1. 742 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 55, NO. 2, FEBRUARY 2017
ToA Ranging and Layer Thickness Computation
in Nonhomogeneous Media
Mohsen Jamalabdollahi, Student Member, IEEE, and Seyed Zekavat, Senior Member, IEEE
Abstract—This paper introduces a novel and effective ranging
approach in nonhomogeneous (NH) media consisting of fre-
quency dispersive submedia via time-of-arrival (ToA). Here, the
NH environment consists of sublayers with a specific thickness
that is estimated throughout the ranging process. First, a novel
technique for ToA estimation in the presence of frequency
dispersive submedia via orthogonal frequency division multi-
ple access subcarriers is proposed. In the proposed technique,
preallocated orthogonal subcarriers are utilized to construct a
ranging waveform that enables high-performance ToA estimation
in dispersive NH media in the frequency domain. The proposed
ToA technique is exploited for multiple ToA measurements at
different carrier frequencies, which leads to a system of linear
equations that can be solved to compute the thickness of the
available submedia and calculate the range. Simulation results for
underwater-airborne media and underground channel confirm
that the proposed technique offers high-resolution ranging at
different signal to noise ratio regimes in the NH media.
Index Terms—Angle-of-arrival, frequency dispersion, nonho-
mogeneous (NH) media, orthogonal frequency division multiple
access (OFDMA), ranging, relative permittivity, time-of-
arrival (ToA).
I. INTRODUCTION
RANGE measurement is desired in a variety of appli-
cations in free space such as digital communications,
wireless sensor network, radar, and wireless indoor and
outdoor localization [1]–[7]. Ranging also has diverse applica-
tions in nonhomogeneous (NH) media, mostly for localization
purposes. Here, the term NH refers to a medium composed
of diverse frequency dispersive submedia such as human
body organs, water, soil, rock, and their combinations with
free space. Capsule endoscopy positioning [8], underwater
localization using airborne sensors [9], or underground target
localization via ground penetrating radar (GPR) [10]–[12]
are a few applications of ranging in NH media.
Range-based localization is possible via received signal
strength [13], the signal time-of-arrival (ToA) [14], time-
difference-of-arrival [15], and direction of arrival (DoA) [16]
or their combinations [17]. Range-free methods utilize connec-
tivity information [18]. Although some range-free approaches
Manuscript received June 9, 2016; revised August 11, 2016; accepted
September 26, 2016. Date of publication October 18, 2016; date of cur-
rent version December 29, 2016. This work was supported by National
Science Foundation under Award ECCS 1101843. (Corresponding author:
Mohsen Jamalabdollahi.)
The authors are with the Department of Electrical and Computer Engineer-
ing, Michigan Technological University, Houghton, MI 49930 USA (e-mail:
mjamalab@mtu.edu; rezaz@mtu.edu).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TGRS.2016.2614263
offer simple methods with acceptable performance, they have
limitations such as network topology and ranging accuracy,
which justify the employment of range-based approaches [19].
Among all the proposed range-based approaches, ToA esti-
mation has received considerable attention because of high
precision and low complexity [13], [14]. ToA estimation
techniques are divided into two different categories, the tra-
ditional matched filter-based techniques, and superresolution
techniques. The former incorporates a predesigned waveform
with autocorrelation properties close to the delta function
at the output of the matched filter. In the latter, ToA is
calculated via maximizing the pseudospectrum of the cor-
responding signal subspace achievable via decomposition of
the matched filter output in the frequency domain [20], [21].
Examples of superresolution techniques are independent com-
ponent analysis [22], maximum likelihood (ML) [23], multiple
signal classification (MUSIC) [24], and estimation of signal
parameter via rotational invariance technique (ESPRIT) [25].
However, the resolution of all the proposed techniques depends
on the transmitted waveform bandwidth [22].
Incorporating wideband waveforms in free space is a feasi-
ble remedy to achieve high ToA resolution. However, increas-
ing the bandwidth of transmitted waveform in frequency
dispersive channels intensifies frequency dispersion [26]–[28].
In the microwave terminology, frequency dispersion refers
to the principle that different frequency components of a
transmitted electromagnetic (EM) waveform propagates with
different velocities due to the frequency dependence of the
medium’s relative permeability and/or permittivity. Therefore,
increasing the transmitted waveform’s bandwidth prompts a
higher frequency dispersion that alleviates the nonlinear dis-
tortion of the transmitted waveform. The nonlinearity nature
of the imposed distortion by frequency dispersion leads to a
received signal that is barely detectable by the matched filter
of transmitted waveform, which dramatically degrades the
ToA estimator performance [27].
To the best of the authors’ knowledge, ranging via high-
resolution ToA estimation exploiting wideband microwave sig-
nals in NH media or single medium in the frequency dispersion
band is an open problem. Many researchers propose acoustics
communication for ToA estimation in underwater or under-
ground media [29]. Although acoustics communication offers
high resolution for underwater ToA estimation, it cannot be
applied to underwater-airborne channels due to strong reflec-
tions and attenuation in the water/air boundary [30]. A few
works [10]–[12], [31]–[33] propose ToA estimation in disper-
sive medium for specific scenarios that cannot be extended
to the NH media. In [10]–[12], time delay estimation of
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2. JAMALABDOLLAHI AND ZEKAVAT: ToA RANGING AND LAYER THICKNESS COMPUTATION IN NH MEDIA 743
buried target echoes for GPR is addressed, while the received
echoes do not represent frequency dispersion assuming
low conductivity and small layer thickness of submedia.
Although these assumption can be feasible for dry media,
when the water content of media exceeds 10% by weight,
the frequency dispersion must be considered due to the
dielectric relaxation of water [26]. In [31], the transmitter
location is estimated by combining the measured ToA data
with the knowledge about the shape and position of the
medium, which is not a feasible assumption in NH media.
Sahota and Kumar [32] propose sensor node localization via
ToA measurements but the procedure of ToA estimation in soil
as a frequency dispersive medium is not discussed. In [33],
ToA estimation of short-range seismic signals in dispersive
environments is addressed; however, the system model does
not represent the frequency dispersion due to low bandwidth
of applied seismic signals.
In this paper, first, high-resolution ToA estimation in
NH media consisting of diverse frequency dispersive submedia
is addressed. The proposed technique exploits preallocated
orthogonal frequency division multiple access (OFDMA) sub-
carriers to construct a wideband ranging signal. Here, we show
that each frequency component of the propagated waveform is
received with different time delays and phases, which dramati-
cally increases the number of unknowns in the received signal
system model. Then, we propose a novel approach that reduces
the number of unknowns by linear approximation of imposed
delays and phases to the allocated OFDMA subcarriers in the
frequency domain. These justified approximations replace the
imposed delays with the delay corresponding to the waveform
spectrum center as the desired unknown, which is estimated
exploiting ML estimator. Then, the proposed ToA measure-
ment technique exploits different carrier frequencies combined
with DoA measurements to construct a system of linear equa-
tions as a function of the thickness of the available submedia.
Once the thicknesses of the available submedia are estimated,
the straight-line range between the transmitter and the receiver
is achievable, accordingly.
Here, underwater-airborne channels and underground chan-
nel consisting of several layers with different water con-
tent exploiting wideband microwave signals are considered
as examples of NH media. The proposed NH media are
simulated according to the state-of-the-art channel and prop-
agation models for seawater [28] and underground [34]–[36].
Simulation results prove the feasibility and efficiency of the
proposed technique in terms of ToA resolution and ranging
error compared with the allocated bandwidth of the transmitted
waveform.
The rest of this paper is organized as follows. Section II
introduces the proposed ToA estimation technique. The
proposed ranging approach is presented in Section III.
Section IV represents the simulation results and discussions
and finally Section V concludes this paper.
II. HIGH-RESOLUTION ToA ESTIMATION
A. System Model
The process of ToA estimation initiates with the transmis-
sion of ranging waveform. Here, 2K OFDMA subcarriers
are utilized to construct the ranging waveform corresponding
to
sn =
1
√
N
2K
k=1
Skej2πmk f nTs
, n = 1, 2, . . . , N (1)
where N and Ts are the total number of available subcarriers
or the number of waveform samples, and sampling interval,
respectively, which leads to a ranging waveform of length NTs.
Moreover, mk, f , and Sk are the frequency index of the
kth subcarrier, the OFDMA subcarrier spacing, and the
kth frequency-domain symbol known at both the transmitter
and receiver, respectively. Here, random Quadrature Phase
Shift Keying symbols are allocated to Sk, which leads to a
feasible peak to average power ratio.
To maintain orthogonality across subcarriers, subcarrier
spacing must satisfy f = 1/(NTs) which leads to
sn =
1
√
N
2K
k=1
Skej
2πnmk
N , n = 1, 2, . . . , N (2)
where N, f , Ts, Sk, and mk and K are defined in (1).
In (1), 2K symmetric subcarriers are allocated to construct
the ranging waveform, where mk is the frequency index of
the kth subcarrier such that
mk =
K, K − 1, . . . , 1 for k = 1, 2, . . . , K
N − K, . . . , N − 1 for k = K + 1, . . . , 2K
(3)
where N is defined in (1) and 2K denotes the number of
allocated subcarriers. Here, symmetric subcarriers are selected
to construct a uniform waveform in the frequency domain,
which enables a linear approximation of the imposed delay
to each subcarrier (see Remark 1). The sequence of baseband
waveform samples detailed in (2) are passed through a dual
channel digital-to-analog converter, with sampling interval Ts
to construct the analog form of ranging waveform. The created
baseband signal is transmitted through the NH channel after
upconversion to the carrier frequency fc. The received signal
corresponding to the transmitted waveform in NH channel
consisting of several frequency dispersive submedia can be
represented by the aggregation of the delayed version of
frequency dispersed waveforms from L propagation paths,
that is
r(t) =
L−1
i=0
hi ˜si (t − τi) + v(t) (4)
where r(t) and ˜si (t) are the received signal and the frequency
dispersed version of the ranging waveform propagated at the
ith path of NH medium with channel impulse response (CIR)
{hi }L−1
i=0 . Moreover, τi and v(t) are the time delay of prop-
agation at ith path and additive white zero mean Gaussian
noise, respectively. Here, the line-of-sight (LoS) path is con-
sidered for the received signal due to very high attenuation
corresponding to the reflected copies of transmitted waveform
through non-LoS paths (see Section IV-A for more details on
simulated media) which leads to
r(t) = h˜s(t − τ0) + v(t) (5)

3. 744 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 55, NO. 2, FEBRUARY 2017
where r(t) and ˜s(t) are the received signal and the frequency
dispersed version of the ranging waveform propagated from
the LoS path of NH medium with complex value CIR, h.
Moreover, it is assumed that the relative permeability of
all submedia is equal to unity; however, the relative permit-
tivity changes with the frequency of transmitted waveform.
Pahlavan et al. [27] have discussed the challenges of ToA esti-
mation in media consisting of frequency dispersive submedia
such as the human body. Regarding the dependence of propa-
gation velocity on relative permittivities of available submedia,
(ν( f ) = c/(εr( f ))1/2
) (where c denotes the universal con-
stant speed of light), frequency dispersion of the transmitted
waveform at receiver is inevitable. This leads to a receiving
waveform with higher frequency components that arrive faster
than lower frequency components as the propagation velocity
is the function of the relative permittivity [27]. Thus, for the
dispersed version of the ranging waveform we can write
˜sl(t) =
1
2|ωK |
ωK
−ωK
S(ω)ejω(t−τ(ω))
dω (6)
where S(ω) is the Fourier transform of the transmitted wave-
form s(t), and τ(ω) and 2|ωK | are the time delays imposed by
the LoS path into the waveform component with a frequency
of ω and the bandwidth of transmitted waveform, respectively.
Applying (5) to the dual channel analog-to-digital converter
with sampling interval Ts, leads to
yn =
h
√
N
2K
l=1
Slej
2πml(nTs −τl)
NTs + vn (7)
where N, Ts, Sl, and ml and K are defined in (1) and (3),
respectively. Moreover, yn and vn are the nth sample of
the baseband received signal and additive, white zero-mean
Gaussian noise, respectively, and τl is the delay corresponding
to the lth frequency component propagated from the LoS path.
According to the ToA definition, it is desired to estimate the
delay corresponding to the first arrived component (ml = K)
of the transmitted signal propagated from the shortest path,
which is equivalent to the value of τK . However, here,
a novel technique is proposed, which exploits frequency
domain analysis to estimate τ0 that represents the delay corre-
sponding to the spectrum center of the transmitted waveform
ml = 0, propagated from the LoS path. In Section III, this
value is exploited to estimate the range between the transmitter
and receiver.
B. Proposed ToA Estimation Technique
This section presents the proposed ToA estimation technique
considering that Nr samples of the received signal proposed
in (7) are available at the receiver. Here, the receiver selects Nr
such that the maximum possible range is covered. Considering
the low range propagation of microwave in dispersive
media such as water and soil, here it is assumed that the
τ0 ≤ (N − 1)Ts, which is a feasible assumption considering
the proposed values of N in Section IV.
Considering N possible samples corresponding to propaga-
tion delay and signal expansion due to frequency dispersion,
the receiver selects Nr = 3N samples from the time origin
and converts them into the frequency domain applying discrete
Fourier transform. Here, the time origin is achievable for
both the transmitter and receiver via clock synchronization or
more feasible techniques such as round-trip [37]. Therefore,
2K samples of the received signal in frequency domain are
achievable via
R = ˜Wy (8)
where ˜W = [W, W, W]2K×3N , for W = [wT
1 ; wT
2 ; . . .;
wT
2K ]2K×N and wk is a column vector such that
wk = [e− j((2πmk)/N), e− j((2π2mk)/N), . . . , e− j((2π Nmk)/N)]T .
Moreover, y is 3N samples of the received signal in time
domain defined by y = [y1, y2, . . . , y3N−1]T where yn is
defined in (7).
Considering (7) as the time domain system model for the
received signal, the system model for kth element of the
received signal at frequency domain achievable via (8) can
be written as
Rk =
h
N
N−1
n=0
2K
l=1
Slej
2πml (nTs−τl)
NTs e− j
2πnmk
N + Vk (9)
where N, Ts, Sk, and mk and K are defined in (1) and (3),
and Vk is the kth sample of additive noise at the frequency
domain. Moreover, τl is the delay corresponding to the lth
transmitted subcarrier received from the LoS path. Considering
the orthogonality of transmitted subcarriers (ej((2πnml)/N)) and
columns of ˜W and applying some mathematical manipulation,
(9) corresponds to
Rk = Skhe− j
2πmkτk
NTs + Vk (10)
where N, Ts, Sk, and mk and K are defined in (1) and (3),
and τk and Vk are defined in (9). The proposed system model
in (10) represents 2M nonlinear equations corresponding to
2M allocated subcarriers, each containing (2M +1) unknowns
(2M for τk, for k ∈ κ and one for h). The very fact that
solving a system of 2M nonlinear equations for (2M + 1)
unknowns leads to infinitely many solutions (or no feasible
solution) demands for reducing the number of unknowns.
Considering (10) as the system model for the received sam-
ples in frequency domain and known transmitted OFDMA
symbols (Sk), the receiver calculates Xk = Yk/Yk+K for
Yk = Rk/Sk, which leads to
Xk =
e− j
2πmkτk
NTs
e− j
2πmk+K τk+K
NTs
+ ˜Vk, ∀k = 1, 2, . . . , K (11)
where N, Ts, and τk are defined in (1) and (10), respectively.
Moreover, ˜Vk is the additive measurement noise defined
by ˜Vk = Vk/e− j((2πmk+K τk+K )/(NTs )). Substituting mk+K by
N − mk according to (3), and applying some mathematical
manipulations, (11) can be written as
Xk = e− j
2πmk(τk+τk+K )
NTs ej
2πτk+K
Ts + ˜Vk (12)
where N, Ts , τk, and ˜Vk are defined in (1), (10), and (11),
respectively. Here, a set of K-independent equations includ-
ing 2K unknowns are available, which leads to infinitely
many solutions (if available). However, it can be shown that

4. JAMALABDOLLAHI AND ZEKAVAT: ToA RANGING AND LAYER THICKNESS COMPUTATION IN NH MEDIA 745
(see Remark 1) the imposed delays to transmitted subcarriers
can be approximated as a line that enables replacing 2M
undesired unknowns with one desired unknown.
Remark 1: The 2M unknowns in (12) corresponding to the
imposed delay by frequency dispersion to the kth subcar-
rier (τk), can be approximated by ˆτk = a(k − 1) + b such
that |τk − ˆτk| ≤ ηTs, k = 0, . . . , 2K − 1, where
a =
1
c
M
m=1
rm(
√
εr,m,K+1 −
√
εr,m,K ) (13a)
b =
1
c
M
m=1
rm
√
εr,m,1 (13b)
and c denotes the universal speed of light, Ts and K are defined
in (1) and (3), respectively, and η is an arbitrary small number
η < 1. Moreover, εr,m,k and rm are the relative permittivity
of the mth media at the kth subcarrier frequency, and its
corresponding propagation range, respectively.
Substituting the actual and approximated values of delays
proposed in Remark 1 into the approximation error bound
(|τk − ˆτk| ≤ ηTs), it can be shown that given any NH media
consisting of M submedia satisfying
1
c
M
m=1
rm
√
εr,m,k −
M
m=1
rm[(
√
εr,m,K+1 −
√
εr,m,K )k
+
√
εr,m,1] ≤ ηTs, k = 0, . . . , 2K − 1. (14)
Remark 1 holds. Here, the slope of approximated line is chosen
according to the slope of the spectrum center of the actual
delays curve τK+1 − τK , while the constant value is selected
to approach the approximated line to the value of τ1. Assuming
all imposed delays (τk) are placed on the approximated line,
the average of all equally spaced delays from line center,
i.e., τk and τk+K are the same and equal to τ0, which is
defined as the delay corresponding to the spectrum center
of the transmitted waveform or the desired ToA. Therefore,
2K unknown delays in (12) can be replaced by applying
(τk + τk+K ) = 2τ0, which leads to
Xk = e− j
4πmkτ0
NTs ej
2πτk+K
Ts + ˜Vk (15)
where N, Ts , τk and ˜Vk are defined in (1), (10), and (11),
respectively. The proposed system model in (15) forms K non-
linear and noisy equations containing one desired unknown τ0,
and K undesired unknowns τk+K for k = 1, 2, . . . , K.
Substituting τk+K with its corresponding coarse and fine
delays such that τk+K = (nk+K + δk+K )Ts, gives
Xk = e− j
4πmkτ0
NTs ej2π(nk+K +δk+K )
+ ˜Vk
= e− j
4πmkτ0
NTs ej2πδk+K + ˜Vk, k = 1, 2, . . . , K (16)
where N and Ts are defined in (1), τ0 is defined in (15), and δk
is the fine delay corresponding to the kth transmitted subcarrier
such that 0 ≤ δk ≤ 1. Although the values of fine delays (δk)
are negligible in terms of delay, ignoring them dramatically
degrades the system model accuracy due to the impact of the
exponential term, or ej2πδk+K . Remark 2 proposes a linear
approximation of the fine delays δk+K for k = 1, 2, . . . , K,
which reduces the number of unknowns proposed in (16).
Here, an approximation in the form of amk + b (for any
constant a and b) is desired, which satisfies an acceptable error
such that |δk+K − ˆδk+K | ≤ ηTs, k = 1, 2, . . . , K. Exploiting
the form amk + b, the proposed system model in (16) is
converted into an exponential term corresponding to a single
frequency that is simply detectable via ML estimator.
Remark 2: The fine delays in (16) can be approximated
linearly as ˆδk+K
∼= −β (mk − 1) + δ2K , ∀k = 1, 2, . . . , K,
such that |δk+K − ˆδk+K | ≤ ηTs, where
β =
1
c
M
m=1
rm(
√
εr,m,K+2 −
√
εr,m,K+1) (17)
and Ts, and K and mk are defined in (1) and (3), respectively,
and c, η, εr,m,k, and rm are defined in (14).
Substituting the actual and approximated values of fine
delays in the approximation error bound regarding Remark 2
(|δk+K − ˆδk+K | ≤ ηTs), it can be shown that given any
NH media consisting M submedia satisfying
τk+K −
τk+K
Ts
Ts +
1
c
M
m=1
rm(
√
εr,m,K+2 −
√
εr,m,K+1)
× (mk − 1) − δ2K ≤ ηTs ∀k = 1, 2, . . . , K (18)
given τk+K = (1/c) M
m=1 rm(εr,m,k+K )1/2
, Remark 2 holds.
Here, the slope of approximated line proposed in Remark 2 is
selected according to the slope of the actual fine delay curve
at the first point δK+2 − δk+1, while the constant value is
selected to approach the approximated line to the last value
of actual fine delay curve δ2K . Exploiting Remark 2, the term
ej2πδk+K in (16), can be approximated by x0e− j((4πmk)/N)β,
for x0 = ej2π(β +δ2K ) and β = ((Nβ )/2). Substituting
x0e− j((4πmk)/N)β into (16), the final approximation of the
proposed system model for the received signal in the frequency
domain corresponds to
Xk = x0e− j
4πmkτ0
NTs e− j
4πmk
N β
+ ˜Vk
= x0e− j
2πmk
N [2(α+β)]
+ ˜Vk ∀k = 1, 2, . . . , K (19)
where α = (τ0)/(Ts), and β, denoting that two unknowns need
to be estimated to reveal τ0.
Defining n(1) = 2(α + β), it can be shown that
(see Appendix A), the ML estimator of n(1) can be
written as
ˆn(1)
= argmax
n
wT
n x (20)
where wn = [ej((2π Kn)/N), ej((2π(K−1)n)/N), . . . , ej((2πn)/N)]T
is K × 1 column vector and x = [X1, X2, . . . , XK ]T for Xk
defined in (19). Exploiting (20), one can estimate (α + β),
however, at least one linearly independent equation in terms of
α and β is required to estimate the desired ToA via τ0 = αTs.
Here, we propose repeating the entire measurement procedure
applying the same number of allocated subcarriers (2K)
and waveform samples (N), while the sampling interval

5. 746 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 55, NO. 2, FEBRUARY 2017
is changed, for instance to T
(p)
s = pTs, where T
(p)
s is the
new sampling interval at the pth (p ≥ 2) measurement.
Considering static channel where the propagation paths and
hence their corresponding delays do not change during P
measurements, the delay corresponding to the spectrum center
of the transmitted waveform τ0, is the same at each mea-
surement, however, the time delays corresponding to other
frequency components change accordingly. This fact reveals
another advantage of estimating the time delay corresponding
to the spectrum center of transmitted waveform in NH media.
Applying T
(p)
s = pTs to (16), the frequency-domain
samples at the pth measurements can be written as
X
(p)
k = x
(p)
0 e
− j
4πmkτ0
NT
(p)
s e− j
4πmk
N β(p)
+ ˜V
(p)
k (21)
where N and τ0 are defined in (1) and (15), respectively,
and T
(p)
s and x
(p)
0 are the applied sample time at transmitter
and receiver, and the constant term x0 = ej2π(β +δ2K ) at the
pth measurement. Writing (17) for β(p) in terms of T
(p)
s
and substituting T
(p)
s = pTs leads to β(p) = β/p. Applying
T
(p)
s = pTs and β(p) = β/p to (21), leads to
X
(p)
k = x
(p)
0 e
− j
4πmkτ0
NpTs e
− j
4πmk
Np β
+ ˜V
(p)
k
= x
(p)
0 e
− j
4πmk
Np α+ β
p
+ ˜V
(p)
k ∀k = 1, 2, . . . , K (22)
where N, K, and mk, and α, β are defined in (9) and (20),
respectively. Therefore, considering P independent measure-
ments applying T
(p)
s = pTs, for p = 1, 2, . . . , P, and
λ = [α, β]T leads to
λ = ( T
)−1 T
n (23a)
=
1 1/2 . . . 1/P
1 1/4 . . . 1/P2 (23b)
n =
ˆn(1)
2
,
ˆn(2)
2
, . . . ,
ˆn(P)
2
T
(23c)
where
ˆn(p)
= argmax
n
wT
n x(p)
(24)
and wT
n is defined in (20).
The performance of the proposed ToA estimator depends on
the precision of the ML estimator in (24), which estimates the
value corresponding to 2(α + (β/p)). However, the returned
value by (24) is the time index and hence 0 ≤ ˆn(p) ≤ N − 1;
meanwhile, the value of 2(α + β) can be a negative num-
ber or larger than N. In this case, the estimated value of
2(α + (β/p)) via (24) would be different from its actual
value. In order to resolve this issue, the estimated ToA via
ˆτ0 = αTs needs to be inspected regarding its acceptable range
0 ≤ ˆτ0 ≤ (N −1)Ts as follows. If 2 (α + β) > N, the returned
value by (24) would be 2 (α + β) − N. Applying this value
to (23c) shifts the estimated α by N point to the left, which
leads to α < 0. Thus, if α < 0, (23a) should be solved again
using
n =
ˆn(1) + N
2
,
ˆn(2)
2
, . . . ,
ˆn(P)
2
T
(25)
Algorithm 1 Summary of the Proposed ToA Estimation
Technique in NH Medium
Require: y, Sk, ∀k = 1, 2, .., 2K
1: return τ0
2: for p = 1, 2, . . . , P do
3: Compute frequency domain samples, R using (8).
4: Compute X
(p)
k = Y
(p)
k /Y
(p)
k+K for Y
(p)
k = R
(p)
k /Sk, ∀k =
1, 2, . . . , K.
5: Compute ˆn(p) using (24).
6: end for
7: Compute λ using (23a)–(23c).
8: if α < 0 then
9: Compute λ using (23a),(23b), and (25).
10: else if α ≥ N then
11: Compute λ using (23a), (23b), and (26).
12: end if
13: ˆτ0 = αTs.
where n
(p)
s and ˆn(p) are defined in (23b) and (24), respectively.
If 2 (α + β) < 0, the returned value by (24) would be
N − 2 (α + β). Applying this value into (23c) shifts the
estimated α by N point to the right which leads to α > N.
Thus, if α > N, (23a) should be solved again using
n =
N − ˆn(1)
2
,
ˆn(2)
2
, . . . ,
ˆn(P)
2
T
(26)
where ˆn(p) is defined in (24). Algorithm 1 details the proposed
high-resolution ToA estimation technique in NH medium
considering ToA measurements applying P different sampling
interval. In the next section, a novel technique exploits the
proposed ToA estimation approach to reveal the range between
the transmitter and receiver nodes.
III. RANGING IN NH MEDIA
Due to varying EM features in NH media, regardless of the
error imposed by EM dispersion, multipath and noise, the ToA
measurement proposed in Section II-B cannot reveal range
between the transmitter and receiver. Fig. 1 depicts an example
where there is an NH channel between the transmitter and
the receiver. Here, the addition of propagation path vectors at
each submedia (ri) is greater than the straight-line propagation
path (r) between the transmitter and receiver.
Moreover, the propagation speed at each medium (vi) is
lower than the free space speed of light (c), which leads to
higher measured ToA. In addition, based on Snell’s law, the
angle of the EM wave changes at the media boundary. Thus,
there is no guarantee that the last measured incident angle at
the receiver ˆθM will be equal to the straight-line propagation
angle, θ. Hence, a straight-line range measurement procedure
needs to be applied, exploiting multiple measured ToA.
A. System Model
Fig. 1 depicts the 3-D model for signal propagation in NH
media. Here, we intend to estimate the range between the
transmitter and receiver, which is the straight-line distance

6. JAMALABDOLLAHI AND ZEKAVAT: ToA RANGING AND LAYER THICKNESS COMPUTATION IN NH MEDIA 747
Fig. 1. 3-D model of signal propagation in NH media.
referred to as r in Fig. 1. The system model for estimated ToA
via Algorithm 1, can be written as a function of propagated
ranges such that
ˆτ0 =
1
c
M
m=1
rm
√
εr,m + vτ (27)
where ˆτ0, rm, εr,m, and vτ are the estimated ToA, the range of
propagation path and the relative permittivity in mth submedia,
and ToA measurement error, respectively, and c and M are
defined in (14). It can be observed that the proposed system
model in (27) cannot be solved for the desired range r,
regarding M available unknowns {rm}M
m=1. In order to address
this problem, we propose to exploit a frequency-test scheme by
repeating the ToA measurement scenario applying Q different
carrier frequencies. Employing this technique, each ToA mea-
surement provides an independent equation as a function of
propagated ranges as the EM features of available submedia
change by the applied carrier frequency. The qth measured
ToA can be written as
ˆτ
(q)
0 =
1
c
M
m=1
r(q)
m ε
(q)
r,m + v(q)
τ (28)
where ˆτ
(q)
0 , r
(q)
m , ε
(q)
r,m, and v
(q)
τ are the estimated ToA,
the range of propagation path and relative permittivity in
mth submedia and ToA measurement error, all at qth mea-
surement, respectively, and c and M are defined in (14).
It can be observed that each equation in (28) contains a
different set of unknowns or r
(q)
m for m = 1, 2, . . . , M and
q = 1, 2, . . . , Q, which leads to a set of Q independent equa-
tions containing M Q unknowns. However, these unknowns
can be replaced by the thickness of available submedia to
construct a set of Q independent equations as a function
of M available unknowns zm for m = 1, 2, . . . , M as follows.
Applying r
(q)
m = zmcos(θ
(q)
m ), where θ
(q)
m is the inci-
dent angle of mth submedia at qth carrier frequency,
Snell’s law (sin(θ
(q)
m )(ε
(q)
r,m+1)
1/2
= sin(θ
(q)
m+1)(ε
(q)
r,m)
1/2
), and
cos(sin−1(x)) = (1 − x2)
1/2
, (28) can be written as
ˆτ
(q)
0 =
1
c
M
m=1
zm ε
(q)
r,mε
(q)
r,M
ε
(q)
r,M − sin2(θ
(q)
M )ε
(q)
r,m
+ v(q)
τ (29)
where ˆτ
(q)
0 , ε
(q)
r,m, and v
(q)
τ are defined in (28), zm denotes
the thickness of mth submedia, and θ
(q)
M denotes the DoA
corresponding to the qth measurement at Mth submedia
(last submedium including receiver) as shown in Fig. 1.
Considering (29), it can be observed that the set Q measured
ToA can be represented as a linear combination of M available
submedia given the measured DoA at each measurement step.
Here, it is assumed that the receiver enjoys array antenna,
which allows DoA estimation to recreate state-of-the art
techniques such as ESPRIT [20] and Root-MUSIC [38].
These approaches are applicable to NH media consisting
of frequency dispersive channels as they utilize a single
subcarrier that is free from frequency dispersion. For more
information about DoA estimation methods and approaches,
we refer the readers to [22].
B. Proposed Technique
Considering Gaussian distribution with zero mean and vari-
ance σ2
v for ToA measurements noise, the ML estimation of
thickness of the available submedia can be derived by solving
the following optimization problem:
argmin
z1,...,zM
Q
q=1
1
σ2
v
ˆτ
(q)
0 −
1
c
M
m=1
zmγ (q)
m
2
(30)
for
γ (q)
m =
ε
(q)
r,mε
(q)
r,M
ε
(q)
r,M − sin2 θ
(q)
M ε
(q)
r,m
(31)
where ˆτ
(q)
0 and ε
(q)
r,m, and θ
(q)
M and zm are defined in
(28) and (29), respectively. The least square solution for the
proposed ML estimator in (30), can be represented by
ˆz = (ϒT
ϒ)−1
ϒT
τ (32)
where the (.)−1 and (.)T operands denote operands for
matrix inversion and transpose, respectively, and ˆz =
[ˆz1, ˆz2, . . . , ˆzM ]T , τ = [ˆτ
(1)
0 , ˆτ
(2)
0 , . . . , ˆτ
(Q)
0 ]T and ϒ =
[ϒ1, ϒ2, . . . , ϒM ]T , for
ϒm = γ (1)
m , γ (2)
m , . . . , γ (Q)
m
T
(33)
where ˆzm denote the estimation of mth submedia’s thickness,
and ˆτ
(q)
0 and γ
(q)
m are defined in (28) and (31), respectively.
Given the estimated thickness of the available submedia
using (32), it can be shown that (see Appendix B), the
propagation range between the transmitter and receiver is
achievable via ˆr = (1/Q)
Q
q=1 ˆr(q) where
ˆr(q)
= D 1 +
⎡
⎣ 1
D
M
m=1
ˆzmsin θ
(q)
M ε
(q)
r,m
ε
(q)
r,M − sin2 θ
(q)
M ε
(q)
r,m
⎤
⎦
2
(34)
for D = M
m=1 zm. Moreover, ε
(q)
r,m and θ
(q)
M , and ˆzm are
defined in (28) and (32), respectively. In the next section, the
performance of high-resolution ToA estimation and range mea-
surement are discussed, considering different NH channels.

7. 748 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 55, NO. 2, FEBRUARY 2017
Fig. 2. Applied NH media. (a) Underwater-airborne considering rough water
surface. (b) Four-layer underground containing different water contents in
volume ρ.
IV. SIMULATIONS RESULT AND DISCUSSIONS
This section discusses the simulation method, parameters,
and results in three parts. First, details of the applied NH media
and transmitted waveform are introduced in Section IV-A.
Then, simulations results for the proposed high resolution ToA
estimation utilizing Algorithm 1 are proposed in Section IV-B.
Finally, the performance of the proposed range measurement
technique is evaluated at different scenarios.
A. Details of Simulated NH Media
In this paper, two different media with diverse physical
characteristics are considered to evaluate the performance of
the proposed ToA and range estimation techniques in different
scenarios. Fig. 2(a) depicts airborne to shallow underwater
as the first NH media. For the second media, a multilayer
underground channel with different water content (ρ) at each
layer is selected as shown in Fig. 2(b). Table I details the
parameters definition, their corresponding symbols, and sim-
ulations applied value for airborne to water and underground
channels. The applied relative permittivities for seawater are
simulated using the Debye model for saltwater [28]. The
applied values of relative permittivities for underground chan-
nel versus carrier frequency are proposed in Table II, using
soil dielectric spectra for different volumetric water contents
in 20 °C proposed in [34].
Although underwater channel modeling including path-loss
measurements and/or multipath characterization are discussed
for underwater acoustic [39] and optical communications [40],
a few works [28], [30] propose channel modeling and path
loss measurements of wideband microwave propagation in
seawater. In [28], the propagation measurements exploiting
TABLE I
SIMULATION PARAMETERS AND APPLIED VALUES
waveform covering the whole band between 800 MHz and
18 GHz is proposed. It can be observed that the attenuation of
the propagated signal in saltwater can be modeled via e−α( f )d,
where α( f ) and d represent the attenuation coefficient as a
function of carrier frequency and propagation range, respec-
tively [28]. Here, the surface of water is considered rough
due to natural effects such as wind-induced waves. As shown
in Fig. 2(a), a rough interface creates multiple resolvable paths
arriving with different incident angles to the receiver. The
paths are resolvable because the difference in ToA is more
than the sampling time. Moreover, a multipath leads to DoA
estimation error. In this paper, we have included the impact
of error in DoA estimation. We consider multipath channels
with five resolvable paths to evaluate the performance of the
proposed technique in the presence of a rough interface. The
delay corresponding to the transmitted waveform spectrum
center of each path is selected uniformly within [τ0, 2τ0],
where τ0 denote the selected delay of the shortest path. The
path attenuation is calculated exploiting the model proposed
in [28].
For underground medium, the LoS path is considered
and the reflected copies from layer boundaries are neglected
due to very high attenuation. Here, the path loss of under-
ground channels is simulated according to the model proposed
in [35] and [36].
B. Performance Analysis
1) ToA Estimation: Fig. 3(a)–(d) depicts the average error
of estimated ToA for underwater-airborne medium consider-
ing multipath channels. Details of simulation parameters and
underwater-airborne channel are discussed in Section IV-A.
In Fig. 3(a)–(d), the average ToA error is evaluated versus
SNR for N = 512, 1024, 2048, and 4096, respectively, where
N denotes the length of the transmitted waveform. As shown
in Fig. 3, increasing the number of allocated subcarriers
improves the average ToA estimation error at medium to high
SNR regimes [see Fig. 3(a)–(d)]. This improvement is acquired
as allocating a larger number of subcarriers increases the
frequency domain samples in (19). In the proposed technique,
ToA is acquired from estimating the frequency of Xk in (19)
using (20). Exploiting more samples improves the precision of
interpolated signal, Xk, and hence, more accurate estimation
can be proposed by applying (20). Moreover, it is observed
that increasing N by applying the same number of allocated
subcarriers K, decreases the performance of the proposed

8. JAMALABDOLLAHI AND ZEKAVAT: ToA RANGING AND LAYER THICKNESS COMPUTATION IN NH MEDIA 749
TABLE II
RELATIVE PERMITTIVITIES AT DIFFERENT APPLIED CARRIER FREQUENCIES [34]
Fig. 3. Impact of the number of allocated subcarriers and transmitted signal
length on the estimated ToA average error in underwater-airborne considering
multipath channels. (a) N = 512. (b) N = 1024. (c) N = 2048. (d) N = 4096.
technique as shown in Fig. 3(a) and (b). This is due to the fact
that increasing N decreases the subcarrier spacing at OFDMA
subcarrier, which leads to a lower bandwidth of transmitted
signal. However, the ToA estimation limit is achievable by
increasing the number of allocated subcarriers K, as shown
in Fig. 3(c) and (d). Moreover, in Fig. 3(a) and (b), it can
be observed that the average ToA estimation error increases
at higher allocated subcarriers and low SNR regimes for
shorter waveform lengths. In addition to the positive impact of
increasing the number of allocated subcarriers at medium to
high SNR regimes, exploiting a larger number of subcarriers
extends the in-band noise power, which affects the ToA
estimator performance at low SNR regimes, specifically at
higher subcarrier spacing values (shorter signal length at fixed
sampling time). This outcome is observed in Fig. 3(a) and (b)
for K = 50 and 25 for −4 dB ≤ SNR ≤ 0. Therefore,
it is concluded that the best ToA estimation is achievable
via increasing the number of allocated subcarriers K, and
Fig. 4. Impact of the number of allocated subcarriers and transmitted
signal length on the estimated ToA average error in underground channels.
(a) N = 512. (b) N = 1024. (c) N = 2048. (d) N = 4096.
the transmitted waveform length N, simultaneously as shown
in Fig. 3(c) and (d) for K = 25 and 50. The same
results are observed for the underground medium depicted in
Fig. 4(a)–(d). Here, the same parameters and channel proposed
in Section IV-A are applied. As shown in Fig. 4(a)–(d), the
proposed technique performs accurately at medium to high
SNR regimes, specifically for large numbers of allocated
subcarriers and waveform lengths. Comparing Figs. 3 and 4 it
is observed that the average ToA error converges to the Ts/2,
which is the minimum average error achievable via coarse
ToA estimation.
Moreover, moving from Figs. 3(a)–(d) and 4(a)–(d) we
observe that increasing the transmitted signal bandwidth
improves the ToA estimation error. This observation is con-
sistent with our expectation that increasing the bandwidth in a
flat fading channel improves the ToA estimation performance.
Here, a one-tap channel is considered as shown in Table I,
because Figs. 3 and 4 represent the simulations for underwater

9. 750 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 55, NO. 2, FEBRUARY 2017
Fig. 5. Impact of the number of applied measurements Q and ToA
measurement error on the estimated range. (a) Underwater-airborne channel.
(b) Underground channel.
and underground, where we only consider one path between
the transmitter and receiver as the effect of other paths due to
path loss is ignorable.
2) Range Estimation: Fig. 5(a) and (b) depicts the impact
of the number of applied measurements (ToA and DoA) and
the average ToA measurement error given perfect DoA mea-
surements on the estimated range for underwater-airborne and
underground channels, respectively, considering N = 4096
and K = 50. It is observed [see Figs. 3(d) and 4(d)] that the
average ToA estimation error is less than or equal to 5 ns for
SNR(dB) ≥ 0 for N = 4096 and K = 50. Therefore, applying
the same parameters (N = 4096 and K = 50), here we
have evaluated the average range error versus the average ToA
measurement error less than or equal to 5 ns. It is observed
that the proposed technique offers the average range error less
that 1 m for underwater-airborne and underground channels
at ToA measurement error around or less than 2 ns, which is
achievable at SNR(dB) ≥ 4. Moreover, it is observed that the
average ranging error decreases by increasing the number of
applied measurements as expected.
Fig. 6(a) and (b) depicts the impact of the number of applied
measurements (ToA and DoA) and average DoA measurement
error given perfect ToA measurements on the estimated range
for underwater-airborne and underground channels, respec-
tively, considering N = 4096 and K = 50. It is observed that
the proposed technique offers proper range estimation even
at imperfect DoA measurements. Furthermore, it is observed
that the average ranging error is decreased by increasing
the number of applied measurements for all the applied
DoA measurement errors as expected.
Comparing Figs. 5 and 6, the feasibility of the proposed
technique on different values of the desired range can be
evaluated. As shown in Fig. 5(a) and (b), the ranging error
for two different range values (30 and 150 m for underwater-
airborne and 10 and 25 m for underground) are the same at
perfect DoA measurements. However, it is observed that the
ranging error corresponding to larger range values are higher
as shown in Fig. 6(a) and (b). This is due to the impact of
DoA on ranging utilizing the estimated thicknesses accord-
ing to the final range estimation equation proposed in (34).
Fig. 6. Impact of the number of applied measurements Q and DoA
measurement error on the estimated range. (a) Underwater-airborne channel.
(b) Underground channel.
Therefore, it should be noted that higher precision DoA
measurements are required in scenarios with a larger range
between the transmitter and receiver.
V. CONCLUSION
In this paper, a novel technique for ToA-based range
measurements in NH media consisting of frequency dispersive
submedia via OFDMA subcarriers is proposed. Here, ToA
is measured in frequency domain utilizing OFDMA subcar-
riers and sampling time diversity. Then, considering multiple
ToA/DoA measurements recruiting different carrier frequen-
cies, a set of linear equations is constructed, which is solved
for available submedia’s thickness and hence, straight-line
range between the transmitter and receiver. The performance
of the proposed technique is evaluated via simulations consid-
ering underwater-airborne and multilayer underground scenar-
ios as two NH media with different physical characteristics
according to realistic propagation and channel models. The
simulation results indicate that the proposed technique offers
high-resolution ToA estimation and hence precise ranging,
especially for a large number of allocated subcarriers with
long lengths for low to high SNR regimes.
The proposed technique opens a new research area in
localization and scanning technologies such as Light Detection
and Ranging and GPR in frequency dispersive NH media,
where no resolution restriction is needed to prevent the fre-
quency dispersion of ultrawideband waveforms. Moreover, the
proposed ToA estimation technique approximates the overall
imposed delays into subcarriers by a line and estimates the
delay corresponding to the spectrum center. Applying the same
idea, the slope of the approximated line is estimated to reveal
the imposed delays into each subcarrier and equalize them in
time domain. This enables very high data rate transmission
exploiting OFDMA subcarriers in frequency dispersive NH
media.
APPENDIX A
DERIVATION OF THE ML ESTIMATOR
Substituting 2(α + β) with n in (19), K-independent
measurements correspond to
Xk = z0e− j
2πmk
N n
+ ˜Vk ∀k = 1, 2, . . . , K (35)

10. JAMALABDOLLAHI AND ZEKAVAT: ToA RANGING AND LAYER THICKNESS COMPUTATION IN NH MEDIA 751
where ˜Vk denote the frequency domain noise. Assuming zero
mean Gaussian distribution for ˜Vk, the likelihood function of
measurements can be written as
f (X1, . . . , XK |z0, n) =
1
2πσ2
V
K/2
K
k=1
e
−
Xk−z0e
− j
2πmk
N
n 2
2σ2
V
(36)
where σ2
V is the variance of frequency domain noise ˜Vk,
∀k = 1, 2, . . . , K and mk, N, and K, and z0 are defined
in (9) and (19), respectively.
The ML estimator of n, can be represented by maximizing
the proposed log-likelihood function with respect to n such
that
ˆn = argmax
z0,n
{ f (X1, . . . , XK |z0, n)} (37)
= argmax
z0,n
ln
1
2πσ2
V
K/2
−
1
2σ2
V
K
k=1
Xk − z0e− j
2πmk
N n 2
(38)
= argmin
z0,n
K
k=1
Xk − z0e− j
2πmk
N n 2
(39)
taking the derivative of (39) and solving it with respect to z0,
leads to
ˆz∗
0 = X∗
k e− j
2πmk
N n
(40)
substituting (40) into (39) and solving for n, leads to
ˆn = argmax
n
2
K
k=1
Xk X∗
k e− j
2πmk
N n
ej
2πmk
N n
(41)
= argmax
n
WT
n x
2
(42)
where wn = [ej((2π Kn)/N), ej((2π(K−1)n)/N), . . . , ej((2πn)/N)]T
is K × 1 column vector and x = [X1, X2, . . . , XK ]T .
APPENDIX B
DERIVING STRAIGHT-LINE RANGE
Considering Fig. 1 we can write r = D/cos(θ), where
D = M
m=1 zm. Therefore, it is desired to find an expression
that substitutes θ as a function of known and estimated
parameters as follows:
tan(θ) =
1
D
M
m=1
zmtan θ(q)
m (43)
where M, zm, θ
(q)
m are defined. Based on Snell’s law, it can
be shown that
θ(q)
m = sin−1
⎛
⎝sin θ
(q)
M
ε
(q)
r,m
ε
(q)
r,M
⎞
⎠. (44)
Substituting (44) into (43) leads to
tan(θ) =
1
D
M
m=1
zmtan
⎛
⎝sin−1
⎛
⎝sin θ
(q)
M
ε
(q)
r,m
ε
(q)
r,M
⎞
⎠
⎞
⎠ (45)
=
1
D
M
m=1
zm
sin θ
(q)
M
ε
(q)
r,m
ε
(q)
r,M
cos sin−1 sin θ
(q)
M
ε
(q)
r,m
ε
(q)
r,M
. (46)
Using cos(sin−1(x)) = (1 − x2)
1/2
and applying some
mathematical manipulations it can be shown that
tan(θ) =
1
D
M
m=1
zmsin θ
(q)
M ε
(q)
r,m
ε
(q)
r,M − sin2 θ
(q)
M ε
(q)
r,m
. (47)
Applying (47) into r = D/cos(θ), and substituting
cos(tan−1(x)) = (1/((1 + x2)
1/2
)), the straight-line range
between the transmitter and receiver as a function of measured
thicknesses and DoAs corresponds to
ˆr(q)
= D 1 +
⎡
⎣ 1
D
M
m=1
zmsin θ
(q)
M ε
(q)
r,m
ε
(q)
r,M − sin2 θ
(q)
M ε
(q)
r,m
⎤
⎦
2
. (48)
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Mohsen Jamalabdollahi (S’15) received the B.Sc.
degree in electrical engineering from the Univer-
sity of Mazandaran, Babol, Iran, in 2003, the
M.Sc. degree from the K. N. Toosi University of
Technology, Tehran, Iran, in 2011, and the Ph.D.
degree from the Department of Electrical and Com-
puter Engineering, Michigan Technological Univer-
sity, Houghton, MI, USA, in 2016.
He was with the Software Radio group of Mobile
Communications Technology Company, Tehran,
from 2006 to 2013. His current research interests
include digital signal processing, wireless communications, and convex
optimization.
Dr. Jamalabdollahi was a recipient of the Matt Wolfe Award for Outstanding
Graduate Research Assistant at the Department of Electrical and Computer
Engineering, Michigan Technological University in 2016.
Seyed (Reza) Zekavat (M’02–SM’10) has been
with Michigan Technological University, Houghton,
MI, USA, since 2002. He is the Editor of the
book Handbook of Position Location: Theory, Prac-
tice, and Advances (Wiley-IEEE). He is the author
of the book Electrical Engineering, Concepts, and
Applications (Pearson) and has also coauthored the
book Multi-Carrier Technologies for Wireless Com-
munications (Kluwer). His current research interests
include wireless communications, positioning sys-
tems, software-defined radio design, dynamic spec-
trum allocation methods, blind signal separation, and MIMO and beam
forming techniques.
Dr. Zekavat was the Founder and Chair of multiple IEEE Space Solar
Power Workshops in 2013–2015. In addition, he is active on the Executive
Committees for several IEEE international conferences. He has served
on the Editorial Board of many journals, including IET Communications,
IET Wireless Sensor System, the International Journal on Wireless Networks
(Springer), and the GSTF Journal on Mobile Communications.