}$ samples; however, using the TD method provides the ACF with
an accuracy of one sample.

##  III Lake experiment

In the previous section, we presented the relevant SCIR characterizations and
the methods we use to estimate them in the experiment. In this section, we
describe the lake experiment and the statistics we obtained.

###  III-A Experimental Set-up

The experiment was performed in Lake Tuscaloosa, AL, USA. Fig. 2 illustrates
the experimental set-up for one side of the UWA-IBFD communication. In this
set-up, the transducer and two hydrophones are installed linearly on a single
pole. The first hydrophone (Hyd-1) is close to the water surface, the second
hydrophone (Hyd-2) is close to the bottom and the transducer is located in the
middle of hydrophones, $7$ m below the water surface. In order to attenuate
the SI, two baffles are installed around the transducer to isolate it from the
hydrophones. At the time of the experiment, the wind speed was very low and
the water surface was quite calm. Fig. 3 also shows the measured sound speed
profile (SSP) of the water at the time of the experiment.

The OFDM signal parameters used in the experiment are given in Table 1. The
transmitted signal contains $N=720$ OFDM blocks with no pulse shaping. This
signal takes about $74$ s for transmission. The two hydrophones shown in Fig.
2 record the received signal with a sampling rate ${f_{\rm{s}}}=512$ kHz. The
inaccuracy of the hydrophones’ sampling rate is about $5$ Hz, which is
compensated at the receiver. In addition, about $0.3$ Hz of carrier frequency
offset (CFO) is removed from the received signals. It should be mentioned that
the cut-off frequency of the low pass filter shown in Fig. 1 is $5$ kHz. After
pre-processing, the resultant signal is applied to estimate the SCIRs using
the FD and TD methods, as in Section II.

Figure 2: Experimental set-up.

Figure 3: Sound speed profile (SSP).

Table 1: OFDM parameters Modulation | BPSK
---|---
Bandwidth |  $B=5$ kHz
Sampling time |  ${f_{\rm{s}}}=512$ kHz
# of OFDM subcarriers | $K=265$
CP length | $\upsilon=256$
OFDM block length | $K+\upsilon=512$
OFDM block duration |  $(K+\upsilon)/B=102.4$ ms
Carrier frequency |  ${f_{\rm{c}}}=28$ kHz
# of OFDM blocks | $N=720$

###  III-B The Estimated PDP

In Fig. 4, we compare the estimated SCIRs and PDPs for Hyd-1 obtained by using
the FD and TD estimators. For the TD method, the observation window size is
chosen equal to the OFDM block length
(${L_{{\rm{win}}}}={L_{{\rm{blk}}}}=512$) and the channel length is equal to
half of the window size (${M}=256$). As can be seen, both methods lead to
approximately the same results. The estimated PDP has three components:

1) A principal path at $3$ ms which, according to the experimental set-up and
the SSP, is caused by the $5$ m long direct path between the transducer and
the hydrophone.

2) The second largest return at $6$ ms, is the first bounce from the water
surface, which travels a total of $9$ m from the transducer to the hydrophone.
The surface bounce path is about $9.3$ dB below the direct path.

3) The remaining indirect paths between $14$ ms and $35$ ms are caused by
reflections from the bottom.

In Fig. 5, we present the results for Hyd-2. The estimated SCIRs and PDPs are
exactly what we expect based on the position of Hyd-2. It is obvious that the
second main return is from the first bounce off the surface, which arrives
after the bottom reflections because Hyd-2 is closer to the bottom. This
result indicates that, although the first surface bounce has more delay than
the bottom reflections, it is still the dominant indirect path of the SCIR. As
a result, one can conclude that, regardless of the hydrophone’s depth, the
most prevalent indirect path is the first bounce from the surface.

Figure 4: The estimated SCIR for Hyd-1 using (a) FD method and (b) TD method.
The estimated PDP using (c) FD method and (d) TD method. In the TD method, the
window size is ${L_{{\rm{win}}}/B}=104.2$ ms. Figure 5: The estimated SCIR for
Hyd-2 using (a) FD method and (b) TD method. The estimated PDP using (c) FD
method and (d) TD method. In the TD method, the window size is
${L_{{\rm{win}}}/B}=104.2$ ms.

###  III-C The Estimated ACF and COT

In this section, we use the estimated SCIRs to obtain the ACF and COT of the
channel during the experiment. For Hyd-1, Fig. 6 indicates the normalized ACF
for the two main paths in Fig. 4, including the direct path (at $3$ ms) and
the first surface bounce path (at $6$ ms). Notice that the estimated ACFs for
each method are identical; however, as discussed in Section II-F, the ACF
obtained from the TD method is more accurate (for the FD method, the accuracy
is ${L_{{\rm{blk}}}}/B=104.4$ ms; while, for the TD method it is $1/B=0.2$
ms). Based on these results, the direct path is stable and almost constant
over the observation period; on the other hand, the surface bounce path is
changing very rapidly. Since the ACF achieved by using the TD method is more
accurate, it can be used to calculate the precise COT. Thus, by looking at
Fig. 6-b, the COT for the surface bounce is ${\eta_{{\rm{co}}}}/B\approx 72$
ms. For Hyd-2, the normalized ACFs are given in Fig. 7. In this case, again
the direct path is stable and the surface bounce path is rapidly time-varying,
with the COT equal to $\eta_{{\rm{co}}}/B\approx 74$ ms.

In summary, the direct path is relatively stable because the transmitted
signal travels a straight and short way to the hydrophone; however, the
surface bounce path travels a longer way and is affected by surface
fluctuations and the inhomogeneity of the water. To obtain more intuition
about the rates of change, Fig. 8 gives a polar plot of the direct and surface
bounce paths in Hyd-1 and Hyd-2, over about $2$ s. Clearly, the direct path
changes very slowly; in contrast, the surface bounce is changing very rapidly.

Figure 6: The estimated normalized ACF of the direct and surface bounce paths
for Hyd-1 using (a) FD method and (b) TD method. In the TD method, the window
size is ${L_{{\rm{win}}}/B}=104.2$ ms. Figure 7: The estimated normalized ACF
of the direct and surface bounce paths for Hyd-2 using the (a) FD method and
(b) TD method. In the TD method, the window size is ${L_{{\rm{win}}}/B}=104.2$
ms. Figure 8: Polar plot of the direct and surface bounce paths for the SCIR
estimated by the FD method (a) Hyd-1 and (b) Hyd-2.

###  III-D Discussion

There are two important points about the results that require some discussion:

1) As mentioned before, the COT of the surface bounce is very short, even
shorter than the OFDM block length in the FD method and the window size in the
TD method (note that we chose ${L_{{\rm{win}}}}={L_{{\rm{blk}}}}$). This means
that the assumption of a constant channel over ${L_{{\rm{blk}}}}$ and
${L_{{\rm{win}}}}$, made in Section II-C and D, is not accurate. However,
recall that we employed these methods just to estimate the statistical
characteristics of the SCIR. Thus, because of long-term averaging, this
inaccuracy does not affect the estimation of PDP, ACF and COT. To show that
the estimated characteristics are not significantly affected by the
aforementioned issue, we shortened the window size in the TD method from
${L_{{\rm{win}}}}=512$ to ${L_{{\rm{win}}}}=180$ (and accordingly, the channel
length is taken as ${M}={L_{{\rm{win}}}}/2=90$). Since ${L_{{\rm{win}}}}/B=36$
ms, it is guaranteed that the surface bounce path remains relatively constant
over the observation window and the assumption of a steady channel over the
observation window is accurate.

The corresponding PDP and ACF results for Hyd-1 are presented in Fig. 9.
According to Fig. 9-a, the PDP is exactly what we expect and in Fig. 9-b the
COT of the surface bounce path is similar to what we previously obtained
${\eta_{{\rm{co}}}}/B\approx 72$ ms. The approaches in Fig. 9 verify the
statement that, although the assumption of a constant channel over the OFDM
block length and the window size in Section III-B and C is not accurate, it
does not affect the SCIR statistical characterization significantly.

2) According to the results in this section, the direct path is strong and
stable and, as such, easily eliminated. However, the second main path, caused
by the surface bounce, is varying rapidly and needs more efforts to track so
that one can cancel the SI. This behavior is also reported for remote UWA
channels in half-duplex systems in [21]; however, it must be noted that in
UWA-IBFD, because the SI is high power (compared to the weak remote signal),
variations in the SCIR are more critical. To investigate how different paths
contribute to the SI power, assume that the transmitted signal, $x[n]$, is
i.i.d. with power $P_{x}=1$. According to (1), the SI signal power is then
given as [22]

$\begin{array}[]{l}{P_{s}}{\rm{=}}{{\rm{E}}}\left\\{{{{\left|{s[n]}\right|}^{2}}}\right\\}=\frac{{{P_{x}}}}{{{M}}}\sum\limits_{m=0}^{{M}-1}{{{\rm{E}}}\left\\{{{{\left|{h[m,n]}\right|}^{2}}}\right\\}}{}{}{}{}{}{}\\\
\,\,\,\,\,\,=\frac{1}{{{M}}}\sum\limits_{m=0}^{{M}-1}{p[m]}.\end{array}$ (14)

where $p[m]$ denotes the PDP of SCIR as defined in Section II. To investigate
the contribution of the SCIR delay paths to the total SI power, as in (14), we
define the accumulated power of SI as

${p_{{\rm{acc}}}}[j]=\frac{1}{M}\sum\limits_{m=0}^{j-1}{p[m]},$ (15)

which denotes the power of SI corresponding to the delay paths from $m=0$ to
$m=j-1$. In this regard, Fig. 10 shows ${p_{{\rm{acc}}}}[j]$ for Hyd-1 using
the estimated PDP from Fig. 4. As can be seen, about $72\%$ of the SI power is
from the direct path. In other words, $72\%$ of the SI is easily eliminated.
On the other hand, the surface bounce path includes $16\%$ of the SI power,
which is difficult to track. These results indicate that, if the UWA-IBFD
system can accurately track just the direct and surface bounce paths, it can
eliminate $88\%$ of the SI. For Hyd-2, this percentage is also about $88\%$.
It is noteworthy that because of the rapid changes, the well-designed half-
duplex UWA OFDM channel estimators in [14, 15, 16, 23] may not lead to an
accurate SCIR estimation in UWA-IBFD because in these methods the channel is
assumed constant over one OFDM block, which we have shown is not correct.

Figure 9: (a) Estimated PDP and (b) ACF of direct and surface bounce paths
using the TD method with window size ${L_{{\rm{win}}}/B}=36$ ms. Figure 10:
Accumulated power of SI signal for Hyd-1.

##  IV Summary

In this paper, we characterize the self-interference channel impulse response
of UWA-IBFD system, using an OFDM signal in both the frequency and time
domains. We use the experimental data recorded in a lake water and determine
the power delay profile, the autocorrelation function and the coherence time.
According to our results, regardless of the depth of the hydrophones, there
are two principal paths in the self-interference channel, including the direct
and surface bounce paths. These two paths contain about $88\%$ of the self-
interference power. We show that the direct path is powerful and stable and
easily eliminated; however, the surface bounce path is weaker and changing
rapidly. Having a very short coherence time, the surface bounce path requires
more efforts for cancellation.

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