shadows MyMdframed[1][]

# Performance Analysis and Window Design for Channel Estimation of OTFS
Modulation

Zhiqiang Wei, Weijie Yuan, Shuangyang Li, Jinhong Yuan, and Derrick Wing Kwan
Ng Zhiqiang Wei, Weijie Yuan, Shuangyang Li, Jinhong Yuan, and Derrick Wing
Kwan Ng are with the School of Electrical Engineering and Telecommunications,
the University of New South Wales, Australia (email: zhiqiang.wei;
weijie.yuan; shuangyang.li; j.yuan; w.k.ng@unsw.edu.au).

###### Abstract

In this paper, we investigate the impacts of transmitter and receiver windows
on orthogonal time-frequency space (OTFS) modulation and propose a window
design to improve the OTFS channel estimation performance. Assuming ideal
pulse shaping filters at the transceiver, we first identify the role of window
in effective channel and the reduced channel sparsity with conventional
rectangular window. Then, we characterize the impacts of windowing on the
effective channel estimation performance for OTFS modulation. Based on the
revealed insights, we propose to apply a Dolph-Chebyshev (DC) window at either
the transmitter or the receiver to effectively enhance the sparsity of the
effective channel. As such, the channel spread due to the fractional Doppler
is significantly reduced, which leads to a lower error floor in channel
estimation compared with that of the rectangular window. Simulation results
verify the accuracy of the obtained analytical results and confirm the
superiority of the proposed window designs in improving the channel estimation
performance over the conventional rectangular or Sine windows.

##  I Introduction

Future wireless networks are expected to provide high-speed and ultra-reliable
communications for a wide range of emerging mobile applications[1, 2, 3, 4, 5,
6, 7], including online video gaming, unmanned aerial vehicles (UAV)[8],
vehicle-to-everything (V2X), high-speed railway systems, etc. In high-mobility
channels, the multipath propagation and the temporal channel variations give
rise to the frequency-selective fading (time dispersion) and time-selective
fading (frequency dispersion), respectively, resulting in the so called
doubly-selective or doubly-dispersive channels[9]. To cope with the channel
dynamics, a new two-dimensional (2D) modulation scheme referred to as the
orthogonal time-frequency space (OTFS) modulation was recently proposed in [1]
and has received an increasing amount of attention in both academia and
industry, e.g. [1, 10, 11, 12, 13].

In OTFS modulation, data symbols are multiplexed in the delay-Doppler (DD)
domain rather than in the time-frequency (TF) domain as in contrast to the
traditional orthogonal frequency division multiplexing (OFDM) modulation[11,
14, 15]. In practice, OTFS modulation effectively transforms the TF domain
time-variant channel into an effective two-dimensional (2D) time-invariant
channel in the DD domain, which exhibits both sparse and stable properties[1,
11]. More importantly, the 2D transformation from the DD domain to the TF
domain employed by an OTFS modulator allows the possibility of each
information symbol to experience the whole TF domain channel over an OTFS
frame. Thus, OTFS enjoys the joint time-frequency diversity[12] (the so-called
full diversity in [1]), which is desirable to provide reliable communications
over doubly dispersive channels. In particular, it has been demonstrated that
OTFS is resilient to severe delay-Doppler shifts and outperforms OFDM
significantly for both uncoded [11] and coded [16, 17] systems.

However, reliable communications with OTFS modulation highly rely on accurate
channel estimation[5], particularly in the presence of fractional Doppler[11],
i.e., the exact Doppler frequency straddles a pair of finite-resolution bins
rather than falls exactly into a bin in the Doppler domain. Yet, most of
existing works only considered integer Doppler for simplicity, e.g. [18, 19,
20]. In fact, ensuring integer Doppler requires a large speed separation among
transceiver and all moving scatters to create a high Doppler resolution, which
is not always possible in practical systems. Although channel acquisition in
the DD domain may be deemed more convenient than that in the TF domain in high
mobility scenarios[21, 20], the effective channel is spread across all the
Doppler bins due to fractional Doppler, which sacrifices the DD domain channel
sparsity. Moreover, the channel estimation performance of OTFS systems is
mainly limited by the inter-Doppler interference (IDI), where a guard space is
usually required to avoid the IDI between data and pilot symbols, either
employing a single pilot symbol [21] or a pilot sequence[20]. Even worse, the
IDI between data and pilot symbols caused by fractional Doppler becomes more
severe leading to an error floor in the effective channel estimation. As a
remedy, to lower the error floor, a much larger guard space inserting between
the data and pilot symbols is required, which causes a higher amount of
signaling overhead. Therefore, a pragmatic approach for reducing the channel
spreading caused by fractional Doppler is desired.

As mentioned in [18, 22], windowing in the TF domain has the potential in
combating the effective channel spreading in the DD domain. Yet, the authors
in [18, 22] did not propose any method for window designs. Moreover, the role
of windowing in OTFS modulation and its impact on the performance of OTFS
channel estimation are not well understood yet. In the literature, to the best
of our knowledge, there is no existing work studying on the window design for
OTFS, which motivates this work.

In this paper, we study the window design for OTFS modulation to improve the
channel estimation performance with the consideration of practical fractional
Doppler. Firstly, the roles of windowing on OTFS systems, such as the
effective channel and the corresponding sparsity, are identified. Secondly, we
analyze the impact of windowing on the effective channel estimation
performance. Thirdly, we propose to employ a Dolph-Chebyshev (DC) window in
the TF domain to facilitate the channel estimation in the DD domain. The
employed DC window is optimal in the sense that it can obtain a predefined
channel sparsity while suppressing the channel spreading caused by the
fractional Doppler to the largest degree. Due to the enhanced channel
sparsity, applying the proposed DC window at either the transmitter or the
receiver can achieve a much lower channel estimation error floor compared with
the conventional rectangular window. Extensive simulations are conducted
verify the analytical results and to demonstrate the substantial performance
gain of the proposed window design over the conventional rectangular and Sine
windows.

Notations: $\mathbb{Z}^{+}$ denotes the set of all non-negative integers;
$\mathbb{C}^{M\times N}$ denotes the set of all $M\times N$ matrices with
complex entries; $\lvert\cdot\rvert$ denotes the absolute value of a complex
scalar; $E\\{\cdot\\}$ denotes the expectation; $\left(\cdot\right)^{*}$
denotes the conjugate operation; $\left(\cdot\right)_{N}$ denotes the modulus
operation with respect to $N$; $\Re\\{\cdot\\}$ returns the real part of the
input complex number; $\lfloor\cdot\rfloor$ is the floor function which
returns the largest integer smaller than the input value; The circularly
symmetric complex Gaussian distribution with mean $\bm{\mu}$ and covariance
matrix $\bm{\Sigma}$ is denoted by ${\cal CN}(\bm{\mu},\bm{\Sigma})$; $\sim$
stands for “distributed as”.

Figure 1: The block diagram of the OTFS transceiver[11].

##  II System Model

###  II-A OTFS Transmitter

A practical implementation of the OTFS transceiver is shown in Fig. 1. Without
loss of generality, we assume that one OTFS frame occupies a bandwidth of
$B_{\mathrm{OTFS}}$ and a time duration of $T_{\mathrm{OTFS}}$. The total
available bandwidth $B_{\mathrm{OTFS}}$ is divided into $M$ subcarriers with
an equal spacing of $\Delta f=\frac{B_{\mathrm{OTFS}}}{M}$. The total time
duration $T_{\mathrm{OTFS}}$ is divided into $N$ time slots with an equal-
length slot duration of $T=\frac{T_{\mathrm{OTFS}}}{N}$. As a result, a grid
of $N\times M$ can be constructed in the TF domain. Note that the delay
resolution is determined by the reciprocal of the system bandwidth, i.e.,
$\frac{1}{M\Delta f}$, while the Doppler resolution is determined by the OTFS
frame duration, i.e., $\frac{1}{NT}$ [11]. Correspondingly, in the DD domain,
$N$ denotes the number of Doppler indices with a Doppler resolution of
$\frac{1}{NT}$ and $M$ denotes the number of delay indices with a delay
resolution of $\frac{1}{M\Delta f}$. Consider a baseband modulated symbol in
the DD domain:

$x\left[{k,l}\right]\in\mathbb{A}=\\{a_{1},\ldots,a_{Q}\\},$ (1)

where $k\in\\{0,\ldots,N-1\\}$ represents the Doppler index,
$l\in\\{0,\ldots,M-1\\}$ represents the delay index, and $\mathbb{A}$ denotes
the constellation set with a size of $Q$. We assume that a normalized
constellation is adopted, i.e.,
$E\left\\{{{{\left|{x\left[{k,l}\right]}\right|}^{2}}}\right\\}=1$, and a
proper scrambler is applied to scramble the output of the encoder such that it
is reasonable to assume
$E\left\\{{{{\left|{x\left[{k,l}\right]}\right|}}}{{{\left|{x\left[{k^{\prime},l^{\prime}}\right]}\right|}}}\right\\}=0$,
$\forall k\neq k^{\prime}$, $\forall l\neq l^{\prime}$. OTFS modulator
performs a 2D transformation which maps the data symbols $x\left[{k,l}\right]$
in the DD domain to $X\left[{n,m}\right]$ in the TF domain. In particular,
such mapping can be realized by the inverse symplectic finite Fourier
transform (ISFFT)[1]:

$X\left[{n,m}\right]=\frac{1}{{\sqrt{NM}}}\sum\nolimits_{k=0}^{N-1}{\sum\nolimits_{l=0}^{M-1}{x\left[{k,l}\right]{e^{j2\pi\left({\frac{{nk}}{N}-\frac{{ml}}{M}}\right)}}}},$
(2)

where $n\in\\{0,\ldots,N-1\\}$ is the time slot index and
$m\in\\{0,\ldots,M-1\\}$ is the subcarrier index.

A TF domain transmitter (TX) window $U\left[{n,m}\right]$ is imposed through a
point-wise multiplication with the TF domain signal $X\left[{n,m}\right]$:

$\widetilde{X}\left[{n,m}\right]=U\left[{n,m}\right]X\left[{n,m}\right],$ (3)

where $U\left[{n,m}\right]\in\mathbb{C}$ denotes the complex-valued TX window
weighted on the point of $\left[{n,m}\right]$ in the TF domain grid. Then, a
multicarrier modulator is adopted to transform the TF domain signal
$\widetilde{X}\left[{n,m}\right]$ to a time-domain signal $s\left(t\right)$,
given by

$s\left(t\right)=\sum\limits_{n=0}^{N-1}{\sum\limits_{m=0}^{M-1}{\widetilde{X}\left[{n,m}\right]{{g_{{\rm{tx}}}}\left({t-nT}\right){e^{j2\pi
m\Delta f\left({t-nT}\right)}}}}},$ (4)

which is referred to as the Heisenberg transform in[1], where $t$ denotes the
continuous time variable. The time domain function
${{g_{{\rm{tx}}}}\left({t}\right)}$ is the pulse-shaping filter of the
multicarrier modulator for the windowed TF domain symbol.

###  II-B DD Domain Channel Response

For a linear time-variant channel, the received signal in the time domain is
given by[11]

$r\left(t\right)=\int{\int{h\left({\tau,\nu}\right)}}{{e^{j2\pi\nu\left({t-\tau}\right)}}}s\left({t-\tau}\right)d\tau
d\nu+w\left(t\right),$ (5)

where $w\left(t\right)$ denotes the noise signal in the time domain following
a stationary Gaussian random process and we have
$w\left(t\right)\sim\mathcal{CN}\left(0,N_{0}\right)$ with $N_{0}$ denoting
the noise variance. In practice, only few reflectors are moving within one
OTFS frame duration and thus only a small number of channel taps are
associated with Doppler shift[1, 11]. Therefore, the resulting channel
response in the DD domain is sparse compared with the whole DD domain grid
spanned by one OTFS frame. In particular, considering a channel consisting of
$P$ independent distinguishable paths, the channel response in the DD domain
can be modeled by

${h\left({\tau,\nu}\right)}=\sum\nolimits_{i=1}^{P}{h_{i}}\delta(\tau-\tau_{i})\delta(\nu-\nu_{i}),$
(6)

where $h_{i}\in\mathbb{C}$, $\tau_{i}$, and $\nu_{i}$ denote the channel
coefficient, delay, and Doppler shift associated with the $i$-th path,
respectively. The variables $\tau_{i}$ and $\nu_{i}$ are defined as
$\tau_{i}=l_{\tau_{i}}\frac{1}{M\Delta f}$ and
$\nu_{i}=\left(k_{\nu_{i}}+\kappa_{\nu_{i}}\right)\frac{1}{NT}$, respectively,
where $l_{\tau_{i}}\in\\{0,\ldots,l_{\mathrm{max}}\\}$,
${k_{{\nu_{i}}}}\in\\{-k_{\mathrm{max}},\ldots,k_{\mathrm{max}}\\}$, and
$-\frac{1}{2}<\kappa_{\nu_{i}}<\frac{1}{2}$ denote the integer delay, integer
Doppler, and fractional Doppler indices, respectively. Variables
$k_{\max}\in\mathbb{Z}^{+}$ and $l_{\max}\in\mathbb{Z}^{+}$ denote the maximum
Doppler and delay indices, respectively.

###  II-C OTFS Receiver

At the receiver side, we first perform a multicarrier demodulation for the
received signal $r\left(t\right)$ with a receiving filter to obtain the TF
domain signal $\widetilde{Y}\left[n,m\right]$, given by:

$\widetilde{Y}\left[n,m\right]=\int{r\left(t\right)g_{{\rm{rx}}}^{*}\left({t-nT}\right){e^{-j2\pi
m\Delta f\left({t-nT}\right)}}dt},$ (7)

which is referred to as the Wigner transform in [1]. A time domain function
${{g_{{\rm{rx}}}}\left({t}\right)}$ serving as a receiving filter for the
multicarrier demodulator is adopted to sample the discrete symbol
$\widetilde{Y}\left[{n,m}\right]$ from the received waveform
$r\left(t\right)$. Substituting (4), (5), and (6) into (7), and assuming ideal
transceiver pulse shaping filters satisfying the bi-orthogonal condition[11],
we have

$\widetilde{Y}\left(n,m\right)={\widetilde{X}\left[{n,m}\right]}\widetilde{H}\left[n,m\right]+\widetilde{Z}\left[n,m\right],$
(8)

where the TF domain effective channel is given by

$\widetilde{H}\left[n,m\right]=\sum\limits_{i=1}^{P}{h_{i}{e^{-j2\pi\frac{\left(k_{\nu_{i}}+\kappa_{\nu_{i}}\right)l_{\tau_{i}}}{NM}}}{e^{j2\pi\left({\frac{{n\left(k_{\nu_{i}}+\kappa_{\nu_{i}}\right)}}{N}-\frac{{ml_{\tau_{i}}}}{M}}\right)}}}.$
(9)

Corresponding to the TX window, we can insert a receiver (RX) window
$V\left[{n,m}\right]$ to the received signal in the TF domain:

${Y}\left[{n,m}\right]=V\left[{n,m}\right]\widetilde{Y}\left[{n,m}\right],$
(10)

where $V\left[{n,m}\right]\in\mathbb{C}$. Then, an OTFS demodulator transforms
the TF domain signals ${Y}\left[{n,m}\right]$ to the DD domain signals
$y\left[{k,l}\right]$ through a symplectic finite Fourier transform (SFFT)
[1]:

$y\left[{k,l}\right]=\frac{1}{{\sqrt{NM}}}\sum\nolimits_{n=0}^{N-1}{\sum\nolimits_{m=0}^{M-1}{Y\left[{n,m}\right]{e^{-j2\pi\left({\frac{{kn}}{N}-\frac{{lm}}{M}}\right)}}}}.$
(11)

##  III The Impact of Windowing on Effective Channel

In this section, we analyze the impacts of windowing on the effective channel
for OTFS modulation.

###  III-A Effective Channel in the DD Domain

According to the OTFS transceiver structure introduced above, the output of
the OTFS demodulator in the DD domain is given by[11]

$\displaystyle y\left[{k,l}\right]$
$\displaystyle=\sum\nolimits_{k^{\prime}=0}^{N-1}{\sum\nolimits_{l^{\prime}=0}^{M-1}{x\left[{k^{\prime},l^{\prime}}\right]}}{h_{w}}\left[{k-k^{\prime},l-l^{\prime}}\right]$
$\displaystyle+\sum\nolimits_{k^{\prime}=0}^{N-1}{\sum\nolimits_{l^{\prime}=0}^{M-1}{z\left[{k^{\prime},l^{\prime}}\right]}}{v_{z}}\left[{k-k^{\prime},l-l^{\prime}}\right].$
(12)

In (III-A), ${h_{w}}\left[{k,l}\right]$ denotes the effective channel in the
DD domain capturing the windows’ effect and it is given by

${h_{w}}\left[{k,l}\right]=\sum\limits_{i=1}^{P}{h_{i}}w(k-k_{\nu_{i}}-\kappa_{\nu_{i}},l-l_{\tau_{i}}){e^{-j2\pi\frac{\left(k_{\nu_{i}}+\kappa_{\nu_{i}}\right)l_{\tau_{i}}}{NM}}},$
(13)

where $w(k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}},l-{l_{{\tau_{i}}}})$ is an
equivalent DD domain filter designed by the TX-RX window which is given by[11]

$\displaystyle w(k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}},l-{l_{{\tau_{i}}}})$
$\displaystyle=\frac{1}{NM}\sum\limits_{n=0}^{N-1}\sum\limits_{m=0}^{M-1}V\left[{n,m}\right]U\left[{n,m}\right]$
$\displaystyle\times{e^{-j2\pi
n\frac{\left({k-k_{\nu_{i}}-\kappa_{\nu_{i}}}\right)}{N}}}{e^{j2\pi
m\frac{\left(l-l_{\tau_{i}}\right)}{M}}}.$ (14)

Also in (III-A), ${v_{z}}\left[{k,l}\right]$ is a DD domain filter induced
only by the RX window and is given by

${v_{z}}\left[{k,l}\right]=\frac{1}{NM}\sum\nolimits_{n=0}^{N-1}\sum\nolimits_{m=0}^{M-1}V\left[{n,m}\right]{e^{-j2\pi\frac{nk}{N}}}{e^{j2\pi\frac{ml}{M}}}.$
(15)

We can observe that different from the original DD domain channel response in
(6), the effective channel in (13) has a circular structure due to
${h_{w}}\left[{\left(k\right)_{N},\left(l\right)_{M}}\right]={h_{w}}\left[{k,l}\right]$.
As such, from (III-A), we can observe that the received signal
$y\left[{k,l}\right]$ is a 2D circular convolution between the data symbols,
${x\left[{k,l}\right]}$, and the effective channel,
${h_{w}}\left[{k,l}\right]$, in the DD domain. Furthermore, as the data and
training symbols are multiplexed in the DD domain [21], the channel estimation
performance and the data detection complexity depend on the effective channel
${h_{w}}\left[{k,l}\right]$ instead of the original channel response
$h\left({\tau,\nu}\right)$. As shown in (13), the effective channel
${h_{w}}\left[{k,l}\right]$ is a summation of the channel spread of each path
where $w(k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}},l-{l_{{\tau_{i}}}})\neq 0$,
$\forall i$, and the spreading pattern can be manipulated by the design of the
DD domain filter $w(k,l)$. In other words, the channel sparsity of the
effective channels can be controlled by the TX and RX windows. In (III-A), we
can observe that imposing a TX window $U\left[{n,m}\right]$ or a RX window
$V\left[{n,m}\right]$ in the TF domain has the same effect on the design of
the DD domain filter $w(k,l)$. In contrast, only the RX window
$V\left[{n,m}\right]$ affects the DD domain filter,
${v_{z}}\left[{k,l}\right]$, which alters the properties of the noise at the
receiver side.

###  III-B Effective Channel Sparsity with Rectangular Window

Since the rectangular window, i.e.,
$V\left[{n,m}\right]=U\left[{n,m}\right]=1$, $\forall n,m$, is the most
straightforward one to be considered[11, 21], we investigate the effective
channel sparsity with rectangular window for both cases of integer and
fractional Doppler111Different from [11, 21], we focus on discussing the
reduced effective channel sparsity due to the existence of fractional Doppler,
which motivates our analysis and design in the following sections.. With
employing the rectangular window, we have the DD domain filters given by[11]

$\displaystyle w(k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}},l-{l_{{\tau_{i}}}})$
$\displaystyle=\mathcal{G}^{\mathrm{Rect}}_{N}\left({k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}}}\right)\mathcal{F}^{\mathrm{Rect}}_{M}\left({l-{l_{{\tau_{i}}}}}\right)$
$\displaystyle\text{and}\;{v_{z}}\left[{k,l}\right]$
$\displaystyle=\mathcal{G}^{\mathrm{Rect}}_{N}\left(k\right)\mathcal{F}^{\mathrm{Rect}}_{M}\left(l\right),$
(16)

respectively. Functions $\mathcal{G}^{\mathrm{Rect}}_{N}\left(k\right)$ and
$\mathcal{F}^{\mathrm{Rect}}_{M}\left(l\right)$ represent the filters in the
delay and Doppler domains, respectively, and they are given by
$\mathcal{G}^{\mathrm{Rect}}_{N}\left(k\right)=\frac{1}{N}\left({e^{-j\left({N-1}\right)\frac{\pi
k}{N}}}\frac{\sin\left(\pi k\right)}{{\sin\left({\frac{{\pi
k}}{N}}\right)}}\right)$ and
$\mathcal{F}^{\mathrm{Rect}}_{M}\left(l\right)=\frac{1}{M}\left({{e^{-j\left({M-1}\right)\frac{\pi
l}{M}}}\frac{{\sin\left({\pi l}\right)}}{{\sin\left({\frac{{\pi
l}}{M}}\right)}}}\right)$, respectively.

For the case of integer Doppler, i.e., $\kappa_{\nu_{i}}=0$, the DD domain
filter is simplified as

$\displaystyle w\left[k-{k_{{\nu_{i}}}},l-{l_{{\tau_{i}}}}\right]$
$\displaystyle=\delta\left[k-{k_{{\nu_{i}}}}\right]\delta\left[l-{l_{{\tau_{i}}}}\right]$
(17)
$\displaystyle=\left\\{{\begin{array}[]{*{20}{c}}{1}&{\left(k-{k_{{\nu_{i}}}}\right)_{N}=0,\left(l-{l_{{\tau_{i}}}}\right)_{M}=0}\\\
0&\mathrm{otherwise}\end{array}}\right.,$ (20)

and the effective channel in the DD domain is given by

${h_{w}}\left[{k,l}\right]=\sum\nolimits_{i=1}^{P}{h_{i}}\delta\left[k-{k_{{\nu_{i}}}}\right]\delta\left[l-{l_{{\tau_{i}}}}\right]{e^{-j2\pi\frac{k_{\nu_{i}}l_{\tau_{i}}}{NM}}}.$
(21)

We can observe that the effective channel ${h_{w}}\left[{k,l}\right]$ in the
DD domain has a response if and only if $k={k_{{\nu_{i}}}}$ and
$l={l_{{\tau_{i}}}}$, i.e., the effective channel shares the same channel
sparsity with the original DD domain channel response in (6). Moreover, the
effective channel ${h_{w}}\left[{k,l}\right]$ is a phase-rotated version of
the original channel response in (6), where the delay and Doppler shift of the
$i$-th path rotates the original channel response ${h_{i}}$ with a phase of
$2\pi k_{\nu_{i}}l_{\tau_{i}}/MN$.

Figure 2: The channel spreading in the Doppler domain with a rectangular
window with/without fractional Doppler, where
$\mathrm{SL}_{w}\approx\frac{1}{N}$ denotes the sidelobe level of the adopted
rectangular window.

For the case of fractional Doppler, i.e., $\kappa_{\nu_{i}}\neq 0$, the
effective channel is given by

${h_{w}}\left[{k,l}\right]=\sum\limits_{i=1}^{P}{h_{i}}\mathcal{G}^{\mathrm{Rect}}_{N}\left({k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}}}\right)\delta\left[l-{l_{{\tau_{i}}}}\right]{e^{-j2\pi\frac{\left(k_{\nu_{i}}+\kappa_{\nu_{i}}\right)l_{\tau_{i}}}{NM}}}.$
(22)

From (22), we can observe that the effective channel in the DD domain
${h_{w}}\left[{k,l}\right]$ contains more “paths” (non-zero entries) than that
of the original channel response in (6), since the Doppler domain filter
$\mathcal{G}^{\mathrm{Rect}}_{N}\left({k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}}}\right)\neq
0$, $\forall k,{k_{{\nu_{i}}}}$, and $\forall\kappa_{\nu_{i}}\neq 0$. In fact,
for each path with a Doppler shift of ${k_{{\nu_{i}}}}+\kappa_{\nu_{i}}$, the
channel coefficient ${h_{i}}$ is spread to all the Doppler indices $k$ in the
Doppler domain. To visualize the channel spreading, we ignore the delay domain
at the moment and plot the Doppler domain filter response
$\left|\mathcal{G}^{\mathrm{Rect}}_{N}\left({k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}}}\right)\right|$
to illustrate the impact of fractional Doppler in Fig. 2. It can be seen that
without the fractional Doppler, the filter
$\left|\mathcal{G}^{\mathrm{Rect}}_{N}\left({k-{k_{{\nu_{i}}}}}\right)\right|$
is a perfect sampling function $\delta\left[k-{k_{{\nu_{i}}}}\right]$, i.e.,
no channel spread. However, the existence of the fractional Doppler shift
$\kappa_{\nu_{i}}$ not only reduces signal power at the sampling point
$k={k_{{\nu_{i}}}}$, but also introduces non-negligible power leakage from the
Doppler shift ${k_{{\nu_{i}}}}$ to $k\neq{k_{{\nu_{i}}}}$. In other words,
with the application of the rectangular window, fractional Doppler sacrifices
the sparsity of the effective channel in the DD domain, which could degrade
the channel estimation performance and increase the complexity of data
detection. Therefore, it is desired to design a window which can null/suppress
the power leakage and improve the effective channel sparsity.

$\displaystyle E\left\\{{{{\left|{I\left[{k,l}\right]}\right|}^{2}}}\right\\}$
$\displaystyle=\sum\limits_{k^{\prime}\notin\mathcal{K}}{\sum\limits_{l^{\prime}=l-l_{\mathrm{max}}}^{{l}}{E\left\\{{{{\left|{x\left[{k^{\prime},l^{\prime}}\right]}\right|}^{2}}}\right\\}E\left\\{{{{\left|{{h_{w}}\left[{{{\left({k-k^{\prime}}\right)}_{N}},{{\left({l-l^{\prime}}\right)}_{M}}}\right]}\right|}^{2}}}\right\\}}}$
$\displaystyle\mathop{=}\limits^{(a)}\sum\limits_{k^{\prime}\notin\mathcal{K}}\sum\limits_{l-l^{\prime}\in[0,l_{\mathrm{max}}],{{\left({l-l^{\prime}}\right)}_{M}}=l_{\tau_{i}}}{{E\left\\{{{{\left|{\sum\limits_{i=1}^{P}{{h_{i}}}w\left({{{\left({k-k^{\prime}}\right)}_{N}}-{k_{{\nu_{i}}}}-{\kappa_{{\nu_{i}}}},0}\right){e^{-j2\pi\frac{{\left({{k_{{\nu_{i}}}}+{\kappa_{{\nu_{i}}}}}\right){l_{{\tau_{i}}}}}}{{NM}}}}}\right|}^{2}}}\right\\}}}$
(24) $\displaystyle\mathrm{MSE}$
$\displaystyle=\sum_{k={k_{p}}-{k_{\max}}-\hat{k}}^{{k_{p}}+{k_{\max}}+\hat{k}}\sum_{l={l_{p}}}^{{l_{p}}+{l_{\mathrm{max}}}}E\left\\{{{{\left|{h_{w}}\left[{\left(k-k_{p}\right)_{N},\left(l-l_{p}\right)_{M}}\right]-{\hat{h}_{w}}\left[{\left(k-k_{p}\right)_{N},\left(l-l_{p}\right)_{M}}\right]\right|}^{2}}}\right\\}$
(27) $\displaystyle\lim\limits_{N_{0}\to 0}\mathrm{MSE}$
$\displaystyle=\sum_{k={k_{p}}-{k_{\max}}-\hat{k}}^{{k_{p}}+{k_{\max}}+\hat{k}}\sum_{l={l_{p}}}^{{l_{p}}+{l_{\mathrm{max}}}}\frac{E\left\\{{{{\left|{I\left[{k,l}\right]}\right|}^{2}}}\right\\}}{\left|x_{p}\right|^{2}}\approx\left(N-4{k_{\max}}-4\hat{k}-1\right)\left(2{k_{\max}}+2\hat{k}+1\right)\left(l_{\mathrm{max}}+1\right)\mathrm{SL}_{w}^{2}$
(28)

##  IV Effective Channel Estimation Performance with Arbitrary Window

In this work, we adopt the channel estimation scheme proposed in [21], where a
single pilot symbol is embedded in the DD domain and a guard space is inserted
between the pilot symbol and data symbols. In fact, to the best of our
knowledge, the channel estimation scheme in [21] is the first DD domain
channel estimation method proposed for OTFS in the literature, which is simple
and practical. In this section, we investigate the impact of windowing on
channel estimation performance based on the scheme in [21]. Let us assume that
the only pilot symbol $x_{p}$ is inserted at the $\left[k_{p},l_{p}\right]$-th
DD grid and data symbols ${x_{d}}\left[{k,l}\right]$ are arranged as
follow[21]

$x\left[{k,l}\right]=\left\\{{\begin{array}[]{*{20}{c}}{{x_{p}}}&{k={k_{p}},l={l_{p}}},\\\
0&\begin{array}[]{l}k\in\mathcal{K},k\neq{k_{p}},l\in\mathcal{L},l\neq{l_{p}},\end{array}\\\
{{x_{d}}\left[{k,l}\right]}&{{\rm{otherwise}}},\end{array}}\right.$ (20)

where
$\mathcal{K}=\\{{k_{p}}-2{k_{\max}}-2\hat{k},\ldots,{k_{p}}+2{k_{\max}}+2\hat{k}\\}$
and
$\mathcal{L}=\\{{l_{p}}-{l_{\mathrm{max}}},\ldots,{l_{p}}+{l_{\mathrm{max}}}\\}$
denotes the index sets of the guard space in the Doppler and delay domains,
respectively. Variable $\hat{k}\in\mathbb{Z}^{+}$ denotes the additional guard
to mitigate the spread due to fractional Doppler and
$\hat{k}\in\left\\{0,\ldots,\lfloor\frac{N-4k_{\max}-1}{4}\rfloor\right\\}$.
Increasing $\hat{k}$ would potentially increase the channel estimation
performance while reduces the spectral efficiency, as the signaling overhead
increases with $\hat{k}$, i.e., the total signaling overhead is
$\left(2{l_{\mathrm{max}}}+1\right)\left(4{k_{\max}}+4\hat{k}+1\right)$.

The estimation of the effective channel is based on the received signals in
the DD domain, which are given by

$\displaystyle y\left[k,l\right]$
$\displaystyle=x_{p}{h_{w}}\left[{\left(k-k_{p}\right)_{N},\left(l-l_{p}\right)_{M}}\right]+I\left[k,l\right]$
$\displaystyle+\sum\nolimits_{k^{\prime}=0}^{N-1}{\sum\nolimits_{l^{\prime}=0}^{M-1}{z\left[{k^{\prime},l^{\prime}}\right]}}{v_{z}}\left[{k-k^{\prime},l-l^{\prime}}\right],$
(21)

where ${k_{p}}-{k_{\max}}-\hat{k}\leq k\leq{k_{p}}+{k_{\max}}+\hat{k}$ and
${l_{p}}\leq l\leq{l_{p}}+{l_{\mathrm{max}}}$. According to [21], the
effective DD domain channel can be estimated by

${\hat{h}_{w}}\left[{\left(k-k_{p}\right)_{N},\left(l-l_{p}\right)_{M}}\right]=\frac{y\left[k,l\right]}{x_{p}},\;\text{if}\left|y\left[k,l\right]\right|\geq
3\sqrt{N_{0}}.$ (22)

In (IV), $I\left[k,l\right]$ denotes the interference spread from data symbols
due to the existence of fractional Doppler, which is given by

$I\left[k,l\right]=\sum_{k^{\prime}\notin\mathcal{K}}\sum_{l^{\prime}=0}^{l_{\mathrm{max}}}x\left[{k^{\prime},\left(l-l^{\prime}\right)_{M}}\right]{h_{w}}\left[{\left(k-k^{\prime}\right)_{N},l^{\prime}}\right].$
(23)

We can observe that in the delay domain, only $l_{\mathrm{max}}+1$ symbols
before $l$ affect the received symbol on $l$. On the other hand, in the
Doppler domain, all the data symbols outside the guard space
$k^{\prime}\notin\mathcal{K}$ affect the received symbol on $k$. Due to the
existence of the interference term $I\left[k,l\right]$, the channel estimation
in (22) suffers from an error floor even increasing the system signal-to-noise
ratio (SNR). Note that when applying the full guard space[21], i.e.,
$4{k_{\max}}+4\hat{k}+1=N$, the interference term in (23) would disappear and
there is no error floor in the effective channel estimation. However, it
requires a higher signaling overhead of $\left(2{l_{\mathrm{max}}}+1\right)N$
compared with that of the scheme in (20).

In the following, we derive the interference power to investigate the impact
of windowing on the effective channel estimation performance. Since the
transmitted data symbols are independent, the interference power can be
calculated as (III-B) at the top of next page, where the equality $(a)$ is
obtained since only the data symbols on
${{\left({l-l^{\prime}}\right)}_{M}}=l_{\tau_{i}}$ in the summation over
$l^{\prime}$ affect the received symbol on $l$ with adopting a rectangular
window in the delay domain, i.e.,
$V\left[{n,m}\right]=V\left[{n,m^{\prime}}\right]=U\left[{n,m}\right]=U\left[{n,m^{\prime}}\right]$,
$\forall n,m,m^{\prime}$, and
$E\left\\{{{{\left|{x\left[{k^{\prime},l^{\prime}}\right]}\right|}^{2}}}\right\\}=1$.
Assuming independent channel coefficients, i.e.,
$E\left\\{{h_{i}}{h^{*}_{j}}\right\\}=0$, $\forall i\neq j$, (III-B) becomes

$E\left\\{{{{\left|{I\left[{k,l}\right]}\right|}^{2}}}\right\\}=\sum\limits_{k^{\prime}\notin\mathcal{K}}\sum\limits_{i=1}^{P}{E\left\\{{{{\left|{{h_{i}}}\right|}^{2}}}\right\\}}{{\left|{w\left({{{\left({k-k^{\prime}}\right)}_{N}}-{k_{{\nu_{i}}}}-{\kappa_{{\nu_{i}}}},0}\right)}\right|}^{2}}.$
(25)

It can be observed that the interference power is determined by the window
response at
${{\left({k-k^{\prime}}\right)}_{N}}-{k_{{\nu_{i}}}}-{\kappa_{{\nu_{i}}}}$.
Thanks to the guard space, the window response
$\left|{w\left({{{\left({k-k^{\prime}}\right)}_{N}}-{k_{{\nu_{i}}}}-{\kappa_{{\nu_{i}}}},0}\right)}\right|$
lies in its sidelobe and becomes almost a constant, as shown in Fig. 2.
Therefore, we assume
$\left|{w\left({{{\left({k-k^{\prime}}\right)}_{N}}-{k_{{\nu_{i}}}}-{\kappa_{{\nu_{i}}}},0}\right)}\right|\approx\mathrm{SL}_{w}$
for ${k^{\prime}\notin\mathcal{K}}$ and ${k_{p}}-{k_{\max}}-\hat{k}\leq
k\leq{k_{p}}+{k_{\max}}+\hat{k}$. Considering a normalized channel power gain,
i.e.,
$\sum\nolimits_{i=1}^{P}{E\left\\{{{{\left|{{h_{i}}}\right|}^{2}}}\right\\}}=1$,
the average interference power can be approximated by

$E\left\\{{{{\left|{I\left[{k,l}\right]}\right|}^{2}}}\right\\}\approx\left(N-4{k_{\max}}-4\hat{k}-1\right)\mathrm{SL}_{w}^{2},$
(26)

where $\mathrm{SL}_{w}$ denotes the sidelobe level of the adopted window. For
instance, as shown in Fig. 2, we have $\mathrm{SL}_{w}\approx\frac{1}{N}$ for
the case of rectangular window. Define the mean squared error (MSE) of the
effective channel estimation in the guard space as (27) at the top of this
page. According to (IV), in high SNR regime, i.e., $N_{0}\to 0$, the MSE of
the effective channel estimation is given by (28) at the top of this page,
which indicates the effective channel estimation error floor. Note that the
analytical result in (28) is applicable to arbitrary window.

Now, we can observe that in the high SNR regime, the error floor level in the
effective channel estimation in (28) depends on the additional guard $\hat{k}$
and the sidelobe level $\mathrm{SL}_{w}$ of the designed window response. It
can be seen that the MSE of the effective channel estimation is a quadratic
function with respect to $\hat{k}$. After some mathematical manipulations, it
can be seen that when $\frac{N-8{k_{\max}-3}}{4}\leq 0$, increasing $\hat{k}$
in the range of
$\left\\{0,\ldots,\lfloor\frac{N-4k_{\max}-1}{4}\rfloor\right\\}$ always
results in a lower error floor level at the expense of more signaling
overhead. On the other hand, when $\frac{N-8{k_{\max}-3}}{4}\geq 1$,
increasing $\hat{k}$ first increases and then decreases the MSE of effective
channel estimation. This is because for large $N$, increasing additional guard
$\hat{k}$ introduces more entries to be estimated within the guard space,
thereby might increasing channel estimation error. Note that further
increasing $\hat{k}$ reduces the IDI caused by the data symbols and thus
reduces the effective channel estimation MSE, but it also consumes more
signaling overhead. More importantly, as shown in (28), a proper design of
window response can achieve a low sidelobe level at the first place, which can
effectively decrease the error floor level with a relatively small $\hat{k}$.
In fact, a window response with a low sidelobe level can enhance the effective
channel sparsity, which can improve the channel estimation performance.
Moreover, as the TX and RX windows have the same impact on the effective
channel in the DD domain in (13), imposing a window at either the transmitter
or the receiver will result in the same channel estimation error floor.

##  V Window Designs for OTFS Channel Estimation

Based on the above discussed properties, we first discuss the ideal window
response, which is not realizable but provides insightful guidelines for
practical window designs. Then, we propose to apply the DC window to enhance
the effective channel sparsity, which will significantly improve the
performance of both channel estimation and data detection.

###  V-A Ideal Window

To facilitate the window design, we consider a separable TF domain window as
follows:

$V\left[{n,m}\right]=V_{\nu}\left[{n}\right]V_{\tau}\left[{m}\right]\;\text{and}\;U\left[{n,m}\right]=U_{\nu}\left[{n}\right]U_{\tau}\left[{m}\right],$
(29)

where $V_{\nu}\left[{n}\right]$ and $U_{\nu}\left[{n}\right]$ denote the RX
and TX windows in the Doppler domain, respectively, and
$V_{\tau}\left[{m}\right]$ and $U_{\tau}\left[{m}\right]$ denote the RX and TX
windows in the delay domain, respectively. As a result, the window response in
the DD domain in (III-A) can be decomposed as

$w(k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}},l-{l_{{\tau_{i}}}})=\mathcal{G}_{N}\left({k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}}}\right)\mathcal{F}_{M}\left({l-{l_{{\tau_{i}}}}}\right),$
(30)

where

$\displaystyle\mathcal{G}_{N}\left({k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}}}\right)$
$\displaystyle=\frac{1}{N}\sum\nolimits_{n=0}^{N-1}V_{\nu}\left[{n}\right]U_{\nu}\left[{n}\right]{e^{-j2\pi
n\frac{\left({k-k_{\nu_{i}}-\kappa_{\nu_{i}}}\right)}{N}}}$
$\displaystyle\text{and}\;\mathcal{F}_{M}\left({l-{l_{{\tau_{i}}}}}\right)$
$\displaystyle=\frac{1}{M}\sum\nolimits_{m=0}^{M-1}V_{\tau}\left[{m}\right]U_{\tau}\left[{m}\right]{e^{j2\pi
m\frac{\left(l-l_{\tau_{i}}\right)}{M}}}.$ (31)

Combining (13) and (30), the effective channel in the DD domain can be
rewritten as

${h_{w}}\left[{k,l}\right]=\sum\nolimits_{i=1}^{P}{h_{i}}\mathcal{G}_{N}\left({k-{k_{{\nu_{i}}}}-\kappa_{\nu_{i}}}\right)\mathcal{F}_{M}\left({l-{l_{{\tau_{i}}}}}\right){e^{-j2\pi\frac{\left(k_{\nu_{i}}+\kappa_{\nu_{i}}\right)l_{\tau_{i}}}{NM}}}.$
(32)

Since the delay resolution is usually sufficient and there is only negligible
channel spread in the delay domain [11], the optimal window in the delay
domain should be maintained as the rectangular window, i.e.,
$\mathcal{F}^{\mathrm{Ideal}}_{M}\left({l}\right)=\mathcal{F}^{\mathrm{Rect}}_{M}\left({l}\right)$.
Besides, with the existence of the fractional Doppler, as
$-\frac{1}{2}<\kappa_{\nu_{i}}<\frac{1}{2}$, the ideal window in the Doppler
domain is given by:

${{\cal
G}^{\mathrm{Ideal}}_{N}}\left(k\right)=\left\\{{\begin{array}[]{*{20}{c}}{1,}&{-0.5\leq
k\leq 0.5,}\\\\[-1.42262pt] {0,}&{{\rm{otherwise.}}}\end{array}}\right.$ (33)

In Fig. 3, we illustrate that the ideal window can tolerate the fractional
Doppler shift without sacrificing any channel gain and causing any channel
spread. However, to implement such an ideal window response, an infinite
length of window in the time domain is needed, i.e., $N\to\infty$. Recall that
the fractional Doppler is caused by the finite $N$. Therefore, the ideal
window in (33) is not realizable in practice.

###  V-B Dolph-Chebyshev Window

In what follows, we propose to apply the Dolph-Chebyshev (DC) window at the
transmitter or the receiver to improve the channel sparsity when channel state
information (CSI) is not available. In fact, it has been proved that the DC
window is effective [23] in the sense that: 1) given the specified sidelobe
level, the width of the mainlobe in the window response is the narrowest; or
2) given the fixed mainlobe width, the sidelobe level is minimized. Note that
the effective channel only has a considerably large entry when it is located
in the mainlobe of the window response function in (V-A). Therefore, given any
channel sparsity requirement, the channel spread to other Doppler indices is
reduced to the largest degree by using a DC window.

Particularly, if the required mainlobe width of the window response in the
Doppler domain is $k_{\mathrm{main}}>1$, the number of non-negligible
effective channel spreading of each path in the Doppler domain is no more than
$k_{\mathrm{main}}$, i.e., the effective channel sparsity is improved. In this
case, the lowest sidelobe level achieved by the DC window is [24]

$\mathrm{SL}_{w}[\mathrm{dB}]=-20\log_{10}{{\cosh\left({\frac{N}{2}{{\cosh}^{-1}}\left(\frac{3-\cos\left(\frac{k_{\mathrm{main}}}{2}\right)}{1+\cos\left(\frac{k_{\mathrm{main}}}{2}\right)}\right)}\right)}}.$
(34)

In this paper, to reveal the insights of employing TX/RX windows, we only
adopt the DC window at the transmitter or the receiver side with the other
side adopting a rectangular window. The TX window $U_{\nu}\left[{n}\right]$ or
the RX window $V_{\nu}\left[{n}\right]$ can be obtained by (16) in [25]
according to the selected $\mathrm{SL}_{w}$. In Fig. 3, for the same setting
with Fig. 2, we employ a DC window at the transmitter side with
$\mathrm{SL}_{w}[\mathrm{dB}]=-40$ dB and $k_{\mathrm{main}}\approx 3$. We can
observe that the resulting effective channel only has approximately $3$
entries with considerable gains and the channel spreading to all the other
Doppler indices has been significantly suppressed due to the $40$ dB
attenuation on the sidelobe introduced by the designed DC window. By putting
$\mathrm{SL}_{w}[\mathrm{dB}]=-40$ dB into (28), we can find that the proposed
DC window can effectively suppress the MSE of effective channel estimation,
compared with the rectangular window.

Figure 3: The effective channel in the Doppler domain with different types of
windows and fractional Dopplers.

##  VI Numerical Results

In this section, we verify the accuracy of the derived analytical results and
the effectiveness of the proposed designs via simulations. For each OTFS
frame, we set $N=20$, $M=30$, carrier frequency $f_{c}=3$ GHz, and the
subcarrier spacing is $\Delta f=5$ kHz. Without loss of generality, we set the
maximum delay index as $l_{\mathrm{max}}=4$ and the maximum Doppler index as
$k_{\mathrm{max}}=3$, corresponding to the relative speed between the
transceiver as $270$ km/h. The number of paths in the DD domain is $P=5$ and
the additional guard space is $\hat{k}=[0,1]$. For each channel realization,
we randomly select the delay and Doppler indices such that we have
$-k_{\mathrm{max}}\leq{k_{{\nu_{i}}}}\leq k_{\mathrm{max}}$ and $0\leq
l_{\tau_{i}}\leq l_{\mathrm{max}}$. The channel coefficients $h_{i}$ are
generated according to the distribution
$h_{i}\sim\mathcal{CN}(0,q^{l_{\tau_{i}}})$, where $q^{l_{\tau_{i}}}$ follows
a normalized exponential power delay profile
$q^{l_{\tau_{i}}}=\frac{\exp(-0.1l_{\tau_{i}})}{\sum_{i}\exp(-0.1l_{\tau_{i}})}$.
The system SNR is defined as $\mathrm{SNR}=\frac{1}{N_{0}}$ and the pilot
power is $\left|x_{p}\right|^{2}=[10,30]$ dBw[21]. The DC window is designed
with $\mathrm{SL}_{w}[\mathrm{dB}]=-40$ dB such that $k_{\mathrm{main}}\approx
3$. All simulation results are averaged over more than $10^{4}$ OTFS frames.

Figure 4: The MSE of effective channel estimation with employing a rectangular
window at the transceiver.

Fig. 4 shows the MSE of effective channel estimation performance when
employing rectangular window at the transceiver. We can observe that the
effective channel estimation suffers from an error floor in all the considered
cases. This is due to the interference spread from data symbols to the guard
space caused by the existence of the fractional Doppler. Moreover, our derived
error floor level in (28) matches closely with the simulation results in the
high SNR regime. Note that a better effective channel estimation performance
can be achieved with a higher pilot power. Besides, as expected, the more
additional guard space $\hat{k}$ inserted, the lower the MSE of channel
estimation will be, at the expense of a higher amount of overhead.

Fig. 5 illustrates the MSE of effective channel estimation when employing the
designed DC window or Sine window at the transmitter and rectangular window at
the receiver. Note that the sine window is generated via
$U_{\nu}\left[{n}\right]=\sin\left(\frac{\pi n}{N-1}\right)$. We can observe
that the derived error floor in (28) is also consistent with the simulation
results in the high SNR regime. Comparing Fig. 4 and Fig. 5, it can be seen
that the employing the designed DC window at the transmitter is able to
achieve a significantly lower MSE in the effective channel estimation than
that of the rectangular and Sine windows. This demonstrates the effectiveness
of the employed DC window in enhancing the effective channel sparsity and
improving the effective channel estimation performance. We also evaluate the
MSE with employing the designed DC window at the receiver and a rectangular
window at the transmitter. The results are identical to those in Fig. 5. This
verifies that employing DC window at either the transmitter or the receiver
can achieve the same error floor for the effective channel estimation in the
high SNR regime, as predicted by our analysis.

Figure 5: The MSE of effective channel estimation with adopting a DC window at
the transmitter and a rectangular window at the receiver.

##  VII Conclusions

In this paper, we investigated the impacts of transmitter and receiver windows
and proposed a window design for OTFS channel estimation. We analyzed and
revealed the insights of the impacts of windowing on the effective channel,
channel sparsity, and the corresponding estimation performance. In particular,
we showed that the existence of fractional Doppler leads to the potential
effective channel spread, which causes an error floor in the effective channel
estimation. We found that adopting a window at the transmitter or the receiver
can obtain an identical performance in the effective channel estimation.
Besides, we proposed to apply a DC window to enhance the channel sparsity,
which improves the performance in channel estimation. We verified the accuracy
of the obtained analytical results and insights and demonstrated the
substantial performance gain of the proposed window designs.

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