# Pilot Design and Doubly-Selective Channel Estimation for Faster-than-Nyquist
Signaling

Simin Keykhosravi and Ebrahim Bedeer

###### Abstract

Being capable of enhancing the spectral efficiency (SE), faster-than-Nyquist
(FTN) signaling is a promising approach for wireless communication systems.
This paper investigates the doubly-selective (i.e., time- and frequency-
selective) channel estimation and data detection of FTN signaling. We consider
the intersymbol interference (ISI) resulting from both the FTN signaling and
the frequency-selective channel and adopt an efficient frame structure with
reduced overhead. We propose a novel channel estimation technique of FTN
signaling based on the least sum of squared errors (LSSE) approach to estimate
the complex channel coefficients at the pilot locations within the frame. In
particular, we find the optimal pilot sequence that minimizes the mean square
error (MSE) of the channel estimation. To address the time-selective nature of
the channel, we use a low-complexity linear interpolation to track the complex
channel coefficients at the data symbols locations within the frame. To detect
the data symbols of FTN signaling, we adopt a turbo equalization technique
based on a linear soft-input soft-output (SISO) minimum mean square error
(MMSE) equalizer. Simulation results show that the MSE of the proposed FTN
signaling channel estimation employing the designed optimal pilot sequence is
lower than its counterpart designed for conventional Nyquist transmission. The
bit error rate (BER) of the FTN signaling employing the proposed optimal pilot
sequence shows improvement compared to the FTN signaling employing the
conventional Nyquist pilot sequence. Additionally, for the same SE, the
proposed FTN signaling channel estimation employing the designed optimal pilot
sequence shows better performance when compared to competing techniques from
the literature.

###### Index Terms:

Channel estimation, faster-than-Nyquist (FTN) signaling, least sum of squared
errors (LSSE), doubly-selective channels, minimum-mean square error (MMSE),
soft-input soft-output (SISO) equalization.

##  I Introduction

The tremendous growth in data traffic in recent years and the increasingly
scarce bandwidth have placed a heavy demand on improving the spectral
efficiency (SE) of wireless communication systems. Introduced in 1970s,
faster-than-Nyquist (FTN) signaling has recently redrawn the interests of
researchers owing to its capability to enhance the SE without necessarily
increasing the bandwidth or the energy per bit [1].

In the conventional Nyquist transmission using $T$-orthogonal pulses, the
Nyquist limit indicates a maximum rate of $1/T$ to have an inter-symbol
interference (ISI)-free communication over frequency flat channels [2].
However, FTN signaling involves the transmission of data symbols using
$T$-orthogonal pulses at every symbol interval $\tau T$, where $0<\tau\leq 1$
is the FTN signaling acceleration parameter. Intentionally violating the
Nyquist limit increases the transmission rate; however, at the price of
unavoidable ISI at the receiver. Mazo inspected the minimum square Euclidean
distance associated with the transmission of uncoded binary sinc pulses and
showed that it remains constant, and thus the asymptotic error rate, when
decreasing $\tau$ to a specific limit of $0.802$ [3]—this limit was later
known as the Mazo limit. Such acceleration results in $25\%$ times higher rate
when compared to the conventional Nyquist transmission employing the same
energy per bit and bandwidth.

Since the optimal detection of FTN signaling for small values of $\tau$, even
in additive white Gaussian noise (AWGN), suffers from high computational
complexity, several works approximated the optimal solution by using reduced
trellis or reduced tree search [4, 5, 6]. These trellis-based equalizers are
too complex to be utilized in applications where the number of bits per symbol
is high. When trellis-based equalizers cannot be used due to complexity,
filter-based equalizers would be a promising option. In the early works of FTN
signaling over AWGN channel, time-domain optimal detectors have been designed
to alleviate the ISI [4], [5]. In mild ISI conditions, i.e., high values of
$\tau$, very low complexity detectors, based on frequency domain equalization
(FDE) [7] or symbol-by-symbol detection [8], are used to strike a balance
between the complexity and detection performance. The detection of FTN
signaling becomes more challenging for high-order and ultra-high order
modulations, and precoding techniques [9] and optimization methods based on
alternating directions multiplier method (ADMM) [10] are effective to reduce
the computational complexity.

A few works considered the detection of FTN signaling over frequency-selective
channels [11, 12, 13, 14]. In particular, in [11], a pilot sequence based on
the Nyquist criterion was inserted preceding the data block to estimate the
channel. Unlike [11], a pilot sequence based on FTN signaling criterion was
employed in the time-domain joint channel estimation and FTN signaling
detection in [12]. To reduce the complexity of the channel estimation and FTN
signaling detection, the work in [13] designed a low complexity joint channel
estimation and data detection for FTN signaling in the frequency-domain.
However, to avoid the ISI from the FTN signaling pilot sequence, a guard block
of the same length as the pilot sequence is inserted before the transmitted
data in the frame. Additionally, a cyclic prefix of the same length as the
pilot sequence and the guard block is added at the end of the transmitted data
in the frame. This results in adding a total overhead of four times the pilot
sequence length within the frame, and eventually, reduced the SE of the design
in [13]. In contrast to [11, 12, 13] where the channel is considered to be a
quasi-static, [14] considered a time-selective channel and proposed a
frequency-domain joint channel estimation and FTN signaling detection
technique based on the message passing algorithm.

This paper contributions can be summarized as:

  * •

This paper scrutinizes the channel estimation and data detection for FTN
signaling over doubly-selective (i.e. time- and frequency-selective) channels.
We adopt a frame structure considering the ISI introduced by both FTN
signaling and the frequency-selective channel. The adopted frame structure
does not suffer from SE loss as it does not require the additional overhead of
a cyclic prefix. It should be mentioned that the detrimental effects of inter-
block interference (IBI) are also considered in the proposed frame structure.

  * •

In order to estimate the complex channel coefficients at the pilot locations
within the frame, we propose a channel estimation approach based on the least
sum of squared errors (LSSE) method. Additionally, we find the optimal FTN
signaling pilot sequence that minimizes the mean square error (MSE) of the
channel estimation. To resolve the time-selective nature of the channel, we
employ a low-complexity linear interpolation to track the channel over the
data symbols part of the frame.

  * •

The proposed approach in this paper is evaluated through extensive simulations
for FTN signaling transmission over a doubly-selective Rayleigh fading
channel. Simulation results show that the MSE of the proposed FTN signaling
channel estimation employing the designed optimal pilot sequence is lower than
its counterpart designed for conventional Nyquist transmission. The bit error
rate (BER) of the FTN signaling employing the proposed optimal pilot sequence
shows improvements compared to the FTN signaling employing the conventional
Nyquist pilot sequence. Simulation results, for the same SE over a frequency-
selective Rayleigh fading channel, show a significant improvement (about 6 dB)
in the MSE of the proposed FTN signaling channel estimation employing the
designed optimal pilot sequence when compared to the technique introduced in
[13].

The remainder of this paper is organized as follows. Section 2 presents the
system model of the FTN signaling in the doubly-selective channel and
introduces the adopted frame structure. Section 3 discusses the channel
estimation, the optimal design of pilot sequence, and the channel tracking.
The data detection of FTN signaling using a linear turbo equalizer is
discussed in Section 4. Section 5 provides the simulation results. Finally,
Section 6 concludes the paper.

Figure 1: Block diagram of a communication system employing FTN signaling.

Throughout the paper, boldface uppercase letters, e.g., $\mathbf{X}$, are used
to denote matrices, boldface lowercase letters, e.g. $\mathbf{x}$, are used to
denote column vectors, and lightface lowercase letters, e.g., $x$, are used to
denote scalers. We use $\mathbf{X}^{-1}$, $\mathbf{X}^{\text{T}}$, and
$\mathbf{X}^{\text{H}}$ to denote the inverse, transpose, complex conjugate
transpose (Hermitian transpose) operations of matrix $\mathbf{X}$,
respectively. We use the notation $x^{\ast}$ to denote the complex conjugate
of a complex number $x$, while the convolution operator between $x$ and $y$ is
denoted by $x\ast y$. We use $\text{Pr}(x=a)$ to denote that the probability
of a discrete random variable $x$ equals $a$, $\text{E}(\mathbf{x})$ to
represent the expectation operator of $\mathbf{x}$, and
$\text{Cov}(\textbf{x},\textbf{y})$ to denote the covariance of x and y. The
operator $\text{Diag}(\textbf{x})$ applied to a vector x of length $N$ denotes
an $N\times N$ diagonal matrix constructed from the vector x. The
$\text{tr}({\mathbf{X}})$ denotes the trace operation of matrix $\mathbf{X}$.
Finally, $\mathbb{C}^{N\times M}$ includes all complex matrices with dimension
${N\times M}$.

##  II System Model

The FTN signaling system employed in this paper is depicted in Fig. 1. On the
transmitter side, each block of independent and identically distributed
(i.i.d.) data bits a is encoded to $\textbf{d}^{\prime}$. To assist in
mitigating the effect of error bursts due to the fading channels, the
interleaver shuffles each block of coded bits to d. Then, each consecutive
${\log_{2}}M$ output bits of the interleaver are modulated onto a complex data
symbol that belongs to an $M$-ary quadrature amplitude modulation ($M$-QAM)
constellation set $\mathcal{S}_{\text{d}}$.

To aid the receiver in the channel estimation, known modulated symbols, i.e.,
pilots, of length $N_{\text{p}}$ are inserted every $N_{\text{d}}$ data symbol
block to form frames of length $N=N_{\text{p}}+N_{\text{d}}$. Pilots and data
symbols in the $i$th frame are denoted as ${s}_{n},\,n=(i-1)N+k$, where
$i=1,\ldots,\infty$, is the frame index and $k=0,1,\ldots,N-1$, is the pilot
and data symbols index within a frame. In particular, within a given frame,
index $k=0,1,\ldots,N_{\text{p}}-1$ denotes the pilot symbols and the index
$k=N_{\text{p}},1,\ldots,N-1$ indicates the data symbols. To simplify the
notation, we drop the frame index $i$, and denote the symbols as
${s}_{n},\,n=0,1,\ldots,\infty$.

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

$\displaystyle x(t)$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{n=0}^{\infty}{s_{n}g(t-n\tau T)}.$ (1)

As discussed earlier, we consider the transmission of FTN signaling over a
doubly-selective fading channel denoted as $c(t,\phi)$; hence, the received
signal can be represented as

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

where $n(t)$ is the zero-mean AWGN with variance $\sigma_{n}^{2}$. The
received signal, $y(t)$, is passed through a filter matched to $g(t)$, and the
signal at the matched filter output can be expressed 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 $s(t)$ from (1), the received signal
in (3) is re-expressed as

$\displaystyle r(t)=\sum\limits_{n=0}^{\infty}{s_{n}h(t-n\tau T)}\ast
c(t,\phi)+z(t),$ (4)

where $h(t)=g(t)\ast g(t)$. Using the tapped delay line channel model [15], a
doubly-selective channel with the impulse response of $c(t,\phi)$ is
considered to have $L_{\text{c}}+1$ channel taps,
$c_{l}(t),\,l=0,1,\ldots,{L_{\text{c}}}$, with the $l$th channel path
$c_{l}(t)$ has a delay of $\phi_{l}$.

The matched-filtered signal $r(t)$ is sampled every $\tau T$ and the $k$th
received sample can be written as

$\displaystyle r_{k}$ $\displaystyle{}={}$ $\displaystyle r(k\tau T)$ (5)
$\displaystyle{}={}$
$\displaystyle\sum\limits_{n=0}^{\infty}{s_{n}}\sum\limits_{l=0}^{L_{\text{c}}}c_{l}(k\tau
T)h\left(\left(k-n-l\right)\tau T\right)+z(k\tau T).$

After a change of variable $n=k-j$, the $k$th received sample in (5) can be
rewritten as

$\displaystyle r_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{l=0}^{L_{\text{c}}}c_{k,l}\sum\limits_{j=-\infty}^{k}s_{k-j}h_{j-l}+z_{k},$
(6)

where $c_{k,l}=c_{l}(k\tau T)$, $h_{j-l}=h\left(\left(j-l\right)\tau
T\right)$, and $z_{k}=z(k\tau T)$. One can show that the noise samples $z_{k}$
are colored, and hence, a discrete-time noise whitening filter can be used
before channel estimation and FTN signaling detection.

In the coming subsection, we adopt a frame structure with reduced overhead.
The received samples introduced in (6) then pass through the whitening filter
and fed into the channel estimation and turbo equalizer. Turbo equalizer also
receives the estimated channel impulse response as an input from the channel
estimation block and outputs the estimated data bits $\hat{a}_{k}$.

Figure 2: The frame structure employed in the paper.

In theory, the FTN signaling ISI length is infinite. However, considering that
the significant power of the raised cosine pulse is concentrated in and around
its main lobe, we truncate the RRC pulse so that it has the length of
$2L_{\text{h}}+1$. That said, the vector
$\mathbf{h}=[h_{-L_{\text{h}}},h_{-L_{\text{h}}+1},\ldots,h_{L_{\text{h}}}]^{\text{T}}$
of length $2L_{\text{h}}+1$ represents the effective taps of the raised cosine
pulse. The parameter $L_{\text{h}}$ is determined based on the roll-off factor
of the RRC pulse and the FTN signaling packing ratio $\tau$.

###  II-A Adopted Frame Structure

In this subsection, we discuss the proposed frame structure for FTN signaling
taking the effects of IBI into consideration as shown in Fig. 2. The sampled
signal after the matched filter goes through a discrete-time whitening filter
to decorrelate the colored noise samples $z_{k}$. In order to build the
whitening filter, we use spectral factorization of the z-transform of
$\mathbf{h}$, i.e., $H(z)=V(z)V(1/z^{\ast})$ [5]. This is equivalent to having
a causal whitening filter v of length $L_{\text{h}}$ such that
$\textbf{v}\ast\underset{\bar{}}{\textbf{v}}=\textbf{h}$. Considering
$\textbf{v}=[v_{0},v_{1},...,v_{L_{\text{h}}-1}]^{\text{T}}$, we have
$\underset{\bar{}}{\textbf{v}}=[v_{L_{\text{h}}-1},v_{L_{\text{h}}-2},...,v_{0}]^{\text{T}}$.

The $k$th received sample in (6) after passing through the whitening filter
can be expressed as

$\displaystyle\tilde{r}_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{l=0}^{L_{\text{c}}}c_{k,l}\sum\limits_{j=-\infty}^{k}s_{k-j}v_{j-l}+w_{k},$
(7)

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

In order not to deteriorate the channel estimation quality, we design the
pilot sequence to have two separated parts. As depicted in Fig. 2, the
$L_{\text{h}}+L_{\text{c}}$ symbols,
${s}_{k},\,k=0,1,\ldots,L_{\text{h}}+L_{\text{c}}-1$, at the beginning of the
pilot sequence in the current frame constitute the first part, and they are
affected by the ISI from the unknown data symbols of the previous frame as
well as the pilot symbols of the current frame. Since, the first part of the
pilot sequence is contaminated by ISI from the unknown data symbols, it will
be discarded from further processing. Each symbol in the pilot sequence is
also contaminated by the ISI from the previous and future pilot symbols.
However, as the pilot symbols are known at the receiver, their ISI
contribution can be easily mitigated, in contrast to the ISI from unknown data
symbols. The symbols,
${s}_{k},\,k=L_{\text{h}}+L_{\text{c}},\ldots,N_{\text{p}}-1$, in the current
frame constitute the second part which is the useful part of pilot sequence
for channel estimation.

We define ${\tilde{p}_{k,j}}$ as the effective ISI resulting from both the
frequency-selective channel and the FTN signaling as

$\displaystyle\tilde{p}_{k,j}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{l=0}^{L_{\text{c}}}{c_{k,l}v_{j-l}}.\quad
j=0,...,L_{\text{h}}+L_{\text{c}}-1.$ (8)

This is as $v_{j-l}$ in (7) is only nonzero where
$(j-l)=0,\ldots,L_{\text{h}}-1$. In other words, $v_{j-l}$ is non-zero for
$j=0,...,L_{\text{h}}+L_{\text{c}}-1$. Thus, for $l=0,\ldots,L_{\text{c}}$,
$h_{j-l}$ is zero for $j$ less than $0$ and $j$ greater than
$L_{\text{h}}+L_{\text{c}}-1$. Thus, the convolution of v of length
$L_{\text{h}}$ and $(L_{\text{c}}+1)$-tap fading channel, results in an
effective ISI of length $L_{\text{eff}}=L_{\text{h}}+L_{\text{c}}$.
Considering (8), we can rewrite (7) for a given frame of received symbols,
i.e., for $k=0,1,\ldots,N-1$, as

$\displaystyle\tilde{r}_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{j=0}^{\min\left(L_{\text{h}}+L_{\text{c}}-1,k\right)}\tilde{p}_{k,j}s_{k-j}+w_{k},\quad
k=0,\ldots,N-1,$ (9)

where the summation limits result from the fact that the effective ISI
$\tilde{p}_{k,j}$ in (8) is zero for $j$ less than $0$ or $j$ greater than
$L_{\text{h}}+L_{\text{c}}-1$.

##  III  Doubly-Selective Channel Estimation For FTN Signaling

This section estimates the unknown doubly-selective channel impulse response
with the aid of pilot sequences known to the receiver. Because of the induced
ISI of FTN signaling, traditional Nyquist-based pilots cannot be directly
applied to the FTN signaling case. We propose a novel channel estimation
approach of FTN signaling to estimate the complex channel coefficients at the
pilot locations within the frame. We also find the optimal pilot sequence that
minimizes the MSE of the channel estimation. Finally, we employ a low-
complexity linear interpolation to track the complex channel coefficients at
the data symbols locations within the frame.

###  III-A Frequency-Selective Channel Estimation for FTN Signaling

In this subsection, we process the received samples in (9) corresponding to
the transmitted pilot symbols to facilitate the frequency-selective channel
estimation. As discussed before and depicted in Fig. 2, the
$L_{\text{h}}+L_{\text{c}}$ symbols at the beginning of the pilot sequence,
$s_{k},k=0,...,L_{\text{h}}+L_{\text{c}}-1$, suffer from the ISI from the
unknown data symbols of the previous frame. Hence, in order to not deteriorate
the channel estimation quality, these pilot symbols are discarded from further
processing. Having said that, we process $r_{k}$,
$k=L_{\text{h}}+L_{\text{c}},...,{N_{\text{p}}-1}$, which are the useful part
of transmitted pilot sequence.

We design the duration of the pilot sequence to be less than the channel
coherence time so that the channel coefficients can be regarded as constants
over this duration. As a result, $c_{k,l}$ and $\tilde{p}_{k,j}$,
$k=L_{\text{h}}+L_{\text{c}},...,{N_{\text{p}}-1}$ are not a function of $k$,
and we can drop the time index $k$. We also design $N_{\text{p}}$ to be
greater than the effective ISI length, i.e., $N_{\text{p}}>L_{\text{eff}}$,
hence, the received samples in (9) corresponding to the useful part of
transmitted pilot symbols can be written as

$\displaystyle\tilde{r}_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{j=0}^{L_{\text{h}}+L_{\text{c}}}\tilde{p}_{j}s_{k-j}+\tilde{z}_{k},\quad
k=L_{\text{h}}+L_{\text{c}},...,{N_{\text{p}}-1},$ (10)

where

$\displaystyle\tilde{p}_{j}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{l=0}^{L_{\text{c}}}{c_{l}v_{j-l}},\quad
j=0,...,L_{\text{h}}+L_{\text{c}}.$ (11)

The received samples in (10) can be rewritten as a vector form

$\displaystyle\mathbf{r}_{\text{p}}$ $\displaystyle{}={}$
$\displaystyle\mathbf{T}\mathbf{V}\mathbf{c}+\tilde{\textbf{z}}_{\text{p}},$
(12)

where
$\textbf{r}_{\text{p}}=[\tilde{r}_{L_{\text{h}}+L_{\text{c}}},\tilde{r}_{L_{\text{h}}+L_{\text{c}}+1},\ldots,\tilde{r}_{N_{\text{p}}-1}]^{\text{T}}$,
$\tilde{\textbf{z}}_{\text{p}}=[\tilde{z}_{L_{\text{h}}+L_{\text{c}}},\tilde{z}_{L_{\text{h}}+L_{\text{c}}+1}...\tilde{z}_{N_{\text{p}}-1}]^{\text{T}}$
is the zero-mean white Gaussian noise with variance $\sigma_{n}^{2}$, and
$\mathbf{V}\in\mathbb{C}^{L_{\text{eff}}\times(L_{\text{c}}+1)}$ is the
circulant ISI matrix constructed from the elements of vector v represented by
$[v_{0},v_{1},...,v_{L_{\text{h}}}]^{\text{T}}$,
$\textbf{c}=[c_{0},c_{1},...,c_{L_{\text{c}}}]^{\text{T}}$ is the unknown
channel to be estimated, and
$\mathbf{T}\in\mathbb{C}^{(N_{\text{p}}-L_{\text{eff}}+1)\times
L_{\text{eff}}}$ is the matrix consists of pilot symbols and can be written as

$\displaystyle\mathbf{T}=\begin{bmatrix}s_{L_{\text{eff}}-1}&s_{L_{\text{eff}}-2}&\dots&\dots&s_{0}\\\
s_{L_{\text{eff}}}&s_{L_{\text{eff}}-1}&\dots&\dots&s_{1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
s_{N_{\text{p}}-1}&s_{N_{\text{p}}-2}&\dots&\dots&s_{N_{\text{p}}-L_{\text{eff}}}\\\
\end{bmatrix}.$ (13)

To estimate the unknown channel $\mathbf{{c}}$ at useful part of the pilot
sequence, i.e., at $k=L_{\text{h}}+L_{\text{c}},...,{N_{\text{p}}-1}$, the
LSSE approach is employed by minimizing the sum of squared errors as

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

The matrix $\mathbf{T}$ in (14) depends only on the pilot sequence which is
known at the receiver side. As the vector $\mathbf{v}$ has constant taps for a
given the roll-off factor of the RRC pulse and a given packing ratio of the
FTN signaling, matrix $\mathbf{V}$ is also known at the receiver side. Thus,
the matrix $(\mathbf{TV})^{-1}$ may be computed offline at the receiver. We
select the pilot sequence in $\mathbf{T}$ that minimizes the MSE of the
channel estimation in (14). As $\tilde{\textbf{z}}_{\text{p}}$ is a zero-mean
additive Gaussian noise vector, then $\text{E}(\mathbf{\hat{c}})=\mathbf{c}$,
and $\mathbf{\hat{c}}$ is an the unbiased estimation of $\mathbf{c}$. Thus,
the MSE of $\mathbf{\hat{c}}$ is equal to its variance and can be written as

MSE $\displaystyle{}={}$
$\displaystyle\text{tr}\left(\sigma_{n}^{2}\mathbf{(TV)}^{-1}\left(\mathbf{(TV)}^{-1}\right)^{\text{H}}\right).$
(15)

To design a pilot sequence suitable for channel estimation that takes the ISI
caused by FTN signaling into account, the MSE can be used as a criterion to
compare the quality of different pilot sequences. To this end, an exhaustive
search can be done to find the pilot sequence,
$\textbf{s}_{\text{p}}=[s_{0},...,s_{N_{\text{p}}-1}]^{\text{T}}$ that
minimizes MSE. This can be formally expressed as

$\displaystyle\textbf{s}_{\text{p}}$ $\displaystyle{}={}$
$\displaystyle\text{arg}\displaystyle\min_{\textbf{s}_{\text{p}}\in\mathcal{S}_{\text{p}}}\text{MSE}.$
(16)

where $\mathcal{S}_{\text{p}}$ indicates constellation set of pilot symbols.

###  III-B Tracking the Time-Selective Channel

Figure 3: Block diagram of a turbo equalizer.

For time-invariant channels, when the channel is estimated with the use of the
pilot sequence in a given frame, the estimation can be considered to be
constant during the entire data block in the same frame. That is to say, the
channel estimation can be done only once for each frame. However, for time-
selective channels, the estimated channel at the pilot sequence location at
the beginning of the frame can become obsoleted during the rest of the frame.
This will eventually increase the mean squared error that accumulates during
the length of the frame. The inaccuracy, resulting from assuming the same
channel coefficients over the entire frame, can be addressed using an
interpolation over the estimated coefficients at the pilot sequence located in
the beginning of the current frame and the pilot sequence of the next frame.

The effects of three interpolation methods on the MSE of estimating the time-
selective channel is investigated in Section V in Fig. 8. As can be seen from
Fig. 8, the MSE is almost the same for the different interpolation methods
(linear, cubic, spline) employed to track the time-selective channel. Having
said that, in our work, we adopt the linear interpolation due to its low-
complexity. Let the channel estimation at the pilot sequence at the beginning
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. The channel estimation can be updated
at each data symbol $k$, $k=0,...,N_{\text{d}}-1$ within the $i$th frame,
using the linear interpolation of $\mathbf{\hat{c}_{i}}$ and
$\mathbf{\hat{c}_{i+1}}$ as

$\displaystyle\mathbf{\hat{c}}_{i,k+1}$ $\displaystyle{}={}$
$\displaystyle\mathbf{\hat{c}}_{i,k}+\frac{(\mathbf{\hat{c}_{i+1}}-\mathbf{\hat{c}_{i}})}{N_{\text{d}}},\quad
k=0,...,N_{\text{d}}-1,$ (17)

where $\mathbf{\hat{c}}_{i,k}$ is the channel estimation for the $k$th data
symbol in the $i$th frame and $\mathbf{\hat{c}}_{i,0}$ is set to be
$\mathbf{\hat{c}_{i}}$.

Figure 4: Block diagram of a SISO equalizer.
$\displaystyle\tilde{\textbf{P}}_{k}$ $\displaystyle{}={}$
$\displaystyle\begin{bmatrix}\tilde{p}_{k+L_{1},0}&\tilde{p}_{k+L_{1},1}&\dots&\tilde{p}_{k+L_{1},L_{\text{h}}+L_{\text{c}}}&0&\dots&0\\\
0&\tilde{p}_{k+L_{1}-1,0}&\dots&\tilde{p}_{k+L_{1}-1,L_{\text{h}}+L_{\text{c}}-1}&\tilde{p}_{k+L_{1}-1,L_{\text{h}}+L_{\text{c}}}&\dots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\dots&\tilde{p}_{k-L_{2},0}&\dots&\tilde{p}_{k-L_{2},L_{\text{h}}+L_{\text{c}}-1}&\tilde{p}_{k-L_{2},L_{\text{h}}+L_{\text{c}}}\end{bmatrix}.$

##  IV FTN Signaling Detection Using Turbo Equalization

This section summarizes the turbo equalizer based on a linear SISO MMSE
equalizer to detect FTN signaling. In this section, we assume that the
effective time-varying ISI introduced in (8) is estimated and thus known at
the receiver using the channel estimation method introduced in this paper.

The block diagram of the turbo equalizer is shown in Fig. 3. The log-
likelihood ratios (LLRs), which are the soft information of the coded bits,
are exchanged between the SISO MMSE equalizer and the SISO decoder through a
deinterleaver/interleaver pair in an iterative manner. After several
equalization and decoding iterations, the quality of the soft information
improves, and hard decisions from the SISO decoder can adequately estimate the
transmitted data bits [16]. The inputs to the SISO MMSE equalizer are the
block of received samples, $\tilde{r}_{k}$, $k=N_{\text{p}},\ldots,N-1$, and
the extrinsic LLRs from the decoder denoted by
$L_{\text{e}}^{\text{D}}(d_{k,j})$, where “e” stands for extrinsic and “D”
stands for decoder. The SISO MMSE equalizer computes the soft information
$L_{\text{e}}^{\text{E}}(d_{k,j})$ of the coded bits, where “E” stands for the
equalizer. Reversing the interleaving process in the transmitter can be done
by deinterleaving $L_{\text{e}}^{\text{E}}(d_{k,j})$ into
$L_{\text{e}}^{\text{E}}({d^{\prime}_{k,j}})$, which is employed as a priori
information to be fed into the decoder [17]. The SISO decoder calculates a
posteriori LLRs, $L_{D}({d^{\prime}_{k,j}})$, which can be used to compute the
extrinsic LLRs as
${L_{\text{e}}}^{\text{D}}({d^{\prime}_{k,j}})=L_{\text{D}}({d^{\prime}_{k,j}})-L_{\text{e}}^{\text{E}}({d^{\prime}_{k,j}})$.
The decoder also outputs $\hat{a}$ as an estimation of the transmitted data
bits. The extrinsic LLRs of the decoder
${L_{\text{e}}}^{\text{D}}({d^{\prime}_{k,j}})$ then interleaved into
$L_{\text{e}}^{\text{D}}(d_{k,j})$ and fed back to the SISO equalizer in the
next iteration [16].

Please note that the a priori LLRs, $L_{\text{e}}^{\text{D}}(d_{k,j})$, input
to the MMSE equalizer are zero in the first equalization and decoding step,
namely the zero-th iteration. The pilot symbols, which are known at the
receiver, are considered to be a perfect a priori information in all
iterations, including the zero-th iteration.

###  IV-A The SISO Equalizer

This subsection reviews the design of the optimal coefficient of a filter-
based SISO MMSE equalizer, illustrated in Fig. 4. The SISO MMSE equalizer is a
time-varying filter
$\tilde{\textbf{q}}_{k}=[\tilde{q}_{L_{1},k},\tilde{q}_{L_{1}-1,k},\ldots,\tilde{q}_{-L_{2},k}]^{\text{T}}$
having the length $L_{\text{eq}}$ such that $L_{\text{eq}}=L_{1}+L_{2}+1$. To
estimate transmitted data symbols of the current frame from the received data
symbols after whitening filter $\tilde{r}_{k}$, $k=N_{\text{p}},...,{N-1}$,
the received samples after whitening filter in (9) corresponding to the data
symbols can be rewritten as

$\displaystyle\tilde{r}_{k}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{j=0}^{L_{\text{h}}+L_{\text{c}}-1}\tilde{p}_{k,j}s_{k-j}+w_{k},\quad
k=N_{\text{p}},\ldots,N-1,$ (18)

where

$\displaystyle\tilde{p}_{k,j}$ $\displaystyle{}={}$
$\displaystyle\sum\limits_{l=0}^{L_{\text{c}}}{c_{k,l}h_{j-l}}.\quad
j=0,\ldots,L_{\text{h}}+L_{\text{c}}-1,$ (19)

Given the length of the equalizer filter, an estimation of transmitted symbol
$\hat{s}_{k}$ can be calculated from the $L_{\text{eq}}$ received samples
$\tilde{r}_{k}$ and a priori information on the $L_{\text{eq}}+L_{\text{eff}}$
transmitted symbols $s_{k}$. Defining
$\tilde{\textbf{r}}_{k}=[\tilde{r}_{k+L_{1}},\tilde{r}_{k+L_{1}-1},\ldots,\tilde{r}_{k-L_{2}}]^{T}$,
as a vector of received symbols at the equalizer introduced in (18),
$\tilde{\textbf{r}}_{k}$ may be written in matrix format as

$\displaystyle\tilde{\textbf{r}}_{k}$ $\displaystyle{}={}$
$\displaystyle\tilde{\textbf{P}}_{k}\textbf{s}_{k}+\textbf{w}_{k},\quad
k=N_{\text{p}},\ldots,N-1,$ (20)

where
$\textbf{s}_{k}=[s_{k+L_{1}+L_{\text{h}}},s_{k+L_{1}+L_{\text{h}}-1},\ldots,s_{k-L_{2}-L_{\text{h}}-L_{\text{c}}}]^{\text{T}}$
is the vector of transmitted symbols and
$\tilde{\textbf{P}}_{k}\in\mathbb{C}^{L_{\text{eq}}\times(L_{\text{eq}}+L_{\text{eff}})}$
is the effective ISI matrix defined in (III-B) at the top of next page. The
$\textbf{w}_{k}=[w_{k+L_{1}},w_{k+L_{1}-1},\ldots,w_{k-L_{2}}]^{\text{T}}$ is
a vector of zero-mean white noise samples with covariance
$\text{Cov}(\textbf{w}_{k},\textbf{w}_{k})=\sigma_{n}^{2}\textbf{I}$. The MMSE
criteria in which the mean square error $\text{E}({|s_{k}-\hat{s_{k}}|}^{2})$
is minimized, is employed to calculate the equalizer coefficients
${\tilde{\textbf{q}}_{k}}$. As introduced in [18], the optimal linear MMSE
estimate $\hat{s}_{k}$ of ${s}_{k}$ is expressed as

$\displaystyle\hat{s}_{k}$ $\displaystyle{}={}$
$\displaystyle{\tilde{\textbf{q}}_{k}}^{\text{H}}(\tilde{\textbf{P}}_{k}-\textbf{r}_{k}\text{E}(\textbf{s}_{k})+\tilde{\textbf{p}}_{k}\text{E}({s}_{k})),$
(22)

where $\tilde{\textbf{p}}_{k}=\tilde{\textbf{P}}_{k}\textbf{u}_{k}$ Thus, the
optimal filter coefficient, $\tilde{\textbf{q}}_{k}$ can be found as [18]

$\displaystyle\tilde{\textbf{q}}_{k}$ $\displaystyle{}={}$
$\displaystyle{\left(\tilde{\textbf{P}}_{k}\Sigma_{k}{\tilde{\textbf{P}}_{k}}^{\text{H}}+\sigma_{n}^{2}\textbf{I}+(1-\sigma_{k})\tilde{\textbf{p}}_{k}{\tilde{\textbf{p}}_{k}}^{\text{H}}\right)}^{-1}\tilde{\textbf{p}}_{k}.$
(23)

where
${\Sigma}_{k}=\text{Cov}(\textbf{s}_{k},\textbf{s}_{k})=\text{Diag}([\sigma_{k+L_{1}+L_{\text{h}}},\sigma_{k+L_{1}+L_{\text{h}}-1},\ldots,\sigma_{k-L_{2}-L_{\text{h}}-L_{\text{c}}}])$,
$\textbf{u}_{k}=[\textbf{0}_{L_{1}+L_{\text{h}}},1,\textbf{0}_{L_{2}+L_{\text{h}}+L_{\text{c}}}]^{\text{T}}$
and variance
$\sigma_{k}=\text{E}({s}_{k}{s}^{*}_{k})-\text{E}({s}_{k})\text{E}({s}^{*}_{k})$.
A priori mean $\text{E}({s}_{k})$ and variance $\sigma_{k}$ can be computed
from the ${L_{\text{e}}}^{\text{D}}(c_{k,j})$ as discussed in [18]. One can
calculate the LLRs $L_{\text{e}}^{\text{E}}(c_{k,j})$ from $\hat{s}_{k}$ [18]
and [17]. It should be noted that calculating the optimal filter coefficient,
$\tilde{\textbf{q}}_{k}$, has a computational complexity per time step
proportional to $N^{3}$. Fortunately, a time-recursive algorithm is introduced
in [16] with a computational complexity per time step proportional to $N^{2}$.

##  V Simulation Results

This section provides the numerical results of the MSE and BER performance to
evaluate the proposed channel estimation and FTN signaling detection schemes.

###  V-A Simulation Setup

On the transmitter side, we set $T$ to $0.41667$ ms and use a rate $1/2$
convolutional code with a constraint length $7$ and the two generator
polynomials 0x5b and 0x79. A code rate of 3/4 from the rate 1/2 code can be
achieved through puncturing the output bits of the encoder with a puncturing
mask of $[111001]$, where a “1” and a “0” indicate that the bit is transmitted
and discarded, respectively. A posteriori probability (APP) decoding algorithm
is employed at the receiver to decode the convolutional code. We consider
binary phase shift keying (BPSK) modulation for both pilot and data symbols
and a RRC pulse with a roll-off factor of $\beta=0.35$ unless otherwise
mentioned.

We consider three channel models: 1) ITU-R Poor channel, a doubly-selective
Rayleigh fading channel with fading rate of $0.0004$ and delay spread of 2.1
ms, and it will be denoted as channel model 1. Hence, we set $N_{\text{p}}$
and $N_{\text{d}}$ to 32 and 256 symbols, respectively. 2) a frequency-
selective Rayleigh fading channel described with similar delay spread of
channel model 1 whose impulse response remains constant over two successive
transmission frames, and it will be denoted as channel model 2. 3) a
frequency-selective Rayleigh fading channel having a delay spread of
$L_{\text{c}}=8$ whose impulse response remains constant over two successive
transmission frames, and it will be denoted as channel model 3. Please note
that channel model 3 is mainly considered to have a fair comparison with [13].
High frequency (HF) communication is an effective method of communicating over
very long distances in rural and remote areas [19]. Having said that we employ
the the HF model with poor channel conditions as recommended by ITU-R. The
ITU-R Poor channel consists of two independent but equal average power
Rayleigh fading paths with a fixed 2 ms delay between paths and with a
frequency Doppler of 1 Hz. We consider the 2.1 ms delay spread for the channel
to have exactly 5 symbol intervals in the Nyquist case having a 2400 symbol
rate which is equivalent to a Watterson model with 6 taps [17]. This is
equivalent to the fading rate of 0.0004. The (channel model 2) is the time-
invariant version of the channel model 1.

We adopt the following notation for the considered cases through the
simulation results section:

  * •

F-F to denote the case of employing the proposed FTN pilot sequence in the FTN
signaling transmission.

  * •

N-F for the case of employing the conventional Nyquist pilot sequence designed
in [20] in the FTN signaling transmission.

  * •

N-N for the case of employing the conventional Nyquist pilot sequence
introduced in [20] in the Nyquist signaling transmission. This case is used as
a benchmark.

  * •

Numerical-F to denote the case of employing the FTN pilot sequence found by
the interior-point algorithm minimizing the MSE in (15) for the FTN signaling
transmission.

Figure 5: The MSE of the proposed FTN signaling channel estimation employing
the designed optimal pilot sequence versus the method in [13] over channel
model 3 for $\tau=0.8$. Figure 6: The MSE of the proposed channel estimation
for FTN signaling with $\tau=0.72$ and $\tau=0.84$ over channel model 2.
Figure 7: The MSE of the proposed channel estimation for FTN signaling with
$\tau=0.72$ and $\tau=0.84$ over channel model 1. Figure 8: The MSE
performance of employing different interpolation methods for the proposed
channel estimation for FTN signaling with $\tau=0.84$ over channel model 1.
Figure 9: The MSE performance of FTN signaling channel estimation employing
the designed optimal pilot sequence found through exhaustive search versus the
case using numerical optimization to find the pilot with $\tau=0.72$ over
channel model 2.

In our simulations, in every simulation run, a superframe consisting of 144
frames is sent over the channel. The MSE in a given simulation run is
calculated 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 averaged over the total number of simulation
runs. The MSE simulation results is also compared to the theoretical MSE in
(15).

###  V-B Evaluation of the MSE of the proposed channel estimation

Fig. 5 depicts the MSE of the proposed channel estimation employing the
designed optimal pilot sequence for FTN signaling having $\tau=0.8$ over
channel model 3. To be consistent with the [13], $\beta=0.5$ is considered for
Fig. 5. In all the simulation results except the ones that we compare our work
with [13] (Fig. 5 and Fig. 9), $\beta$ is set to $0.35$ which is a practical
value in communication systems. To have a fair comparison with [13], we fix
the SE of both transmission to 0.7407 bits/s/Hz. This is achieved as follows.
The SE in our work is defined as
$\text{SE}=\frac{N_{\text{d}}}{(N_{\text{d}}+N_{\text{p}})}\cdot\frac{{\log_{2}}M}{\tau(1+\beta)}=\frac{256}{(256+32)}\cdot\frac{{\log_{2}}2}{0.8(1+0.5)}=0.7407$
bits/s/Hz. To achieve the same SE, we set the pilot sequence length in [13] to
$8$. This is because a two pilot sequence length is added at the beginning and
at the end of each data block to construct a frame in [13]. That is to say
taking the pilot length to be 8 in [13] results in having 32 known symbols
(overhead of 32) in each frame, the same as the overhead in a given frame in
our paper. The MSE of [13] is expressed in the frequency domain ([13], Eq.
(38)), and hence, to have a fair comparison of the MSE in our work, we convert
the MSE in [13] to the time-domain. It should be noted that the time-domain
MSE is also 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. 5, the proposed FTN signaling channel estimation employing
the designed optimal pilot sequence in this paper results in a significantly
better (about 6 dB) MSE when compared to the MSE resulting from the channel
estimation method employing designed pilot sequence in [13]. It is worth
mentioning that the emphasis in this paper is on the pilot design and the
channel estimation rather than iterative channel estimation and FTN signaling
detection.

Fig. 6 depicts the MSE of the proposed channel estimation for FTN signaling
having $\tau=0.72$ and $\tau=0.84$ over channel model 2. Particularly, the
F-F, N-F cases for $\tau=0.72$ and F-F case for $\tau=0.84$ are considered.
Fig. 6 depicts the simulation results for MSE,
$\sum\nolimits_{l=0}^{L_{c}}({\hat{\textbf{c}}_{l}}-\textbf{c}_{l})^{\text{H}}({\hat{\textbf{c}}_{l}}-\textbf{c}_{l})$,
and the theoretical MSE in (15). As illustrated in Fig. 6, the MSE resulting
from the simulation for all the employed pilot sequences matches the
theoretical results of the MSE in (15). Fig. 6 compares F-F for the FTN
signaling system having $\tau=0.72$ with $\tau=0.84$. As expected, the MSE for
$\tau=0.84$ is lower compared with $\tau=0.72$. As it is shown in Fig. 6, by
increasing the $E_{\text{b}}/N_{0}$, the MSE decreases. As depicted in Fig. 6
for $\tau=0.72$, the MSE for F-F case is about 2 dB better than N-F case. That
is to say, the optimal pilot sequence designed in this paper achieves a
significantly better MSE compared to the Nyquist pilot sequence used in the
FTN transmission signaling.

The MSE performance of the proposed channel estimation for FTN signaling
having $\tau=0.72$ and $\tau=0.84$ over channel model 1 is depicted in Fig. 7.
Particularly, the F-F and N-F cases for $\tau=0.72$ and F-F case for
$\tau=0.84$ are considered. As depicted in Fig. 7 for the $\tau=0.72$, the MSE
for F-F cases are about 2 dB (for MSE of $9\times 10^{-3}$) better than N-F
case. That is to say, the optimal pilot sequence designed in this paper
achieves a significantly better MSE compared to the Nyquist pilot sequence
used in the FTN transmission signaling. Fig. 7 compares F-F for the FTN
signaling system having $\tau=0.72$ with $\tau=0.84$ over channel model 1. As
expected, the simulation shows that the MSE for $\tau=0.84$ is lower compared
with $\tau=0.72$ case. As it is shown in Fig. 6 for FTN signaling over channel
model 2, by increasing the $E_{\text{b}}/N_{0}$, the MSE decreases. However,
as channel model 1 a doubly-selective channel, is employed in Fig. 7, the MSE
is saturated in high $E_{\text{b}}/N_{0}$. This is because of the
interpolation error to track the time-selective channel.

Fig. 8 depicts the MSE of the proposed channel estimation for FTN signaling
with $\tau=0.84$ over channel model 1 in case of employing different
interpolation methods. As it is shown in Fig. 8, the MSE does not change
significantly for the different interpolation methods employed to track the
time-selective channel.

In the above MSE performance simulations, we employed an exhaustive search to
find the optimal pilot minimizing the MSE in (15). This is mainly as pilot
design can be carried out offline and only once for a given channel.
Additionally, the pilot design optimization problem is non-convex, and low-
complexity solutions are not necessarily guaranteed to reach the global
optimal solution which may harm the channel estimation quality and degrate the
error rate. To strike a balance between MSE performance and computational
complexity, we relax the integer constraint of the problem in (15) and solve
the relaxed optimization problem numerically using the interior-point
algorithm [21] to obtain a sub-optimal solution. This sub-optimal solution is
then rounded to find the final pilot sequence. Please note that due rounding
the continuous sub-optimal solution, optimality is no longer guaranteed.
Additionally, given the non-convex nature of the optimization problem, the
interior-point algorithm may stuck at a local optimal solution. To increase
our chances of getting good quality local sub-optimal solution, we initialize
the interior-point algorithm with value of $m$ uniformly distributed random
pilot sequences and calculated the MSE and then choose the sub-optimal pilot
sequence with the lowest MSE value. The simulation is done for $m=1$ and $10$.
As depicted in the Fig. 9, by increasing the $m$, the MSE approaches the
optimal MSE calculated using the exhaustive search. That is to say the
interior-point algorithm, which has polynomial time complexity [21], seems a
promising approach to find sub-optimal pilots.

###  V-C BER Performance Evaluation

Figure 10: The BER performance of FTN signaling channel estimation employing
the designed optimal pilot sequence versus the method in [13] with $\tau=0.8$
over channel model 3. Figure 11: The BER performance of FTN signaling with
$\tau=0.72$ over channel model 2. Figure 12: The BER performance of FTN
signaling with $\tau=0.72$ over channel model 2. Figure 13: The BER
performance of FTN signaling with $\tau=0.72$ over channel model 1.

Figure 14: The BER performance of FTN signaling with $\tau=0.72$ over channel
model 1

.

Fig. 10 depicts the BER performance of the proposed channel estimation
employing the designed optimal pilot sequence of the FTN signaling detection
using F-F having $\tau=0.8$ over channel model 3. To be consistent with [13],
$\beta=0.5$ is considered. To guarantee a fair comparison with [13], we fix
the SE of both transmission to 0.7407 bits/s/Hz. The channel estimation method
in [13] is implemented and the BER output of the SISO decoder is measured to
have a fair comparision. As depicted in Fig. 10, the proposed FTN signaling
channel estimation employing the designed optimal pilot sequence in this paper
shows better BER compared to the channel estimation method employing the
designed pilot sequence in [13]. More specifically, after two iterations, the
BER achieved employing the proposed channel estimation method using the
designed optimal pilot sequence shows about 3.5 dB improvement compared to the
case of [13].

Fig. 11 compares the BER performance of the proposed channel estimation of the
FTN signaling detection using F-F and N-F with $\tau=0.72$ over channel model
2. As a benchmark, the Nyquist case, N-N, with perfectly known channel at the
receiver is plotted. As depicted in Fig. 11, the BER for F-F case is lower
than N-F. That is to say, the designed optimal pilot sequence in this paper
achieves a better BER compared to the Nyquist pilot sequence used in the FTN
transmission signaling. More specifically, after two iterations, the BER
achieved using the designed optimal pilot sequence shows about 0.5 dB
improvement compared to the case employing the Nyquist pilot sequence. As
illustrated in Fig. (11), after two iterations, the BER achieved using the
proposed channel estimation and designed optimal pilot sequence for FTN
signaling system F-F shows about 3 dB difference with the Nyquist scenario N-N
where channel is completely known at the receiver.

Fig. 12 compares the BER performance of the proposed channel estimation for
F-F and the case with perfectly known channel at the receiver in a FTN
signaling system with $\tau=0.72$ over channel model 2. As shown in Fig. 12,
after only one iteration, the BER achieved by the proposed channel estimation
employing the designed optimal pilot sequence in this paper shows about 2 dB
difference with the case where channel is perfectly known at the receiver.

Fig. 13 compares the BER performance of F-F and N-F scenarios in a FTN
signaling system with $\tau=0.72$ over channel model 1. As a benchmark, the
N-N case with perfectly known channel at the receiver is also illustrated. It
is shown in Fig. 11 that F-F achieves better BER compared to the N-F. That is
to say, the designed optimal pilot sequence achieves better BER comparing to
the Nyquist pilot sequence. As depicted in Fig. 13, after two iterations, the
BER achieved of the proposed channel estimation employing designed pilot
sequence for FTN signaling system F-F shows about 2 dB difference with the
Nyquist scenario N-N where channel is completely known at the receiver.

Fig. 14 compares the BER performance of the proposed channel estimation for
F-F and the case with perfectly known channel at the receiver in FTN signaling
system with $\tau=0.72$ over channel model 1. As depicted in Fig. 14, for the
FTN signaling system, after two iterations, the BER achieved employing the
proposed channel estimation and pilot design method in the paper shows less
than 2 dB difference with the case where channel is perfectly known at the
receiver.

##  VI Conclusion

In this paper, we proposed a channel estimation and data detection for FTN
signaling over doubly-selective channels without imposing additional SE loss.
To improve the channel estimation quality, we adopted a frame structure
considering the ISI introduced by both FTN signaling and the frequency-
selective channel. We proposed a LSSE-based channel estimation approach for
FTN signaling to estimate the complex channel coefficients at the pilot
locations within the frame. To this end, the optimal FTN signaling pilot
sequence that minimized the MSE of the channel estimation is found. A low-
complexity linear interpolation was employed to track the time-selective
complex channel coefficients at the data symbols locations within the frame. A
turbo equalization technique based on a linear SISO MMSE equalizer was
employed to detect the data symbols of FTN signaling. Simulation results
showed that employing the designed optimal pilot sequence resulted in a better
MSE of the channel estimation (up to 2 dB for both doubly-selective and
frequency-selective channel models) compared to the Nyquist pilot sequence
when used in the FTN signaling. Additionally, as the simulation results
indicated, the BER performance of the FTN signaling employing the designed
optimal pilot sequence showed improvements compared to the FTN signaling
employing the conventional Nyquist pilot sequence. Simulation results, at the
same SE, showed a significant improvement (about 6 dB) in the MSE of the
channel estimation and (about 3.5 dB) BER of FTN signaling employing the
designed optimal pilot sequence compared to the one introduced in [13] over
frequency-selective channels.

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