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Transactions on Communications
1
High Resolution ToA Estimation via Optimal
Waveform Design
Mohsen Jamalabdollahi, Student Member, IEEE, Seyed (Reza) Zekavat, Senior Member, IEEE,
Abstract—This paper introduces a novel method to improve the
Time of Arrival (ToA) estimation resolution for a fixed available
bandwidth in the presence of unknown multipath frequency se-
lective (MPFS) channels. Here, the maximum rising level detector
(MRLD) technique is proposed which utilizes oversampling and
multiple correlation paths to evaluate with high resolution the
path corresponding to the maximum rising level of matched
filters output. However, employing such technique demands for
transmission of waveform that creates a very high rising level at
autocorrelation center. This paper proposes an efficient technique
to design proper waveforms (very high rising level at autocorrela-
tion center) via minimization of weighted integrated sidelobe level
(WISL), exploiting the trust-region algorithm. The performance
of the proposed technique is evaluated via simulations of the ToA
mean square error (MSE), and compared to the state-of-the-art
approaches considering the same bandwidth, and Cramer-Rao
lower bound (CRLB) as benchmark. Simulations confirm that the
ToA resolution is improved as the number of correlation paths
increases and verify the feasibility of the proposed technique
compared to the available approaches for the MPFS channels.
Index Terms—Time-of-arrival, matched-filter, autocorrelation,
integrated sidelobe level, trust-region algorithm, waveform.
I. INTRODUCTION
TIME-of-arrival (TOA) estimation is desired in variety
of applications such as digital communications, wireless
sensor network, radar and sonar and wireless indoor and
outdoor localization [1]-[7]. The ToA estimation techniques
are divided into two different categories, the traditional tech-
niques that are based on the matched filter output, and the
super resolution techniques. The former incorporates a pre-
designed waveform with autocorrelation properties close to
the delta function at the output of matched filter. Although
correlation based techniques propose excellent performance
in the presence of unknown multipath frequency selective
(MPFS) channels, however, their proposed ToA resolution
is limited by transmitted waveform bandwidth. In later, the
ToA is calculated via maximizing the pseudo-spectrum of
the corresponding signal sub-space. This is achievable via
matched filter output decomposition in frequency domain [8]
and [9]. Examples of super resolution techniques are inde-
pendent component analysis (ICA) [10], maximum likelihood
(ML) [11], multiple signal classification (MUSIC) [12]-[14]
and estimation of signal parameter via rotational invariance
M. Jamalabdollahi is with the Department of Electrical and Computer
Engineering, Michigan Technological University, Houghton, MI, 49930 USA
e-mail: mjamalab@mtu.edu
S. Zekavat is with the Department of Electrical and Computer Engineer-
ing, Michigan Technological University, Houghton, MI, 49930 USA e-mail:
rezaz@mtu.edu
This Paper has been partially supported by NSF Award ECCS 1101843
technique (ESPRIT) [15]. However, the frequency domain
techniques only improve the ToA resolution in flat fading
channels or in the presence of multiple resolvable paths which
are not feasible assumptions in many ToA applications.
Designing a waveform with good autocorrelation properties
(close to delta function) has been discussed in literature
[16]-[19]. In [16] and [18], authors propose several cyclic
algorithms such as CA-pruned (CAP), CAN (CA-new) and
weighted CAN (WeCAN) where the latter offers the best
performance in terms of minimum integrated sidelobe level
(ISL) of the waveform’s autocorrelation. In [19], authors
propose polyphase sequences and compare achieved merit
factor (MF) with several traditional sequences such as the
Golomb and Chu sequences. More information on waveform
design can be found in [17] and [20]. Exploiting the proposed
waveforms, one can estimate the ToA with a resolution that is
limited to the sampling interval.
Higher resolution is achievable via decreasing sampling in-
terval, however, this increases the waveform bandwidth which
is not feasible and/or affordable in many wireless communi-
cations and wireless sensor network applications. Some prior
works [10], propose fine ToA estimation in addition to the
traditional coarse ToA approaches, however, these techniques
consider a known channel impulse response (CIR) or prior
knowledge of statistical model for multipath channels [21]
which are not feasible prior to time synchronization.
Despite the impact of bandwidth on the performance of
designed waveform in ToA estimation, a few works discuss
the problem of the ToA estimation exploiting bandlimited
waveforms. In [17], authors propose the stop-band CAN
(SCAN) and Weighted stop-band CAN (WeSCAN) which
aims to design a band-limited waveform with good autocor-
relation properties. Exploiting this technique one can design
a waveform with good autocorrelation properties considering
smaller sampling interval (higher ToA resolution) at fixed
bandwidth, however, in the proposed technique the ISL in-
creases while a feasible attenuation in stop-band is interested.
Authors in [22] and [23] propose different criteria for sparse
spectrum limitation, however, these techniques cannot impose
acceptable attenuation into stop-band while the waveform still
performs good autocorrelation in terms of ISL.
In this paper, a novel approach is proposed to improve the
resolution of the ToA estimation for a fixed available band-
width in the presence of unknown the MPFS channels. First,
the low resolution of the ToA estimator imposed by limited
bandwidth of the designed waveform is addressed by propos-
ing the maximum rising level detector (MRLD) technique via
oversampling and parallel correlation paths. Here, the ToA

2
resolution improves via finding the path corresponding to the
maximum rising level of the matched filters output of distinct
correlation paths at their maximum absolute values. The idea
of exploiting oversampling to improve the resolution of ToA
[24]-[26], range [27]-[29] and synchronization [30] has been
discussed in the literature.
In [24], the fast parabolic method is proposed which
improves the ToA resolution by processing the maximum
value of correlator and two samples in the maximum value
neighborhood. The resolution improvement of the proposed
method in [24] is limited to half of sampling time. In [25]
interpolating function that incorporates oversampling is pro-
posed. Here, the interpolating function is designed such that
the matched filter output approaches delta function. However,
due to the transmitted signal limited bandwidth, the output
of matched filter cannot approach to the delta function. This
increases ambiguity function’s sensitivity to noise and the ToA
estimation error. Authors in [26] propose the Line-of-Sight
(LoS) ToA estimation via oversampling. Here, the system
model assumes that coarse time synchronization has been
achieved and aims to increase the ToA estimation resolution in
frequency domain. The proposed approach is only applicable
to flat fading channels, as in realistic MPFS channels, the
information of fine ToA cannot be separated from estimated
channel impulse response. Authors in [27]-[29] exploit linear
array processing using pulse based waveform to improve
the ToA resolution via oversampling in flat fading channels,
however the case of MPFS channels is not considered (see
(2) in [28] and [29]). In [30], non-synchronized sampling is
exploited to increase time synchronization resolution, however,
the proposed method is only applicable to low sampling rates.
In the MRLD technique, the sidelobe levels of matched
filter output corresponding to the samples generated by over-
sampling process are extremely larger than sidelobe levels of
matched filter output of the original samples. However, the
rising level created by oversampling at the autocorrelation
center would be high as well. To this end, a novel technique
exploiting the trust-region algorithm [31] is proposed which
designs waveforms with the aforementioned properties via
minimizing the weighted ISL (WISL) objective function. Here,
the trust-region technique is exploited [31]. Trust-region is a
powerful and popular technique for solving system of non-
linear equations. Its popularity has been gained due to global
convergence to extreme points for a system of non-linear
equations.
The performance of the proposed technique is evaluated
by simulating the estimated ToA mean square error (MSE),
and compared to the state-of-the-art approaches SCAN and
WeSCAN [17], considering the same bandwidth. Moreover,
the ToA Cramer-Rao lower bound (CRLB) is calculated in the
presence of MPFS as evaluation benchmark. Simulation results
verify the feasibility of the proposed technique compared to
the SCAN and WeSCAN waveforms specially over unknown
MPFS channels. Furthermore, it is observed that increasing the
oversampling rate or equivalently the number of correlation
paths, improves the ToA resolution. In addition, increasing
the oversampling rate reduces the gap between the MSE
of ToA and its corresponding CRLB imposed by lack of
fine ToA estimation. Finally, the complexity and performance
of the proposed technique is discussed theoretically and via
simulations for different numbers of correlation paths and
waveform lengths.
The rest of paper is organized as follow. Section II in-
troduces the system model and problem formulation. The
proposed MRLD technique and the algorithm for waveform
design is presented in Section III. Section IV represents
the simulation results and discussions, and finally Section V
concludes the paper.
II. SYSTEM MODEL AND PROBLEM FORMULATION
To perform the ToA estimation, receiver should detect the
arrival of the transmitted signal instantaneously, and measure
the signal propagation time with respect to a reference time.
The system model for the baseband received signal corre-
sponds to:
r(t) =
L−1
l=0 h(lTs)s(t − τl) + v(t), τ0 ≤ t ≤ τL−1 + (N − 1)Ts
v(t), elsewhere
(1)
where s(t), h(lTs), τl and v(t) represent the transmitted
baseband signal with length N, gain and delay of the lth
tap of wireless channel impulse response, and additive, white,
zero mean Gaussian noise, respectively. Applying the proposed
baseband received signal in (1) to the analog to digital con-
verter (ADC), with the sampling frequency of fs = 1/Ts,
where Ts is the sample interval of the baseband signal, leads
to:
r(k) =



L−1
l=0 h(lTs)s(kTs − τl) + v(kTs),
kτ0
≤ k ≤ kτL−1
+ N − 1
v(kTs), elsewhere
, (2)
where kτl
is the closest integer number to τl/Ts, and L,
h(lTs), τl and N are defined in (1), and s(kTs) and v(kTs)
represent the kth
sample of the transmitted waveform and
additive zero mean Gaussian noise, respectively.
The ToA estimation is equal to the estimation of the time
delay corresponding to the received signal from the shortest
path, τ0. Considering τ0 = kτ0
Ts + δτ0
, the ToA can be
introduced via coarse (kτ0 Ts) and fine (δτ0 ) components. In
the correlation based ToA techniques, the problem of the ToA
estimation is represented by estimating coarse-ToA via solving
the following objective function:
ˆkτ0 = argmax
k
k+N−1
n=k
rns∗
n−k
2
, (3)
where rn and s∗
n represent the nth
sample of received
signal defined in (2), and the complex conjugate of the
transmitted waveform, respectively. This technique proposes
the ToA resolution limited to the sample interval Ts. Given
N+kτ0 −1
n=kτ0
rks∗
n−k
2
>
N+k−1
n=k rks∗
n−k
2
, ∀k, k = kτ0 ,
the maximum of (3) occurs at k = kτ0
. Considering a constant
energy signal, the aforementioned inequality holds via mini-
mizing of the energy of sidelobe levels,
N+k−1
n=k rks∗
n−k
2
,

3
∀k, k = kτ0
. This is equivalent to minimization of ISL defined
as [16], [17]:
ISL =
N−1
k=1
|yk|
2
, for, yk =
N−1
n=k
sns∗
N+n−k, (4)
where sn and s∗
n represent the nth
sample of the transmitted
waveform and its complex conjugate, respectively. Therefore,
minimization of ISL defined in (4) leads to a waveform
with good autocorrelation properties and consequently, lower
ToA estimation error. In [16], the weighted-ISL (WISL) is
introduced where sidelobe levels are weighted to form a
desired autocorrelation, such that:
WISL =
N−1
k=1
w2
k |yk|
2
, wk ≥ 0 (5)
where yk is the kth
autocorrelation sidelobe defined in (4), and
wk represents its corresponding weight. Exploiting such crite-
ria allows minimization of the sidelobes around to the center
of autocorrelation, which reduces the interference imposed by
multipath channels [16].
Improving the ToA estimation via optimal bandlimited
signal is defined as designing a waveform with minimum
WISL and bandwidth. However, in [16], [17], it is shown that
minimization of (W)ISL, leads to a sequence with near wight
spectrum which uniformly includes all frequencies within
[−fs/2, fs/2], where fs represents the sampling frequency.
This indicates that restricting signal bandwidth to any spec-
trum lower than [−fs/2, fs/2], dramatically increases the
(W)ISL which leads to the ToA estimation error.
Next section introduces the the MRLD technique which
tremendously improves the low resolution of the ToA es-
timation imposed by bandwidth limitations of the designed
waveform via oversampling and parallel processing(see section
III-A).
III. PROPOSED TECHNIQUE
In this section enhancing the ToA resolution using the
MRLD technique is proposed. Then, a novel procedure for
optimization of the weighted ISL (WISL) is proposed which
leads to a waveform with very high rising level at its autocor-
relation.
A. Maximum rising level Detector
Figure 1 shows the proposed transmitter (a), and receiver
(b) block diagrams corresponding to the proposed MRLD
technique. Applying the baseband received signal in (1) to the
analog to digital convertor (ADC) with sampling time interval
˜Ts = Ts/M, where Ts represents the sample interval of the
transmitted waveform. Considering ML resolvable paths for
the proposed CIR in (1) and (2) due to oversampling with rate
M, the baseband signal at the output of the ADC corresponds
to:
r(k ˜Ts) =



ML−1
l=0 h(l ˜Ts)s(k ˜Ts − τl) + v(k ˜Ts),
Mkτ0 + kδ ≤ k ≤ M kτL−1
+ N + kδL−1
− 1
v(k ˜Ts), elsewhere
,
(6)
where s(k ˜Ts), r(k ˜Ts) and v(k ˜Ts) represent the samples of
transmitted and received signals, and zero-mean complex addi-
tive Gaussian noise at t = k ˜Ts, and h(l ˜Ts), l = 0, 1, ..., ML−
1, denote the complex value CIR. Furthermore, M, kτl
and kδl
are the oversampling rate and sample indices corresponding to
the coarse and fine of lth
path delay, respectively. Reformu-
lating the ToA equation (τ0 = kτ0 Ts + δτ0 ) in terms of new
sample time, ˜Ts leads to:
τ0 = (Mkτ0
+ kδ) ˜Ts + ˜δ0, (7)
where kτ0
and kδ0
are the sample indices corresponding to the
coarse and the fine ToA, respectively. Moreover, ˜δ0 represents
the remnant of the fine ToA such that ˜δ = δτ0 − kδ
˜Ts for
0 ≤ kδ ≤ M − 1.
The proposed MRLD receiver oversamples the baseband
received signal with rate M, then segregates the output of the
ADC into M parallel correlation paths each corresponding to a
consecutive sample time offset 0, ˜Ts, ..., (M −1) ˜Ts. Exploiting
this technique, the sequence of existing samples in the mth
correlation path corresponds to the samples observed at times
[(m − 1) ˜Ts, (M + m − 1) ˜Ts, (2M + m − 1) ˜Ts, ...], ∀m =
0, 1, ..., M − 1.
Assuming zero-Inter Symbol Interference (ISI) pulse shap-
ing filter at transmitter, the kth
sample at the mth
correlation
path can be represented by:
rm(kTs) =



L−1
l=0 h(l ˜Ts)s (Mk + m) ˜Ts − τl +
v (Mk + m) ˜Ts , kτ0
≤ k ≤ kτL−1
+ N
v (Mk + m) ˜Ts , elsewhere
,
(8)
where M is the number of correlation path or oversampling
rate, and h(lTs), s(k ˜Ts), and v(k ˜Ts) are defined in (6).
Here, estimating the ToA is equivalent to discovering the
index corresponding to τ0 in terms of new sampling time, i.e.
(Mkτ0 + kδ) ˜Ts. Considering (8), it can be show that matched
filter output at the mth
correlation path at receiver can be
written as (see Appendix A for details):
gm(kTs) =



ML−1
l=0 h l ˜Ts
N+k−1
n=k s n − kτ0
− l
M Ts
s∗
((n − k) Ts) + ˜vm(kTs), m = kδ
ML−1
l=0 h l ˜Ts
N+k−1
n=k s n − kτ0 − m−kδ−l
M Ts
s∗
((n − k) Ts) + ˜vm(kTs), m = kδ
,
(9)
where h(lTs) and s(k ˜Ts), kδ and M are defined in (6), (7)
and (8) respectively. Moreover, gm(kTs) and ˜vm(kTs) are the
kth
sample at the mth
correlation path of matched filter out-
put and additive colored noise, respectively. Considering the
resolvable channel paths corresponding to oversampling, i.e.,
h l ˜Ts , ∀l = 1, 2, ..., ML − 1, l = pM, p = 0, 1, ..., L − 1,
are much smaller than original resolvable path of channel,
i.e., h l ˜Ts , ∀l = 0, M, 2M, ..., (L − 1)M, due to limited
bandwidth of transmitted signal, (9) can be simplified to:

4
gm(kTs) =



L−1
l=0 h (lTs)
N+k−1
n=k s ((n − kτ0
) Ts)
s∗
((n − k) Ts) + ˜vm(kTs), m = kδ
L−1
l=0 h (lTs)
N+k−1
n=k s n − kτ0
− m−kδ
M Ts
s∗
((n − k) Ts) + ˜vm(kTs), m = kδ
,
(10)
where h(lTs) and s(k ˜Ts), kδ, M and ˜vm(kTs) are defined
in (6), (7), (8) and (9), respectively. Here, it is observed that
the matched filter output corresponding to the correlation path
including the original waveform samples, i.e., m = kδ can be
simplified to:
gm,k =
L−1
l=0
hlyk−kτ0
−l + ˜vm,k, for m = kδ, (11)
where yk represents the autocorrelation function of the trans-
mitted waveform defined in (4). Considering transmission of
waveform with very low ISL obtained in Section III-B, (11),
can be written as:
gm,k =
hl + ˜vm,k kτ0
≤ k ≤ kτ0
+ L − 1
˜vm,k elsewhere
(12)
Here, it is observed that the matched filter output for m = kδ
contains the noisy version of channel impulse response hl, for
kτ0
≤ k ≤ kτ0
+L−1, and colored noise elsewhere. However,
for m = kδ the sidelobe corresponding to oversampled
version of waveform s n − kτ0 − m−kδ
M Ts is available.
The absence of sidelobes in the correlation path corresponding
to m = kδ, is key to reveal higher ToA resolution exploiting
the MRLD technique. Therefore, the MRLD receiver finds
the maximum absolute values of the matched filters output
at each correlation path, i.e. ˆkm, m = 1, 2, , M. Then it
estimates the ToA via finding the correlation path (m∗
) by
maximizing the rising levels cm,ˆkm
/cm,ˆkm−1
, among the
obtained maximum values of matched filter outputs. Thus, the
MRLD technique is efficiently implementable based on three
simple steps: (1) searching for the peak of correlation levels
at each path using the transmitted waveform matched filter
(s∗
(N − n)), (2) measuring the rising level corresponding to
the revealed peaks at each path, and (3) searching for the
maximum among the measured rising levels. Therefore, the
path containing the maximum rising level (m∗
), among M
available correlation paths is achievable via:
m∗
= argmax
m
cm,ˆkm
cm,ˆkm−1
,
s.t. cm,ˆkm
= argmax
k
k+N−1
n=k
rm(nTs)s∗
N+n−k
2
,
(13)
where rm(kTs) and s∗
N−k are the kth
samples of received
sequence at mth
correlation path and the matched filter to
the transmitted waveform, respectively. Moreover, M and N
represent the number of correlation paths or oversampling rate,
Fig. 1. Block diagram of the proposed MRLD transmitter (a) and receiver
(b)
and the length of received signal, respectively. Using (13), the
estimated ToA can be represented by:
ˆτ0 = Mˆkm∗ + m∗ ˜Ts, (14)
where m∗
and ˆkm∗ denote the correlation path number con-
taining the maximum rasing level and its corresponding index
of maximum correlation level, respectively, achievable via
(13). The performance of the proposed technique is evalu-
ated by comparing the MSE of the estimated ToA over K
independent run via:
MSEˆτ =
1
K
K
k=1
τ
(k)
0 − ˆτ
(k)
0
2
, (15)
where τ
(k)
0 and ˆτ
(k)
0 represent the kth
ToA value and its
estimation, respectively. Figure 2 depicts the correlation levels
corresponding to the MRLD receiver for M = 4, considering
the transmitted waveform with very high rising level at its
autocorrelation center. Here, (without loss of generality) it is
assumed τ0 = (k0 +0.75)Ts where τ0, Ts and k0 represent the
ToA, sampling interval and an arbitrary integer. Exploiting the
traditional correlation techniques, the estimated ToA would be
ˆτ0 = (k0 + 1)Ts. Considering (8) , the sequence of existing
samples in the third correlation path corresponds to:
r3(kTs) =h0s (4k + 3) ˜Ts − τ0 +
L−1
l=1
h(lTs)s (4k + 3) ˜Ts − τl
+ v (4k + 3) ˜Ts , k0 ≤ k ≤ kτL−1
+ N,
(16)
where h(lTs), s(k ˜Ts) and v(k ˜Ts), and rm(kTs) are defined
in (6) and (8), respectively. Substituting τ0 and ˜Ts with (T0 +

5
Fig. 2. Autocorrelation level corresponding to the over-sampled signal with
M = 4 and fine-resolution kτ0 = 3.
0.75)Ts and Ts/4, respectively, leads to:
r3(kTs) =h0s (k − k0)Ts +
L−1
l=1
hls (4k + 3) ˜Ts − τl +
v (4k + m) ˜Ts , k0 ≤ k ≤ kτL−1
+ N,
(17)
The proposed sequence in (17) is the convolution of the
transmitted waveform and the CIR which maximizes the
matched filter output at k = k0. Although, the matched filter
outputs corresponding to the other pathes is maximum at the
same index, however, their corresponding rising levels of the
offered indices are much smaller than the rising level of the
third path (5, 17 and 15 dB shown in Fig. (2)-(a), (b) and (d)
compared to 275 dB as shown in Fig. (2)-(c)). This is the result
of designed waveform with a very high rising level proposed
in section III-B, where the sidelobe levels corresponding to
the correlation of the designed waveform matched filter with
samples generated by oversampling are extremely larger than
the sidelobe levels corresponding to its autocorrelation. Next
section discusses designing a waveform with maximum rising
level exploited in the MRLD technique.
B. Waveform Design
1) Forming the objective function: The desired waveform
sequence {sn}
N
n=1 = ejϕn
N
n=1
can be acquired as the
solution of the following optimization problem:
ˆφ = argmin
φ
N−1
n=k
wk |yk|
2
s.t. yk =
N−1
n=1
sns∗
N+n−k, for: sn = ejϕn
(18)
where φ = [ϕ1, ϕ2, ..., ϕN ]T
, and N, and yk and wk are
defined in (1) and (5), respectively. Considering the Parseval
energy equality in time-frequency domain and defining the
2N point discrete Fourier transform of weighted sidelobes
{wnyn}
N−1
p=1 as {Sp}
2N
p=1, it can be shown that [17], [16]:
WISL =
1
4N
2N
p=1
[Sp − N]
2
,
for, Sp =
1
√
2N
N−1
n=−(N−1)
w2
kyke−j 2πkp
2N ,
(19)
where N, and yk and wk are defined in (1) and (5), respec-
tively. Substituting yk with (4) leads to [17], [16]:
Sp =
1
√
2N
N−1
k=−(N−1)
N−1
n=k
sns∗
N+n−kw2
ke−j 2πkp
2N
=
N
l=1
N
k=1
e−jϕk
ej πkp
N Γk,lejϕl
e−j πlp
N
(20)
where (.)∗
indicates the complex conjugate notation and Γk,l
represents the elements at the kth
row and lth
column of
squared weights matrix Γ, defined such that:
Γ =





1 γ1 ... γN−1
γ1 1 ... γN−2
...
...
...
...
γN−1 γN−2 ... 1





(21)
Here, Γk,l denotes the weight matrix as its elements represent
the squared values of ISL weights, i.e. w2
k as shown in (20).
This leads to the final format of waveform design objective
function as follow:
ˆφ = argmin
φ
2N
p=1
[Sp − N]
2
for, Sp =
N
l=1
N
k=1
e−jϕk
ej πkp
N Γk,lejϕl
e−j πlp
N
(22)
where N and yk are defined in (1) and (5), respectively,
and Γk,l represents the element at the kth
row and lth
column of Γ defined in (21). Next section introduces a novel
optimization approach that exploits the trust-region algorithm
to optimize (22) via solving the system of nonlinear equations
corresponding to φf = 0 for f =
2N
p=1 [Sp − N]
2
, where
Sp is defined in (22).

6
2) The Trust-region Algorithm: In order to find the op-
timum solution for the proposed objective function in (22),
the trust-region algorithm is exploited to solve φf = 0 for
f =
2N
p=1 [Sp − N]
2
, where Sp is defined in (22). The trust-
region algorithm approximates the objective function via a
quadratic function within a trusted region and solves it for
the next iteration. The selected trust region can be extended
if computations verify that the approximation model fits the
objective function properly within the extended region. This
prevents the algorithm step to be too far from available extreme
points. Here, we define g = φf, and H = 2
φf as the
gradient vector and Hessian matrix of the proposed objective
function in (22). The analytical expressions for the gradient
vector and Hessian matrix are discussed in Appendices B and
C, respectively.
Algorithm 1 proposes the trust-region method to solve the
system of nonlinear equations corresponding to φf = 0.
Trust-region and Levenberg-Marquardt algorithms are the
leading state-of-the-art approaches to solve a system of non-
linear and non-convex equations (regarding H is negative
definite). The trust-region technique aims to find the opti-
mum step size s without exceeding the trust-region radius,
δ which leads to quick convergence to a more interesting
area [31], and therefore much better performance compare
to Levenberg-Marquardt approach. This interesting property
demands for finding the optimum step size at each iteration
which is not a trivial problem regarding its quadratic form
subject to one non-linear constraint (see Algorithm 1 line 4).
However, the off-line procedure of waveform design justifies
selecting the approach with higher computational complexity
for better performance. Here, the Dogleg approach [31] is
introduced to find the optimum step size at each iteration (see
Algorithm 2). Next section discusses the impact of several
parameters such as signal length and oversampling rate via
simulations. It is observed that the resolution of the proposed
ToA estimation technique significantly improves by increasing
the oversampling rate or equivalently the number of correlation
paths and transmitted waveform length.
Figure 3 (a) and (b) depicts the autocorrelation levels of the
designed waveforms exploiting Algorithm 1 for N = 128 and
N = 512, respectively. Here, the exploited squared weights
are defined in (23) as follow:
γn =



8 n = N
0.6 N − No ≤ n ≤ N − 1
0 elsewhere
(23)
where No = 35 and 140 for Fig. 3 (a) and (b), respectively.
Moreover, the initial phase values φ0 are chosen exploiting
the Golomb sequence [17] which leads to ϕn = πn(n −
1)/N, ∀n = 1, 2, ..., N, where N denote the waveform length.
The designed waveforms samples with different length exploit-
ing the proposed technique can be found online at [32].
IV. SIMULATION RESULTS AND DISCUSSION
Simulations are conducted to investigate the performance of
the proposed ToA estimation method. Here, the mean square
error (MSE) of the proposed ToA estimation in diverse scenar-
ios is calculated and compared to SCAN and WeSCAN [17]
Algorithm 1 Trust-region algorithm to solve φf = 0 for
f =
2N
p=1 [Sp − N]
2
, where Sp is defined in (22)
Require: Γ, φ0,
1: return φ
2: Initialize η1 = 0.05, η2 = 0.9, γ1 = 2.5, γ2 = 0.25, δ =
1, = 1e − 6 .
3: while HT
g > do
4: find: s = argmin
s
sT
HT
g + 1
2 sT
HT
Hs via Alg. 2.
5: pred = − sT
HT
g + 1
2 sT
HT
Hs .
6: ared = gT
φ+sgφ+s − gT
φ gφ.
7: if (ared/pred) < η1 then
8: δ = δγ1, repeat step 4.
9: else
10: φ = φ + s.
11: if (ared/pred) > η2 then
12: δ = max{δ, γ2 s },
13: end if
14: end if
15: end while
Algorithm 2 The Dogleg approach that find the optimum step
size at each step
Require: H, g, δ
1: return sdl
2: sc = −δ HT
g
HT g
, sn = −H−1
g.
3: sc = τcsc for: τc = min 1, HT
g
3
/ δgT
HT
Hg
4: a = sc − sn
2
.
5: b = 0.5( sc
2
+ sn
2
− a),
6: α = sn
2
−δ2
sn
2
−b+
√
b2− sc
2
sn
2
+δ2a
,
7: sdl = αsc + (1 − α)sn
Fig. 3. Autocorrelation level of designed waveform, (a) N = 128, (b),
N = 512.

7
TABLE I
Simulation parameters and applied values
Sym. Definition Value
Ts Sampling time 1µs
N Length of waveform 128, 256, 512, 1024
M Number of correlation paths 2/8/16/32/64/128
Ls Length of received signal 5000
L Taps of channel M(indoor), 10M (outdoor)
Tm Channel delay spread 500ns, 10µs
K No. of independent runs 104
and the ToA CRLB as benchmark. Moreover, the complexity
of the proposed technique is investigated vs performance
via discussing the impacts of the number of parallel paths
or oversampling rate M, and waveform length N, on the
total required operations and the ToA MSE at different SNR
regimes.
A. Simulation Parameters and Methods
In this sub-section, the details of simulated system model
which is used for performance analysis are introduced. Table
I shows the definitions and the values of parameters used to
simulate the system model. In order to simulate the over-
sampled baseband signal at receiver as described in (8), the
oversampled version of waveform is generated incorporating
an ISI free pulse shaping filter such as raised cosine. The
oversampled signal is passed through the flat and the MPFS
channels and added by independent samples of zero mean
circularly complex white Gaussian noise. In order to verify
the performance and stability of the proposed technique K
independent Monte-Carlo simulations are performed.
Here, indoor and outdoor channel models are considered
to evaluate the performance of the proposed technique in
MPFS channels. For indoor scenario the power delay profile
is generated according to the model proposed in [33] and [34]
with delay spread limited to 500ns. Figure 4(a) shows three
examples of discrete-time impulse response model of applied
channels power delay profile (PDP). For outdoor scenario, the
PDP is generated according to the model proposed in [35] and
[36] with delay spread limited to 10µsec. Figure 4(b) shows
three examples of discrete-time impulse response model of
applied the channel PDP. In Figs. 4(a) and (b), the time axis
is divided into small bins that contain the multipath component
corresponding to the bin size of 1µsec.
128 = 7.8nsec (maximum
available resolution corresponding to M = 128). Next sub-
section details the simulation results.
B. The ToA Estimation Performance
As discussed earlier in section III, the proposed MRLD
technique improves the ToA estimation resolution incorpo-
rating a fixed bandwidth signal. Figure 5 depicts the spec-
tral density of exploited waveforms designed by the SCAN,
WeSCAN approaches [17], and the proposed technique. The
spectral density of waveforms are calculated considering Ts =
0.25µs, for the SCAN and WeSCAN, and Ts = 1µs, for
the proposed technique, respectively. As shown in Fig. 5, the
WeSCAN approach generates a waveform with better stop-
band attenuation at the cost of an increased correlation levels.
Fig. 4. discrete-time impulse response model of applied channel channel, (a)
indoor, (b) outdoor.
Fig. 5. Spectral density of exploited Waveforms, SCAN, WeSCAN and
proposed technique.
However, the spectral density of the WeSCAN approach still is
much higher than proposed technique considering Ts = 1µs.
Figure 6(a) represents the MSE of estimated ToA exploiting
the designed waveform in Section III-B for N = 512, and
the MRLD receiver in the presence of flat fading channel.
Here, different numbers of correlation paths or oversampling
rates are applied to investigate its impact on the proposed
ToA estimator. The performance of proposed method is com-
pared exploiting the waveforms corresponding to SCAN and
WeSCAN approaches where Ts = 0.25µs and the stop-band
equals to [−500KHz, 500KHz], as shown in Fig. 5. Although
the acquired attenuation for the frequencies outside the stop-
band is around 25 and 30 dB for the SCAN and WeSCAN
approaches, respectively, (which are not sufficient for many
applications), it is observed that exploiting a band limited
waveform with lower sampling time (higher resolution) offers
stable results even at the low SNR regimes. However, as shown
in Fig. 6(a), applying the MRLD approach, higher resolution is
achievable for medium to the high SNR regimes (SNR>0dB)
compared to the SCAN and WeSCAN.
Figures 6(b) and 7(a) represent the MSE of the estimated
ToA exploiting the designed waveform in Section III-B for
N = 512, and the MRLD receiver in the presence of unknown

8
MPFS channels according to the channel PDPs proposed in
Section IV-A. Here, different numbers of correlation paths are
applied to the MRLD receiver to investigate its impact on the
proposed ToA estimator and compare them with the SCAN
and WeSCAN approaches.
As shown in Fig. 6(b), resolution of the ToA estimator
improves by increasing the number of correlation paths in
the MRLD receiver specifically at higher SNR regimes. This
is due to the impact of extra resolvable paths created by
increasing oversampling rate as shown in Fig. 4 (a). Therefore,
according to the simulation results proposed in Fig. 6(b), it is
concluded that for indoor channels, lower oversampling rate
should be selected at low SNR regimes, while for higher SNR
regimes, higher oversampling rate is feasible. For outdoor sce-
nario, resolution of the ToA estimator improves by increasing
the number of correlation paths in the MRLD receiver as
shown in Fig. 7(a), however, it is observed that performance of
the proposed MRLD technique is more reliable even at lower
SNR regimes.
In general, the ToA error is due to two components of
noise and resolution limit. The CRLB offers ToA error limit
corresponding to noise only, assuming resolution approaches
to zero. The proposed technique estimates the ToA with
resolution limited to Ts/M. Thus, to reach the CRLB, we
need to select M equal to infinity.
Figure 7b depicts the MSE of estimated ToA exploiting the
designed waveform in Section III-B for N = 512, and the
MRLD receiver in the presence of flat fading channel assuming
no fine ToA is available, or τ0 = k ˜Ts. As shown in Fig. 7b, the
MSE of ToA increases exploiting higher number of correlation
paths. This indicates that in the proposed MRLD receiver
the probability of correct coarse ToA estimation decreases by
increasing the correlation paths or oversampling rate. However,
the overall performance of the ToA estimation of the MRLD
receiver outperforms the traditional matched filter techniques
(exploiting full band or limited bandwidth waveforms such as
the SCAN and WeSCAN) regarding the error floor imposed
by the absence of fine ToA in these approaches.
C. Complexity Analysis vs Performance
Figure 8(a) represents the number of complex operations
(summation and multiplication) required by the proposed
technique vs different numbers of correlation paths M, and
waveform length N compared with the (We)SCAN approach.
Considering Lr as length of the received signal, (N +2)(Lr −
N) complex multiplications and N(Lr − N) summations are
required at each correlation path. Here, it is observed that
increasing the number of correlation paths leads to higher
computational complexity compared to single correlation ap-
proaches such as the SCAN or WeSCAN. This is the cost of
obtaining higher ToA resolution achievable via oversampling.
As shown in Fig. 8(a), for a fix value of total required
operations, multiple choices as a function of correlation path
number or waveform length are available. Figure 8(b) depicts
the MSE of the estimated ToA incorporating different numbers
of waveform length and correlation paths. Here, it is aimed
to reveal the impacts of regulating correlation path number
Fig. 6. The MSE of ToA exploiting different numbers of correlation paths
M vs SCAN and WeSCAN [17] with the same bandwidth (1MHz) (a) flat
fading channel, (b) indoor MPFS channels.
Fig. 7. (a) The MSE of ToA exploiting different numbers of correlation paths
M vs SCAN and WeSCAN [17] with the same bandwidth (1MHz) in outdoor
MPFS channels, (b) Impact of the proposed MRLD technique on coarse ToA
estimation.
or waveform length on the performance of the proposed
method. As shown in Fig. 8(b), increasing the waveform length
improves the MSE at the low SNR values. For instance,
the MSE represented by N = 1024, M = 16 is lower
than the MSE proposed by N = 128, 256, M = 64 for
−4dB ≤ SNR ≤ 6dB. However, for the higher SNR regimes
increasing M corresponds to the lower MSE. Therefore, it
is observed that in the low SNR regimes increasing the
waveform length (N), is more beneficial; however, at the
higher SNR regimes expanding the number of correlation paths
or oversampling rate (M), is found more advantageous.
D. ToA CRLB
The CRLB of the estimated ToA MSE defined in (15) is
proposed in Figures 6 and 7. Although approaching to the ToA

9
Fig. 8. (a) Impact of increasing the number of parallel paths M vs waveform
length N, on the performance of proposed technique, (b) Total required
complex operations by the proposed technique.
CRLB is too optimistic specifically at the low SNR regimes
[37], it is observed that increasing the correlation paths in
the MRLD receiver alleviates this gap at the cost of higher
computational complexity and hardware cost. Considering
x = d, h
(I)
0 , h
(Q)
0 , ..., h
(I)
L−1, h
(Q)
L−1 as unknown vector, where
d = [τ0, ..., τL−1], the log-likelihood function of probability
distribution function of the received signal proposed in (8), r,
can be represented by:
Λr|x = Λ0 −
1
σ2
(r − Ah)
H
(r − Ah) , (24)
The CRLB requires the computation of the Fisher Information
Matrix (FIM), that corresponds to:
J
∆
= Er|x xΛr|x. xΛr|x
T
, (25)
then, the CRLB for MSE of ToA can be represented by:
CRLBτ0
= Jinv(1, 1), (26)
where Jinv
∆
= J−1
. Applying L = 1 and h = 1, the CRLB of
the estimated ToA over the AWGN channel can be represented
by formerly discussed equation in [3] and [7] as follow (see
Appendix D for details):
CRLBτ =
3σ2
8π2NTsσ2
s (f3
2 − f3
1 )
, (27)
where N, Ts are defined in (8), and σ2
and σ2
s denote
the variance of noise and transmitted waveform, respectively.
Moreover, f2 and f1 represent the effective waveform’s spec-
trum such that BW = |f2 − f1|.
V. CONCLUSION
In this paper, the MRLD technique is proposed which
improves ToA resolution for a fixed available bandwidth
in the presence of unknown MPFS channels. The proposed
MRLD technique enjoys exploiting waveform with very high
rising level at its correlation center and oversampling in
several correlation paths at receiver to reveal higher resolution
than sampling interval. Simulation results confirm that the
resolution of estimated ToA can be improved via increasing
the number of correlation paths in the MRLD receiver, to a
great extent in medium to high SNR regimes and increasing
the waveform length at the lower SNR values. In order to
mitigate sidelobes corresponding to the ISI in the MPFS
channels, adaptive whitening filters can be applied at each
correlation path prior to matched filters. Optimum criteria
for such adaptive filters design and possible modifications on
applied waveform will be addressed in future works.
The very fact that MRLD technique exploits parallel pro-
cessing to improve the ToA resolution makes this technique a
proper candidate for entities recruiting multi-core processors
such as cell phones or base-station receivers. Moreover, the
significant desire on utilized bandwidth reduction in wireless
technology endorses exploiting the MRLD receiver to increase
ranging resolution and/or time synchronization while low
bandwidth signal are exploited. Furthermore, applying the
MRLD receiver improves the ToA resolution in the presence
of frequency dispersive channel such as ionosphere layers
and under-ground/water communications, where band limited
waveforms must be applied to avoid signal dispersion.
APPENDIX A
MATCHED FILTER OUTPUT DERIVATION
Defining gm(kTs) as the kth
sample of matched filter output
at mth
correlation path leads to:
gm(kTs) =
N+k−1
n=k
rm (nTs) s∗
((n − k) Ts)+vm(kTs), k ≤ N,
(28)
where rm (nTs) and vm(kTs) are defined in (8). Substituting
rm (nTs) with (8) leads to:
gm(kTs) =
N+k−1
n=k
ML−1
l=0
h l ˜Ts s (MK + m) ˜Ts − τl
s∗
((n − k) Ts) + vm(kTs),
(29)
where h(lTs) and s(k ˜Ts), M, and ˜vm(kTs) are defined in
(6), (8) and (9) respectively. Considering τl = τ0 + l ˜Ts and
substituting τ0 by (7), leads to:
gm(kTs) =
ML−1
l=0
h l ˜Ts
N+k−1
n=k
s (Mn + m − (Mkτ0
+ kδ + l)) ˜Ts
s∗
((n − k) Ts) + vm(kTs),
(30)
where h(lTs) and s(k ˜Ts), kδ, M, and ˜vm(kTs) are defined in
(6), (7), (8) and (9) respectively. Using ˜Ts = Ts
M , and applying

10
some mathematical manipulations leads to:
gm(kTs) =



ML−1
l=0 h l ˜Ts
N+k−1
n=k s n − kτ0 − l
M Ts
s∗
((n − k) Ts) + ˜vm,k, m = kδ
ML−1
l=0 h l ˜Ts
N+k−1
n=k s n − kτ0 − m−kδ−l
M Ts
s∗
((n − k) Ts) + ˜vm,k, m = kδ
,
(31)
APPENDIX B
FIRST ORDER DERIVATIVE OF THE WISL OBJECTIVE
FUNCTION
Considering (19), we define:
WISL =
1
4N
2N
p=1
Cp, (32)
for Cp = (Sp − Γ1,1N)
2
and Sp = ˜sp
†
Γ ˜sp, where,
˜sp = ejϕ1
e−j πp
N , ..., ejϕN
e−j πNp
N
T
, (33)
where {ϕn}N
n=1 is the desired phase sequence of waveform
with length N defined in (18), and Γ is defined in (21).
Substituting (33) into (32) leads to:
Sp =
N
l=1
N
k=1
e−jϕk
ej πkp
N Γk,lejϕl
e−j πlp
N
=
N
l=1
ejϕl
e−j πlp
N
N
k=1,k=l
e−jϕk
ej πkp
N Γk,l +
N
l=1
Γl,l
(34)
The real part of (34) can be represented by:
Re {Sp} =
Re



N
l=1
ejϕl
e−j πlp
N
N
k=1,k=l
e−jϕk
ej πkp
N Γk,l +
N
l=1
Γl,l



=
N
l=1

cosαl
N
k=1,k=l
cosαkΓk,l + sinαl
N
k=1,k=l
sinαkΓk,l


(35)
where αl = ϕl − πlp
N . Substituting (35) into:
∂WISL
∂ϕm
=
∂
∂ϕm
|Sp|
2
− 2NRe {SpΓ1,1} (36)
leads to:
∂WISL
∂ϕm
=
= 2Sp
∂
∂ϕm
N
l=1
N
k=1
e−jϕk
ej πkp
N Γk,lejϕl
e−j πlp
N
∗
− 2N
∂
∂ϕm
N
l=1
cosαl
N
k=1,k=l
cosαkΓk,l
+ sinαl
N
k=1,k=l
sinαkΓk,l
(37)
the first term in (37) can be mathematically manipulated into:
∂
∂ϕm
N
l=1
N
k=1
e−jϕk
ej πkp
N Γk,lejϕl
e−j πlp
N
∗
= −je−jϕm
ej πmp
N
N
k=1,k=m
ejϕk
e−j πkp
N Γk,m + jejϕm
e−j πmp
N
N
k=1,k=m
e−jϕk
ej πkp
N Γm,k = −2
N
k=1,k=m
sin (αk − αm) Γm,k
(38)
Applying the same procedure, the second term in (37) can be
written by:
∂
∂ϕm
N
l=1
cosαl
N
k=1,k=l
cosαkΓk,l
+ sinαl
N
k=1,k=l
sinαkΓk,l = −sinαm
N
k=1,k=m
cosαkΓk,m
+ cosαm
N
k=1,k=m
sinαkΓk,m − sinαm
N
l=1,l=m
cosαlΓm,l
+ cosαm
N
l=1,l=m
sinαlΓm,l = 2
N
k=1,k=m
sin (αk − αm) Γk,m
(39)
substituting (38) and (39) into (37), leads to the final equation
for the first order derivative of WISL as follow:
∂Cp
∂ϕm
=
Sp
N
− Γ1,1
N
k=1,k=m
sin (αk − αm) Γk,m (40)
APPENDIX C
SECOND ORDER DERIVATIVE OF THE WISL OBJECTIVE
FUNCTION
The Hessian matrix corresponding the WISL can be repre-
sented by:
Hl,m =
∂
∂ϕl
∂WISL
∂ϕm
=
∂
∂ϕl



2N
p=1
Sp
N
− Γ1,1
N
k=1,k=m



=
2N
p=1
Sp
N
− Γ1,1
∂
∂ϕl
N
k=1,k=m
+
N
k=1,k=m
∂
∂ϕl
2N
p=1
Sp
N
− Γ1,1
(41)
where αl = ϕl − πlp
N for {ϕn}N
n=1 as the desired phase
sequence of waveform with length N defined in (18), and Γ is

11
defined in (21). The first term in (41) can be mathematically
manipulated into:
∂
∂ϕl
N
k=1,k=m
sin (αk − αm) Γk,m =
cos (αk − αm) Γm,l, m = l
−
N
k=1,k=m cos (αk − αm) Γm,k, m = l
(42)
Applying the same procedure, the second term in (41) can be
written by:
∂
∂ϕl
2N
p=1
Sp
N
− Γ1,1 =
1
N
2N
p=1
∂
∂ϕl
Sp =
2
N
2N
p=1
N
n=1
sin (αn − αm) Γn,l
(43)
Therefore, the final equation for the Hessian matrix of the
WISL is achievable by substituting (42) and (43) into (41):
∂
∂ϕl
∂WISL
∂ϕm
=
2
N
2N
p=1
N
n=1
sin (αn − αm) Γn,l
N
k=1,k=m
sin (αk − αm) Γm,k+



2N
p=1
Sp
N − Γ1,1 cos (αk − αm) Γm,l, m = l
−
2N
p=1
Sp
N − Γ1,1
N
k=1,k=m cos (αk − αm) Γm,k, m = l
(44)
APPENDIX D
DETAILS OF THE TOA CRLB CALCULATION
The derivatives of log-likelihood function with respect to the
elements of unknown vector, x can be represented by [38]:
d =
2
σ2
Re diag {h} A
H
(r − Ah) , (45)
where A is defined by:
A =




∂s(Ts−τ0)
∂τ0
... ∂s(Ts−τL−1)
∂τL−1
...
...
...
∂s(NTs−τ0)
∂τ0
... ∂s(NTs−τL−1)
∂τL−1



 (46)
where,
∂s(nTs − τl)
∂τl
=
∂
∂τl
S(ω)ejω(t−τl)
dω, (47)
where S(ω) represents the Fourier transform of s(t). Further-
more
h(I) =
2
σ2
Re AH
(r − Ah) , (48)
h(Q) =
2
σ2
Im AH
(r − Ah) , (49)
Applying the expectation into (45), (48) and (49) leads to:
E dΛr|x. T
d Λr|x =
2
σ2
Re diag (h∗
) A
H
A diag (h) ,
(50)
E dΛr|x. T
h(I) Λr|x =
2
σ2
Re diag (h∗
) A
H
A , (51)
E dΛr|x. T
h(Q) Λr|x =
−2
σ2
Im diag (h∗
) A
H
A , (52)
E h(I) Λr|x. T
h(I) Λr|x =
E h(Q) Λr|x. T
h(Q) Λr|x =
2
σ2
Re AH
A ,
(53)
E h(I) Λr|x. T
h(Q) Λr|x =
E h(Q) Λr|x. T
h(I) Λr|x =
−2
σ2
Im AH
A ,
(54)
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Vol.5.
Mohsen Jamalabdollahi (S’15) received the B.Sc.
degree in electrical engineering from University 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 at the
department of Electrical and Computer Engineering,
Michigan Technological University, Houghton, MI,
USA at 2016. He also was with the Software Ra-
dio group of Mobile Communications Technology
Co. Tehran, Iran from 2006 to 2013. His research
interests include digital signal processing, wireless
communications and convex optimization. Dr. Jamalabdollahi was awarded
the Matt Wolfe Award for Outstanding Graduate Research Assistant at the
department of Electrical and Computer Engineering, Michigan Technological
University at 2016.
Seyed (Reza) Zekavat (M’02–SM’10) has been
with Michigan Technological University, Houghton,
MI, USA, since 2002. He is the Editor of Handbook
of Position Location: Theory, Practice and Advances
(Wiley/IEEE). He has also co-authored the book
Multi-Carrier Technologies for Wireless Communi-
cations (Kluwer). He is also the author of the book
Electrical Engineering, Concepts and Applications
(Pearson). His research interests are in wireless com-
munications, positioning systems, software-deﬁned
radio design, dynamic spectrum allocation methods,
blind signal separation, and MIMO and beam forming techniques. He is
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, and
Springers International Journal on Wireless Networks,and the GSTF Journal
on Mobile Communications. He has been on the Executive Committee of
multiple IEEE conferences.

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