# IM-based Pilot-assisted Channel Estimation for FTN Signaling HF
Communications

Simin Keykhosravi and Ebrahim Bedeer
Department of Electrical and Computer Engineering, University of Saskatchewan,
Saskatoon, Canada
Emails:{sik904<EMAIL_ADDRESS>
###### Abstract

This paper investigates doubly-selective (i.e., time- and frequency-selective)
channel estimation in faster-than-Nyquist (FTN) signaling HF communications.
In particular, we propose a novel IM-based channel estimation algorithm for
FTN signaling HF communications including pilot sequence placement (PSP) and
pilot sequence location identification (PSLI) algorithms. At the transmitter,
we propose the PSP algorithm that utilizes the locations of pilot sequences to
carry additional information bits, thereby improving the SE of HF
communications. HF channels have two non-zero independent fading paths with
specific fixed delay spread and frequency spread characteristics as outlined
in the Union Radio communication Sector (ITU-R) F.1487 and F.520. Having said
that, based on the aforementioned properties of the HF channels and the
favorable auto-correlation characteristics of the optimal pilot sequence, we
propose a novel PSLI algorithm that effectively identifies the pilot sequence
location within a given frame at the receiver. This is achieved by showing
that the square of the absolute value of the cross-correlation between the
received symbols and the pilot sequence consists of a scaled version of the
square of the absolute value of the auto-correlation of the pilot sequence
weighted by the gain of the corresponding HF channel path. Simulation results
show very low pilot sequence location identification errors for HF channels.
Our simulation results show a 6 dB improvement in the MSE of the channel
estimation as well as about 3.5 dB BER improvement of FTN signaling along with
an enhancement in SE compared to the method in [1]. We also achieved an
enhancement in SE compared to the work in [2] while maintaining comparable MSE
of the channel estimation and BER performance.

###### Index Terms:

Channel estimation, faster-than-Nyquist (FTN) signaling, HF communications,
index modulation (IM).

##  I Introduction

Diverse applications of wireless communications require transmitting
information over long distances in rural and remote areas. High-frequency (HF)
communications represent an effective method of communicating with long-
distance users in remote or isolated locations without access to other
wireless services [3]. The frequencies between 3 MHz and 30 MHz are denoted as
the HF band, which represents a major part of the shortwave band. Refraction
of radio waves from the ionosphere back to earth known as skywave propagation
facilitates long-distance communications[4]. Usually, the allocated bandwidth
for the HF communication channels is limited to 3 kHz [5]. Hence, HF
communications have a bottleneck because of its small bandwidth and limited
data rate. Having limited bandwidth in HF places a significant demand on
enhancing the spectral efficiency (SE) for long-distance communications.

Introduced in the 1970s, faster-than-Nyquist (FTN) signaling has increasingly
gained researchers’ interest as a potential candidate to enhance the SE for a
given bandwidth and energy per bit [6, 7]. The conventional Nyquist
transmission employing $T$-orthogonal pulses specifies a maximum rate of $1/T$
to have inter-symbol interference (ISI)-free communications over frequency
flat channels known as the Nyquist limit [8]. FTN signaling, on the other
hand, involves the transmission of data symbols utilizing $T$-orthogonal
pulses at symbol intervals of $\tau T$, where $0<\tau<1$ is the FTN signaling
acceleration parameter. Exceeding the Nyquist limit results in a higher
transmission rate, albeit at the expense of inevitable ISI at the receiver.

The detection of FTN signaling is addressed in several works, e.g.,[9, 10, 11,
12, 13, 14, 15, 16, 7, 17, 1, 18, 2]. The ISI, introduced by the uncoded
transmission of FTN signaling, has been demonstrated to exhibit a trellis
structure [9]. Therefore, in order to mitigate the effects of trellis-
structured ISI, one can utilize a trellis decoder for data detection. Due to
the high computational complexity associated with the optimal FTN signaling
detection for small values of $\tau$ even in the presence of additive white
Gaussian noise (AWGN), various studies have proposed reduced trellis or
reduced tree search methods to approximate the optimal solution [9, 10, 11].
Tree search and trellis-based equalizers become challenging to employ in
scenarios with high modulation orders due to their excessive complexity.

Early studies of FTN signaling over AWGN channel employ time-domain optimal
detectors to address the ISI [10]. In scenarios with high values of $\tau$,
methods such as frequency domain equalization (FDE) [13, 12, 14] or symbol-by-
symbol detection [15] are used to design low complexity detectors for FTN
signaling. For high-order and ultra-high order modulations where FTN signaling
detection becomes even more challenging, precoding techniques [7] and
optimization methods based on the alternating directions multiplier method
(ADMM) [16] prove effective in reducing the computational complexity.

The channel estimation of FTN signaling across frequency-selective channels
has been studied in a few papers [17, 18, 1, 2]. In order to estimate the
channel in [17], a Nyquist-based pilot sequence was inserted before the data
block. In contrast to [17], in [18] an FTN signaling-based pilot sequence was
utilized for time-domain joint channel estimation and FTN signaling detection.
In [1], a low-complexity joint channel estimation and data detection for FTN
signaling was developed in the frequency domain to reduce the complexity. In
particular, a guard block with the same length as the pilot sequence is
introduced ahead of the transmitted data within the frame to prevent the ISI
from affecting the FTN signaling pilot sequence. Similarly, a cyclic prefix of
the same length as the pilot sequence and the guard block was inserted at the
end of the transmitted data in the frame. Such overhead which is four times
the length of the pilot sequence within the frame will eventually reduce the
SE in [1].

In most works on the detection of FTN signaling over frequency-selective
channels, the channel is regarded as quasi-static. While [17, 18, 1] assume a
quasi-static channel, a time-selective channel was considered in [14]. In
[14], a technique for frequency domain joint channel estimation and FTN
signaling detection is proposed, relying on the implementation of the message-
passing algorithm. Additionally, the paper in [2] proposed a novel channel
estimation technique for FTN signaling over doubly-selective channels based on
the least sum of squared errors (LSSE) approach which takes the ISI caused by
both FTN signaling and the frequency-selective channel into account. The
optimal pilot sequence that minimizes the mean square error (MSE) of the
channel estimation is found in [2]. For the time-selective nature of the
channel, a low-complexity linear interpolation is employed in [2] to track
complex channel coefficients at data symbol locations within the frame. In
contrast to [1], the implemented frame structure in [2] is more SE efficient
since it eliminates the necessity for an extra cyclic prefix, thereby reducing
the overhead.

In both Nyquist and FTN signaling studies, inserting a pilot sequence for the
channel estimation purpose results in a SE reduction. Thus, there is a need to
use methods to further improve the SE. Due to the restricted bandwidth in HF
communication, it is crucial to improve the SE. The index modulation (IM)
technique has drawn the interest of researchers owing to its capability of
enhancing the transmission rate by activating a selection of indices to carry
extra information [19]. IM is also employed in the frequency domain to convey
additional bits over the activated subcarrier indices in conjunction with the
$M$-ary modulation in [20]. The work in [20] proposes a low-complexity
frequency-domain joint channel estimation and equalization algorithm for
spectrally efficient frequency division multiplexing (SEFDM) based on IM
systems communicating over frequency selective fading channels.

Inspired by IM techniques, in this paper, we propose a novel IM-based channel
estimation algorithm including pilot sequence placement (PSP) and pilot
sequence location identification (PSLI) algorithms. The main contributions of
this paper can be outlined as follows:

  * •

At the transmitter, we propose the PSP algorithm, which makes use of the pilot
sequence locations to transmit extra information bits. That is to say, the
location of the pilot sequence in each frame is not fixed and it is determined
based on the information bits.

  * •

At the receiver, we propose a novel PSLI algorithm to detect the incorporated
information bits from the pilot sequence location. Pilot sequence location
identification is achieved by taking the inherent characteristics of the HF
channel as well as the autocorrelation properties of the optimal pilot
sequence that minimizes the mean square error (MSE) of the channel estimation
into account. On one hand, based on our analysis of the comprehensive array of
test HF channels introduced in the International Telecommunication Union Radio
Communication Sector (ITU-R) F.1487 document, we make use of the fact that
each HF channel comprises two non-zero independent fading paths with specific
fixed delay spread and frequency spread attributes. On the other hand, our
observation of the square of the absolute value of the autocorrelation
exhibited by the optimal pilot sequence reveals favorable local
characteristics. In particular, we show that for the received symbols
corresponding to the pilot sequence, the square of the absolute value of the
cross-correlation between the received symbols and the pilot sequence contains
a scaled version of the square of the absolute value of the auto-correlation
of the pilot sequence weighted by the gain corresponding to the HF channel
path. We exploit these identified characteristics to find the location of the
pilot sequence within a frame.

  * •

The simulation results of the proposed IM-based channel estimation algorithm
for FTN signaling demonstrate exceptionally low pilot sequence location
identification error (less than $2\times 10^{-6}$ and $10^{-3}$ for
$E_{\text{b}}/N_{0}$ values higher than 6 dB and 12 dB for time-invariant and
time-variant HF Poor channel, respectively, for $\tau=0.84$). Our simulation
results indicate achieving comparable performance in terms of MSE and BER of
the channel estimation while having an improvement in SE when compared to the
work in [2] over frequency-selective channels. Simulation results also
demonstrate an enhancement in SE along with noteworthy improvement (about 6
dB) in the MSE of the channel estimation as well as (about 3.5 dB) in BER of
FTN signaling compared to the approach introduced in [1] when dealing with
frequency-selective channels.

The following sections of this paper are structured as follows. Section II
introduces the system model of the proposed IM-based HF channel estimation FTN
signaling system and familiarizes the reader with the adopted frame structure.
Section III proposes a PSP algorithm to place the pilot sequence in each frame
at the transmitter in a way that facilitates the detection in the receiver.
Section IV elucidates the proposed PSLI algorithm specifically tailored for
the HF channel. Section IV presents the simulation results, emphasizing the
outcomes achieved through the conducted simulations. Section V serves as the
final section, offering concise conclusions derived from the preceding
sections of the paper.

In this paper, column vectors are denoted by boldface lowercase letters, such
as $\mathbf{x}$, while scalars are represented by lightface lowercase letters,
such as $x$. The complex conjugate of a complex number $x$ is indicated by
$x^{\ast}$, and the convolution operator between two variables $x$ and $y$ is
denoted as $x\ast y$. The expectation operator for a vector $\mathbf{x}$ is
symbolized as $\text{E}(\mathbf{x})$, and we use $\backslash$ to exclude an
element from a set. We use $\mathbf{x}^{\text{H}}$ to denote the complex
conjugate transpose (Hermitian transpose) operations of vector $\mathbf{c}$.
The ceil operator represented as $\lceil{x}\rceil$ rounds the real value $x$
up to the nearest integer greater than or equal to $x$. The floor operator
represented as $\lfloor{x}\rfloor$ rounds the real value $x$ down to the
nearest integer less than or equal to $x$. The boldface uppercase letters,
e.g., $\mathbf{X}$, are employed to represent matrices. Additionally, the
$\mathbb{C}^{N\times M}$ encompasses all complex matrices of complex matrices
of size ${N\times M}$. To represent the inverse, transpose, and complex
conjugate transpose operations of a matrix $\mathbf{X}$, we employ
$\mathbf{X}^{-1}$, $\mathbf{X}^{\text{T}}$, and $\mathbf{X}^{\text{H}}$,
respectively. Finally, the cardinality of a set $\mathbb{P}$ is defined by
$\text{Card}\\{\mathbb{P}\\}$.

Figure 1: Block diagram of a communication system employing IM-based HF
channel estimation FTN signaling system.

##  II System Model

In this paper, we employ an IM-based HF channel estimation FTN signaling
system for long-distance communications as depicted in Fig. 1. Each block of
independent and identically distributed (i.i.d.) $N_{\text{d}}^{\prime}$ data
bits, denoted as $\textbf{a}^{\prime}={[a,b]}^{\text{T}}$, consists of a block
of $N_{\text{d}}$ regular information bits
$\textbf{a}={[a_{0},a_{1},...,a_{N_{\text{d}}-1}]}^{\text{T}}$ along with
$N_{b}$ additional bits
$\textbf{b}={[b_{0},b_{1},...,b_{N_{b}-1}]}^{\text{T}}$ that will be used to
specify the pilot sequence location in the frame.

A block of $N_{\text{d}}$ regular information bits, a, to be transmitted in a
given frame is first encoded to $\textbf{d}^{\prime}$ and then is shuffled by
an interleaver to d. Each ${\log_{2}}M$ interleaver’s output bits are then
modulated onto a complex data symbol belonging to an $M$-ary quadrature
amplitude modulation ($M$-QAM) constellation set $\mathcal{S}_{\text{d}}$. A
frame of length $N=N_{\text{p}}+{{N_{\text{s}}}}$ is constructed by inserting
known pilot sequence
$\textbf{p}=[p_{0},p_{1},...,p_{N_{\text{p}}-1}]^{\text{T}}$ of length
$N_{\text{p}}$ in the location determined by the $N_{b}$ information bits to
assist the receiver in the channel estimation. Please note that
$N_{\text{s}}=\frac{N_{\text{d}}}{{\log_{2}}M}$ is the number of symbols in a
frame. As mentioned earlier, the location of the pilot sequence,
$n_{s_{p}}^{(i)}$, is not fixed in a given frame $i$ and conveys additional
$N_{b}$ data bits. At the transmitter, the pilot sequence could be located at
any location belonging to $n_{s_{p}}^{(i)}=\\{0,1,...,N_{\text{s}}\\}$, and
hence, one has $N_{\text{s}}+1$ choices to place the pilot in a frame, which
is equivalent to carrying $\lfloor{\log_{2}}{(N_{s}+1)}\rfloor$ extra bits. It
is worth mentioning that if the ${\log_{2}}{(N_{s}+1)}$ is not an integer
number, the number of bits carried by the location of pilot sequence is
$\lfloor{\log_{2}}{(N_{s}+1)}\rfloor<{{\log_{2}}{(N_{s}+1)}}$ and this is
equivalent to $2^{\lfloor{\log_{2}}{(N_{s}+1)}\rfloor}$ (which is less than
${N_{s}+1}$) choices for the pilot sequence to be located to construct the
frame and thus the pilot sequence location is
$n_{s_{p}}^{(i)}=\\{0,1,...,2^{\lfloor{\log_{2}}{(N_{s}+1)}\rfloor}-1\\}$. In
Section III, we describe the proposed PSP algorithm for transmitting the
$N_{b}$ additional bits through the pilot sequence location in each frame at
the transmitter and a low complexity pilot sequence identification algorithm
for identification of the pilot sequence location and thus detecting the
$N_{b}$ bits for each frame at the receiver in Section IV.

The pilot sequence and data symbols within a frame are transmitted every
symbol interval $\tau T$ through a $T$-orthogonal root raised cosine (RRC)
pulse, $g(t)$, where $g(t)$ is a unit energy pulse, i.e.,
$\int_{-\infty}^{\infty}|g(t)|^{2}\,dt=1$. Having said that, the baseband
transmitted signal can be expressed as

$\displaystyle x(t)$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{p}^{(i)}\in\Omega_{\text{p}}^{(i)}}{s_{n_{p}^{(i)}}g(t-n_{p}^{(i)}\tau
T)}$ (1)
$\displaystyle{}+\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{d}^{(i)}\in\Omega_{\text{d}}^{(i)}}{s_{n_{d}^{(i)}}g(t-n_{d}^{(i)}\tau
T)},$

where $N_{i}$ is the total number of frames,
$\Omega_{\text{p}}^{(i)}=\\{n_{s_{p}}^{(i)},n_{s_{p}}^{(i)}+1,\ldots,n_{s_{p}}^{(i)}+N_{p}-1\\}$
is the set of pilot sequence symbols’ locations within the $i$th frame
starting from the $n_{s_{p}}^{(i)}$th symbol of pilot sequence in the frame
$i$ as can be seen in Fig. 2. That is to say for the frame $i$, we have
$[s_{n_{s_{p}}^{(i)}},s_{n_{s_{p}}^{(i)}+1},\ldots,s_{n_{s_{p}}^{(i)}+N_{p}-1}]^{\text{T}}=[p_{0},p_{1},...,p_{N_{\text{p}}-1}]^{\text{T}}$.
We define $\Omega_{\text{d}}^{(i)}=\Omega\backslash\Omega_{\text{p}}^{(i)}$ as
the set of data symbols’ locations within the $i$th frame where
$n_{d}^{(i)}\in\Omega_{\text{d}}^{(i)}$, and ${s_{n_{d}^{(i)}}}$ is the data
symbol within the frame $i$. We have $\Omega=\\{0,1,,\ldots,N-1\\}$.

Considering the transmission of FTN signaling over an HF doubly-selective
fading channel denoted as $c(t,\phi)$, the received signal can be expressed as

$\displaystyle y(t)$ $\displaystyle{}={}$ $\displaystyle x(t)\ast
c(t,\phi)+n(t),$ (2)

where $n(t)$ is the zero-mean AWGN with double-sided power spectral density
$N_{0}/2$. The received signal, $y(t)$, is passed through a filter matched to
$g(t)$, $g^{\ast}(-t)=g(t)$, and the output of the matched filter is
represented as

$\displaystyle r(t)$ $\displaystyle{}={}$ $\displaystyle x(t)\ast
c(t,\phi)\ast g(t)+z(t),$ (3)

where $z(t)=n(t)\ast g(t)$. Substituting $x(t)$ from (1), the received signal
in (3) can be re-expressed as

$\displaystyle r(t)$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{p}^{(i)}\in\Omega_{\text{p}}^{(i)}}{s_{n_{p}^{(i)}}h(t-n_{p}^{(i)}\tau
T)}\ast c(t,\phi)$ (4)
$\displaystyle{}+\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{d}^{(i)}\in\Omega_{\text{d}}^{(i)}}{s_{n_{d}^{(i)}}h(t-n_{d}^{(i)}\tau
T)}\ast c(t,\phi)$ $\displaystyle{}+z(t),$

where $h(t)=g(t)\ast g(t)$ is the RC pulse. While the ISI length of the FTN
signaling is theoretically infinite, we consider the fact that the power of
the raised cosine pulse, $h(t)$, is concentrated in and around its main lobe.
Consequently, we limit the length of the raised cosine pulse, $h(t)$, to
$2L_{\text{h}}+1$. In other words, we truncate the pulse to have
$2L_{\text{h}}+1$ effective taps, which can be represented by the vector
$\mathbf{h}=[h_{-L_{\text{h}}},h_{L_{\text{h}}+1},\ldots,h_{L_{\text{h}}}]^{\text{T}}$.
The parameter $L_{\text{h}}$ depends on the FTN signaling packing ratio $\tau$
and the roll-off factor of the RRC pulse $g(t)$.

Being widely employed in the simulation of the HF ionospheric channel, the
Watterson model [21] defines a tapped delay line, with $L_{\text{c}}+1$ taps,
each one models an ionospheric propagation path. Hence, we utilize the widely
adopted tapped delay line channel model to represent a doubly-selective
channel. In this model, the impulse response of the channel, denoted as
$c(t,\phi)$, consists of a total of $L_{\text{c}}+1$ channel taps, namely
$c_{l}(t)$ where $l$ ranges from 0 to ${L_{\text{c}}}$. Each channel path
$c_{l}(t)$ within this model corresponds to a specific delay $\phi_{l}(t)$.
The output of the matched filter in (4) is sampled at intervals of $\tau T$,
which can be mathematically represented as

$\displaystyle r_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{p}^{(i)}\in\Omega_{\text{p}}^{(i)}}{s_{n_{p}^{(i)}}\sum\limits_{l=0}^{L_{\text{c}}}c_{k,l}h_{k-l-
n_{p}^{(i)}}}$ (5)
$\displaystyle{}+\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{d}^{(i)}\in\Omega_{\text{d}}^{(i)}}{s_{n_{d}^{(i)}}\sum\limits_{l=0}^{L_{\text{c}}}c_{k,l}h_{k-l-
n_{d}^{(i)}}}+z_{k},$

Figure 2: The frame structure with dynamic pilot sequence location in frames
employed in the paper.

In order to decorrelate the colored noise samples $z_{k}$, the sampled signal
in (5) passes through a discrete-time whitening filter,
$\textbf{v}=[v_{0},v_{1},...,v_{L_{\text{h}}-1}]^{\text{T}}$ of length
$L_{\text{h}}$ as explained in [2]. The $k$th output sample of the whitening
filter can be written as

$\displaystyle\tilde{r}_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{p}^{(i)}\in\Omega_{\text{p}}^{(i)}}{s_{n_{p}^{(i)}}\sum\limits_{l=0}^{L_{\text{c}}}c_{k,l}v_{k-l-
n_{p}^{(i)}}}$ (6)
$\displaystyle{}+\sum\limits_{i=1}^{N_{i}}\sum\limits_{n_{d}^{(i)}\in\Omega_{\text{d}}^{(i)}}{s_{n_{d}^{(i)}}\sum\limits_{l=0}^{L_{\text{c}}}c_{k,l}v_{k-l-
n_{d}^{(i)}}}+w_{k},$

where $w_{k}$ is the zero-mean white Gaussian noise with variance
$\sigma_{n}^{2}$.

The output of the whitening filter, as introduced in (6), is then fed into the
PSLI block to identify the pilot sequence’s location within a given frame.
Once the pilot sequence’s location is determined, the receiver employs the
least sum of squared errors (LSSE) based approach, as proposed in [2], to
estimate the complex channel coefficients at the pilot sequence locations
within the frame. The turbo equalizer receives the samples output of the
whitening filter, alongside the estimated channel impulse response as an input
from the channel estimation block, and outputs the estimated data blocks
$\hat{\textbf{a}}$. Finally, the estimated regular bits output of the turbo
equalizer, along with the bits representing the pilot sequence’s location in
the frame, are multiplexed in the MuX block to generate the estimated data
block $\hat{\textbf{a}^{\prime}}$.

Figure 3: Example for the square of the absolute value of the cross-
correlation of the received signal and the pilot sequence. Figure 4: Example
for PSP algorithm having pilot sequence located in position 8 over HF channel
with 7 taps.

##  III Pilot sequence placement algorithm

In this section, we introduce our PSP algorithm to incorporate additional
information bits within the pilot sequence location of each frame. Not solely
restricted to HF channels, the PSP algorithm at the transmitter can be applied
to any channel. The frame structure employed in this paper is illustrated in
Fig. 2. This frame structure considers the detrimental effects of inter-block
interference (IBI) which is discussed in detail in [2]. As described in detail
in the system model, the pilot sequence can be located at
$n_{s_{p}}^{(i)}=\\{0,1,...,2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-1\\}$
in a given frame to carry ${\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}$
additional bits.

Identifying the location of the pilot sequence, $\hat{n}_{s_{p}}^{(i)}$, at
the receiver by the pilot sequence identification algorithm depends on the
magnitude of the received signal for decision-making. This could be
problematic for example when the first tap of the $(L_{\text{c}}+1)$-tap HF
channel is in a deep fade. In a multipath channel, the received signal
consists of a summation of different versions of the transmitted signal
received from different paths of the channel. Thus, if the first path is in a
deep fade, the first version of the signal becomes very weak at the receiver
and the strong version of the signal coming from the strong tap becomes
dominant in the receiver. Having said that, at the receiver, employing a
correlation-based technique, the location of the peak that corresponds to the
pilot sequence received through the strongest tap of the channel could be
identified and not the location that corresponds to the first tap,
$n_{s_{p}}^{(i)}$. To address this issue, in our proposed PSP algorithm, we
restrict $n_{s_{p}}^{(i)}$ to some predefined locations. In other words, we do
not allow $n_{s_{p}}^{(i)}$ to take all possible values in
$\\{0,1,...,2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-1\\}$. We
define a set, denoted as $\mathbb{P}$, to represent the predefined locations
$\\{0,1,\ldots,2^{\lfloor\log_{2}({{N_{\text{s}}}}+1)\rfloor}-1\\}$ which are
multiples of $2^{\lceil\log_{2}(L_{c}+1)\rceil}$. The cardinality of
$\mathbb{P}$, $\text{Card}\\{\mathbb{P}\\}$, is equal to
$2^{{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}$.
This represents the number of choices for $n_{s_{p}}^{(i)}$ at the
transmitter. Consequently, the location of the pilot sequence could carry an
additional
$N_{\text{b}}={{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}$
bits. As $n_{s_{p}}^{(i)}$, a member of predefined locations set $\mathbb{P}$,
is a multiple of $2^{\lceil\log_{2}(L_{c}+1)\rceil}$, can be represented with
${\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}$ bits where the
${\lceil\log_{2}(L_{c}+1)\rceil}$ least significant bits of its binary
representation are zero. Thus, $N_{\text{b}}$ additional data bits that are
carried through the location of the pilot in a given frame can be mapped
uniquely to a location in $\mathbb{P}$ where $N_{\text{b}}$ most significant
bits (MSB) of the binary representation specify the location. Consider a
common HF channel having $L_{\text{c}}+1$ taps with two of them to be non-zero
($c_{0}\ \text{and}\ c_{L_{\text{c}}}$). For a given $n_{s_{p}}^{(i)}$ in a
given frame $i$ at the transmitter belongs to the predefined locations set,
$\mathbb{P}$, we expect to have spikes at the locations ${n}_{s_{p}}^{(i)}$
and $n_{s_{p}}^{(i)}+L_{\text{c}}$ in the square of the absolute value of the
cross-correlation of the received signal with the pilot sequence,
$|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$ for $\delta=0,...,N-1$, as defined in
Section IV at the receiver. Thus, the set of expected pilot sequence locations
at the receiver, denoted as $\hat{\mathbb{P}}$, consists of
$\\{{n}_{s_{p}}^{(i)},n_{s_{p}}^{(i)}+L_{\text{c}}\\}$ for all
${n}_{s_{p}}^{(i)}\in\mathbb{P}$. Consequently, an estimated pilot sequence
location at the receiver, $\hat{n}_{s_{p}}^{(i)}$, should belong to
$\hat{\mathbb{P}}$ to be valid. As for each predefined pilot sequence location
at the transmitter ${n}_{s_{p}}^{(i)}\in\mathbb{P}$, we have two expected
pilot sequence locations the receiver,
$\\{{n}_{s_{p}}^{(i)},n_{s_{p}}^{(i)}+L_{\text{c}}\\}\in\hat{\mathbb{P}}$, the
cardinality of $\hat{\mathbb{P}}$,
$\text{Card}\\{\hat{\mathbb{P}}\\}=2\times\text{Card}\\{{\mathbb{P}}\\}$.
Thus, we have $\text{Card}\\{\hat{\mathbb{P}}\\}=2\times
2^{{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}$.

It is best to explain the idea behind the proposed PSP algorithm with the help
of the following example. Consider a system with $N_{\text{p}}=32$,
${{N_{\text{s}}}}=256$, and a channel with 7 taps ($L_{\text{c}}=6$) with two
non-zero taps ($c_{0}\ \text{and}\ c_{L_{\text{c}}}$) having a fixed delay
spread with the first tap has lower strength. For this example, we investigate
two cases:

Case 1: Consider the case that we allow $n_{s_{p}}^{(i)}$ to take all possible
values in
$\\{0,1,...,2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-1\\}$. As in
this example ${\log_{2}}{({{N_{\text{s}}}}+1)}=8.0056$ is not an integer
number and thus ${\lfloor{{\log_{2}}{({{N_{\text{s}}}}+1)}}\rfloor}=8$ bits
could be carried through the pilot sequence location. This is equivalent to
$2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}=256$ (which is less than
${{{N_{\text{s}}}}+1}=257$) choices for the pilot sequence to be located to
construct the frame, and thus, the pilot sequence location
$n_{s_{p}}^{(i)}=\\{0,1,...,2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-1=255\\}$.
Let us for the first case, consider the pilot sequence’s location at the
transmitter $n_{s_{p}}^{(i)}$ to be $10$, the receiver receives the summation
of two versions of the signal corresponding to each non-zero channel tap. Fig.
3 depicts the square of the absolute value of the cross-correlation of the
received signal with the pilot sequence, i.e.,
$|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$, $\delta=0,...,N-1$, as defined in
Section IV, under the presumption that the pilot sequence has perfect auto-
correlation properties. As shown in Fig. 3 there are two spikes in
$|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$, one at the location
$n_{s_{p}}^{(i)}=10$ and the other at the location
$n_{s_{p}}^{(i)}+L_{\text{c}}=16$ where the second is the strongest. Hence,
the location of the pilot sequence $\hat{n}_{s_{p}}^{(i)}$ will be incorrectly
identified to be $n_{s_{p}}^{(i)}+L_{\text{c}}=16$ if the PSP algorithm is not
designed carefully, i.e. $n_{s_{p}}^{(i)}$ does not belong to $\mathbb{P}$. As
mentioned before, to address this issue, we do not allow $n_{s_{p}}^{(i)}$ to
take all possible values in $\\{0,1,...,255\\}$. The predefined locations
should belong to $\mathbb{P}$. In this example, $\mathbb{P}$ consisted of
integer values in
$\\{0,1,...,2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-1=255\\}$ which
are multiples of
$2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}=2^{\lceil{{\log_{2}}{7}}\rceil}=8$.
Thus, in this example, $n_{s_{p}}^{(i)}$ should have been chosen to belong to
the set
$\mathbb{P}=\\{0,2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil},2\times
2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil},3\times
2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil},...,31\times
2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}\\}=\\{0,8,2\times 8,3\times
8,...,248\\}$. Having said that, as $n_{p}^{(i)}=10$ does not belong to
$\mathbb{P}$, thus we can not set $n_{p}^{(i)}$ to be $10$ at the transmitter.

Fig 4 further illustrates an example of pilot locations allowed by PSP
algorithm. Predefined pilot sequence locations, where the transmitter can
place the pilot sequence within a frame, are represented in green. Blocked
locations, which are unavailable for the transmitter to use for pilot sequence
placement, are depicted in pink. For a given $n_{s_{p}}^{(i)}$ in a given
frame $i$ at the transmitter, we expect to have spikes at the locations
${n}_{s_{p}}^{(i)}$ and $n_{s_{p}}^{(i)}+L_{\text{c}}$ in
$|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$ at the receiver. Thus the estimated
pilot sequence location at the receiver $\hat{n}_{s_{p}}^{(i)}$ should belong
to the set
$\hat{\mathbb{P}}=\\{\\{0,6\\},\\{8,14\\},\\{16,22\\},\\{24,30\\},...,\\{248,256\\}\\}$.

Case 2: Let us consider $n_{s_{p}}^{(i)}=16$ belongs to $\mathbb{P}$ at the
transmitter. Now having the described channel, at the receiver we expect two
spikes $n_{s_{p}}^{(i)}=16$ and $n_{s_{p}}^{(i)}+L_{\text{c}}=22$ (as shown in
Fig. 4) with the second spike to be stronger. Based on
$|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$, the location of the pilot sequence
location is identified to be $n_{s_{p}}^{(i)}+L_{\text{c}}=22$ but we know
that the location of the pilot sequence $n_{s_{p}}^{(i)}$ is a multiple of 8
and thus $n_{s_{p}}^{(i)}$ module 8 should be zero. Thus, as 22 belongs to the
expected locations $\\{16,22\\}$ we can identify $\hat{n}_{s_{p}}^{(i)}=16$.
In other words, 22 module 8 is 6, and by subtracting 6 from 22, we can
correctly identify $\hat{n}_{s_{p}}^{(i)}=16$. In this example, there are
$2^{{\lfloor{\log_{2}}{256+1}\rfloor}-\lceil{{\log_{2}}{(6+1)}}\rceil}=2^{8-3}=32$
choices for $n_{s_{p}}^{(i)}$ at the transmitter and thus the location of
pilot sequence could carry
$N_{\text{b}}={{\lfloor{\log_{2}}{256+1}\rfloor}-\lceil{{\log_{2}}{(6+1)}}\rceil}=8-3=5$
additional bits in each frame.

The SE measured in bits/sec/Hz in a Nyquist system, FTN signaling system and
our proposed IM-based channel estimation for FTN signaling system is denoted
respectively as $\gamma_{\text{Nyq}}$, $\gamma_{\text{FTN}}$, and
$\gamma_{\text{IM-FTN}}$ and is defined as

$\displaystyle\gamma_{\text{Nyq}}$ $\displaystyle{}={}$
$\displaystyle\frac{{{N_{\text{s}}}}}{({{N_{\text{s}}}}+N_{\text{p}})}\cdot\frac{{\log_{2}}M}{(1+\beta)}{{R_{\text{c}}}},$
(7) $\displaystyle\gamma_{\text{FTN}}$ $\displaystyle{}={}$
$\displaystyle\frac{{{N_{\text{s}}}}}{({{N_{\text{s}}}}+N_{\text{p}})}\cdot\frac{{\log_{2}}M}{\tau(1+\beta)}R_{\text{c}},$
(8) $\displaystyle\gamma_{\text{IM-FTN}}$ $\displaystyle{}={}$
$\displaystyle{\frac{N_{\text{s}}\cdot
R_{\text{c}}\cdot{{\log_{2}}M}+{N_{\text{b}}}}{(N_{\text{s}}+N_{\text{p}})}\cdot\frac{1}{\tau(1+\beta)}},$
(9)

where $\beta$ is the roll-off factor of the RRC pulse shaping filter and
$R_{\text{c}}$ is the code rate. It should be noted that in our proposed IM-
based channel estimation for the FTN signaling system, $N_{\text{b}}$
additional bits are transmitted in a given frame. The improvement percentage
of the SE of the proposed IM-based channel estimation for the FTN signaling
over Nyquist signaling and over FTN signaling is defined, respectively, as

$\displaystyle g_{\text{IM-FTN,Nyq}}$ $\displaystyle{}={}$
$\displaystyle\frac{(\gamma_{\text{IM-
FTN}}-\gamma_{\text{Nyq}})}{\gamma_{\text{Nyq}}}$ $\displaystyle{}={}$
$\displaystyle{{\frac{{({N_{\text{s}}}\cdot
R_{\text{c}}\cdot{{\log_{2}}M})}(\frac{1}{\tau}-1)+\frac{N_{\text{b}}}{\tau}}{N_{\text{s}}\cdot
R_{\text{c}}\cdot{{\log_{2}}M}}}}\times 100\%,$ $\displaystyle g_{\text{IM-
FTN,FTN}}$ $\displaystyle{}={}$ $\displaystyle\frac{(\gamma_{\text{IM-
FTN}}-\gamma_{\text{FTN}})}{\gamma_{\text{FTN}}}=\frac{N_{\text{b}}}{{{N_{\text{s}}\cdot
R_{\text{c}}}}\cdot{{\log_{2}}M}}\times 100\%.$

Table I shows the spectral efficiencies $\gamma_{\text{Nyq}}$,
$\gamma_{\text{FTN}}$, and $\gamma_{\text{IM-FTN}}$ and the percentage of SE
improvements for our proposed IM-based channel estimation for the FTN
signaling compared to Nyquist signaling $g_{\text{IM-FTN,Nyq}}$ and the FTN
signaling $g_{\text{IM-FTN,FTN}}$ for practical values of ${{N_{\text{s}}}}$
and $N_{\text{p}}$ employed in HF systems for $\beta=0.35$ and $\tau=0.72$,
when considering the $R_{\text{c}}=3/4$. As shown in Table I, there could be
more than 40% improvement in SE when utilizing the proposed IM-based channel
estimation for the FTN signaling compared to the traditional Nyquist signaling
for some typical values ${{N_{\text{s}}}}$ and $N_{\text{p}}$ for HF systems.
As well as that Table I shows that there could be between approximately 1.3%
and 5.55% improvement in SE when utilizing the proposed IM-based channel
estimation for FTN signaling compared to the FTN signaling with the
conventional pilot sequences (pilot sequence location being fixed in the
frames) for some typical values ${{N_{\text{s}}}}$ and $N_{\text{p}}$ for HF
systems.

TABLE I: SE gains employing the proposed IM-based channel estimation for FTN
signaling system for $\beta=0.35$, $\tau=0.72$, and $R_{\text{c}}=3/4$

.  $N_{\text{p}}\>(\text{Symbols})$ ${{N_{\text{s}}}}\>(\text{Symbols})$ $M$
$\gamma_{\text{Nyq}}\>({\text{bits/sec/Hz}})$ $\gamma_{\text{FTN}}$
(bits/sec/Hz) $\gamma_{\text{IM-FTN}}$ (bits/sec/Hz) $g_{\text{IM-FTN,Nyq}}$
$g_{\text{IM-FTN,FTN}}$ 48 48 2 0.2778 0.3858 0.4072 46.60% 5.55% 32 96 2
0.4167 0.5787 0.6028 44.67% 4.16% 32 256 2 0.4938 0.6859 0.7037 42.50% 2.60%
32 256 4 0.9877 1.3717 1.3896 40.70% 1.30%

###  III-A Pilot sequence Placement Algorithm

The proposed pilot sequence placement algorithm is summarized as:

Algorithm 1 Pilot sequence Placement Algorithm

Input: ${{N_{\text{s}}}}$ and $L_{\text{c}}$
1) Construct the set of predefined pilot sequence locations allowed at the
transmitter, ${\mathbb{P}}$, consisting of integer multiples of
$2^{\lceil\log_{2}(L_{\text{c}}+1)\rceil}$ belonging to the set
$\\{0,1,...,2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-1\\}$.
2) Calculate the number of bits that can be carried through the pilot sequence
location
$N_{\text{b}}={{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}$.
3) Map the $N_{\text{b}}$ additional data bits to a predefined location in
${\mathbb{P}}$, $n_{s_{p}}^{(i)}$, where the $N_{\text{b}}$ MSB of the binary
representation specify the location.
4) Place the pilot sequence of length $N_{\text{p}}$ in the location
$n_{s_{p}}^{(i)}$ of the $i$th frame of length $N$.
Output: $n_{s_{p}}^{(i)}$

##  IV Pilot Sequence Location Identification

In this section, we discuss the proposed low-complexity pilot sequence
identification algorithm to identify the location of the pilot sequence in
each frame at the receiver. As mentioned earlier, this paper focuses on the HF
channel model recommended by the International Telecommunications Union Radio
Communication Sector (ITU-R). The ITU-R has introduced a comprehensive set of
recommended HF channel models widely utilized for simulating the performance
of HF systems. ITU-R F.520 [22] recommends three distinct test channels,
namely the ITU-R good, ITU-R moderate, and ITU-R poor channel, which serve to
classify the channel conditions into varying levels based on their individual
characteristics. Although these channels are still commonly utilized for
assessing the performance of HF systems, it is important to note that the
corresponding recommendation has become outdated and has been replaced by
ITU-R F.1487 [23]. ITU-R F.1487 introduces a comprehensive set of 10 test
channels, which include various latitude regions and levels of ionospheric
disturbance. Notably, each of the three test channels specified in F.520
aligns with at least one of the test channels defined in F.1487.

The ITU-R F.520 [22] and ITU-R F.1487 [24] recommendations describe all test
channels as tapped delay line models comprising only two non-zero taps. These
taps exhibit independent fading, undergoing a Rayleigh probability density
function. Additionally, both paths experience equal average power with the
same Doppler frequency. In this paper, the ITU-R poor channel is utilized for
simulation purposes to represent worst case scenario on channel estimation
performance. The channel condition labeled as poor channel is in accordance
with the ITU-R F.1487 specifications for the mid-latitude-disturbed channel.
It comprises two non-zero independent paths having a fixed delay of 2 ms and a
frequency Doppler of 1 Hz. The channel is assumed to have a delay spread of
2.1 ms, allowing for precisely 5 symbol intervals in the Nyquist case.
Considering a symbol rate of 2400 symbols per second, this channel can be
represented as a Watterson model with $6$ taps.

To determine the pilot sequence’s location within a given receiver frame, we
employ a cross-correlation technique. This approach serves two primary
purposes: firstly, it avoids the excessive complexity associated with the
maximum likelihood (ML) technique required to identify the pilot sequence
location, and secondly, it utilizes the intrinsic characteristics of both the
HF channel and the favorable local auto-correlation of the optimal pilot
sequence designed to minimize the mean square error (MSE) of channel
estimation. The proposed pilot sequence identification algorithm involves
correlating the output of the whitening filter in the receiver,
$\tilde{r}_{k}$, with the known pilot sequence, which has been subject to FTN-
induced ISI and the whitening filter. In this study, we utilize an optimized
pilot sequence that minimizes the MSE of channel estimation for FTN signaling
over a doubly-selective channel, as proposed in [2]. The pilot sequence
affected by FTN-induced ISI and a whitening filter can be characterized as

$\displaystyle\tilde{p}_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{j=0}^{N_{p}-1}{p_{j}v_{k-j}},\quad
k=0,...,N_{p}-1.$ (11)

Figure 5: The square of the absolute value of the autocorrelation of the pilot
sequence of length $N_{\text{p}}=32$ affected by FTN-induced ISI and whitening
filter.

The value of the autocorrelation of the designed pilot sequence affected by
FTN-induced ISI and whitening filter indicated in (11) can be expressed as

$\displaystyle R_{\tilde{p}\tilde{p}}(\delta)$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{k=0}^{N_{p}-1}{\tilde{p}_{k}\tilde{p^{\ast}}_{\delta+k}},\quad\delta=0,...,N_{p}-1.$
(12)

Fig. 5 presents the square of the absolute value of the autocorrelation
function ${|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$ of the pilot sequence
designed for channel estimation of FTN signaling with $\tau=0.84$ over a
discretized doubly-selective HF channel. The choice of $\tau=0.84$ ensures
that the number of taps in the tapped delay line channel model is an integer.
As can be seen in Fig. 5, the optimal pilot sequence has desirable local auto-
correlation characteristics that assist in the pilot sequence identification.
The local auto-correlation characteristics $\psi(\delta)$ at a given location
$\delta$, $\delta=0,...,N_{p}-1.$ is defined as the ratio of
${|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$ and the normalized sum of
${|R_{\tilde{p}\tilde{p}}(j)|}^{2}$ for adjacent locations $j$ within a given
radius $r_{0}$, and the value of $r_{0}$ should be chosen to be less than
$L_{\text{c}}$ to avoid taking into account the contribution of the second
channel tap. Hence, the local auto-correlation characteristics $\psi(\delta)$
is formally defined as

$\displaystyle\psi(\delta)$ $\displaystyle{}={}$
$\displaystyle\frac{{|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}}{\lambda(\delta)},\quad\delta=0,...,N_{p}-1,$
(13)

where

$\displaystyle\lambda(\delta)$ $\displaystyle{}={}$
$\displaystyle\frac{{\sum\limits_{j=\max(\delta-
r_{0},\delta_{\text{min}})}^{\min(\delta+r_{0},\delta_{\text{max}})}{|R_{\tilde{p}\tilde{p}}(j)|}^{2}-{|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}}}{{\min(\delta+r_{0},\delta_{\text{max}})-\max(\delta-
r_{0},\delta_{\text{min}})}},$ (14)

where $\delta_{\text{min}}=0$ and $\delta_{\text{max}}={{N_{\text{s}}}}$ are
the lower and upper limits of $\delta$ in
${|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$, respectively.

We utilize the characteristics of $|R_{\tilde{p}\tilde{p}}(\delta)|^{2}$ to
define a measure for determining the pilot sequence’s location within a given
frame based on the square of the absolute of the cross-correlation between the
received signal $\tilde{r}_{k}$ and the pilot sequence affected by FTN-induced
ISI and the whitening filter, denoted as $\tilde{p}_{k}$. Considering the
received signal for a given frame $i$ in (6) as a sequence of
$\tilde{r}_{k},k=0,...,{{N_{\text{s}}}}-1$, the cross-correlation of
$\tilde{r}_{k}$ and $\tilde{p}_{k}$ can be written as

$\displaystyle R_{\tilde{r}\tilde{p}}(\delta)$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{m=0}^{N_{p}-1}{\tilde{r}_{\delta+m}\tilde{p}_{m}^{\ast}},\quad\delta=0,...,N_{d}-1,$
(15) $\displaystyle{}={}$
$\displaystyle\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{i=1}^{N_{i}}\left(\tilde{p}_{m}^{\ast}\mkern-12.0mu\sum\limits_{n_{p}^{(i)}\in\Omega_{\text{p}}^{(i)}}\mkern-18.0mu{s_{n_{p}^{(i)}}\sum\limits_{l=0}^{L_{\text{c}}}c_{\delta+m,l}v_{\delta+m-l-
n_{p}^{(i)}}}\right)$
$\displaystyle{}+\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{i=1}^{N_{i}}\left(\tilde{p}_{m}^{\ast}\mkern-18.0mu\sum\limits_{n_{d}^{(i)}\in\Omega_{\text{d}}^{(i)}}\mkern-18.0mu{s_{n_{d}^{(i)}}\sum\limits_{l=0}^{L_{\text{c}}}c_{\delta+m,l}v_{\delta+m-l-
n_{d}^{(i)}}}\mkern-8.0mu\right)$
$\displaystyle{}+{}\sum\limits_{m=0}^{{{N_{\text{s}}}}-1}w_{\delta+m}\tilde{p}_{m}^{\ast}.$

To simplify the notation, let us consider the transmission of one frame and
drop the frame index $i$. Having the cross-correlation of $\tilde{r}_{k}$ and
$\tilde{p}_{k}$ in (15), we could write the square of the absolute value of
the $|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$ over a general frequency selective
time-invariant channel as

$\displaystyle\small{|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ $\displaystyle=$
$\displaystyle\mkern-12.0mu\small\begin{cases}\small{|c_{l}|}^{2}{|R_{\tilde{p}\tilde{p}}(0)|}^{2}\\!+\\!\zeta(\delta),\delta=n_{s_{p}}+l,l=\\{0,1,...,L_{\text{c}}\\},\vspace{12pt}\mkern-2.0mu\\\
\small\zeta(\delta),\qquad\qquad\qquad\ \ \text{otherwise.}\end{cases}$

The proof of (IV) is provided in the appendix. Hence, for the received symbols
corresponding to the pilot sequence, the square absolute value of the cross-
correlation of received symbols and pilot sequence, i.e.,
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$, for
$\delta=n_{s_{p}}+l,\>l=\\{0,1,...,L_{\text{c}}\\}$ is the
${|R_{\tilde{p}\tilde{p}}(0)|}^{2}$ weighted by the gain of corresponding
channel path. In other words, the value of
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ at $\delta=n_{s_{p}}$ is equal to
${|R_{\tilde{p}\tilde{p}}(0)|}^{2}$ scaled by ${|c_{0}|}$ and an interfere
term $\zeta(\delta)$.

For the HF poor channel having $c_{0}$ and $c_{L_{\text{c}}}$ to be the two
non-zero taps, we rewrite (IV) as

$\displaystyle{|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ $\displaystyle{}={}$
$\displaystyle\begin{cases}{|c_{0}|}^{2}{|R_{\tilde{p}\tilde{p}}(0)|}^{2}+\zeta(\delta),&\qquad\delta=n_{s_{p}},\\\
{|c_{L_{\text{c}}}|}^{2}{|R_{\tilde{p}\tilde{p}}(0)|}^{2}+\zeta(\delta),&\qquad\delta=n_{s_{p}}+c_{L_{\text{c}}},\\\
\zeta(\delta),&\qquad\text{otherwise}.\end{cases}$

To explain the proposed algorithm, we initially describe the proposed
algorithm through different time-invariant HF channel realizations. It is
important to highlight that a potential approach for determining the pilot
sequence location involves considering only the maximum value of
$|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$. However, it fails to utilize the
inherent characteristics of the HF channel delay spread and does not lead to
improved performance. In order to enhance the accuracy of pilot sequence
location identification, we propose an algorithm that includes a measure that
accounts for the unique characteristics of the HF channel and the favorable
square of the absolute value of the auto-correlation of the optimal pilot
sequence as discussed in the next subsection.

Figure 6: $|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$ with $N_{\text{p}}=32$,
${{N_{\text{s}}}}=256$, $\textbf{b}=[0,1,1,1,0]$,
${|\textbf{c}|}^{2}=[0.30,0,0,0,0,0,0.699]^{\text{T}}$, and possible pilot
sequence location from 0 to 256.

###  IV-A Pilot Sequence Location Identification for Time-Invariant HF Channel

In this subsection, we introduce the proposed algorithm for pilot sequence
location identification in HF channels. To better illustrate the algorithm, we
provide examples using some selected realizations of the time-invariant HF
channels.

Fig. 6 illustrates $|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$ for the scenario
where the transmit frame has $N_{\text{p}}=32$ and ${{N_{\text{s}}}}=256$.
When considering a typical symbol rate of 2400 in HF systems, the delay spread
of 2.1 ms in the FTN signaling system, with a value of $\tau=0.84$, can be
equivalently represented by a discretized channel with 7 taps.
$N_{\text{b}}={{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}={{\lfloor{\log_{2}}{(256+1)}\rfloor}-\lceil{{\log_{2}}{(6+1)}}\rceil}=5$.
Thus, as explained in detail in Section III, in this example,
$n_{s_{p}}^{(i)}$ is predefined to be multiples of
$2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}$ belong to
$\\{0,1,...,2^{\lfloor{\log_{2}}{({{N_{\text{s}}}}+1)}\rfloor}-1\\}$. Thus,
$n_{s_{p}}^{(i)}$ belongs to the set
$\mathbb{P}=\\{0,2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil},2\times
2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil},3\times
2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil},...,31\times
2^{\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil}\\}=\\{0,8,2\times 8,3\times
8,...,248\\}$. As the length of the predefined locations to choose
$n_{s_{p}}^{(i)}$ from is $\text{Card}\\{\mathbb{P}\\}=32$ (designed to be
$2^{N_{\text{b}}}$), we map each bit sequence of length $N_{\text{b}}=5$ to
one predefined location in $\mathbb{P}$ and place the pilot sequence in that
location to construct the frame. At the transmitter, we consider the example
when $\textbf{b}=[0,1,1,1,0]$ and map it to a decimal equivalent of 112 (a
multiple of 8 whose $N_{\text{b}}=5$ MSB of the binary representation is equal
to b), belongs to the predefined pilot sequence set of $\mathbb{P}$. Consider
the channel realization given by
${|\textbf{c}|}^{2}=[0.30,0,0,0,0,0,0.699]^{\text{T}}$, with the 7th channel
tap being the strongest. For a given $n_{s_{p}}^{(i)}$ in a given frame $i$ at
the transmitter, we expect to have spikes at the locations
$\hat{n}_{s_{p}}^{(i)}$ and $n_{s_{p}}^{(i)}+L_{\text{c}}$ in the square of
the absolute value of the cross-correlation at the receiver. Thus
$\hat{n}_{s_{p}}^{(i)}$ belongs to
$\hat{\mathbb{P}}=\\{\\{0,6\\},\\{8,14\\},...,\\{112,118\\},...,\\{248,256\\}\\}$.
Having $n_{s_{p}}^{(i)}=112$, at the receiver we expect two spikes
$n_{s_{p}}^{(i)}=112$ and $n_{s_{p}}^{(i)}+L_{\text{c}}=118$ with the second
spike to be stronger. The peak location of
${|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$ in Fig. 6 is found to be 118. We know
that the location of the pilot sequence $n_{s_{p}}^{(i)}$ is a multiple of 8
and thus $n_{s_{p}}^{(i)}$ module 8 should be zero. Thus, as 118 belongs to
the expected locations $\\{112,118\\}$ we can identify
$\hat{n}_{s_{p}}^{(i)}=112$. In other words, 118 module 8 is 6, and by
subtracting 6 from 118, we can correctly identify $\hat{n}_{s_{p}}^{(i)}=112$.

Figure 7: $|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$ with $N_{\text{p}}=32$ and
${{N_{\text{s}}}}=256$, $\textbf{b}=[1,1,1,0,1]^{\text{T}}$,
${|\textbf{c}|}^{2}=[0.655,0,0,0,0,0,0.345]^{\text{T}}$, and possible pilot
sequence location from 0 to 256.

Fig. 7 depicts the value of $|R_{\tilde{r}\tilde{p}}(\delta)|^{2}$ for the
scenario where the transmit frame has $N_{\text{p}}=32$ and
${{N_{\text{s}}}}=256$. At the transmitter, we consider the example when
$\textbf{b}=[1,1,1,0,1]^{\text{T}}$ and map to its decimal worth of location
232 (a multiple of 8 whose $N_{\text{b}}=5$ MSB of the representation is equal
to b), belongs to the predefined pilot sequence set of $\mathbb{P}$. Consider
the channel realization given by
${|\textbf{c}|}^{2}=[0.655,0,0,0,0,0,0.345]^{\text{T}}$, with the first
channel tap being the strongest. Similar to the previous example, the set of
all expected pilot sequence locations at the receiver is
$\hat{\mathbb{P}}=\\{\\{0,6\\},\\{8,14\\},...,\\{112,118\\},...,\\{248,256\\}\\}$.
The peak location of ${|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$ in Fig. 7 is
found to be 206. We consider all expected locations where the
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ surpasses a certain threshold
$r_{\text{th}_{1}}$ which is equal to a constant value of $c_{1}$ multiply by
the peak location of ${|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$. Thus, the
algorithm identifies locations 206, 232, and 238 from the set of expected
pilot sequence locations of $\hat{n}_{s_{p}}^{(i)}$ with a higher
$|R_{\tilde{p}\tilde{p}}(\delta)|^{2}$ value than $r_{\text{th}_{1}}$. As 232
and 238 belong to $\\{232,238\\}$ in the set for $\hat{n}_{s_{p}}^{(i)}$ and
corresponds to ${n}_{s_{p}}^{(i)}=232$, we need to consider only one of them,
let us consider 232 between these two. Subsequently, the proposed measure is
calculated for locations 206 and 232 to select the location with a better
local characteristic of the ${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$. More
specifically, we introduce a measure that considers the unique characteristics
of the HF channel, characterized by only two non-zero taps separated by a
specific delay spread. For any given location, the local characteristic of the
square of the absolute value of the autocorrelation can be quantified as
defined in (13) by comparing the ratio of the
${|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$ of that location to the normalized
sum of ${|R_{\tilde{p}\tilde{p}}(j)|}^{2}$ for $j$ adjacent locations within a
radius. To accomplish this, we establish a radius, denoted as $r_{0}$ centered
around potential locations identified by the algorithm. Based on the
specifications of the system, we choose $r_{0}$ to be less than the length of
the channel ($L_{\text{c}}+1$). This precaution is taken due to the
expectation that the second non-zero tap will fall within a radius equal to
the length of the channel ($L_{\text{c}}+1$).

In order to exploit the unique characteristics of the HF channel, we define
the measure by considering the local characteristic of the
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ for both non-zero taps jointly. To
end this, we determine $\delta_{c_{0}}$ and $\delta_{c_{L_{c}}}$ as the
corresponding locations for the two spikes in
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ corresponding to two versions of the
signal received from the two non-zero taps of the channel. As
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ for $\delta_{c_{0}}$ and
$\delta_{c_{L_{c}}}$ have high values, we add these two values in the
numerator of the proposed measure $\mu(\delta_{c_{0}},\delta_{c_{L_{c}}})$.
The normalized sum of ${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ for the
adjacent locations within a radius $r_{0}$ around location $\delta_{c_{0}}$
and $\delta_{c_{L_{c}}}$ is defined as $\lambda({\delta_{c_{0}}})$ and
$\lambda({\delta_{c_{L_{c}}}})$, respectively, as defined in (14). Hence, the
proposed measure $\mu(\delta_{c_{0}},\delta_{c_{L_{c}}})$ is defined as
follows

$\displaystyle\mu(\delta_{c_{0}},\delta_{c_{L_{c}}})=\frac{{|R_{\tilde{p}\tilde{r}}(\delta_{c_{0}})|}^{2}+{|R_{\tilde{p}\tilde{r}}(\delta_{c_{L_{c}}})|}^{2}}{\lambda({\delta_{c_{0}}})+\lambda({\delta_{c_{L_{c}}}})}.$
(18)

For the given realization illustrated in Fig. 7, upon examining location 206,
we determine the congruence modulo $8$ to be 6. Consequently, we subtract the
congruence modulo value from the location obtained through the measure,
resulting in $\delta_{c_{0}}$ and $\delta_{c_{L_{c}}}$ being identified as 200
and 206, respectively. Following this, we proceed to calculate the proposed
measure. Notably, since location 206 corresponds to the peak of
${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}$, we derive $r_{\text{th}_{2}}$ based
on this particular observation as $r_{\text{th}_{2}}=c_{2}\cdot\mu(200,206)$.
Where $c_{2}$ is a constant value. Afterward, for location 232, within the
same realization depicted in Fig. 7 we ascertain $\delta_{c_{0}}$ and
$\delta_{c_{L_{c}}}$ to be 232 and 238, respectively, and subsequently compute
the measure in (18). The measure calculated for location 232 exceeds the
established threshold. Consequently, the proposed algorithm successfully
identifies 232 as the pilot sequence location within the current frame. It is
important to note that relying only on the identification of the maximum of
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ would lead to an incorrect
determination of the pilot sequence location for this particular channel
realization. However, through the utilization of the proposed algorithm, we
successfully address this issue and achieve precise identification of the
pilot sequence location.

Following the identification of the pilot sequence’s location, the receiver
proceeds to utilize the LSSE-based approach for estimating the complex channel
coefficients of FTN signaling at the designated pilot sequence locations
within the frame. Let us consider the received samples in (6) corresponding to
transmitted pilot sequence symbols for a given frame $i$. As explored in [2],
$L_{\text{h}}+L_{\text{c}}-1$ symbols at the start of the $N_{\text{p}}$
length pilot sequence experience interference from the unknown data symbols of
the previous frame. To prevent a degradation in the quality of channel
estimation, we need to exclude these pilot symbols from any further
processing. With that in mind, we focus our processing efforts on the symbols
$\tilde{r}_{k},k=n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}-1,...,{n_{s_{p}}^{(i)}+N_{\text{p}}-1}$,
which corresponds to the useful segment of the transmitted pilot sequence. The
$\textbf{r}_{\text{p}}=[\tilde{r}_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}-1},\tilde{r}_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}+1},\ldots,\tilde{r}_{n_{s_{p}}^{(i)}+N_{\text{p}}-1}]^{\text{T}}$
derived from received samples corresponding to the useful part of transmitted
pilot sequence in (6) can be expressed in a vector format as

$\displaystyle\mathbf{r}_{\text{p}}$ $\displaystyle{}={}$
$\displaystyle\mathbf{T}\mathbf{V}\mathbf{c}+\textbf{w}_{\text{p}},$ (19)

where
$\textbf{w}_{\text{p}}=[w_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}-1},w_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}+1}...,w_{n_{s_{p}}^{(i)}+N_{\text{p}}-1}]^{\text{T}}$
represents the zero-mean white Gaussian noise with variance $\sigma_{n}^{2}$
and
$\mathbf{V}\in\mathbb{C}^{(L_{\text{h}}+L_{\text{c}})\times(L_{\text{c}}+1)}$
is the circulant ISI matrix formed using the elements of vector v. The channel
coefficients that need to be estimated are denoted by the vector c, which is
defined as $\textbf{c}=[c_{0},c_{1},\ldots,c_{L_{\text{c}}}]^{\text{T}}$. The
matrix
$\mathbf{T}\in\mathbb{C}^{(N_{\text{p}}-L_{\text{h}}-L_{\text{c}}+1)\times(L_{\text{h}}+L_{\text{c}})}$
is composed of pilot sequence symbols and can be expressed as introduced in
[2] as follows

$\displaystyle\mathbf{T}$ $\displaystyle{}={}$
$\displaystyle\\!\\!\\!\begin{bmatrix}s_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}-1}&s_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}-2}&\dots&s_{n_{s_{p}}^{(i)}}\\\
s_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}}&s_{n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}-1}&\dots&s_{n_{s_{p}}^{(i)}+1}\\\
\vdots&\vdots&\vdots&\vdots\\\
s_{n_{s_{p}}^{(i)}+N_{\text{p}}-1}&s_{n_{s_{p}}^{(i)}+N_{\text{p}}-2}&\dots&s_{n_{s_{p}}^{(i)}+N_{\text{p}}-L_{\text{h}}+L_{\text{c}}}\\\
\end{bmatrix}.$

For estimating the unknown channel $\mathbf{c}$ within the useful segment of
the pilot sequence (namely, when $k$ lies in the range of
$n_{s_{p}}^{(i)}+L_{\text{h}}+L_{\text{c}}$ to
$n_{s_{p}}^{(i)}+N_{\text{p}}-1$), we employ the LSSE approach, which involves
minimizing the sum of squared errors as minimizing the sum of squared errors
as

$\displaystyle\mathbf{\hat{c}}$ $\displaystyle{}={}$
$\displaystyle\text{arg}\displaystyle\min_{\mathbf{c}}(\mathbf{r}_{\text{p}}-\mathbf{T}\mathbf{V}\mathbf{c})^{\text{H}}(\mathbf{r}_{\text{p}}-\mathbf{T}\mathbf{V}\mathbf{c})$
(21) $\displaystyle{}={}$
$\displaystyle(\mathbf{TV})^{-1}\mathbf{r}_{\text{p}}.$

The matrix $(\mathbf{TV})^{-1}$ can be precomputed offline at the receiver.
This is because, the matrix $\mathbf{T}$ is derived from the known pilot
sequence, and given the constant taps of the vector $\mathbf{v}$ for an RRC
pulse’s roll-off factor and the FTN signaling’s packing ratio, the matrix
$\mathbf{V}$ is also known to the receiver. We choose the pilot sequence in
$\mathbf{T}$ that minimizes the MSE for channel estimation in (21) as

MSE $\displaystyle{}={}$
$\displaystyle\text{tr}\left(\sigma_{n}^{2}\mathbf{(TV)}^{-1}\left(\mathbf{(TV)}^{-1}\right)^{\text{H}}\right).$
(22)

The equation provided above emerges directly from the fact that the estimate
$\mathbf{\hat{c}}$ serves as an unbiased estimator of $\mathbf{c}$ in the
presence of zero-mean additive Gaussian noise and
$\text{E}(\mathbf{\hat{c}})=\mathbf{c}$. Hence, the MSE of $\mathbf{\hat{c}}$
is equal to its variance. Considering MSE as a metric assessing the
performance of various pilot sequences, an exhaustive search can be conducted
to identify the pilot sequence denoted as
$\textbf{p}=[s_{n_{s_{p}}^{(i)}},s_{n_{s_{p}}^{(i)}+1},\ldots,s_{n_{s_{p}}^{(i)}+N_{\text{p}}-1}]^{\text{T}}$
which minimizes the MSE. As represented in [2], we have

p $\displaystyle{}={}$
$\displaystyle\text{arg}\displaystyle\min_{\textbf{p}\in\mathcal{S}_{\text{p}}}\biggl{(}\text{tr}\left(\sigma_{n}^{2}\mathbf{(TV)}^{-1}\bigl{(}\mathbf{(TV)}^{-1}\bigr{)}^{\text{H}}\right)\biggr{)}.$
(23)

Here, $\mathcal{S}_{\text{p}}$ represents the constellation set of pilot
symbols.

###  IV-B Pilot Sequence Location Identification Algorithm

The proposed pilot sequence location identification algorithm is summarized
as:

Algorithm 2 Pilot Sequence Location Identification Algorithm

Input: $\tilde{r}$, $c_{1}$, and $c_{2}$
Initialize: Initialize $\hat{n}_{s_{p}}$ and $r_{\text{th}_{2}}$ to zero.
1) Calculate ${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}$ according to (15) for
the expected pilot sequence locations $\hat{\mathbb{P}}$.
2) Sort ${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}$ for the expected pilot
sequence locations, $\hat{\mathbb{P}}$, in descending order and find the
maximum value $m$.
3) Select the candidate locations from the set $\hat{\mathbb{P}}$, for which
${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}$ exceeds
$r_{\text{th}_{1}}=c_{1}\times m$.
4) For each candidate do
Find the corresponding $\delta_{c_{0}}$ and $\delta_{c_{L_{c}}}$.
Calculate the measure, $\mu(\delta_{c_{0}},\delta_{c_{L_{c}}})$ according to
(18).
If the calculated $\mu(\delta_{c_{0}},\delta_{c_{L_{c}}})>r_{\text{th}_{2}}$,
then,
update the
$r_{\text{th}_{2}}=c_{2}\times\mu(\delta_{c_{0}},\delta_{c_{L_{c}}})$ and
$\hat{n}_{s_{p}}=\delta_{c_{0}}$
EndIf
EndFor

Output: $\hat{n}_{s_{p}}$

TABLE II: Complexity comparison of the PSLI algorithm and channel estimation in this paper with the works in [1] and [2].  Algorithm | Computational complexity
---|---
MMSE-based channel estimation for FTN signaling system in [1] | $\mathcal{O}\Big{(}{N_{\text{p}}\log}(N_{\text{p}})+N_{\text{p}}\Big{)}$
LSSE-based channel estimation for FTN signaling system in [2] | $\mathcal{O}\Big{(}(N_{\text{p}}-L_{\text{h}}-L_{\text{c}}+1)\cdot(L_{\text{c}}+1)\Big{)}$
|
$\mathcal{O}\Big{(}(N_{\text{p}}-L_{\text{h}}-L_{\text{c}}+1)\cdot(L_{\text{c}}+1)+$
PSLI algorithm and LSSE-based IM-based channel estimation for FTN signaling system | $N_{\text{p}}\times\text{Card}\\{\hat{\mathbb{P}}\\}+\text{Card}\\{\hat{\mathbb{P}}\\}\times{\log}{\big{(}\text{Card}\\{\hat{\mathbb{P}}\\}\big{)}}$
| $+{\log}{\big{(}\text{Card}\\{\hat{\mathbb{P}}\\}\big{)}}\small{\times
4\times\text{Card}\\{\hat{\mathbb{P}}\\}}\normalsize{}\times
L_{\text{c}}\Big{)}$

###  IV-C Complexity Analysis

This subsection presents a complexity analysis of the proposed PSLI algorithm.
In the first step, we calculate ${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}$ using
(15) for the $\hat{\mathbb{P}}$ locations. The computational complexity of
this step is
$\mathcal{O}\big{(}N_{\text{p}}\cdot\text{Card}\\{\hat{\mathbb{P}}\\}\big{)}$.
To sort the ${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}$ for the expected pilot
sequence locations, $\hat{\mathbb{P}}$, in the second step, one requires a
complexity of
$\mathcal{O}\big{(}\text{Card}\\{\hat{\mathbb{P}}\\}\cdot{\log}{(\text{Card}\\{\hat{\mathbb{P}}\\}}\big{)}$.
As in the third step the ${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}$ values for
the $\hat{\mathbb{P}}$ locations are sorted, finding the location for which
${|R_{\tilde{p}\tilde{r}}(\delta)|}^{2}>r_{\text{th}_{1}}$ requires
$\mathcal{O}\big{(}{\log}{(\text{Card}\\{\hat{\mathbb{P}}\\})}\big{)}$
computational complexity. In the fourth step, first, we need to calculate the
congruence modulo to find $\delta_{c_{0}}$ and $\delta_{c_{L_{c}}}$. This step
has a constant computational complexity. Secondly, calculating the
$\mu(\delta_{c_{0}},\delta_{c_{L_{c}}})$ according to (18) involves
calculation of $\lambda(\delta_{c_{0}})$ and $\lambda(\delta_{c_{L_{c}}})$
according to (14) which requires summation over $2r_{0}$ elements where
$r_{0}<L_{\text{c}}$. Thus, one requires $\mathcal{O}(4\cdot L_{\text{c}})$.
The third part in the fourth step has a constant computational complexity.
Thus, the fourth step requires the complexity of
$\mathcal{O}\big{(}4\cdot\text{Card}\\{\hat{\mathbb{P}}\\}\cdot
L_{\text{c}}\big{)}$. Having said that, the total computational complexity of
the proposed PSLI algorithm can be written as $\begin{aligned}
&\left.\\!\small{\mathcal{O}\Big{(}N_{\text{p}}\times\text{Card}\\{\hat{\mathbb{P}}\\}+\text{Card}\\{\hat{\mathbb{P}}\\}\times{\log}{\big{(}\text{Card}\\{\hat{\mathbb{P}}\\}\big{)}}+{\log}{\big{(}\text{Card}\\{\hat{\mathbb{P}}\\}\big{)}}}\right.\\\
&\left.\small{\times
4\times\text{Card}\\{\hat{\mathbb{P}}\\}}\normalsize{}\times
L_{\text{c}}\Big{)}\text{, where we have $\text{Card}\\{\hat{\mathbb{P}}\\}$
is equal to }\right.\\\
&\left.2^{{\lfloor{\log_{2}}{(N_{d}+1)}\rfloor}-\lceil{{\log_{2}}{(L_{\text{c}}+1)}}\rceil+1}\text{.}\right.\end{aligned}$

The complexity of channel estimation can be calculated from (21). As mentioned
before, the matrix $(\mathbf{TV})^{-1}$ can be precomputed offline at the
receiver. Having said that, the computational complexity of (21) is
$\mathcal{O}\big{(}(N_{\text{p}}-L_{\text{h}}-L_{\text{c}}+1)\cdot(L_{\text{c}}+1)\\}\big{)}$.
This is the same as the complexity of channel estimation in [2].

The computational complexity of the proposed channel estimation in [1] can be
computed based on (32) and (35) in their work. The channel estimation in (32)
of [1] requires, FFT and inverse FFT operation and diagonal matrix
multiplication as well as calculation of minimum mean squared error (MMSE)
weights matrix according to (35) in [1]. Thus the computational complexity of
the channel estimation in [1] is
$\mathcal{O}\big{(}{N_{\text{p}}\log}(N_{\text{p}})\big{)}+\mathcal{O}\big{(}N_{\text{p}}\big{)}$.

Table II summarizes the computational complexity of the PSLI algorithm and
channel estimation in this paper compared to the channel estimation methods in
[2] and [1]. It should be noted as the pilot location is fixed in [2] and [1],
hence there is no need to employ an algorithm to identify the location of the
pilot in these papers.

###  IV-D Pilot Sequence Location Identification for Doubly-selective HF
Channel

In the context of time-invariant channels, a single estimation of the channel
suffices for the entire frame. More specifically, once the pilot sequence is
employed to estimate the channel within a given frame, that estimation remains
consistent throughout the entire data block of that frame. Conversely, when
dealing with doubly-selective channels, the initially estimated channel at the
pilot sequence location in the frame’s beginning may become outdated as the
frame progresses. Consequently, this can lead to an accumulation of mean
squared error over the duration of the frame. To mitigate the inaccuracies
arising from assuming identical channel coefficients across the entire frame,
an interpolation technique can be employed. By utilizing these techniques, the
coefficients can be estimated by interpolating between the pilot sequence of
the current frame and the pilot sequence of the next frame. Performing
interpolation alleviates the impact of outdated channel estimates, thereby
enhancing the accuracy of the estimation process.

The investigation conducted in [2] explores the impact of three interpolation
methods on the MSE associated with estimating the doubly-selective channel.
The study reveals that the MSE remains relatively consistent across the
various interpolation techniques employed, namely linear, cubic, and spline
interpolation. That being stated, we opt for the utilization of linear
interpolation due to its inherent advantage of low computational complexity.

Let the channel estimation at the pilot sequence of the current frame, i.e.,
$i$th frame, and the pilot sequence of the next frame, i.e., $(i+1)$th frame
be $\mathbf{\hat{c}_{i}}$ and $\mathbf{\hat{c}_{i+1}}$, respectively. To
update the channel estimation at each data symbol $k$ within the $i$th frame,
where $k=0,...,{{N_{\text{s}}}}-1$, we employ a linear interpolation technique
that leverages the values $\mathbf{\hat{c}_{i}}$ and $\mathbf{\hat{c}_{i+1}}$.

$\displaystyle\mathbf{\hat{c}}_{i,k+1}$ $\displaystyle{}={}$
$\displaystyle\mathbf{\hat{c}}_{i,k}+\frac{(\mathbf{\hat{c}_{i+1}}-\mathbf{\hat{c}_{i}})}{{{N_{\text{s}}}}},\quad
k=0,...,{{N_{\text{s}}}}-1,$ (24)

where $\mathbf{\hat{c}}_{i,k}$ denote the channel estimation for the $k$th
data symbol within the $i$th frame. Additionally, $\mathbf{\hat{c}}_{i,0}$ is
initialized as $\mathbf{\hat{c}_{i}}$, representing the channel estimation at
the pilot sequence of the current frame.

##  V Simulation Results

In this section, numerical results of the performance evaluation of the
proposed IM-based channel estimation algorithm in terms of pilot sequence
location identification error (PSLIE), MSE and BER are presented.

###  V-A Simulation Setup

We select a typical symbol rate of 2400 symbols/sec on the transmitter side.
We employ a rate $1/2$ convolutional code characterized by a constraint length
of $7$ and utilizing the two generator polynomials 0x5b and 0x79. By
puncturing the bits of a rate 1/2 code with a puncturing mask of
$[1,1,1,0,0,1]$, a code rate of $R_{\text{c}}=3/4$ can be attained. In this
context, a “1” signifies that the corresponding bit is transmitted, while a
“0” implies that the bit is discarded. At the receiver, a decoding algorithm
based on a posteriori probability (APP) is utilized to decode the
convolutional code. Binary phase shift keying (BPSK) and quadrature phase
shift keying (QPSK) modulations are used for pilot and data symbols,
respectively unless explicitly specified otherwise. We conducted an exhaustive
search to identify the pilot sequence, p, which minimizes the MSE as outlined
in (22). It’s worth noting that, the optimal pilot sequence is determined by
implementing (23).

In our MSE performance simulations described in this paper, we employed an
exhaustive search to find the optimal pilot sequence for minimizing the MSE as
outlined in (23). This choice was made primarily because pilot sequence design
is typically a one-time, offline task for a given channel. However, for pilot
sequences of higher modulation order, the exhaustive search process becomes
impractical. Therefore, we opted for a sub-optimal approach to tackle this
issue. This sub-optimal solution involves relaxing the integer constraint of
(23) and solving the relaxed optimization problem corresponding to (23)
numerically using the interior-point algorithm [25]. The ultimate pilot
sequence is then determined by rounding this sub-optimal solution. It’s
important to note that we can no longer assure optimality as a result of this
rounding procedure over the continuous suboptimal. It’s important to note that
since the optimization problem is non-convex, the interior-point algorithm may
converge to a local optimal solution, which may not necessarily be the global
optimal solution. To enhance the likelihood of obtaining a high-quality local
sub-optimal solution, we initialized the interior-point algorithm with
uniformly distributed random pilot sequences and repeated the simulation 100
times. We then computed the MSE for each of these sequences and selected the
sub-optimal pilot sequence with the lowest MSE value. Having said that we
demonstrate the effectiveness of our algorithm with higher-order modulations
by employing 8-PSK modulation, which is utilized in HF transmissions, for both
pilot and data symbols. An RRC pulse shaping filter having a roll-off factor
of $\beta=0.35$ is employed unless stated otherwise.

Figure 8: The PSLIE rate of the proposed IM-based channel estimation algorithm
for FTN signaling with $\tau=0.8$ over channel model 3. Figure 9: The MSE of
the proposed FTN the proposed IM-based channel estimation algorithm for FTN
signaling employing the designed optimal pilot sequence versus the methods in
the literature over channel model 3 for $\tau=0.8$. Figure 10: The PSLIE rate
of the proposed IM-based channel estimation algorithm for FTN signaling with
$\tau=0.72$, $\tau=0.84$ , and $\tau=1$ over channel model 2. Figure 11: The
MSE of the proposed IM-based channel estimation algorithm for FTN signaling
with $\tau=0.72$, $\tau=0.84$, and $\tau=1$ over channel model 2. Figure 12:
The PSLIE rate of the proposed IM-based channel estimation algorithm for FTN
signaling with $\tau=0.72$, $\tau=0.84$, and $\tau=1$ over channel model 1.
Figure 13: The MSE of the proposed IM-based channel estimation algorithm for
FTN signaling with $\tau=0.72$, $\tau=0.84$, and $\tau=1$ over channel model
1. Figure 14: The PSLIE rate of the proposed IM-based channel estimation
algorithm for FTN signaling with $\tau=0.84$ over channel model 2 where pilot
and data symbols are modulated with 8-PSK. Figure 15: The MSE of the proposed
IM-based channel estimation algorithm for FTN signaling with $\tau=0.84$ over
channel model 2 where pilot and data symbols are modulated with 8-PSK. The
pilot sequence was found using numerical optimization.

We investigate three distinct channel models in our study. Firstly, we examine
the ITU-R Poor channel, which is characterized as a doubly-selective Rayleigh
fading channel. This channel exhibits a fading rate of $0.0004$ (Doppler
frequency of 1 Hz) and a delay spread of 2.1 ms [22]. We refer to it as
channel model 1. We set the values of $N_{\text{p}}$ and ${{N_{\text{s}}}}$ to
32 and 256 symbols, respectively. Secondly, we consider a frequency-selective
Rayleigh fading channel with a delay spread similar to that of channel model
1. This particular channel, referred to as channel model 2 which can be
regarded as the time-invariant version of channel model 1. Lastly, we examine
a frequency-selective Rayleigh fading channel denoted as channel model 3. It
possesses a channel of $L_{\text{c}}=7$ and maintains a constant impulse
response over two successive transmission frames. It is important to note that
the inclusion of channel model 3 primarily serves the purpose of facilitating
a fair comparison with the work presented in [1].

For each run of the simulation, a superframe comprising 144 frames is
transmitted through the channel. The PSLIE rate represents the percentage of
packets in which the receiver fails to correctly identify the pilot sequence
location. The MSE is computed for each individual simulation run as
$\sum\nolimits_{l=0}^{L_{c}}({\hat{\textbf{c}}_{l}}-\textbf{c}_{l})^{\text{H}}({\hat{\textbf{c}}_{l}}-\textbf{c}_{l})$,
and the final simulated MSE is obtained by taking the average of the MSE
values calculated across all simulation runs.

###  V-B PSLIE and MSE results

Fig. 8 depicts the PSLIE of the proposed IM-based channel estimation algorithm
for FTN signaling with $\tau=0.8$ over channel model 3. As shown in Fig. 8,
increasing the $E_{\text{b}}/N_{0}$ leads to a decrease in PSLIE. The PSLIE
for $E_{\text{b}}/N_{0}$ as low as 3 dB is shown to be about $10^{-4}$. As
$E_{\text{b}}/N_{0}$ increases to 8 dB, the PSLIE becomes even lower, reaching
approximately $2\times 10^{-7}$. This striking result clearly demonstrates
that the proposed IM-based channel estimation algorithm enables precise
identification of the pilot sequence location.

Fig. 9 presents a comparison of the MSE of the proposed IM-based channel
estimation algorithm for FTN signaling with $\tau=0.8$ over channel model 3
with the MSE of the work in [1] having $\beta=0.5$. The MSE for the case of
using the Nyquist pilot sequence instead of the optimal pilot sequence
designed for channel estimation over FTN signaling in our work and the case of
using the LS method in the frame structure model of [1] is illustrated in Fig.
9. It should be noted that in all these cases except our proposed work
illustrated in Fig. 9, the pilot sequence location is fixed in the frame, and
thus no additional information bits are conveyed through the location of the
pilot. In order to maintain equal overhead between our work and the work in
[1], the pilot sequence length in Ishihara et al.’s work [1] is set to 8. This
choice is made because, in their approach, a two-pilot sequence length is
added at the beginning and end of each data block to form a frame. By setting
the pilot length to 8 in [1], they achieve a total of 32 known symbols
(overhead of 32) in each frame, which is equivalent to the overhead in a given
frame in our paper. This consistency of overhead ensures a fair and consistent
comparison between the two methods in terms of their pilot usage and overhead.
The MSE in Ishihara et al.’s work [1] is formulated in the frequency domain,
as indicated in Eq. (38) of their paper. To ensure a meaningful and equitable
comparison of MSE with our own work, we perform a conversion of the MSE
expression from the frequency domain to the time domain. This conversion
allows for a consistent assessment of MSE between the two approaches, despite
their different domain representations, and facilitates a comprehensive
evaluation of the performance of both methods in a unified framework. The work
in [1] is based on the MMSE method to estimate the channel. We used the LS
method in the frame structure introduced in [1] to find the optimal pilot and
estimate the channel for FTN signaling in the frequency domain. We also
compared our work with the case when the Nyquist pilot sequence (32 which is
of the same length as our work) is employed instead of the optimal pilot
sequence designed for channel estimation over FTN signaling. A noteworthy
point is that the time-domain MSE can be equivalent to calculating
$\sum\nolimits_{l=0}^{L_{c}}({\hat{\textbf{c}}_{l}}-\textbf{c}_{l})^{\text{H}}({\hat{\textbf{c}}_{l}}-\textbf{c}_{l})$.
As depicted in Fig. 9, the proposed IM-based channel estimation algorithm in
this paper results in a significantly better (about 6 dB) MSE when compared to
the MSE resulting from the channel estimation method in [1]. The SE in our
work can be written as
$\text{SE}={\frac{{{N_{\text{s}}}}\cdot{{R_{\text{c}}}}\cdot{{\log_{2}}M}+{N_{\text{b}}}}{({N_{\text{s}}}+N_{\text{p}})}\cdot\frac{1}{\tau(1+\beta)}=\frac{256{\cdot\frac{3}{4}}\cdot{\log_{2}}4+5}{(256+32)}\cdot\frac{1}{0.8(1+0.5)}=1.1256}$
bits/s/Hz for $\tau=0.8$. For the same value of $\tau=0.8$, the SE in [1] can
be written as
$\text{SE}=\frac{{{N_{\text{s}}}}}{({{N_{\text{s}}}}+N_{\text{p}})}\cdot\frac{{\log_{2}}M}{\tau(1+\beta)}{{R_{\text{c}}}}=\frac{256}{(256+32)}\cdot\frac{{\log_{2}}4}{0.8(1+0.5)}{\cdot{\frac{3}{4}}}={{1.1111}}$
bits/s/Hz. That is a 1.3% improvement in SE when utilizing the proposed IM-
based channel estimation as well as improvement (about 6 dB) in the MSE. As
simulation results in Fig. 9 show, the proposed IM-based channel estimation
exhibits more than a 6 dB improvement in MSE compared to the LS channel
estimation in the frequency domain while gaining SE improvement. Additionally,
there is approximately 0.5 dB improvement when using the optimal pilot
sequence for the proposed IM-based channel estimation of FTN signaling
compared to the case of using the Nyquist pilot while achieving SE
improvement.

Fig. 10 portrays the PSLIE of the proposed IM-based channel estimation
algorithm for FTN signaling with $\tau=0.72$, $\tau=0.84$, and $\tau=1$ over
channel model 2. As expected, the PSLIE for $\tau=0.84$ is lower compared to
$\tau=0.72$. As evidenced by Fig. 10, an increase in the $E_{\text{b}}/N_{0}$
results in a corresponding reduction in the PSLIE. For $E_{\text{b}}/N_{0}$ as
low as 4 dB, considering $\tau=0.72$, $\tau=0.84$, and $\tau=1$, the PSLIE
stands approximately at $10^{-3}$, $3\times 10^{-5}$ and $1.3\times 10^{-5}$,
respectively. The PSLIE continues to decrease as the $E_{\text{b}}/N_{0}$ is
increased. Notably, for $E_{\text{b}}/N_{0}$ of 8 dB, the PSLIE reaches
approximately $4\times 10^{-5}$, $4.5\times 10^{-7}$ and $1.4\times 10^{-7}$
for $\tau=0.72$, $\tau=0.84$, and $\tau=1$, respectively. This remarkable
outcome shows the efficiency of our proposed algorithm, enabling precise and
accurate identification of pilot sequence locations within the transmitted
frames.

Fig. 11 offers comparative results of the MSE for our proposed IM-based
channel estimation algorithm for FTN signaling with $\tau=0.72$, $\tau=0.84$,
and $\tau=1$ over channel model 2 with [2]. As clearly illustrated in Fig. 11,
the MSE maintains consistent with the MSE results presented in [2], even for
$E_{\text{b}}/N_{0}$ as low as 2 dB. As shown in Table I, the SE employed in
our proposed IM-based channel estimation for the FTN signaling having the code
rate of $3/4$ equals $1.3896$ bits/s/Hz for $\tau=0.72$. The SE in [2] for the
same value of $\tau$ amounts to 1.3717 bits/sec/Hz, indicating about 1.3%
improvement in SE when utilizing the proposed IM-based channel estimation in
the HF channel while maintaining the same MSE as [2]. The SE for the
traditional Nyquist case is equal to $0.9877$. This indicates a $40.70\%$
improvement in SE when utilizing the proposed IM-based channel estimation for
the FTN signaling compared to the traditional Nyquist signaling.

In Fig. 12, the PSLIE of the proposed IM-based channel estimation algorithm of
doubly-selective HF channel (channel model 1) for FTN signaling with
$\tau=0.72$, $\tau=0.84$, and $\tau=1$ is presented. For $E_{\text{b}}/N_{0}$
of 10 dB, the PSLIE is observed to be less than $3\times 10^{-3}$ for
$\tau=0.72$ and less than $2\times 10^{-3}$ for $\tau=0.84$, and $\tau=1$
cases. For the aforementioned values of $\tau$, as $E_{\text{b}}/N_{0}$
increased to 12 dB, the PSLIE experiences a further reduction, approaching or
falling below a value of $10^{-3}$. This outcome underscores the capability of
the proposed algorithm to achieve precise identification of the pilot sequence
location within the transmitted frames.

Fig. 13 provides a comparative examination of the MSE for our proposed IM-
based channel estimation algorithm for FTN signaling with $\tau=0.72$,
$\tau=0.84$, and $\tau=1$ over channel model 1 with [2]. For the three values
of $\tau$, Fig. 13 reveals a degradation less than 0.5 dB for the
$E_{\text{b}}/N_{0}$ of 14 dB for the proposed IM-based channel estimation
when compared to the MSE in [2]. Thus, the increase in SE in this scenario
comes at the price of degradation in the MSE.

The PSLIE for the proposed IM-based channel estimation algorithm of FTN
signaling with $\tau=0.84$ over channel model 2 is illustrated in Fig. 14. In
this scenario, 8-PSK modulation is employed for both the pilot sequence and
data symbols. As depicted in Fig. 14, an increase in $E_{\text{b}}/N_{0}$
corresponds to a reduction in PSLIE. Notably, even at a relatively low
$E_{\text{b}}/N_{0}$ value of 4 dB, the PSLIE is as low as $5\times 10^{-5}$.
With the $E_{\text{b}}/N_{0}$ increasing to 8 dB, the PSLIE exhibits an even
more significant reduction, reaching approximately $10^{-6}$. This impressive
result unequivocally attests to the precision enabled by the proposed IM-based
channel estimation algorithm in identifying the pilot sequence location for
higher-order modulations for both data and pilot symbols within the frames.

In Fig. 15, a comparison is shown between the MSE of the proposed IM-based
channel estimation for FTN signaling with $\tau=0.84$ over channel model 2
with its counterpart in [2]. To illustrate the efficiency of when dealing with
higher-order modulations, we have adopted 8-PSK modulation, which is commonly
employed in HF transmissions, for both pilot and data symbols. Fig. 15
demonstrates that, even with a low $E_{\text{b}}/N_{0}$ value of 2 dB, the MSE
consistently matches the MSE reported in [2]. For the case of our proposed IM-
based channel estimation for the FTN signaling, the SE as defined in (7)
equals to ${\gamma_{\text{IM-FTN}}=1.7790}$ bits/s/Hz when $\tau=0.84$. In
[2], the SE for the same value of $\tau$ is reported as 1.7637 bits/sec/Hz.
This demonstrates a 0.87 percent enhancement in SE when employing the proposed
IM-based channel estimation in the HF channel, a spectrum-scarce environment.
This improvement in SE is achieved while keeping the MSE at the same level as
reported in [2].

Figure 16: The BER performance of the proposed IM-based channel estimation
algorithm for FTN signaling employing the designed optimal pilot sequence
versus the method in [1] over channel model 3 for $\tau=0.8$. Figure 17: The
BER performance of the proposed IM-based channel estimation algorithm for FTN
signaling with $\tau=0.84$ over channel model 2. Figure 18: The BER
performance of the proposed IM-based channel estimation algorithm for FTN
signaling with $\tau=0.72$, $\tau=0.84$ and $\tau=1$ over channel model 2.
Figure 19: The BER performance of the proposed IM-based channel estimation
algorithm for FTN signaling with $\tau=0.84$ over channel model 1. Figure 20:
The BER performance of the proposed IM-based channel estimation algorithm for
FTN signaling with $\tau=0.72$, $\tau=0.84$, and $\tau=1$ over channel model
1.

###  V-C BER Results

Fig. 16 illustrates a comparison of the BER performance of the proposed IM-
based channel estimation algorithm for FTN signaling with $\tau=0.8$ over
channel model 3 with the BER of the work in [1] considering $\beta=0.5$. To
ensure consistent overhead between the two studies, the pilot sequence length
in the work in [1] is configured to be 8 as discussed in detail in detail in
subsection B of section V. In order to maintain a fair comparison, we
implemented the channel estimation method in [1] and measured the BER output
of the decoder. Fig. 16 reveals that the proposed IM-based channel estimation
algorithm in this paper achieves a significantly better BER when compared to
the BER resulting from the channel estimation approach in [1]. After two
iterations, the BER achieved with the proposed IM-based channel estimation
algorithm exhibits about a 3.5 dB improvement compared to the case of [1].

Fig. 17 illustrates the BER performance of our proposed IM-based channel
estimation algorithm designed for FTN signaling with $\tau=0.84$ over channel
model 2 with [2]. Fig. 17 shows that the BER of our proposed method remains
the same as [2]. As mentioned before, our proposed IM-based channel estimation
algorithm for FTN signals has a $1.3\%$ SE gain compared to [2].

Fig. 18 presents comparative results of the BER performance for our proposed
IM-based channel estimation algorithm designed for FTN signaling with
$\tau=0.72$, $0.84$, and 1 over channel model 2 with [2]. Fig. 18 shows that
the BER of our proposed method remains the same as [2]. As shown in Fig. 18
the BER of our proposed method remains the same as [2] for the case of
$\tau=0.84$ and 1. Our proposed IM-based channel estimation algorithm for FTN
signaling of $\tau=0.84$ has a $1.3\%$ SE gain compared to [2] and a $20.60\%$
SE gain compared to the Nyquist case. For the case of $\tau=0.72$, after two
iterations, a degradation of 0.5 dB for $E_{\text{b}}/N_{0}$ of about 8 dB for
the proposed IM-based channel estimation is observed when compared to the BER
in [2]. Therefore, the enhancement in SE in this scenario is accompanied by a
degradation in the BER.

Fig. 19 compares the BER performance of our proposed IM-based channel
estimation algorithm designed for FTN signaling with $\tau=0.84$ over channel
model 1 with the method in [2]. As shown in Fig. 19, the BER of our proposed
method stays consistent with that of the method presented in [2] for BER in
the order of $10^{-4}$. However, there is about 1 dB degradation in the
$E_{\text{b}}/N_{0}$ above 8 dB. This is because at high $E_{\text{b}}/N_{0}$
when the order of BER is less than $10^{-5}$, even a small PSLIE resulting in
an error in the identification of pilot sequence location in a frame will lead
to an error in data detection.

Figure 20 provides a comparative evaluation of the BER performance of our
proposed IM-based channel estimation algorithm designed for FTN signaling.
Different values of $\tau$ (0.72, 0.84, and 1) using channel model 1 are
considered, and the comparison is made with the approach presented in [2]. As
depicted in Fig. 20, our proposed method maintains a comparable BER to the
approach outlined in [2] for BER in the order of $10^{-3}$. Nevertheless, a
degradation of approximately 1 dB is observed in the $E_{\text{b}}/N_{0}$
above 8 dB. This occurs due to the impact of PSLIE becoming apparent at high
$E_{\text{b}}/N_{0}$, particularly when BER order is less than $10^{-5}$.

##  VI Conclusion

In this paper, we proposed a novel IM-based channel estimation algorithm
including PSP and PSLI algorithms to improve the SE for doubly-selective
channel estimation in HF communications. The proposed PSP algorithm
proficiently conveys additional information within the pilot sequence location
of each frame for FTN signaling over frequency-selective channels. By
harnessing the inherent characteristics of the HF channel and the advantageous
auto-correlation properties of the optimal pilot sequence, we proposed the
PSLI algorithm that effectively identifies pilot sequence locations within a
given frame at the receiver. This is accomplished by demonstrating that the
squared absolute value of the cross-correlation between the received symbols
and the pilot sequence is a scaled representation of the squared absolute
value of the auto-correlation of the pilot sequence, weighted by the gain of
the corresponding HF channel path. In this paper, we adopted an LSSE-based
channel estimation approach for FTN signaling, aiming to estimate the complex
channel coefficients at the pilot sequence locations within the frame. The
proposed IM-based channel estimation algorithm of FTN signaling shows very low
pilot sequence location identification error (less than $2\times 10^{-6}$ and
$10^{-3}$ for $E_{\text{b}}/N_{0}$ values higher than 6 dB and 12 dB for time-
invariant and time-variant HF Poor channel, respectively, while considering
$\tau=0.84$). Simulation results showed reaching similar performance in terms
of the MSE and BER of the channel estimation compared to [2] over frequency-
selective channels while gaining SE improvement. Simulation results
demonstrated an enhancement in SE along with noteworthy improvement (about 6
dB) in the MSE of the channel estimation as well as (about 3.5 dB) in BER of
FTN signaling compared to the algorithm proposed in [1] when dealing with
frequency-selective channels. Simulation results for a higher order modulation
(8-PSK modulation for both pilot sequence and data symbols) showed very low
pilot sequence location identification error (less than $10^{-5}$ for
$E_{\text{b}}/N_{0}$ higher than 6 dB for time-invariant HF Poor channel).

## Derivation of (IV)

In this appendix, we provide details on how we calculated the (IV). For
simplification, we consider the transmission of one frame and drop the frame
index $i$ over a general frequency selective time-invariant channel. Having
the auto-correlation of $\tilde{p}_{k}$ in (11), we could write the square of
the absolute value of the $R_{\tilde{p}\tilde{p}}(\delta)$ for
$\delta=0,...,N_{p}-1$ as

$\displaystyle{|R_{\tilde{p}\tilde{p}}(\delta)|}^{2}$ $\displaystyle{}={}$
$\displaystyle
R_{\tilde{p}\tilde{p}}(\delta)\times{{R^{\ast}}_{\tilde{p}\tilde{p}}(\delta)}$
(25) $\displaystyle{}={}$
$\displaystyle\sum\limits_{k=0}^{N_{p}-1}\sum\limits_{k^{\prime}=0}^{N_{p}-1}{\tilde{p}_{k}\tilde{p^{\ast}}_{\delta+k}\tilde{p}_{k^{\prime}}\tilde{p^{\ast}}_{\delta+k^{\prime}}}.$

Thus, for $\delta=0$, we can write ${|R_{\tilde{p}\tilde{p}}(0)|}^{2}$ as

$\displaystyle{|R_{\tilde{p}\tilde{p}}(0)|}^{2}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{k=0}^{N_{p}-1}\sum\limits_{k^{\prime}=0}^{N_{p}-1}{\tilde{p}_{k}\tilde{p^{\ast}}_{k}\tilde{p}_{k^{\prime}}\tilde{p^{\ast}}_{k^{\prime}}}$
(26) $\displaystyle{}={}$
$\displaystyle\sum\limits_{k=0}^{N_{p}-1}\sum\limits_{k^{\prime}=0}^{N_{p}-1}{|\tilde{p}_{k}|}^{2}{|\tilde{p}_{k^{\prime}}|}^{2}.$

We can write a vector of the output samples of the whitening filter in the (6)
over a general frequency selective time-invariant channel for a sequence of
$\tilde{\textbf{r}}={[\tilde{r}_{0},\tilde{r}_{1},...,\tilde{r}_{\text{N}-1}]}^{\text{T}}$
as

$\displaystyle\tilde{\textbf{r}}$ $\displaystyle{}={}$
$\displaystyle\textbf{s}\ast\textbf{c}\ast\textbf{v}+\textbf{w},$ (27)
$\displaystyle{}={}$
$\displaystyle\textbf{c}\ast\tilde{\textbf{s}}+\textbf{w},$

where $\textbf{s}={[s_{0},s_{1},...,s_{\text{N}-1}]}^{\text{T}}$ is a vector
of data and pilot symbols in a transmitted frame,
$\textbf{c}={[c_{0},c_{1},...,c_{L_{\text{c}}}]}^{\text{T}}$ is a vector of
channel taps, $\textbf{v}={[v_{0},v_{1},...,v_{L_{\text{h}}-1}]}^{\text{T}}$
is the whitening filter,
$\textbf{w}={[w_{0},w_{1},...,w_{\text{N}-1}]}^{\text{T}}$ is the AWGN noise
vector and $\tilde{\textbf{s}}=\textbf{s}\ast\textbf{v}$. Thus, the $k$th
output sample of the whitening filter can be written as

$\displaystyle\tilde{r}_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{l=0}^{L_{\text{c}}}c_{l}\tilde{s}_{k-l}+w_{k},\quad
k=0,...,{{N_{\text{s}}}}-1.$ (28)

Considering the received symbols in (28) as a sequence of
$\tilde{r}_{k},k=0,...,{{N_{\text{s}}}}-1$, the cross-correlation of
$\tilde{r}_{k}$ and $\tilde{p}_{k}$ can be written as

$\displaystyle R_{\tilde{r}\tilde{p}}(\delta)$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{m=0}^{N_{p}-1}{\tilde{r}_{\delta+m}\tilde{p}_{m}^{\ast}},\quad\delta=0,...,{{N_{\text{s}}}}-1,$
(29) $\displaystyle{}={}$
$\displaystyle\sum\limits_{m=0}^{N_{p}-1}\tilde{p}_{m}^{\ast}\left(\sum\limits_{l=0}^{L_{\text{c}}}c_{l}{\tilde{s}_{\delta+m-l}}\right)+\sum\limits_{m=0}^{N_{p}-1}w_{\delta+m}\tilde{p}_{m}^{\ast},$
$\displaystyle{}={}$
$\displaystyle\sum\limits_{l=0}^{L_{\text{c}}}\sum\limits_{m=0}^{N_{p}-1}c_{l}\tilde{p}_{m}^{\ast}{\tilde{s}_{\delta+m-l}}+\sum\limits_{m=0}^{N_{p}-1}w_{\delta+m}\tilde{p}_{m}^{\ast}.$

Having the cross-correlation of $\tilde{r}_{k}$ and $\tilde{p}_{k}$ in (29),
we could write the square of the absolute value of the
$R_{\tilde{r}\tilde{p}}(\delta)$ as

$\displaystyle{|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ $\displaystyle{}={}$
$\displaystyle
R_{\tilde{r}\tilde{p}}(\delta)\times{{R^{\ast}}_{\tilde{r}\tilde{p}}(\delta)}$
(30) $\displaystyle{}={}$
$\displaystyle\sum\limits_{l^{\prime}=0}^{L_{\text{c}}}\sum\limits_{l^{\prime\prime}=0}^{L_{\text{c}}}\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{m^{\prime}=0}^{N_{p}-1}c_{l^{\prime}}c_{l^{\prime\prime}}^{\ast}\tilde{p}_{m}^{\ast}\tilde{p}_{m^{\prime}}{\tilde{s}_{\delta+m-l^{\prime}}}{\tilde{s}_{\delta+m^{\prime}-l^{\prime\prime}}^{\ast}}$
$\displaystyle{}+\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{m^{\prime}=0}^{N_{p}-1}w_{\delta+m}w_{\delta+m^{\prime}}^{\ast}\tilde{p}_{m}^{\ast}\tilde{p}_{m^{\prime}}$
$\displaystyle{}+\sum\limits_{l^{\prime}=0}^{L_{\text{c}}}\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{m^{\prime}=0}^{N_{p}-1}c_{l^{\prime}}\tilde{p}_{m}^{\ast}{\tilde{s}_{\delta+m-l^{\prime}}}w_{\delta+m^{\prime}}^{\ast}\tilde{p}_{m^{\prime}}$
$\displaystyle{}+\sum\limits_{l^{\prime\prime}=0}^{L_{\text{c}}}\sum\limits_{m^{\prime}=0}^{N_{p}-1}\sum\limits_{m=0}^{N_{p}-1}c_{l^{\prime\prime}}\tilde{p}_{m^{\prime}}^{\ast}{\tilde{s}_{\delta+m^{\prime}-l^{\prime}}}w_{\delta+m}^{\ast}\tilde{p}_{m}.\
$

For $\delta=n_{s_{p}}+l,l=\\{0,1,...,L_{\text{c}}\\}$, we can rewrite the
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ in (30) using the (26) as

$\displaystyle{|R_{\tilde{r}\tilde{p}}(\delta=n_{s_{p}}+l)|}^{2}$
$\displaystyle{}={}$
$\displaystyle\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{m^{\prime}=0}^{N_{p}-1}{|c_{l}|}^{2}{|\tilde{p}_{m}|}^{2}{|\tilde{p}_{m^{\prime}}|}^{2}+\zeta(\delta)$
(31) $\displaystyle{}={}$
$\displaystyle{|c_{l}|}^{2}{|R_{\tilde{p}\tilde{p}}(0)|}^{2}+\zeta(\delta).$

It should be noted that for the
$\delta=n_{s_{p}}+l,l=\\{0,1,...,L_{\text{c}}\\}$ and $l^{\prime}=l$ and
$l^{\prime\prime}=l$, we have
$\tilde{s}_{\delta+m-l^{\prime}}=\tilde{s}_{n_{s_{p}}+m}=\tilde{p}_{m}$ and
$\tilde{s}_{\delta+m^{\prime}-l^{\prime\prime}}=\tilde{s}_{n_{s_{p}}+m^{\prime}}=\tilde{p}_{m^{\prime}}$.
It should be noted that $\zeta(\delta)$ is the interfered term and can be
written as

$\displaystyle\mkern-12.0mu\zeta(\delta)$ $\displaystyle={}$
$\displaystyle\small{\mkern-8.0mu\sum\limits_{\begin{subarray}{c}l^{\prime}=0\\\
l^{\prime}\neq
l\end{subarray}}^{L_{\text{c}}}\mkern-2.0mu\sum\limits_{\begin{subarray}{c}l^{\prime\prime}=0\\\
l^{\prime\prime}\neq
l\end{subarray}}^{L_{\text{c}}}\mkern-5.0mu\sum\limits_{m=0}^{N_{p}-1}\mkern-5.0mu\sum\limits_{m^{\prime}=0}^{N_{p}-1}\mkern-8.0muc_{l^{\prime}}c_{l^{\prime\prime}}^{\ast}\tilde{p}_{m}^{\ast}\tilde{p}_{m^{\prime}}{\tilde{s}_{n_{s_{p}}+m+l-l^{\prime}}}{\tilde{s}_{n_{s_{p}}+m^{\prime}+l-l^{\prime\prime}}^{\ast}}}$
(32)
$\displaystyle{}+\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{m^{\prime}=0}^{N_{p}-1}w_{\delta+m}w_{\delta+m^{\prime}}^{\ast}\tilde{p}_{m}^{\ast}\tilde{p}_{m^{\prime}}$
$\displaystyle{}+\sum\limits_{l^{\prime}=0}^{L_{\text{c}}}\sum\limits_{m=0}^{N_{p}-1}\sum\limits_{m^{\prime}=0}^{N_{p}-1}c_{l^{\prime}}\tilde{p}_{m}^{\ast}{\tilde{s}_{\delta+m-l^{\prime}}}w_{\delta+m^{\prime}}^{\ast}\tilde{p}_{m^{\prime}}$
$\displaystyle{}+\sum\limits_{l^{\prime\prime}=0}^{L_{\text{c}}}\sum\limits_{m^{\prime}=0}^{N_{p}-1}\sum\limits_{m=0}^{N_{p}-1}c_{l^{\prime\prime}}\tilde{p}_{m^{\prime}}^{\ast}{\tilde{s}_{\delta+m^{\prime}-l^{\prime}}}w_{\delta+m}^{\ast}\tilde{p}_{m}.$

Thus, for all values of $\delta=n_{s_{p}}+l$ for which $l>L_{\text{c}}$, the
$c_{l}=0$ and thus the first term in (31) is zero. We can write the
${|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ as

$\displaystyle\small{|R_{\tilde{r}\tilde{p}}(\delta)|}^{2}$ $\displaystyle=$
$\displaystyle\mkern-12.0mu\small\begin{cases}\small{|c_{l}|}^{2}{|R_{\tilde{p}\tilde{p}}(0)|}^{2}\\!+\\!\zeta(\delta),\delta=n_{s_{p}}+l,l=\\{0,1,...,L_{\text{c}}\\},\\!\vspace{12pt}\\\
\small\zeta(\delta),\qquad\qquad\qquad\ \ \text{otherwise.}\end{cases}$

Thus, (IV) is proved.

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