by $2/N_{\rm BS}$. According to the angular resolution, we further define the
transmit angular index by
${a_{i,p}}\buildrel\Delta\over{=}{\left[{{N_{{\rm{BS}}}}-\frac{{\sin\left({{\varphi_{i,p}}}\right){N_{{\rm{BS}}}}}}{2}}\right]_{{N_{{\rm{BS}}}}}}+1$
and the receive angular index by
${{\tilde{a}}_{i,p}}\buildrel\Delta\over{=}{\left[{{N_{{\rm{BS}}}}+\frac{{\sin\left({{\varphi_{i,p}}}\right){N_{{\rm{BS}}}}}}{2}}\right]_{{N_{{\rm{BS}}}}}}+1$.
Based on these, we have the following corollary.

Corollary 7-1 (_Asymptotical Orthogonality_): With a sufficiently large number
of antennas at the BS, the angular index ${a_{i,p}}$ is of an integer value.
In this case, we have ${{h}}_{i,p}^{\rm{A}}\left[l\right]=0$, for
$l\neq{{a_{i,p}}}$, while
${{h}}_{i,p}^{\rm{A}}\left[l\right]=\sqrt{\alpha_{l}}$, for $l={{a_{i,p}}}$.
Furthermore, we have ${{\tilde{H}}}_{i,p}^{\rm{A}}\left[k,l\right]=0$, for
$k\neq{\tilde{a}}_{i,p},l\neq{{a_{i,p}}}$, while
${{\tilde{H}}}_{i,p}^{\rm{A}}\left[k,l\right]=\sqrt{\alpha_{l}}$, for
$k={{\tilde{a}}_{i,p}},l={a_{i,p}}$.

_Proof_ : The proof is given in the Appendix of this chapter.

Figure 7.4: Absolute values of ${\bf{\tilde{H}}}_{i,p}^{\rm{A}}$ without
spreading and de-spreading. Figure 7.5: Absolute values of
${\bf{\tilde{H}}}_{i,p}^{\rm{A}}$ with spreading and de-spreading.

As indicated from Lemma 7-1, we notice that the interference for both the
communication and radar channels due to the multipath and multiuser
transmissions can be approximately eliminated by spatial spreading and de-
spreading, given a sufficient number of antennas. For a better understanding,
we provide an example of the equivalent angular domain radar channel matrix
with $\varphi_{i,p}=\pi/4$ and $N_{\rm BS}=128$ in Fig. 7.4 and Fig. 7.5. As
indicated by the figure, the channel matrix with spatial spreading and de-
spreading is much more sparse, i.e., only the grids around the $84$-th
transmitted antenna and the $46$-th received antenna are of values, which is
consistent with our analysis. In what follows, we assume that $N_{\rm BS}$ is
sufficiently large such that the angular indices are of integer values777Given
the accuracy of AoD and AoA estimates for practical systems, this assumption
is valid for practical BS setups of ISAC transmissions [59, 61]. .
Particularly, notice that when $N_{\rm BS}$ is sufficiently large, the angular
resolution is sufficiently high. Hence, we further assume that all the paths
can be fully separated by its angular domain features, i.e.,
${{a_{i,p}}\neq{a_{i^{\prime},p^{\prime}}}}$, for $i\neq i^{\prime}$ or $p\neq
p^{\prime}$. Consequently, both the communication and radar models in (7.15)
and (7.22) can be further simplified, yielding

$\displaystyle{{\bf{y}}_{i}}=\sum\limits_{p=1}^{P}{\sqrt{\alpha_{a_{i,p}}}}{\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){\bf{H}}_{i,p}^{{\rm{TD}}}{{\bf{W}}_{{a_{i,p}}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right){\bf{x}}}+{{\bm{\eta}}_{i}},$
(7.27)

and

$\displaystyle{\bf{\tilde{z}}}={\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}{\bf{z}}+{\bm{\tilde{\eta}}},$
(7.28)

respectively, where ${\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}$ is the TDA
domain effective radar sensing matrix. In particular, it can be shown that
${\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}$ is a block matrix of size
$N_{\rm BS}MN\times N_{\rm BS}MN$, whose $({\tilde{a}}_{i,p},a_{i,p})$-th sub-
block is given by
${\sqrt{\alpha_{a_{i,p}}}}{\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}$ and the rest
sub-blocks are given by ${{\bf{0}}_{MN}}$.

Remark 7-1: It is of importance to discuss the connections and differences
between the spatial spreading/de-spreading and conventional beamforming. In
conventional beamforming, the beams are formed delicately according to the
exact AoDs or AoAs. In contrast, the proposed spatial spreading/de-spreading
can be viewed as a special type of beamforming, where the beams are formed
according to a pre-defined angular grid according to the DFT/inverse DFT
matrices, as implied by (7.23) and (7.24). Thanks to the perfect orthogonality
among the rows/columns of the the DFT/inverse DFT matrices, there is no inter-
beam interference when applying spatial spreading/de-spreading, which is
different from the conventional beamforming. However, there could be a
potential beam alignment issue for spatial spreading/de-spreading, when the
AoDs/AoAs associated to the paths are not perfectly aligned with the pre-
defined angular grid. In this case, there may be a power reduction due to the
misalignment. Note that the misalignment penalty could be mitigated by
introducing a phase offset before the spatial spreading/de-spreading according
to the _a priori_ knowledge of AoDs and AoAs, i.e., rotate the whole pre-
defined angular grid with respect to a certain angle, such that the beams are
more likely to be aligned with the AoDs or AoAs associated to the paths. Some
recently proposed methods based on machine learning could also be useful for
this problem [120, 123].

Remark 7-2: As indicated by the above discussion, the proposed spatial
spreading and de-spreading leads to the discretization of angular features,
which simplifies the input-output relationship for both communication and
radar channels. In particular, the angular domain discretization enables
direct interactions between the transmitted signal on a specific antenna and
the channel distortion associated to a specific path. Therefore, it is
suitable for ISAC transmissions888It is worth mentioning that the proposed
framework can naturally be extended to the general case of multi-user MIMO by
introducing the _indicator matrices_ that assign the data stream for each UE
to the corresponding antenna before spatial spreading. Some related details
could be found in [119]..

###  7.2 Radar Sensing Designs based on SS-OTFS Modulation

In this section, we are focusing on the radar sensing designs based on the SS-
OTFS modulation. According to the angular discretization enabled by spatial
spreading and de-spreading, we aim to design practical beam tracking and AoA
estimation algorithms based on the previous AoA estimates and radar reflection
coefficients. It should be noted that, from the radar point of view, not all
the targets are of interest, because they may introduce clutter interference
that can potentially undermine the sensing performance. However, some of the
clutter might come from scatterers that can significantly contribute to the
total received power for communication transmissions. Therefore, it is still
necessary to estimate the parameters corresponding all the scatterers for ISAC
systems [59].

####  7.2.1 Beam Tracking

Let us focus on the radar channel model given in (7.28). In particular, with a
sufficiently large number of antennas, the effective radar sensing matrix
${\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}$ is a sparse block matrix as
discussed in Section 7.1. Thanks to the sparsity enabled by the SS-OTFS
transmission, the angular domain is sufficiently discretized. Therefore, the
signals can be transmitted towards the desired directions by simply assigning
power to the corresponding antennas without sophisticated precoding designs.

For the case of beam tracking, the BS knows the previous AoA estimates
associated to the different paths of the UEs from the previous time instant.
Therefore, a common approach for beam tracking is to send relatively wider
beams towards the AoAs estimated from the previous time instant. To achieve
this, we only need to allocate power on the corresponding antennas. Denote by
${\theta_{{\rm{range}}}}$ the desired beam width. According to the angular
resolution, we need to allocate powers on
${N_{{\rm{range}}}}\buildrel\Delta\over{=}\frac{{{\theta_{{\rm{range}}}}}}{2}{N_{{\rm{BS}}}}$
antennas. In specific, let ${{\mathbb{U}}_{i,p}}$ denote the antenna set
corresponding to those antennas for the $p$-th paths of the $i$-th UE and it
is given by

$\displaystyle{{\mathbb{U}}_{i,p}}\\!\buildrel\Delta\over{=}$
$\displaystyle\left\\{{{{\left[{{N_{{a_{i,p}}}}\\!\\!-\\!1\\!-\\!\frac{{{N_{{\rm{range}}}}}}{2}}\right]}_{{N_{{\rm{BS}}}}}}\\!\\!\\!\\!\\!+\\!1,{\left[{{N_{{a_{i,p}}}}\\!\\!\\!-\\!\frac{{{N_{{\rm{range}}}}}}{2}}\right]_{{N_{{\rm{BS}}}}}}\\!\\!\\!\\!\\!+\\!1,}\right....,\left.{{{\left[{{N_{{a_{i,p}}}}\\!\\!-\\!1+\frac{{{N_{{\rm{range}}}}}}{2}}\right]}_{{N_{{\rm{BS}}}}}}\\!\\!\\!\\!\\!+\\!1}\right\\},$
(7.29)

where we assume that ${N_{{\rm{range}}}}$ is an even number. Given the high
angular resolution, we further assume that the AoAs are sufficiently
separated, such that
${{\mathbb{U}}_{i,p}}\cap{{\mathbb{U}}_{i^{\prime},p^{\prime}}}=\varnothing,\forall
i\neq i^{\prime},p\neq p^{\prime}$. To ensure the reliability of radar sensing
and communication, we propose to send the precoded signals with equal powers
towards all the antennas in ${{\mathbb{U}}_{i,p}}$, such that the ISAC signal
will be guaranteed to be reflected/received by the corresponding UE.

####  7.2.2 AoA Estimation

Now let us focus on the AoA estimation. According to the radar model in
(7.28), we notice that the TDA domain equivalent radar sensing matrix is a
sparse block matrix, where only the sub-blocks related to the angular indices
${\tilde{a}}_{i,p}$ are of non-zero values. This observation motivates us to
consider a correlation-based AoA estimation method. According to (7.28), the
covariance matrix of the TDA domain radar received symbol vector
${\bf{\tilde{z}}}$ can be calculated by
${{\bf{R}}_{{\bf{\tilde{z}}}}}\buildrel\Delta\over{=}{\mathbb{E}}\left[{{\bf{\tilde{z}}}{{{\bf{\tilde{z}}}}^{\rm{H}}}}\right]={\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}{\left({{\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}}\right)^{\rm{H}}}+{{\tilde{N}}_{0}}{{\bf{I}}_{{N_{{\rm{BS}}}}MN}}$.
Furthermore, since ${\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}$ is a block
matrix, ${{\bf{R}}_{{\bf{\tilde{z}}}}}$ is also a block matrix with $N_{\rm
BS}$ row groups and $N_{\rm BS}$ column groups. In specific, it can be shown
that the $n_{\rm r}$-th diagonal sub-block of ${{\bf{R}}_{{\bf{\tilde{z}}}}}$
is given by

$\displaystyle{{\bf{R}}_{{\bf{\tilde{z}}}}}\left[{{n_{\rm{r}}},{n_{\rm{r}}}}\right]=\left\\{\begin{array}[]{l}{\alpha_{{a_{i,p}}}}\\!{\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}{{\bf{W}}_{{a_{i,p}}}}\\!\\!{\bf{W}}_{{a_{i,p}}}^{\rm{H}}{\left({{\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}}\right)^{\rm{H}}}\\!\\!+\\!{{\tilde{N}}_{0}}{{\bf{I}}_{MN}},\quad\;\;\\!\\!{n_{\rm{r}}}={{\tilde{a}}_{i,p}}.\\\
{{\tilde{N}}_{0}}{{\bf{I}}_{MN}},\\!\\!\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{n_{\rm{r}}}\neq{{\tilde{a}}_{i,p}},\end{array}\right.$
(7.32)

where we assume that ${\bf{x}}{{\bf{x}}^{\rm{H}}}={{\bf{I}}_{MN}}$.
Furthermore, since
${{\bf{W}}_{{{a}_{i,p}}}}{\bf{W}}_{{{a}_{i,p}}}^{\rm{H}}={{\bf{I}}_{MN}}$
according to our definition, (7.32) can be further simplified by

$\displaystyle{{\bf{R}}_{{\bf{\tilde{z}}}}}\left[{{n_{\rm{r}}},{n_{\rm{r}}}}\right]=\left\\{\begin{array}[]{l}\left({{\alpha_{{a_{i,p}}}}{{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}+{{\tilde{N}}_{0}}}\right){{\bf{I}}_{MN}},\\!\quad{n_{\rm{r}}}={{\tilde{a}}_{i,p}},\\\
{{\tilde{N}}_{0}}{{\bf{I}}_{MN}},\quad\quad\quad\quad\quad\quad\quad\quad\;\;{n_{\rm{r}}}\neq{{\tilde{a}}_{i,p}}.\end{array}\right.$
(7.35)

As implied by (7.35), it can be shown that the diagonal sub-blocks of
${{\bf{R}}_{{\bf{\tilde{z}}}}}$ are also of diagonal structure. Furthermore,
by noticing that the value of
${\alpha_{{a_{i,p}}}}{{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}$ can be much
larger than ${{\tilde{N}}_{0}}$ in the high SNR regime, we propose to
calculate the traces of the diagonal sub-blocks of
${{\bf{R}}_{{\bf{\tilde{z}}}}}$ in order to find ${{\tilde{a}}_{i,p}}$. In
specific, the AoA estimation can be carried out by calculating the traces of
$N_{\rm BS}$ diagonal sub-blocks first and then finding the corresponding
indices associated to the $KP$ largest of the trace values and finally
calculating the values of the AoDs based on the indices and the angular
resolution as discussed in Section 7.1. An important issue at this point is
how to associate the estimated AoAs with the paths of different UEs.
Unfortunately, we do not have enough space to discuss this issue. We refer the
interested readers to [59] for more information.

####  7.2.3 Power Allocation for Radar Sensing

As indicated by (7.35), the power allocation will affect estimation
performance. In particular, the value of
${{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}$ can be largely different for
different paths. Therefore, it is important to assign different powers to the
related antennas according to the value of
${{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}$. Let us assume that the value of
${{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}$ of the current time instant is
accurately derived based on the corresponding estimates from the previous time
instant. As the above AoA estimation is based on calculation of the
correlation matrix, we aim to maximize the minimum effective radar SNR among
all the paths, such that the radar can provide a relatively accurate sensing
performance for each path. Notice that ${\tilde{N}}_{0}$ is assumed to be
equal to each UE. Therefore, to maximize the minimum radar SNR is equivalent
to maximize the received power
${{\alpha_{{a_{i,p}}}}{{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}}$ associated
to each path. Therefore, our power allocation is designed as follows

$\displaystyle\mathop{\max}\limits_{{\alpha_{{a_{1,1}}}},...,{\alpha_{{a_{K,P}}}}}$
$\displaystyle\quad{\left({{\alpha_{{a_{i,p}}}}{{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}}\right)_{\min}}$
(7.36) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}$
$\displaystyle\quad\left({\sum\limits_{i=1}^{K}{\sum\limits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}}}\right)\left({{N_{{\rm{range}}}}+1}\right)={\alpha_{{\rm{total}}}},$
$\displaystyle\quad\quad{{a_{i,p}}}\geq 0.$ (7.37)

It can be shown that the maximum of (7.36) is achieved when all the received
power are of the same value, i.e.,
${\alpha_{{a_{1,1}}}}{\left|{{{\tilde{h}}_{1,1}}}\right|^{2}}={\alpha_{{a_{1,2}}}}{\left|{{{\tilde{h}}_{1,2}}}\right|^{2}}=...={\alpha_{{a_{K,P}}}}{\left|{{{\tilde{h}}_{K,P}}}\right|^{2}}$.
Therefore, according to (7.37), the power allocation should satisfy

$\displaystyle{\alpha_{{a_{i,p}}}}={{\left({\frac{{{\alpha_{{\rm{total}}}}}}{{{N_{{\rm{range}}}}+1}}\frac{1}{{{{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}}}}\right)}\mathord{\left/{\vphantom{{\left({\frac{{{\alpha_{{\rm{total}}}}}}{{{N_{{\rm{range}}}}+1}}\frac{1}{{{{\left|{{{\tilde{h}}_{i,p}}}\right|}^{2}}}}}\right)}{\left({\sum\limits_{j=1}^{K}{\sum\limits_{p^{\prime}=1}^{P}{\frac{1}{{{{\left|{{{\tilde{h}}_{j,p^{\prime}}}}\right|}^{2}}}}}}}\right)}}}\right.\kern-1.2pt}{\left({\sum\limits_{j=1}^{K}{\sum\limits_{p^{\prime}=1}^{P}{\frac{1}{{{{\left|{{{\tilde{h}}_{j,p^{\prime}}}}\right|}^{2}}}}}}}\right)}}.$
(7.38)

Remark 7-3: Intuitively speaking, the proposed power allocation aims to assign
a larger power towards to the direction where the power of the radar echo is
small. This is contradict to the conventional water-filling principle from the
communication point of view, where more power should be assigned towards the
direction with a larger channel gain in order to improve the achievable rate.
However, since the communication fading coefficients cannot be directly
obtained based on the radar sensing estimates, it may be difficult for the BS
to design a suitable power allocation that can provide a good trade-off for
both radar sensing and communication performances. Consequently, the best we
can do is to adapt the proposed power allocation with respect to the
statistical distribution of the communication fading coefficients. The related
issues will be discussed in detail in Section 7.3.3.

###  7.3 Sensing-Assisted Communication Design

In this section, we will develop communication schemes based on the estimated
parameters from radar sensing. We notice that there is a deterministic
relationship between the round-trip delay and Doppler from radar sensing and
the delay and Doppler for communication. Furthermore, given the AoA estimates
from radar, we can also determine the corresponding transmit angular indices
${a_{i,p}}$. Unfortunately, there is no direct relationship between the
communication fading coefficients and radar reflection coefficients.
Therefore, our design criterion is to minimize the PEP for communication with
respect to the _a priori_ AoA, delay and Doppler estimates that are obtained
from radar sensing. According to the quasi-static property of the DD domain
channel response [14, 49, 7], we assume that the minor changes of the related
parameters from the previous time instant are well-compensated [59], including
the delay shifts, and Doppler shifts associated to the corresponding paths. On
the other hand, we note that the AoAs at the current time instant may be
different from the estimates from the previous time instant, due to the high
angular resolution and mobility. However, as will be explained in detail
later, our proposed design can be easily combined with the beam tracking
scheme introduced in the previous section. Therefore, we first assume that the
AoA estimates are accurate in this section for the ease of derivation.

####  7.3.1 Pair-wise Error Probability Analysis

Recalling (7.27), we define the effective DD domain communication channel
matrix for the $i$-th UE by

$\displaystyle{\bf{H}}_{i}^{{\rm{DD}}}\buildrel\Delta\over{=}$
$\displaystyle\sum\limits_{p=1}^{P}{{\sqrt{\alpha_{a_{i,p}}}}{h_{i,p}}}\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){{\bf{\Pi}}^{{l_{i,p}}}}{{\bf{\Delta}}^{{k_{i,p}}+{\kappa_{i,p}}}}{{\bf{W}}_{{a_{i,p}}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right).$
(7.39)

Then, (7.27) can be rewritten by

$\displaystyle{{\bf{y}}_{i}}={\bf{H}}_{i}^{{\rm{DD}}}{\bf{x}}+{{\bm{\eta}}_{i}}.$
(7.40)

In what follows, we will study the PEP performance with the ML detection999We
note that the channel estimation and channel equalization for SS-OTFS can be
carried out in a similar way to the conventional OTFS. Some related
discussions can be found in [14, 52, 47].  in order to facilitate our
precoding design. For the ease of presentation, let us define the following
vectors and matrices for related parameters. We define the effective fading
coefficients by
${\bf{h}}_{i}^{{\rm{eff}}}\buildrel\Delta\over{=}{\left[{{h_{i,1}},{h_{i,2}},...,{h_{i,P}}}\right]^{\rm{T}}}$,
the delay shifts by
${\bm{\omega}}_{i}^{\tau}\buildrel\Delta\over{=}{\left[{{l_{i,1}},{l_{i,2}},...,{l_{i,P}}}\right]^{\rm{T}}}$,
the Doppler shifts by
${\bm{\omega}}_{i}^{\nu}\buildrel\Delta\over{=}{\left[{{k_{i,1}}+{\kappa_{i,1}},{k_{i,2}}+{\kappa_{i,2}},...,{k_{i,P}}+{\kappa_{i,P}}}\right]^{\rm{T}}}$,
the precoding matrices by
${\bf{W}}_{i}^{{\rm{com}}}\buildrel\Delta\over{=}{\left[{{\bf{W}}_{{a_{i,1}}}^{\rm{H}},{\bf{W}}_{{a_{i,2}}}^{\rm{H}},...,{\bf{W}}_{{a_{i,P}}}^{\rm{H}}}\right]^{\rm{H}}}$,
and the allocated power for the $i$-th UE by
${\bm{\alpha}}_{i}^{{\rm{com}}}\buildrel\Delta\over{=}{\rm{diag}}\left\\{{\sqrt{{\alpha_{{a_{i,1}}}}},\sqrt{{\alpha_{{a_{i,2}}}}},...,\sqrt{{\alpha_{{a_{i,P}}}}}}\right\\}$,
respectively. According to [9] and [48], (7.40) can be rewritten by

$\displaystyle{{\bf{y}}_{i}}={\bf{\Phi}}_{i}^{\omega_{i}^{\tau},\omega_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{x}}\right){\bm{\alpha}}_{i}^{{\rm{com}}}{\bf{h}}_{i}^{{\rm{eff}}}+{{\bm{\eta}}_{i}},$
(7.41)

where
${\bf{\Phi}}_{i}^{\omega_{i}^{\tau},\omega_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{x}}\right)$
is referred to as the _equivalent codeword matrix_ and it is a concatenated
matrix of size $MN\times P$, constructed by the column vector
${{\bf{\Xi}}_{i,p}}{\bf{x}}$ , i.e.,

${\bf{\Phi}}_{i}^{\omega_{i}^{\tau},\omega_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{x}}\right)=\left[{{{\bf{\Xi}}_{i,1}}{\bf{x}}\quad{{\bf{\Xi}}_{i,2}}{\bf{x}}\quad\cdots\quad{{\bf{\Xi}}_{i,P}}{\bf{x}}}\right],$
(7.42)

and ${{\bf{\Xi}}_{i,p}}$ is given by

${{\bf{\Xi}}_{i,p}}\buildrel\Delta\over{=}\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){{\bm{\Pi}}^{{l_{i,p}}}}{{\bm{\Delta}}^{{k_{i,p}}+{\kappa_{i,p}}}}{{\bf{W}}_{{a_{i,p}}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right).$
(7.43)

To analyze the PEP performance for communication, we start from the study of
the conditional PEP based on (7.41). In particular, let us define the
_conditional Euclidean distance_
${d_{{\bf{h}}_{i}^{{\rm{eff}}},{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}},{{\bm{\alpha}}_{i}^{{\rm{com}}}},{\bf{W}}_{i}^{{\rm{com}}}}^{2}\left({{\bf{x}},{\bf{x^{\prime}}}}\right)}$
between a pair of codewords ${\bf{x}}$ and ${\bf{x^{\prime}}}$
(${\bf{x}}\neq{\bf{x^{\prime}}}$) by

$\displaystyle
d_{{\bf{h}}_{i}^{{\rm{eff}}},{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bm{\alpha}}_{i}^{{\rm{com}}},{\bf{W}}_{i}^{{\rm{com}}}}^{2}\left({{\bf{x}},{\bf{x^{\prime}}}}\right)=$
$\displaystyle
d_{{\bf{h}}_{i}^{{\rm{eff}}},{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bm{\alpha}}_{i}^{{\rm{com}}},{\bf{W}}_{i}^{{\rm{com}}}}^{2}\left({\bf{e}}\right)$
$\displaystyle\buildrel\Delta\over{=}$
$\displaystyle{\left\|{{\bf{\Phi}}_{i}^{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{e}}\right){\bm{\alpha}}_{i}^{{\rm{com}}}{{\bf{h}}_{i}^{{\rm{eff}}}}}\right\|^{2}}$
$\displaystyle=$
$\displaystyle{\left({{\bf{h}}_{i}^{{\rm{eff}}}}\right)^{\rm{H}}}{\left({{\bm{\alpha}}_{i}^{{\rm{com}}}}\right)^{\rm
H}}{\bf{\Omega}}_{i}^{{}_{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}}\left({\bf{e}}\right){{\bm{\alpha}}_{i}^{{\rm{com}}}}{\bf{h}}_{i}^{{\rm{eff}}}$
$\displaystyle=$
$\displaystyle{\left({{\bf{h}}_{i}^{{\rm{eff}}}}\right)^{\rm{H}}}{\bf{\tilde{\Omega}}}_{i}^{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{\nu},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{e}}\right){\bf{h}}_{i}^{{\rm{eff}}},$
(7.44)

where ${\bf{e}}={\bf{x}}-{\bf{x^{\prime}}}$ is the corresponding codeword
difference (error) sequence. In particular, we refer to
${\bf{\Omega}}_{i}^{{}_{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}}\left({\bf{e}}\right)\buildrel\Delta\over{=}{\left({{\bf{\Phi}}_{i}^{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{e}}\right)}\right)^{\rm{H}}}{\bf{\Phi}}_{i}^{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{e}}\right)$
in (7.44) as the _codeword difference matrix_ , while to
${\bf{\tilde{\Omega}}}_{i}^{{}_{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}}\left({\bf{e}}\right)\buildrel\Delta\over{=}{\left({{\bm{\alpha}}_{i}^{{\rm{com}}}}\right)^{\rm
H}}{\bf{\Omega}}_{i}^{{}_{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}}\left({\bf{e}}\right){{\bm{\alpha}}_{i}^{{\rm{com}}}}$
as the _weighted codeword difference matrix_. For notational simplicity, we
henceforth drop the superscript of
${\bf{\Omega}}_{i}^{{}_{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}}\left({\bf{e}}\right)$
and
${\bf{\tilde{\Omega}}}_{i}^{{}_{{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}}\left({\bf{e}}\right)$,
and we have

$\displaystyle{\bf{\Omega}}_{i}\left({\bf{e}}\right)=\left[{\begin{array}[]{*{20}{c}}{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,1}^{\rm{H}}{{\bf{\Xi}}_{i,1}}{\bf{e}}}&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,1}^{\rm{H}}{{\bf{\Xi}}_{i,2}}{\bf{e}}}&\cdots&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,1}^{\rm{H}}{{\bf{\Xi}}_{i,P}}{\bf{e}}}\\\
{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,2}^{\rm{H}}{{\bf{\Xi}}_{i,1}}{\bf{e}}}&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,2}^{\rm{H}}{{\bf{\Xi}}_{i,2}}{\bf{e}}}&{}\hfil&\vdots\\\
\vdots&{}\hfil&\ddots&\vdots\\\
{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,P}^{\rm{H}}{{\bf{\Xi}}_{i,P}}{\bf{e}}}&\cdots&\cdots&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,P}^{\rm{H}}{{\bf{\Xi}}_{i,P}}{\bf{e}}}\end{array}}\right].$
(7.49)

According to [48], the conditional PEP is upper-bounded by

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{\bf{h}}_{i}^{{\rm{eff}}}},{{\bm{\omega}}_{i}^{\tau}},{{\bm{\omega}}_{i}^{\nu}}},{\bf{W}}_{i}^{{\rm{com}}},{\bm{\alpha}}_{i}^{{\rm{com}}}\right)\leq$
$\displaystyle\exp\left({-\frac{{{1}}}{{4{N_{0}}}}{d_{{\bf{h}}_{i}^{{\rm{eff}}},{\bm{\omega}}_{i}^{\tau},{\bm{\omega}}_{i}^{{}_{\nu}},{\bm{\alpha}}_{i}^{{\rm{com}}},{\bf{W}}_{i}^{{\rm{com}}}}^{2}\left({{\bf{x}},{\bf{x^{\prime}}}}\right)}}\right)$
$\displaystyle=$
$\displaystyle\exp\left({-\frac{{{1}}}{{4{N_{0}}}}}{\left({{\bf{h}}_{i}^{{\rm{eff}}}}\right)^{\rm{H}}}{\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right){\bf{h}}_{i}^{{\rm{eff}}}\right).$
(7.50)

To further simplify (7.50), let us focus on the structures of both
${\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right)$ and
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$. We observe that both
${\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right)$ and
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ are positive-semidefinite Hermitian
matrices by their definitions. Furthermore, with a proper design of power
allocation, i.e., ${{\bm{\alpha}}_{i}^{{\rm{com}}}}$ is of full-rank, both
${\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right)$ and
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ share the same rank. Based on this
observation, we consider the eigenvalue decomposition to further our
derivation. Let $r_{i}$ denote the rank of both
${\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right)$ and
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$, where $r_{i}\leq P$. Furthermore,
let us denote by
$\left\\{{{{\bf{u}}_{i,1}},{{\bf{u}}_{i,2}},...,{{\bf{u}}_{i,P}}}\right\\}$
the eigenvectors of ${\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right)$ and
$\left\\{{{\lambda_{i}}\left[1\right],{\lambda_{i}}\left[2\right],...,{\lambda_{i}}\left[P\right]}\right\\}$
the corresponding nonnegative real eigenvalues sorted in the descending order,
where ${{\lambda_{i}\left[j\right]}}>0$ for $1\leq j\leq r_{i}$ and
${{\lambda_{i}\left[j\right]}}=0$ for $r_{i}+1\leq j\leq P$. Then, (7.50) can
be further expanded by [98]

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{\bf{h}}_{i}^{{\rm{eff}}}},{{\bm{\omega}}_{i}^{\tau}},{{\bm{\omega}}_{i}^{\nu}}},{\bf{W}}_{i}^{{\rm{com}}},{\bm{\alpha}}_{i}^{{\rm{com}}}\right)\leq\exp\left({-\frac{1}{{4{N_{0}}}}\sum\limits_{j=1}^{{r_{i}}}{{\lambda_{i}}\left[j\right]}{{\left|{\bar{h}_{i}^{{\rm{eff}}}\left[j\right]}\right|}^{2}}}\right),$
(7.51)

where
$\bar{h}_{i}^{{\rm{eff}}}\left[j\right]={{\bf{u}}_{i,j}}{\bf{h}}_{i}^{{\rm{eff}}}$,
for $1\leq j\leq r_{i}$. Note that the exact values of the elements in
${\bf{h}}_{i}^{\rm{eff}}$ are unknown to the BS. Therefore, we need to
consider the distributions of those elements in order to further our
derivation. It can be shown that
$\left\\{{\bar{h}_{i}^{{\rm{eff}}}\left[1\right],\bar{h}_{i}^{{\rm{eff}}}\left[2\right],...,\bar{h}_{i}^{{\rm{eff}}}\left[r_{i}\right]}\right\\}$
are independent complex Gaussian random variables with zero mean and variance
$1/(2P)$ per real dimension. Consequently,
${\small|\bar{h}_{i}^{{\rm{eff}}}\left[j\right]\small|}$ follows the Rayleigh
distribution [98], whose PDF is given by
${f_{{\rm{PDF}}}}\left(x\right)=2Px\exp\left({-P{{x}^{2}}}\right)$. With the
uniform power delay and Doppler profile, we can get rid of the influence of
effective fading coefficients in (7.51) by averaging
${\small|\bar{h}_{i}^{{\rm{eff}}}\left[j\right]\small|}$ term by term,
yielding

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{\bm{\omega}}_{i}^{\tau}},{{\bm{\omega}}_{i}^{\nu}}},{\bf{W}}_{i}^{{\rm{com}}},{\bm{\alpha}}_{i}^{{\rm{com}}}\right)\leq\prod\limits_{j=1}^{{r_{i}}}{\frac{1}{{1+\frac{{{\lambda_{i}[j]}}}{{4{N_{0}}P}}}}}\leq\frac{1}{{\prod\limits_{j=1}^{{r_{i}}}{{\lambda_{i}[j]}}}}{\left({\frac{1}{{4{N_{0}}P}}}\right)^{-{r_{i}}}}.$
(7.52)

As indicated by (7.52), the PEP decreases exponentially with an order of
${r_{i}}$ with the reduction of the noise PSD. In fact, this exponent is the
diversity gain of the transmission [98, 99, 106].

In order to enable reliable transmissions, we aim to minimize the upper-bound
in (7.52) by designing suitable precoding matrices and power allocation. To
facilitate our design, let we first assume that there is a set of precoding
matrices ${\bf{W}}_{i}^{{\rm{com}}}$ and a specific power allocation
${\bm{\alpha}_{i}^{{\rm{com}}}}$ that can minimize the PEP upper-bound in
(7.52) for any given $\bf e$ with respect to all possible delay and Doppler
shifts, i.e., ${\bm{\omega}}_{i}^{\tau}$ and ${\bm{\omega}}_{i}^{\nu}$. Then,
we will develop practical precoding designs and power allocation in the next
subsection, such that this lowest PEP upper-bound is approachable. Notice that
both ${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ and
${\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right)$ are _Gram_ matrices [105] of
size $P\times P$ and thus the maximum value of the ranks of both
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ and
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ is $P$. In particular, when
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ is of full-rank, (7.52) can be
further simplified by

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{\bm{\omega}}_{i}^{\tau}},{{\bm{\omega}}_{i}^{\nu}}},{\bf{W}}_{i}^{{\rm{com}}},{\bm{\alpha}}_{i}^{{\rm{com}}}\right)\leq$
$\displaystyle\frac{1}{{\prod\limits_{j=1}^{{P}}{{\lambda_{i}[j]}}}}{\left({\frac{1}{{4{N_{0}}P}}}\right)^{-{P}}}=\frac{1}{{\det\left({{{{\bf{\tilde{\Omega}}}}_{i}\left({\bf{e}}\right)}}\right)}}{\left({\frac{1}{{4{N_{0}}P}}}\right)^{-P}}$
$\displaystyle=$
$\displaystyle\frac{1}{{\det\left({{{\bf{\Omega}}_{i}\left({\bf{e}}\right)}}\right)}}{\left({\frac{{{{\left({\prod\limits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}}\right)}^{\frac{1}{P}}}}}{{4{N_{0}}P}}}\right)^{-P}}.$
(7.53)

Based on (7.53), we notice that the determinant of
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ is related to the delay shifts
${{\bm{\omega}}_{i}^{\tau}}$, the Doppler shifts ${{\bm{\omega}}_{i}^{\nu}}$,
and the precoding matrices ${\bf{W}}_{i}^{{\rm{com}}}$, but it is independent
from the power allocation ${\bm{\alpha}}_{i}^{{\rm{com}}}$. This fact
indicates that the we can analyze the influence from the precoding matrices
and power allocation on the error performance separately. In particular,
(7.53) indicates that in the case of full-rank, the precoding scheme should be
designed to maximize the determinant of the codeword difference matrix
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$. To maximize the determinant of
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$, let us consider the following
theorem.

Theorem 7-1 (_Upper-bound on the determinant of
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$_): The determinant of the codeword
difference matrix ${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ can be upper-
bounded by

$\det\left({{{\bf{\Omega}}_{i}}\left({\bf{e}}\right)}\right)\leq{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{P}},$
(7.54)

where ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ denotes the Euclidean distance
of the error sequence $\bf e$, i.e.,
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}\buildrel\Delta\over{=}{\bf e}^{\rm
H}{\bf e}$. Furthermore, a sufficient condition for achieving the equality is
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ being a diagonal matrix.

_Proof_ : The proof is given in the Appendix of this chapter.

In fact, Theorem 7-1 indicates that the PEP upper-bound can be minimized if
the received signals from different paths are orthogonal to each other [49,
48]. Notice that Theorem 7-1 provides an upper-bound of the determinant that
is independent from the delay and Doppler shifts. Therefore, in the best case
scenario, i.e., orthogonal paths, by substituting (7.54) into (7.53), we
arrive at

$\displaystyle\Pr\left(\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{\bf{W}}_{i}^{{\rm{com}}},{\bm{\alpha}}_{i}^{{\rm{com}}}\right)\leq\frac{1}{{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{P}}}{\left({\frac{{{{\left({\prod\limits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}}\right)}^{\frac{1}{P}}}}}{{4{N_{0}}P}}}\right)^{-P}}={\left({\frac{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{P}}\right)^{-P}}{\left({\frac{{{{\left({\prod\limits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}}\right)}^{\frac{1}{P}}}}}{{4{N_{0}}}}}\right)^{-P}}.$
(7.55)

According to (7.55), we refer to the term
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}/P$ as the maximum coding gain of the
underlying transmission [49, 48], which indicates how much can the error
performance be possibly improved by varying $\bf e$, for all possible values
of the delay and Doppler shifts. In the next subsection, we will design
suitable precoding matrices based on (7.55).

####  7.3.2 Precoding Design

We notice that the delay and Doppler indices among different paths will affect
the PEP performance. Motivated by this observation, we propose our precoding
design by considering the concepts of virtual delay index and virtual Doppler
index, whose definitions are given as follows.

Definition 7-1 (_Virtual Delay and Doppler Indices_): The virtual delay and
Doppler indices are defined by $0\leq{{{\dot{l}}_{p}}}\leq M-1$ and
$0\leq{{{\dot{k}}_{p}}}\leq N-1$, for $1\leq p\leq P$, where
${{{\dot{l}}_{p}}}$ and ${{{\dot{k}}_{p}}}$ are of integer values, for $1\leq
p,p^{\prime}\leq P$.

Recalling the discussions in previous subsections, we note that our precoding
design is to shape the codeword difference matrix
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$, such that it can be a diagonal
matrix, for any possible ${\bm{\omega}}_{i}^{\tau}$,
${\bm{\omega}}_{i}^{\nu}$, and $\bf e$. By observing the structure of (7.49)
and according to Theorem 7-1, we notice that the aforementioned design
criterion for precoding matrices is satisfied if

$\displaystyle{\bf e}^{\rm
H}{\bf{\Xi}}_{{}_{i,p}}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf e}=0,$ (7.56)

and

$\displaystyle{\bf{W}}_{{a_{i,p}}}^{\rm{H}}{{\bf{W}}_{{a_{i,p}}}}={{\bf{I}}_{MN}},$
(7.57)

for any possible $\bf e$, and any $1\leq p,p^{\prime}\leq P$ and
$p^{\prime}\neq p$. Corresponding to both (7.56) and (7.57), the following
lemma shows an interesting fact of the precoding design problem.

Lemma 7-3 (_Determinant Dilemma_): The precoding matrices cannot satisfy
${\bf{W}}_{{a_{i,p}}}^{\rm{H}}{{\bf{W}}_{{a_{i,p}}}}={{\bf{I}}_{MN}}$ and
${\bf{W}}_{{a_{i,p}}}^{\rm{H}}{{\bf{W}}_{{a_{i,p^{\prime}}}}}={{\bf{0}}_{MN}}$
at the same time, for $1\leq p,p^{\prime}\leq P$ and $p^{\prime}\neq p$.

_Proof_ : The proof is given in the Appendix of this chapter.

As indicated by Lemma 7-3, we note that an explicit design algorithm of the
precoding matrices satisfying the above criteria is not realizable. Therefore,
we consider a relaxation for the precoding design, where
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ is a diagonally-dominant matrix [124]
instead of a strict diagonal matrix. To achieve this, we require that both the
virtual delay and Doppler indices are different for different paths, i.e.,
${{{\dot{l}}_{p}}}\neq{{{\dot{l}}_{p^{\prime}}}}$ and
${{{\dot{k}}_{p}}}\neq{{{\dot{k}}_{p^{\prime}}}}$ for any $1\leq
p,p^{\prime}\leq P$ and $p^{\prime}\neq p$. Therefore, for given _a priori_
AoA, delay and Doppler estimates from radar sensing, the proposed precoding
matrices are of the form

$\displaystyle{{\bf{W}}_{{a_{i,p}}}}\buildrel\Delta\over{=}{{\bf{\Delta}}^{-{{\hat{k}}_{i,p}}-{{\hat{\kappa}}_{i,p}}}}{{\bf{\Pi}}^{-{{\hat{l}}_{i,p}}}}{{\bf{\Pi}}^{{{\dot{l}}_{p}}}}{{\bf{\Delta}}^{{{\dot{k}}_{p}}}},$
(7.58)

where ${{\hat{l}}_{i,p}}$ and ${{{\hat{k}}_{i,p}}+{{\hat{\kappa}}_{i,p}}}$ are
the estimated delay and Doppler indices, while ${{{\dot{l}}_{p}}}$ and
${{{\dot{k}}_{p}}}$ are the _virtual_ delay and Doppler indices with different
values for different paths.

The rationale of the proposed precoding design is to improve the orthogonality
devised by OTFS transmissions based on the nature of delay and Doppler shifts
of the channel. Note that the delay and Doppler shifts are two physical
parameters that determines the distortion characteristics of the resolvable
path for transmission and different resolvable paths cannot share the same
delay and Doppler shifts according to the definition [8, 7]. However, it is
possible that different resolvable paths share the same delay or Doppler shift
and in this case the natural orthogonality among different paths may be
undermined. In the following proposition, we prove that the proposed precoding
scheme can improve the orthogonality in the case where the delay or Doppler
shifts associated to different paths are of the same values.

Proposition 7-1 (_Diagonal Dominance_): Let ${\varepsilon}$ denote the event
where at least two paths share the same delay or Doppler indices. Then, it can
be shown that

$\displaystyle\Pr\left({\left.{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p}}{\bf{e}}}\right|\geq\sum\limits_{\scriptstyle{p^{\prime}}=1\hfill\atop\scriptstyle{p^{\prime}}\neq
p\hfill}^{P}{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}}\right|}}\right|{{\bf{W}}_{{a_{i,p}}}},\varepsilon}\right)\geq\Pr\left({\left.{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p}}{\bf{e}}}\right|\geq\sum\limits_{\scriptstyle{p^{\prime}}=1\hfill\atop\scriptstyle{p^{\prime}}\neq
p\hfill}^{P}{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}}\right|}}\right|\varepsilon}\right).$
(7.59)

_Proof_ : The proof is given in the Appendix of this chapter.

The definition of diagonally-dominant matrix suggests that the amplitude of
the diagonal elements is larger than the sum of the amplitudes of the other
elements in the same row. According to Proposition 7-1, we can infer that the
proposed precoding can improve the probability of the codeword difference
matrix ${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ having a diagonally-dominant
structure. Generally speaking, diagonally-dominant matrix are well-structured,
whose determinant value is close to the corresponding diagonal matrix [124].
To further qualify the effectiveness of the proposed precoding, we compare the
average determinant of ${{\bf{\Omega}}_{i}\left({\bf{e}}\right)}$ with and
without precoding by numerical simulations. Without loss of generality, we
consider $M=8,N=8$ in Fig. 7.6, where the maximum delay and Doppler indices
are set to be $l_{\rm{max}}=2$ and $k_{\rm{max}}=2$, respectively. Since
numerically emulating all the error sequences with such a frame size is
generally intractable in a reasonable time frame even with BPSK mapping, we
consider the comparison between the average determinant values of
${{\bf{\Omega}}_{i}\left({\bf{e}}\right)}$ with and without precoding for
given error sequences101010We use the same error sequence as in [48], i.e.,
${\bf e}=[2,0,-2,2,0,-2,...,0...0]^{\rm T}$. [49, 48], with respect to all
possible channel realizations111111Without loss of generality, we require the
absolute value of the difference between any two Doppler indices no smaller
than $0.2$.. For a better illustration, we also plot the determinant upper-
bound in (7.54) for comparison. As indicated by the figure, the proposed
precoding can indeed increase the determinant value of
${{\bf{\Omega}}_{i}\left({\bf{e}}\right)}$ compared to the case without
precoding, where the determinant value with precoding aligns well with the
upper-bound, especially for small values of
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$. In particular, we observe that the
improvement becomes more obvious with more resolvable paths, which indicates
that the proposed precoding is more helpful for communication transmissions in
rich scattering scenarios. On the other hand, it has reported in the
literature that the fractional Doppler may potentially degrade the error
performance [52]. Therefore, we have also plotted the curves of the average
determinant values corresponding to the case of different delay indices and
fractional Doppler indices in Fig. 7.6, for both $P=4$ and $P=5$. As indicated
by the figure, the determinant values increase slightly compared to the non-
precoded case (random delay and fractional Doppler indices), but still shows a
noticeable gap compared to the precoded case. This observation agrees with the
previous conclusions in [52].

Figure 7.6: Average determinant values of
${{\bf{\Omega}}_{i}\left({\bf{e}}\right)}$ with (in dash-dotted lines) and
without precoding (in solid lines for random delay indices and in dotted lines
for different delay indices), comparing with the determinant bound in (7.54)
(in dashed lines), where $M=8,N=8,l_{\rm max}=2$, and $k_{\rm max}=2$,
respectively.

Remark 7-4: It should also be noted that the effective DD domain channel
matrix for the case of fractional Doppler shifts is generally dense and
complex [14]. Consequently, the detection complexity required for fractional
Doppler case is quite high [25]. Therefore, our proposed precoding design can
also reduce the detection complexity at the UE side.

Remark 7-5: We briefly discuss the extension of our communication design in
the case of inaccurate AoA estimates. According to the proposed beam tracking
scheme, it is suggested to allocate power to all the antennas in
${{\mathbb{U}}_{i,p}}$. Hence, given that the minor changes of the related
parameters from the previous time instant are well-compensated [59], we can
apply the proposed precoding design to all the antennas in
${{\mathbb{U}}_{i,p}}$, and evenly assign the power among related antennas. We
note that this will inevitably lead to an SNR reduction for communication.
However, this reduction is small if the beam width is small.

####  7.3.3 Power Allocation for Communication

According to (7.55), the power allocation problem for communication is
formulated by

$\displaystyle\mathop{\rm
minimize}\limits_{{\alpha_{{a_{i,1}}}},...,{\alpha_{{a_{i,P}}}}}$
$\displaystyle{\left({\frac{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{P}}\right)^{-P}}{\left({\frac{{{{\left({\prod\limits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}}\right)}^{\frac{1}{P}}}}}{{4{N_{0}}}}}\right)^{-P}}$
(7.60) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}\quad$
$\displaystyle\sum\limits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}={{\tilde{\alpha}}_{i}}\quad{\rm
and}\quad{{a_{i,p}}}\geq 0,$ (7.61)

where ${{\tilde{\alpha}}_{i}}$ denotes the allocated power for the $i$-th UE.
As indicated by the above optimization problem, the power allocation should be
designed to maximize the power product
${{\prod\nolimits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}}}$, i.e., the geometric mean
of the allocated power associated to the corresponding paths in order to
minimize the PEP. According to the arithmetic mean-geometric mean (AM-GM)
inequality, it is not hard to show that with a total power constraint
${{\tilde{\alpha}}_{i}}$, the optimal power allocation for minimizing PEP is
the equal power allocation. This is not unexpected, because the communication
fading coefficients are of the same distribution and therefore the power
allocation should not provide any bias to any path.

Remark 7-6: It is interesting to see that both radar sensing and communication
require different power allocations. Therefore, we in the following briefly
discuss how to adapt those two allocations in practical systems. Let us first
consider the total power constraint for the $i$-th UE for the communication
transmission, i.e., $\sum\nolimits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}$. According
to the AM-GM inequality, we have
$\prod\nolimits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}\leq{\left({\frac{{\sum\nolimits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}}}{P}}\right)^{P}}$.
Notice that there is an exponent $P$ on the right hand side. Therefore, the
communication performance degradation due to unfit power allocation may become
more severe with a larger number of resolvable paths. However, it should be
noted that the communication fading coefficients are not known at the BS.
Therefore, the random nature of the communication channel may also mitigate
the performance degradation induced by the undesirable power allocation. On
the other hand, since the proposed AoA estimation algorithm in Section 7.2.2
is based on the principle of matched-filtering, where the received power will
be the key factor determining the estimation performance. Different from the
communication counterpart, the radar reflection coefficients are assumed to be
known at the BS. Therefore, a suitable power allocation that is designed
specifically for each signal transmission can largely improve the sensing
performance. Therefore, in practical ISAC scenarios, radar sensing performance
should be the priority for power allocation designs.

It should be noticed that a precise analysis on the relationship of the power
allocation between radar sensing and communication performances requires
detailed statistical models for both the radar reflection coefficients and
communication fading coefficients, which is also related to system settings,
e.g., the frame size. Since the major focus of this chapter is to propose the
ISAC transmission framework based on SS-OTFS modulation, we leave this
interesting issue for our future work.

###  7.4 Numerical Results

We demonstrate the numerical results for the proposed ISAC transmissions in
this section, where the SPA detection [42] for OTFS equalization is adopted at
the UE side. In specific, we set $N_{\rm BS}=128$, $M=32$, $N=16$,
$\tau_{\max}=10$, and $\nu_{\max}=6$, respectively, unless specified
otherwise, where the transmitted signals are BPSK modulated. To evaluate the
communication performance, we define the average symbol SNR by
$\frac{{{E_{s}}}}{{{N_{0}}}}\buildrel\Delta\over{=}\frac{\beta_{i}}{{{N_{0}}}}$,
where $\beta_{i}$ denotes the average power assigned to each antenna for the
$i$-th UE, i.e.,
${\beta_{i}}=\sum\nolimits_{p=1}^{P}{{\alpha_{{a_{i,p}}}}}/P$. Meanwhile, we
define the radar SNR by the ratio between the total power $\alpha_{\rm total}$
and the radar noise PSD ${\tilde{N}}_{0}$. Without loss of generality, we
assume that the reflection coefficient coefficients follow a uniform complex
Gaussian distribution, i.e., ${{\tilde{h}}_{i,p}}\sim{\cal
CN}\left({0,1}\right)$.

####  7.4.1 Beam Tracking Performance

We present the AoA estimation performance for the proposed ISAC transmission
with various beam widths in Fig. 7.7, where we assume that there are in total
$K=4$ UEs and each UE has $P=2$ paths. In specific, we consider a radar SNR at
$5$ dB in Fig. 7.7, where the proposed power allocation in Section 7.2.3 is
applied and beam widths are controlled by the value of $N_{\rm range}$ as
discussed in Section 7.2.1. In particular, the amplitude in the figure
represents the normalized trace of (7.35) with respect to the number of
transmitted symbols. As can be observed from the figure, the proposed beam
tracking and AoA estimation can provide an accurate estimation performance for
the considered scenario with various beam widths. Note that the received echo
power is reduced with the increase of the beam widths. In specific, it can be
shown the average received power with $N_{\rm range}=4$ is only $1/5$ of that
with $N_{\rm range}=0$, since we evenly assign the transmit power among all
$N_{\rm range}+1$ antennas. Therefore, this observation also indicates that
the proposed power allocation is suitable for the considered radar sensing
issues.

Figure 7.7: AoA estimation performance with various beam widths for $K=4$ UEs,
where each UE has $P=2$ paths.

####  7.4.2 Precoding Performance

We verify the effectiveness of the proposed precoding scheme in Fig. 7.8,
where the FER performance for a specific UE with only integer delay and
Doppler indices is illustrated. In particular, we consider the coded BPSK
signals with $P=8$ for transmission. Due to the high detection/decoding
complexity, we consider a smaller frame size for simulation, where we have
$N=8$, $M=16$. Without loss of generality, we apply the terminated (7, 5)
convolutional code as the channel code, and the virtual delay and Doppler
indices are randomly generated for precoding. Meanwhile, we consider the equal
power allocation in this example. As can be observed from the figure, with the
same channel coding and power allocation, the transmission with precoding has
a roughly $1.7$ dB gain in terms of average bit SNR compared to the
transmission without precoding at FER $4\approx\times 10^{-4}$. Furthermore,
we also notice that the FER slope for precoded transmission is steeper than
the transmission without precoding. This observation indicates that the
proposed precoding can also improve the diversity gain, which is due to the
fact that the codeword difference matrix is more likely to have full-rank when
different delay and Doppler indices are of different values.

Figure 7.8: FER comparisons between the proposed ISAC transmissions with and
without precoding, where (7, 5) convolutionally coded BPSK signals are
considered with $P=8$.

We demonstrate the FER performances of SS-OTFS121212According to Remark 7-1,
we introduce a phase offset according to the AoD associated to the first path
of the first UE before spatial spreading. for the first UE with $P=4$ with
respect to different numbers of antennas in Fig. 7.9, where a rate-half (3, 2)
convolutional code is used for ISAC transmission. To show the superior
performance of SS-OTFS for ISAC transmissions, we also provide the FER
performances for the beamforming counterparts, including the beamforming-based
OTFS (BF-OTFS) and the beamforming-based OFDM (BF-OFDM), where the proposed
precoding are applied for both SS-OTFS and BF-OTFS. In particular, we do not
assume perfect beam separation by using spatial spreading, as mentioned in our
system model. Instead, we take into account of the interferences from
different beams and study the potential performance improvements with
increased number of antennas, while the perfect beam separation cases for both
OTFS and OFDM are served as the corresponding error performance lower-bounds
for references. Furthermore, two ISAC transmission cases are considered for a
fair comparison. In the first case, the AoDs associated to different paths are
randomly generated from $\left[{-{{90}^{\circ}},{{90}^{\circ}}}\right]$ with a
uniform distribution, while in the second case, the AoDs are uniformly
generated at random from $\left[{-{{90}^{\circ}},{{90}^{\circ}}}\right]$ with
a uniform distribution but we require an at least ${{5^{\circ}}}$ separation
between any two paths, which is more reasonable in practical ISAC
transmissions. As can be observed from Fig. 7.9, SS-OTFS outperforms BF-OTFS
and BF-OFDM counterparts for both the two ISAC cases with the same number of
antennas. Also, the FER performances for SS-OTFS, BF-OTFS and BF-OFDM improve
with more antennas and the AoD separation. This is not unexpected, because the
shaped beams become narrower with more antennas and the AoD separation can
effectively mitigate the level of crosstalk between different beams in
multipath transmissions. Moreover, we notice that the FER performances for SS-
OTFS approach the error performance lower-bound, i.e., perfect beam separation
case, with an increased number of antennas, which implies that SS-OTFS can
effectively mitigate the interferences from different beams.

Figure 7.9: FER performance comparisons between SS-OTFS, BF-OTFS, and BF-OFDM
for the first UE with $P=4$.

####  7.4.3 Radar Sensing and Communication Performances with Power
Allocations

As discussed in Remark 7-6, both radar sensing and communication requires
different power allocations. Therefore, we evaluate the performances of radar
sensing and communication with different power allocations in this subsection.
In particular, we consider both the equal power allocation (designed for
communication, referred to as “communication power allocation" in the figures)
and the power allocation (designed for radar sensing, referred to as “radar
power allocation" in the figures) proposed in Section 7.2.3. Without loss of
generality, we set $N_{\rm range}=0$.

To demonstrate the advantages of the proposed ISAC transmission, we consider a
strict performance metric for radar sensing. As the angular domain is
discretized, we are interested in the “miss-detection probability" which is
defined by the ratio between the times when the radar does not accurately
detect the receive antenna indices ${\tilde{a}}_{i,p}$ for $1\leq i\leq K$ and
$1\leq p\leq P$ and the total number of ISAC transmissions. Without loss of
generality, we consider two cases for radar sensing in Fig. 7.10, where the
number of UEs are $K=4$ and $K=2$, and the number of paths are $P=2$ and
$P=1$, respectively. As indicated by the figure, suitable power allocation can
provide significant performance improvements for radar sensing, especially
when the number of targets is large. This indicates that the proposed power
allocation is indeed suitable for radar sensing.

Figure 7.10: Miss-detection probability for radar sensing with
radar/communication power allocation for different number of UEs and paths.
Figure 7.11: Comparisons between the FER performances for a specific UE with
communication power allocation and with radar power allocation.

We show the FER performances for a specific UE with different power
allocations for communications in Fig. 7.11, where the proposed precoding
scheme is applied. As shown in the figure, power allocation designed for
communication can provide a better error performance for communication, and
the performance improvement becomes larger with more paths, which are
consistent with our discussions in Section 7.3.3. On the other hand, we also
notice that with the power allocation designed for radar sensing, the
communication performance degrades. However, this performance degradation is
relatively small compared to the performance improvement for radar sensing.
Therefore, it is desirable to design the power allocation with a priority for
improving the radar sensing performance.

###  7.5 Summary of the Chapter

In this chapter, we proposed a novel framework for ISAC transmissions based on
the SS-OTFS modulation by considering the mismatch of the reflection strengths
between the radar sensing and communication. We first derived the channel
models for both radar sensing and communication, which are then simplified
based on the properties of SS-OTFS modulation. Based on the radar model, we
proposed simple beam tracking, AoA estimation algorithms, and power allocation
for radar sensing. Furthermore, we carried out a detailed analysis on the PEP
for communication, where we showed that the PEP can be minimized with equal
power allocation if the received signals from different paths are orthogonal
to each other. Based on this conclusion, we proposed a symbol-wise precoding
design to improve this orthogonality by introducing virtual delay and Doppler
indices. We also noticed that radar sensing and communication require
different power allocations. To facilitate the power allocation design for
practical ISAC systems, we briefly discussed the radar sensing and
communication performances with different power allocations and concluded that
the power allocation should be designed leaning towards radar sensing.
Finally, the effectiveness of the proposed framework is verified by numerical
results.

###  7.6 Appendices of the Chapter

In this section, we provide the related proofs from this chapter.

####  7.6.1 Proof of Lemma 7-1

The proof of this lemma is based on the calculations of (7.23) and (7.24). Let
us start with the calculation for the equivalent angular domain channel vector
for the communication channel. For $1\leq l\leq N_{\rm BS}$, it can be shown
that the $l$-th column of ${\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}$ is given by

$\displaystyle\left[{\frac{1}{{\sqrt{{N_{{\rm{BS}}}}}}},\frac{1}{{\sqrt{{N_{{\rm{BS}}}}}}}\exp\left({j2\pi\frac{1}{{{N_{{\rm{BS}}}}}}}\right),...,}\right.{\left.{\frac{1}{{\sqrt{{N_{{\rm{BS}}}}}}}\exp\left({j2\pi\frac{{{N_{{\rm{BS}}}}-1}}{{{N_{{\rm{BS}}}}}}}\right)}\right]^{\rm{T}}}.$
(7.62)

Furthermore, notice that $\bm{\alpha}$ is a diagonal matrix, whose $l$-th
diagonal element is $\alpha_{l}$. Thus, according to the rule of matrix
multiplication, we have

$\displaystyle
h_{i,p}^{\rm{A}}\left[l\right]=\frac{{\sqrt{{\alpha_{l}}}}}{{{N_{{\rm{BS}}}}}}\sum\limits_{{n_{\rm{t}}}=0}^{{N_{{\rm{BS}}}}-1}{\exp\left({j{n_{\rm{t}}}\pi\left({\sin{\varphi_{i,p}}+\frac{{2\left({l-1}\right)}}{{{N_{{\rm{BS}}}}}}}\right)}\right)}.$
(7.63)

The radar channel elements can be calculated in a similar way to the above
calculation for communication channel elements, according to the rule of
matrix multiplication. This completes the proof of Lemma 7-1. $\blacksquare$

####  7.6.2 Proof of Corollary 7-1

Let $\sin{\varphi_{i,p}}=q\frac{2}{{{N_{{\rm{BS}}}}}}$, where $q$ is an
arbitrary integer number. Then, (7.23) becomes

$\displaystyle
h_{i,p}^{\rm{A}}\left[l\right]=\frac{{\sqrt{{\alpha_{l}}}}}{{{N_{{\rm{BS}}}}}}\sum\limits_{{n_{\rm{t}}}=0}^{{N_{{\rm{BS}}}}-1}{\exp\left({j2\pi\frac{{{n_{\rm{t}}}}}{{{N_{{\rm{BS}}}}}}\left({l+q-1}\right)}\right)}.$
(7.64)

Notice that

$\displaystyle\sum\limits_{{n_{\rm{t}}}=0}^{{N_{{\rm{BS}}}}-1}{\exp\left({j2\pi\frac{{{n_{\rm{t}}}}}{{{N_{{\rm{BS}}}}}}l}\right)}=\left\\{\begin{array}[]{l}{N_{{\rm{BS}}}},\quad\quad
l=0,\\\ 0,\quad\quad\quad\;l\neq 0.\end{array}\right.$ (7.67)

Therefore, it can be shown that (7.64) only has non-zero values if
$l={a_{i,p}}={\left[{{N_{{\rm{BS}}}}-\frac{{\sin\left({{\varphi_{i,p}}}\right){N_{{\rm{BS}}}}}}{2}}\right]_{{N_{{\rm{BS}}}}}}+1$,
where the modulo operation is due to the fact that $\sin{\varphi_{i,p}}$ could
take negative values while the antenna index has to be positive. Similarly,
(7.24) can be expressed by

$\displaystyle\tilde{H}_{i,p}^{\rm{A}}\left[{k,l}\right]=$
$\displaystyle\frac{{\sqrt{{\alpha_{l}}}}}{{{{\left({{N_{{\rm{BS}}}}}\right)}^{2}}}}\left({\sum\limits_{{n_{\rm{r}}}=0}^{{N_{{\rm{BS}}}}-1}\\!{\exp\left({j2\pi\frac{{{n_{\rm{r}}}}}{{{N_{{\rm{BS}}}}}}\left({q-k+1}\right)}\right)}}\right)\left({\sum\limits_{{n_{\rm{t}}}=0}^{{N_{{\rm{BS}}}}-1}\\!{\exp\left({j2\pi\frac{{{n_{\rm{t}}}}}{{{N_{{\rm{BS}}}}}}\left({l+q-1}\right)}\right)}}\right).$
(7.68)

Thus, (7.68) only has non-zero values if $l={a_{i,p}}$ and
$k={{\tilde{a}}_{i,p}}={\left[{{N_{{\rm{BS}}}}+\frac{{\sin\left({{\varphi_{i,p}}}\right){N_{{\rm{BS}}}}}}{2}}\right]_{{N_{{\rm{BS}}}}}}+1$.
On the other hand, the values of $h_{i,p}^{\rm{A}}\left[l\right]$ and
$\tilde{H}_{i,p}^{\rm{A}}\left[{k,l}\right]$ for $l={a_{i,p}}$ and
$k={{\tilde{a}}_{i,p}}$ can be easily calculated based on (7.64) and (7.68).
This completes the proof of Corollary 7-1. $\blacksquare$

####  7.6.3 Proof of Theorem 7-1

According to (7.49), we notice that the codeword difference matrix
${{\bf{\Omega}}_{i}\left({\bf{e}}\right)}$ is a Gram matrix corresponding to
the vectors
$\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,P}}}\right\\}$,
where ${{\bf{u}}_{i,j}}\buildrel\Delta\over{=}{{\bf{\Xi}}_{i,j}}{\bf{e}}$. In
particular, the determinant of the Gram matrix
${{\bf{\Omega}}\left({\bf{e}}\right)}$, i.e., the Gram determinant, is equal
to the square of the $P$-dimensional volume of the parallelotope constructed
on
$\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,P}}}\right\\}$.
Let us refer to
${\rm{GD}}\left(\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,P}}}\right\\}\right)$
as the Gram determinant of ${{\bf{\Omega}}_{i}\left({\bf{e}}\right)}$. It can
be shown that the Gram determinant can be calculated recursively, such as
[105]

$\displaystyle{\rm{GD}}\left(\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,j}}}\right\\}\right)={\rm{GD}}\left(\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,j-1}}}\right\\}\right){\left\|{{{{\bf{\tilde{u}}}}_{i,j}}}\right\|^{2}},$
(7.69)

where the term ${{{{\bf{\tilde{u}}}}_{i,j}}}$ denotes the _orthogonal
projection_ of ${{{{\bf{u}}}_{i,j}}}$ onto the _orthogonal complement_ of
${\rm{span}}\left({{{{\bf{u}}}_{i,1}},{{{\bf{u}}}_{i,2}},\ldots{{{\bf{u}}}_{i,j-1}}}\right)$.
Considering the property of orthogonal projection, we have

$\displaystyle{\rm{GD}}\left(\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,j}}}\right\\}\right)=$
$\displaystyle{\rm{GD}}\left(\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,j-1}}}\right\\}\right){\left\|{{{{\bf{\tilde{u}}}}_{i,j}}}\right\|^{2}}$
$\displaystyle\leq$
$\displaystyle{\rm{GD}}\left({\left\\{{{{\bf{{u}}}_{i,1}},{{\bf{{u}}}_{i,2}},\ldots{{\bf{{u}}}_{i,j-1}}}\right\\}}\right){\left\|{{{{\bf{u}}}_{i,j}}}\right\|^{2}},$
(7.70)

where the equality holds if ${{{\bf{u}}_{i,j}}}$ is orthogonal to
${{{\bf{u}}_{i,j^{\prime}}}}$, for $1\leq j^{\prime}<j$, i.e.,
${\left({{{\bf{\Xi}}_{i,j^{\prime}}}{\bf{e}}}\right)^{\rm{H}}}\left({{{\bf{\Xi}}_{i,j}}{\bf{e}}}\right)=0$,
for $1\leq j^{\prime}<j$. Hence, by considering (7.70), the Gram determinant
can be upper-bounded by

$\displaystyle\det\left({{{\bf{\Omega}}_{i}}\left({\bf{e}}\right)}\right)=$
$\displaystyle{\rm{GD}}\left({\left\\{{{{\bf{u}}_{i,1}},{{\bf{u}}_{i,2}},\ldots{{\bf{u}}_{i,P-1}}}\right\\}}\right){\left\|{{{{\bf{\tilde{u}}}}_{i,P}}}\right\|^{2}}$
$\displaystyle\leq$
$\displaystyle{\rm{GD}}\left({\left\\{{{{\bf{u}}_{i,1}},{{\bf{u}}_{i,2}},\ldots{{\bf{u}}_{i,P-1}}}\right\\}}\right){\left\|{{{\bf{u}}_{i,P}}}\right\|^{2}}$
$\displaystyle\leq$
$\displaystyle\prod\limits_{j=1}^{P}{{{\left\|{{{{\bf{u}}}_{i,j}}}\right\|}^{2}}}$
$\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{P}{{\bf{e}}^{\rm{H}}}\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){\bf{W}}_{{a_{i,j}}}^{\rm{H}}{{\bf{W}}_{{a_{i,j}}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right){\bf{e}}.$
(7.71)

By noticing that ${\bf{W}}_{{a_{i,j}}}^{\rm{H}}{{\bf{W}}_{{a_{i,j}}}}={\bf
I}_{MN}$, (7.71) can be further derived as (7.54). This completes the proof of
Theorem 7-1.

####  7.6.4 Proof of Lemma 7-3

Assuming that
${\bf{W}}_{{a_{i,p}}}^{\rm{H}}{{\bf{W}}_{{a_{i,p^{\prime}}}}}={{\bf{0}}_{MN}}$,
for $1\leq p,p^{\prime}\leq P$ and $p^{\prime}\neq p$. Then, it is obvious
that there exists an index $p$, where $1\leq p\leq P$, such that
${\bf{W}}_{{a_{i,p}}}$ has a zero determinant, which is contradict to
${\bf{W}}_{{a_{i,p}}}^{\rm{H}}{{\bf{W}}_{{a_{i,p}}}}={{\bf{I}}_{MN}}$. This
completes the proof of Lemma 7-3.

####  7.6.5 Proof of Proposition 7-1

Let ${\varepsilon^{\prime}}$ denote the event that different paths do have the
same delay or Doppler indices. To prove (7.59), we only need to prove that

$\displaystyle\Pr\left({\left.{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p}}{\bf{e}}}\right|\geq\sum\limits_{\scriptstyle{p^{\prime}}=1\hfill\atop\scriptstyle{p^{\prime}}\neq
p\hfill}^{P}{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}}\right|}}\right|\varepsilon^{\prime}}\right)$
$\displaystyle\geq$
$\displaystyle\Pr\left({\left.{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p}}{\bf{e}}}\right|\geq\sum\limits_{\scriptstyle{p^{\prime}}=1\hfill\atop\scriptstyle{p^{\prime}}\neq
p\hfill}^{P}{\left|{{{\bf{e}}^{\rm{H}}}{{\bf{\Xi}}_{i,p}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}}\right|}}\right|\varepsilon}\right).$
(7.72)

To this end, let us focus on the $(p,p^{\prime})$-th element of
${{\bf{\Omega}}_{i}}\left({\bf{e}}\right)$, and it is rewritten by

$\displaystyle{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,p}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}=$
$\displaystyle{{\bf{e}}^{\rm{H}}}\\!\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right)\\!{{\bm{\Delta}}\\!^{-{{k}_{p}}-{\kappa_{p}}}}{{\bm{\Pi}}^{-{{l}_{p}}}}{{\bm{\Pi}}^{{{l}_{p^{\prime}}}}}\\!{{\bm{\Delta}}^{{{k}_{p^{\prime}}+{\kappa_{p^{\prime}}}}}}\\!\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right)\\!{\bf{e}}$
$\displaystyle=$
$\displaystyle{{{\bf{\tilde{e}}}}^{\rm{H}}}{{\bm{\Delta}}^{-{{k}_{p}}-{\kappa_{p}}}}{{\bm{\Pi}}^{{{l}_{p^{\prime}}}-{{l}_{p}}}}{{\bm{\Delta}}^{{{k}_{p^{\prime}}+{\kappa_{p^{\prime}}}}}}{\bf{\tilde{e}}},$
(7.73)

where
${\bf{\tilde{e}}}\buildrel\Delta\over{=}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right){\bf{e}}$
is the TDA domain error sequence. Furthermore, let
$n^{\prime}={\left[{n-1-\left({{l_{p^{\prime}}}-{l_{p}}}\right)}\right]_{MN}}+1$,
(7.73) can be further derived by

$\displaystyle{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,p}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}=\sum\limits_{n=1}^{MN}{{e^{j\frac{{2\pi}}{{MN}}\left({\left({n^{\prime}-1}\right)\left({{k_{p^{\prime}}}+{\kappa_{p^{\prime}}}}\right)-\left({n-1}\right)\left({{k_{p}}+{\kappa_{p}}}\right)}\right)}}{{\tilde{e}}^{*}}\left[n\right]\tilde{e}\left[{n^{\prime}}\right]}.$
(7.74)

In particular, it can be shown that for $p=p^{\prime}$, (7.74) becomes
${{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}$, which is independent with the delay
and Doppler distributions. Thus, we notice that (7.72) can be verified if the
absolute value of
${{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,p}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}},\forall
p\neq p^{\prime}$ becomes smaller when different paths share the same
delay/Doppler indices. However, the values of the elements in ${\bf{e}}$ vary
with respect to different transmitted symbol vectors and detection outputs,
which are usually studied statically for the related analysis [47]. Note that
each TDA domain OTFS symbol is a superposition of $N$ DD domain OTFS symbols
with specific phase rotations according to the spreading effect of IFFT. Thus,
in practical systems with a sufficiently large $N$, the elements in
${\bf{\tilde{e}}}$ behave like i.i.d. Gaussian variable due to the law of
large numbers [47]. In the case where $l_{p}=l_{p^{\prime}}$, i.e., different
paths share the same delay index, (7.74) can be simplified as

$\displaystyle{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,p}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}={{{\bf{\tilde{e}}}}^{\rm{H}}}{{\bm{\Delta}}^{{{k}_{p^{\prime}}+{\kappa_{p^{\prime}}}}-{{k}_{p}-{\kappa_{p}}}}}{\bf{\tilde{e}}}=\sum\limits_{n=1}^{MN}{{e^{j\frac{{2\pi}}{{MN}}\left({n-1}\right)\left({{k_{p^{\prime}}}+{\kappa_{p^{\prime}}}-{k_{p}}-{\kappa_{p}}}\right)}}{{\left|{\tilde{e}\left[n\right]}\right|}^{2}}}.$
(7.75)

Comparing (7.74) and (7.75), we observe that the absolute value of (7.75) is
larger than that of (7.74) with a sufficiently large $MN$, because the
expectation of the term
${{{\tilde{e}}^{*}}\left[n\right]\tilde{e}\left[{n^{\prime}}\right]}$ is zero
based on the i.i.d. assumption [47], while the term
${{\left|{\tilde{e}\left[n\right]}\right|}^{2}}$ is strictly non-negative. On
the other hand, when
${{k_{p}}+{\kappa_{p}}}={{k_{p^{\prime}}}+{\kappa_{p^{\prime}}}}$, i.e.,
different paths share the same Doppler index, (7.74) can be simplified as

$\displaystyle{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,p}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}=\sum\limits_{n=1}^{MN}{{e^{j\frac{{2\pi}}{{MN}}\left({n^{\prime}-n}\right)\left({{k_{p}}+{\kappa_{p}}}\right)}}{{\tilde{e}}^{*}}\left[n\right]\tilde{e}\left[{n^{\prime}}\right]}\approx{e^{j\frac{{2\pi}}{{MN}}\left({{l_{p}}-{l_{p^{\prime}}}}\right)\left({{k_{p}}+{\kappa_{p}}}\right)}}\sum\limits_{n=1}^{MN}{{{\tilde{e}}^{*}}\left[n\right]\tilde{e}\left[{n^{\prime}}\right]},$
(7.76)

whose absolute value is given by
$\left|{\sum\limits_{n=1}^{MN}{{{\tilde{e}}^{*}}\left[n\right]\tilde{e}\left[{n^{\prime}}\right]}}\right|$.
Therefore, it can be shown that the absolute value of (7.74) is no larger than
the absolute value of (7.76), due to
$\left|{\sum\limits_{n=1}^{MN}{a_{n}b_{n}}}\right|\leq\sum\limits_{n=1}^{MN}{\left|a_{n}\right|\left|b_{n}\right|}$,
for any two arbitrary complex vectors $\bf a$ and $\bf b$. Based on the above
discussions, we have shown that the absolute value of
${{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,p}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}},\forall
p\neq p^{\prime}$ generally becomes smaller when different paths share the
same delay/Doppler indices, while the value of
${{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i,p}^{\rm{H}}{{\bf{\Xi}}_{i,p^{\prime}}}{\bf{e}}$
remains constant with different delay and Doppler distributions. Thus, (7.72)
holds and this completes the proof of Proposition 7-1.

##  Chapter 8 Conclusions and Future Work

In this thesis, we discussed the related subjects of OTFS modulation.
Specifically, we have studied the topics of wireless channels, OTFS concepts,
OTFS detection, OTFS performance analysis, and the application of OTFS in ISAC
transmissions.

In Chapter 1, we reviewed the motivation and needs for the application of OTFS
modulation. We also provided a literature review for OTFS modulation. In
specific, the literature review covered the related studies of OTFS modulation
including the concept and implementation, channel estimation, signal
detection, and applications. The main structure of this thesis was highlighted
and the main research contributions were summarized.

In Chapter 2, the properties wireless channels were reviewed and the related
discussions on the channel responses in different domains were presented. We
also adopted several examples to describe the characteristics of the channel
responses in different domains.

In Chapter 3, the concept of OTFS modulation was derived by using ZT and DZT.
In particular, we offered DD domain interpretations of key components of
modulation, such as pulse shaping and matched-filtering. Furthermore, the
input-output relationship for rectangular pulse-shaped OTFS modulation was
highlighted.

In Chapter 4, the hybrid-MAP-PIC detection for OTFS modulation was studied. We
presented the derivations of the hybrid-MAP-PIC detection according to the SPA
framework and also discussed its error performance and parameter selection.
Our numerical results implied that the hybrid-MAP-PIC detection outperforms
the conventional message passing algorithm and approaches the optimal
detection performance with a reduced complexity.

In Chapter 5, we introduced the cross domain iterative detection for OTFS
modulation. The main novelty of this detection is that the extrinsic
information is passed between different domains via the corresponding unitary
transformation. Error performance analysis of this detection was conducted
based on the state evolution. Our numerical results have demonstrated the
superior performance of the cross domain iterative detection over the
conventional OTFS detection approaches.

In Chapter 6, we studied the error performance of coded OTFS systems based on
the PEP analysis. In particular, our derivations have shown that there is a
trade-off between the diversity gain and the coding gain, i.e., the diversity
gain of OTFS systems improves with the number of resolvable paths, while the
coding gain declines. Rule-of-thumb code design guidelines were also provided
according to the trade-off. Numerical results agreed with our analysis and
clearly demonstrated the trade-off between diversity gain and coding gain.

In Chapter 7, the application of OTFS modulation in ISAC transmissions was
discussed. To facilitate the ISAC designs, we introduced the concept of
spatial-spreading, which leads to simple and insightful input-output
relationships for both communications and radar sensing. A framework of ISAC
transmission using OTFS waveform was then investigated and low-complexity
radar estimation and communication precoding schemes were proposed. Finally,
we applied numerical approaches to verify the effectiveness of the considered
scheme.

Although we have provided some related discussions of OTFS modulation, it
should be noted that the related development of OTFS modulation is still at an
early stage and some important issues of OTFS modulation still require to
prompt attention. In the following, we list several related future works for
OTFS modulation.

###  8.1 Future Work

####  8.1.1 Exploiting DD Domain Channel Characteristics

The application of OTFS modulation gives a direct way of exploiting the DD
domain channel characteristics. However, several appealing DD domain channel
characteristics have not yet been fully exploited. For example, the commonly
adopted channel model for OTFS modulation assumes that the channel geometry
remains unchanged for a period of time. However, this assumption may not also
hold in practical scenarios, especially for complex communication
applications, such as high-speed train communication, where practical issues,
such as path live-or-die, need to be considered. Another example may be the
case of channel inference based on DD domain channel reciprocity. This problem
is of practical importance for a cellular system, where the uplink and
downlink channels may share the same channel geometry over a period of time
that is longer than the conventional coherence time. However, how to exploit
the DD domain channel reciprocity still needs investigation. As such, there is
still a challenge for related OTFS designs to fully copilot the DD domain
channel characteristics, and this challenge could be solved by conducting
extensive real-world channel measurements for various wireless channels.

####  8.1.2 Fundamental Limits for DD Domain Communications

Although the DD domain communication has shown appealing advantages over the
conventional TF domain counterpart, its fundamental limits are still unclear
in the literature. Despite the earlier works on the achievable rate analysis
for OTFS modulation, the information-theoretical understanding of OTFS
modulation still needs further investigations. For example, the DD domain has
the abilities to fully localize a pulse without violating the Heisenberg’s
uncertainty principle [6, 23]. But how much information we can gain from the
DD domain localization is still unclear. Furthermore, the capacity scaling law
for MIMO-OTFS has not been fully studied in the literature, despite the fact
that it is crucial for many practical applications. On top of that, security
and privacy performances are also important aspects of future wireless
networks. However, the related studies on security and privacy performances
for DD domain communications still require further investigations.

####  8.1.3 DD Domain Communication Designs

Although most of the related works on OTFS modulation have been focused on the
system design, there are still many important aspects of DD domain
communication designs that need to be studied. For instance, the long latency
may be an issue for OTFS transmission. In particular, the DD domain received
symbols can only be obtained once all the information symbols in the TF domain
are received, which is generally time-consuming compared to the conventional
OFDM. Therefore, it is important to design OTFS receivers with a low latency.
Another issue could be the code design for OTFS. We have studied the error
performance for coded OTFS systems in Chapter 7, but the detailed design
criteria still need to be investigated. In specific, how to design good
channel codes for OTFS systems, such as LDPC codes [125, 126, 127, 128],
coupled codes [129, 130, 131], Bose-Chaudhuri-Hocquenghem (BCH) codes [132],
could be an important issue for practical application of OTFS modulation.
Furthermore, the combination of OTFS with emerging new technologies [133, 134,
135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149,
150] is also of importance to exploit. On the other hand, machine learning has
shown great potential to facilitate the design of wireless systems [151, 152,
104, 101, 102], and some preliminary works on machine learning-based OTFS
designs appeared in [153, 154]. However, there are still many aspects of OTFS
modulation that could be improved with the assistance of machine learning.

## References

  * [1] G. Meyer and S. Beiker, _Road Vehicle Automation_. Springer International Publishing, 2019.
  * [2] Y. Cai, Z. Wei, R. Li, D. W. K. Ng, and J. Yuan, “Joint trajectory and resource allocation design for energy-efficient secure UAV communication systems,” _IEEE Trans. Commun._ , vol. 68, no. 7, pp. 4536–4553, Mar. 2020.
  * [3] W. Yuan, C. Liu, F. Liu, S. Li, and D. W. K. Ng, “Learning-based predictive beamforming for UAV communications with jittering,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 11, pp. 1970–1974, Jul. 2020.
  * [4] A. Fotouhi, H. Qiang, M. Ding, M. Hassan, L. G. Giordano, A. Garcia-Rodriguez, and J. Yuan, “Survey on UAV cellular communications: Practical aspects, standardization advancements, regulation, and security challenges,” _IEEE Commun. Sur. & Tut._, vol. 21, no. 4, pp. 3417–3442, 2019.
  * [5] T. Hwang, C. Yang, G. Wu, S. Li, and G. Y. Li, “OFDM and its wireless applications: A survey,” _IEEE Trans. Veh. Technol._ , vol. 58, no. 4, pp. 1673–1694, May 2008.
  * [6] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, and R. Calderbank, “Orthogonal time frequency space modulation,” in _Proc. IEEE Wireless Commun. Net. Conf._ , 2017, pp. 1–6.
  * [7] F. Hlawatsch and G. Matz, _Wireless Communications over Rapidly Time-varying Channels_. Academic Press, 2011.
  * [8] D. Tse and P. Viswanath, _Fundamentals of Wireless Communication_. Cambridge university press, 2005.
  * [9] P. Raviteja, Y. Hong, E. Viterbo, and E. Biglieri, “Effective diversity of OTFS modulation,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 2, pp. 249–253, Feb. 2020.
  * [10] E. Biglieri, P. Raviteja, and Y. Hong, “Error performance of orthogonal time frequency space (OTFS) modulation,” in _IEEE Int. Conf. on Commun. Workshops (ICC Workshops)_ , May 2019, pp. 1–6.
  * [11] G. D. Surabhi, R. M. Augustine, and A. Chockalingam, “On the diversity of uncoded OTFS modulation in doubly-dispersive channels,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 6, pp. 3049–3063, Jun. 2019.
  * [12] J. Wu and P. Fan, “A survey on high mobility wireless communications: Challenges, opportunities and solutions,” _IEEE Access_ , vol. 4, pp. 450–476, 2016.
  * [13] R. Hadani and A. Monk, “OTFS: A new generation of modulation addressing the challenges of 5G,” _arXiv preprint arXiv:1802.02623_ , 2018.
  * [14] Z. Wei, W. Yuan, S. Li, J. Yuan, G. Bharatula, R. Hadani, and L. Hanzo, “Orthogonal time-frequency space modulation: A promising next-generation waveform,” _IEEE Wireless Commun._ , vol. 28, no. 4, pp. 136–144, Aug. 2021\.
  * [15] S. Li, W. Yuan, Z. Wei, R. He, B. Ai, B. Bai, and J. Yuan, “A tutorial to orthogonal time frequency space modulation for future wireless communications,” in _2021 IEEE/CIC Int. Conf. Commun. China (ICCC Workshops)_ , 2021, pp. 439–443.
  * [16] J. Zak, “Finite translations in solid-state physics,” _Phys. Rev. Lett._ , vol. 19, no. 24, p. 1385, 1967.
  * [17] A. J. Janssen, “The Zak transform: A signal transform for sampled time-continuous signals.” _Philips J. Res._ , vol. 43, no. 1, pp. 23–69, 1988.
  * [18] S. K. Mohammed, “Derivation of OTFS modulation from first principles,” _IEEE Trans. Veh. Technol._ , vol. 70, no. 8, pp. 7619–7636, Aug. 2021.
  * [19] F. Lampel, A. Alvarado, and F. M. Willems, “Orthogonal time frequency space modulation: A discrete Zak transform approach,” _arXiv preprint arXiv:2106.12828_ , 2021.
  * [20] H. Bölcskei, _Gabor Expansion and Frame Theory_. Vienna Univ. of Technol., 1994.
  * [21] H. Bölcskei and F. Hlawatsch, “Discrete Zak transforms, polyphase transforms, and applications,” _IEEE Trans. signal process._ , vol. 45, no. 4, pp. 851–866, Apr. 1997.
  * [22] Y. Hong, T. Thaj, and E. Viterbo, _Delay Doppler Communications: Principles and Applications_. Elsevier, 2022.
  * [23] R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, C. Ibars, J. Delfeld, Y. Hebron, A. J. Goldsmith, A. F. Molisch, and R. Calderbank, “Orthogonal time frequency space modulation,” _arXiv preprint arXiv:1808.00519_ , 2018.
  * [24] A. RezazadehReyhani, A. Farhang, M. Ji, R. R. Chen, and B. Farhang-Boroujeny, “Analysis of discrete-time MIMO OFDM-Based orthogonal time frequency space modulation,” in _Proc. IEEE Int. Conf. Commun._ , May 2018, pp. 1–6.
  * [25] P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, “Interference cancellation and iterative detection for orthogonal time frequency space modulation,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 10, pp. 6501–6515, Oct. 2018.
  * [26] P. Raviteja, Y. Hong, E. Viterbo, and E. Biglieri, “Practical pulse-shaping waveforms for reduced-cyclic-prefix OTFS,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 1, pp. 957–961, Jan. 2019.
  * [27] B. C. Pandey, S. K. Mohammed, P. Raviteja, Y. Hong, and E. Viterbo, “Low complexity precoding and detection in multi-user massive MIMO OTFS downlink,” _IEEE Trans. Veh. Technol._ , vol. 70, no. 5, pp. 4389–4405, Feb. 2021.
  * [28] A. Nimr, M. Chafii, M. Matthé, and G. Fettweis, “Extended GFDM framework: OTFS and GFDM comparison,” in _IEEE Global Commun. Conf._ IEEE, 2018, pp. 1–6.
  * [29] H. Lin and J. Yuan, “Multicarrier modulation on delay-Doppler plane: Achieving orthogonality with fine resolutions,” __to appear in_ IEEE Int. Commun. Conf._, pp. 1–6, 2022.
  * [30] C. Shen, J. Yuan, and H. Lin, “Error performance of rectangular pulse-shaped OTFS with practical receivers,” __submitted to_ IEEE Wireless Commun. Lett._, 2022.
  * [31] Y. Ge, Q. Deng, P. C. Ching, and Z. Ding, “OTFS signaling for uplink NOMA of heterogeneous mobility users,” _IEEE Trans. Commun._ , vol. 69, no. 5, pp. 3147–3161, Feb. 2021.
  * [32] A. M. Sayeed, “How is time frequency space modulation related to short time fourier signaling?” _arXiv preprint arXiv:2109.06047_ , 2021.
  * [33] P. Raviteja, K. T. Phan, and Y. Hong, “Embedded pilot-aided channel estimation for OTFS in delay-Doppler channels,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 5, pp. 4906–4917, May 2019.
  * [34] M. Kollengode Ramachandran and A. Chockalingam, “MIMO-OTFS in high-doppler fading channels: Signal detection and channel estimation,” in _IEEE Global Commun. Conf. (GLOBECOM)_ , 2018, pp. 206–212.
  * [35] W. Yuan, S. Li, Z. Wei, J. Yuan, and D. W. K. Ng, “Data-aided channel estimation for OTFS systems with a superimposed pilot and data transmission scheme,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 9, pp. 1954–1958, Sep. 2021.
  * [36] W. Shen, L. Dai, J. An, P. Fan, and R. W. Heath, “Channel estimation for orthogonal time frequency space (OTFS) massive MIMO,” _IEEE Trans. Signal Process._ , vol. 67, no. 16, pp. 4204–4217, May 2019.
  * [37] M. Li, S. Zhang, F. Gao, P. Fan, and O. A. Dobre, “A new path division multiple access for the massive MIMO-OTFS networks,” _IEEE J. Sel. Areas Commun._ , vol. 39, no. 4, pp. 903–918, Aug. 2021.
  * [38] L. Zhao, W.-J. Gao, and W. Guo, “Sparse bayesian learning of delay-Doppler channel for OTFS system,” _IEEE Commun. Lett._ , vol. 24, no. 12, pp. 2766–2769, Sept. 2020.
  * [39] Z. Wei, W. Yuan, S. Li, J. Yuan, and D. W. K. Ng, “Off-grid channel estimation with sparse Bayesian learning for OTFS systems,” __to apppear in_ IEEE Trans. Wireless Commun._, 2022.
  * [40] ——, “A new off-grid channel estimation method with sparse bayesian learning for OTFS systems,” in _IEEE Global Commun. Conf. (GLOBECOM)_ , 2021, pp. 01–07.
  * [41] W. Yuan, Z. Wei, J. Yuan, and D. W. K. Ng, “A simple variational Bayes detector for orthogonal time frequency space (OTFS) modulation,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 7, pp. 7976–7980, Apr. 2020.
  * [42] S. Li, W. Yuan, Z. Wei, J. Yuan, B. Bai, D. W. K. Ng, and Y. Xie, “Hybrid MAP and PIC detection for OTFS modulation,” _IEEE Trans. Veh. Technol._ , vol. 70, no. 7, pp. 7193–7198, Jul. 2021.
  * [43] Z. Yuan, F. Liu, W. Yuan, Q. Guo, Z. Wang, and J. Yuan, “Iterative detection for orthogonal time frequency space modulation using approximate message passing with unitary transformation,” _arXiv preprint arXiv:2008.06688_ , 2020.
  * [44] G. D. Surabhi and A. Chockalingam, “Low-complexity linear equalization for OTFS modulation,” _IEEE Commun. Lett._ , vol. 24, no. 2, pp. 330–334, Feb. 2020.
  * [45] T. Thaj and E. Viterbo, “Low complexity iterative rake decision feedback equalizer for zero-padded OTFS systems,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 12, pp. 15 606–15 622, Dec. 2020.
  * [46] Q. Li, J. Yuan, and H. Lin, “Iterative MMSE detection for orthogonal time frequency space modulation,” __to appear in_ IEEE Int. Commun. Conf. Workshop_, pp. 1–6, 2022.
  * [47] S. Li, W. Yuan, Z. Wei, and J. Yuan, “Cross domain iterative detection for orthogonal time frequency space modulation,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 4, pp. 2227–2242, Apr. 2022.
  * [48] S. Li, J. Yuan, W. Yuan, Z. Wei, B. Bai, and D. W. K. Ng, “Performance analysis of coded OTFS systems over high-mobility channels,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 9, pp. 6033–6048, Apr. 2021.
  * [49] ——, “On the performance of coded OTFS modulation over high-mobility channels,” _IEEE. Int. Commun. Conf. Workshop_ , pp. 1–6, 2021.
  * [50] M. Liu, S. Li, C. Zhang, B. Wang, and B. Bai, “Coded orthogonal time frequency space modulation,” _ZTE Communications_ , vol. 19, no. 4, pp. 54–62, Jan. 2022.
  * [51] C. Zhang, W. Xing, J. Yuan, and Y. Zhou, “Performance of LDPC coded OTFS systems over high mobility channels,” _ZTE Communications_ , vol. 19, no. 4, pp. 45–53, 2022.
  * [52] Z. Wei, W. Yuan, S. Li, J. Yuan, and D. W. K. Ng, “Transmitter and receiver window designs for orthogonal time frequency space modulation,” _IEEE Trans. Commun._ , vol. 69, no. 4, pp. 2207–2223, Apr. 2021.
  * [53] ——, “Performance analysis and window design for channel estimation of OTFS modulation,” in _IEEE Int. Conf. Commun._ IEEE, 2021, pp. 1–7.
  * [54] L. Gaudio, G. Colavolpe, and G. Caire, “OTFS vs. OFDM in the presence of sparsity: A fair comparison,” _to appear in IEEE Trans. Wireless Commun._ , 2021.
  * [55] G. D. Surabhi, R. M. Augustine, and A. Chockalingam, “Peak-to-average power ratio of OTFS modulation,” _IEEE Commun. Lett._ , vol. 23, no. 6, pp. 999–1002, Jun. 2019.
  * [56] G. Surabhi, R. M. Augustine, and A. Chockalingam, “Multiple access in the delay-doppler domain using OTFS modulation,” _arXiv preprint arXiv:1902.03415_ , 2019.
  * [57] R. Chong, S. Li, J. Yuan, and D. W. K. Ng, “Achievable rate upper-bounds of uplink multiuser otfs transmissions,” _to appear in _IEEE Wireless Commun. Lett.__ , pp. 1–5, 2022.
  * [58] R. Chong, S. Li, W. Yuan, and J. Yuan, “Outage analysis for OTFS-based single user and multi-user transmissions,” in _IEEE Int. Conf. Commun. Workshops (ICC Workshops)_ , 2022, pp. 746–751.
  * [59] F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, “Joint radar and communication design: Applications, state-of-the-art, and the road ahead,” _IEEE Trans. Commun._ , vol. 68, no. 6, pp. 3834–3862, Jun. 2020\.
  * [60] F. Liu, C. Masouros, A. Li, H. Sun, and L. Hanzo, “MU-MIMO communications with MIMO radar: From co-existence to joint transmission,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 4, pp. 2755–2770, Apr. 2018.
  * [61] W. Yuan, Z. Wei, S. Li, J. Yuan, and D. W. K. Ng, “Integrated sensing and communication-assisted orthogonal time frequency space transmission for vehicular networks,” __to appear in_ IEEE J. Sel. Topics. Signal Process._, vol. 15, no. 6, pp. 1515–1528, Oct. 2021.
  * [62] L. Gaudio, M. Kobayashi, G. Caire, and G. Colavolpe, “On the effectiveness of OTFS for joint radar parameter estimation and communication,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 9, pp. 5951–5965, Sep. 2020.
  * [63] L. Gaudio, M. Kobayashi, G. Caire, and G. Colavolpe, “Hybrid digital-analog beamforming and MIMO radar with OTFS modulation,” _arXiv preprint arXiv:2009.08785_ , 2020.
  * [64] P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, “Orthogonal time frequency space (OTFS) modulation based radar system,” in _IEEE Radar Conf. (RadarConf)_ , 2019, pp. 1–6.
  * [65] W. Yuan, S. Li, Z. Wei, J. Yuan, and D. W. Kwan Ng, “Bypassing channel estimation for OTFS transmission: An integrated sensing and communication solution,” in _IEEE Proc. Wireless Commun. Net. Conf. Workshop_ , 2021, pp. 1–5.
  * [66] A. F. Molisch, _Wireless Communications_. John Wiley & Sons, 2012, vol. 34.
  * [67] _Evolved Universal Terrestrial Radio Access: Further Advancements for E-UTRA Physical Layer Aspects_. document 3GPP TR 36.814, Oct. 2010.
  * [68] P. Bello, “Characterization of randomly time-variant linear channels,” _IEEE Trans. Commun. Syst._ , vol. 11, no. 4, pp. 360–393, Dec. 1963.
  * [69] W. C. Jakes and D. C. Cox, _Microwave Mobile Communications_. Wiley-IEEE press, 1994.
  * [70] G. Matz, “On non-WSSUS wireless fading channels,” _IEEE Trans. Wireless Commun._ , vol. 4, no. 5, pp. 2465–2478, Jul. 2005.
  * [71] D. Gabor, “Theory of communication. Part 1: The analysis of information,” _J. IEE_ , vol. 93, no. 26, pp. 429–441, 1946.
  * [72] J. B. Anderson, F. Rusek, and V. Öwall, “Faster-than-Nyquist signaling,” _Proc. IEEE_ , vol. 101, no. 8, pp. 1817–1830, Aug. 2013.
  * [73] S. Li, J. Yuan, B. Bai, and N. Benvenuto, “Code based channel shortening for faster-than-Nyquist signaling: Reduced-complexity detection and code design,” _IEEE Trans. Commun._ , vol. 68, no. 7, pp. 3996–4011, Jul. 2020\.
  * [74] S. Li, B. Bai, J. Zhou, P. Chen, and Z. Yu, “Reduced-complexity equalization for faster-than-Nyquist signaling: New methods based on Ungerboeck observation model,” _IEEE Trans. Commun._ , vol. 66, no. 3, pp. 1190–1204, Mar. 2017.
  * [75] S. Li, W. Yuan, J. Yuan, B. Bai, D. W. K. Ng, and L. Hanzo, “Time-domain vs. frequency-domain equalization for FTN signaling,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 8, pp. 9174–9179, Jun. 2020.
  * [76] S. Li, B. Bai, J. Zhou, Q. He, and Q. Li, “Superposition coded modulation based faster-than-Nyquist signaling,” _Wireless Commun. Mobile Comput._ , vol. 2018, May 2018.
  * [77] S. Li, J. Yuan, and B. Bai, “Code based channel shortening for faster-than-Nyquist signaling,” in _IEEE Int. Conf. Commun. (ICC)_ , 2020, pp. 1–6.
  * [78] F. Rusek and J. B. Anderson, “Constrained capacities for faster-than-Nyquist signaling,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 2, pp. 764–775, Feb. 2009.
  * [79] S. Li, Z. Wei, W. Yuan, J. Yuan, B. Bai, D. W. K. Ng, and L. Hanzo, “Faster-than-Nyquist asynchronous NOMA outperforms synchronous NOMA,” __to appear in_ IEEE J. Sel. Areas Commun., arXiv preprint arXiv:2111.08258v2_, 2022.
  * [80] S. K. Mohammed, “Time-domain to delay-Doppler domain conversion of OTFS signals in very high mobility scenarios,” _IEEE Trans. Veh. Technol._ , vol. 70, no. 6, pp. 6178–6183, Jun. 2021.
  * [81] A. Pfadler, P. Jung, and S. Stanczak, “Pulse-shaped OTFS for V2X short-frame communication with tuned one-tap equalization,” in _Int. ITG Workshop on Smart Antennas_ , 2020, pp. 1–6.
  * [82] Z. Zhu and M. B. Wakin, “On the asymptotic equivalence of circulant and Toeplitz matrices,” _IEEE Trans. Inf. Theory_ , vol. 63, no. 5, pp. 2975–2992, Mar. 2017.
  * [83] Cohere, “6G technology,” https://www.cohere-technologies.com/technology/6g/.
  * [84] F. R. Kschischang, B. J. Frey, and H. -A. Loeliger, “Factor graphs and the sum-product algorithm,” _IEEE Trans. Inf. Theory_ , vol. 47, no. 2, pp. 498–519, Feb. 2001.
  * [85] J. Ma and L. Ping, “Orthogonal AMP,” _IEEE Access_ , vol. 5, pp. 2020–2033, 2017.
  * [86] Q. Guo and J. Xi, “Approximate message passing with unitary transformation,” _arXiv preprint arXiv:1504.04799_ , 2015.
  * [87] Z. Yuan, Q. Guo, and M. Luo, “Approximate message passing with unitary transformation for robust bilinear recovery,” _IEEE Trans. Signal Process._ , vol. 69, pp. 617–630, Dec. 202.
  * [88] M. Luo, Q. Guo, M. Jin, Y. C. Eldar, D. D. Huang, and X. Meng, “Unitary approximate message passing for sparse Bayesian learning,” _arXiv preprint arXiv:2101.09954_ , 2021.
  * [89] J. Ma, X. Yuan, and L. Ping, “Turbo compressed sensing with partial DFT sensing matrix,” _IEEE Signal Process. Lett._ , vol. 22, no. 2, pp. 158–161, 2015.
  * [90] H. Wen, W. Yuan, and S. Li, “Downlink OTFS non-orthogonal multiple access receiver design based on cross-domain detection,” in _IEEE Int. Conf. Commun. Workshops (ICC Workshops)_ , 2022, pp. 928–933.
  * [91] Q. Guo and D. D. Huang, “A concise representation for the soft-in soft-out LMMSE detector,” _IEEE Commun. Lett._ , vol. 15, no. 5, pp. 566–568, May 2011.
  * [92] S. M. Kay, _Fundamentals of Statistical Signal Processing_. Prentice Hall PTR, 1993.
  * [93] G. Forney, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” _IEEE Trans. Inf. Theory_ , vol. 18, no. 3, pp. 363–378, May 1972.
  * [94] G. Ungerboeck, “Adaptive maximum-likelihood receiver for carrier-modulated data-transmission systems,” _IEEE Trans. Commun._ , vol. 22, no. 5, pp. 624–636, May 1974.
  * [95] A. Lozano, A. M. Tulino, and S. Verdú, “Mercury/waterfilling: Optimum power allocation with arbitrary input constellations,” in _IEEE Proc. Int. Sym. Inf. Theory_ , 2005, pp. 1773–1777.
  * [96] E. Kılıç and P. Stanica, “The inverse of banded matrices,” _J. Comput. Applied Math._ , vol. 237, no. 1, pp. 126–135, 2013.
  * [97] P. Bickel and M. Lindner, “Approximating the inverse of banded matrices by banded matrices with applications to probability and statistics,” _Theory Prob. & Its App._, vol. 56, no. 1, pp. 1–20, 2012.
  * [98] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” _IEEE Trans. Inf. Theory_ , vol. 44, no. 2, pp. 744–765, Mar. 1998.
  * [99] B. Vucetic and J. Yuan, _Space-time coding_. John Wiley & Sons, 2003.
  * [100] B. Lu and X. Wang, “Space-time code design in OFDM systems,” in _IEEE. Global Telecommun. Conf._ , vol. 2, 2000, pp. 1000–1004.
  * [101] X. Liu, C. Liu, Y. Li, B. Vucetic, and D. W. K. Ng, “Deep residual learning-assisted channel estimation in ambient backscatter communications,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 2, pp. 339–343, Feb. 2021.
  * [102] C. Liu, X. Liu, D. W. K. Ng, and J. Yuan, “Deep residual learning for channel estimation in intelligent reflecting surface-assisted multi-user communications,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 2, pp. 898–912, Feb. 2022.
  * [103] C. Liu, Z. Wei, D. W. K. Ng, J. Yuan, and Y. C. Liang, “Deep transfer learning for signal detection in ambient backscatter communications,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 3, pp. 1624–1638, Mar. 2021.
  * [104] C. Liu, J. Wang, X. Liu, and Y.-C. Liang, “Deep CM-CNN for spectrum sensing in cognitive radio,” _IEEE J. Sel. Areas Commun._ , vol. 37, no. 10, pp. 2306–2321, Oct. 2019.
  * [105] M. Rothstein, _The Gram matrix, orthogonal projection, and volume_.  _Tutorial_ , Dept. of Math., Georgia University.
  * [106] J. Yuan, Z. Chen, B. Vucetic, and W. Firmanto, “Performance and design of space-time coding in fading channels,” _IEEE Trans. Commun._ , vol. 51, no. 12, pp. 1991–1996, Dec. 2003.
  * [107] A. Papoulis and S. U. Pillai, _Probability, Random Variables, and Stochastic Processes_. Tata McGraw-Hill Education, 2002.
  * [108] W. Ryan and S. Lin, _Channel Codes: Classical and Modern_. Cambridge university press, 2009.
  * [109] J. Ventura-Traveset, G. Caire, E. Biglieri, and G. Taricco, “Impact of diversity reception on fading channels with coded modulation. I. coherent detection,” _IEEE Trans. Commun._ , vol. 45, no. 5, pp. 563–572, May 1997\.
  * [110] A. Stefanov and T. M. Duman, “Performance bounds for space-time trellis codes,” _IEEE Trans. Inf. Theory_ , vol. 49, no. 9, pp. 2134–2140, Sep. 2003\.
  * [111] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Information-theoretic and communications aspects,” _IEEE Trans. Inf. Theory_ , vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
  * [112] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate (corresp.),” _IEEE Trans. Inf. Theory_ , vol. 20, no. 2, pp. 284–287, Mar. 1974.
  * [113] J. Yuan and J. Choi, “Channel adaptive space-time (CAST) coding and precoding for wireless downlink packet services,” Interim Report, 2006.
  * [114] Y. Mostofi and D. C. Cox, “ICI mitigation for pilot-aided OFDM mobile systems,” _IEEE Trans. Wireless Commun._ , vol. 4, no. 2, pp. 765–774, Mar. 2005.
  * [115] 3GPP, “NR; Multiplexing and channel coding,” Release 15, Technical Specification (TS) 38.212, 2017.
  * [116] ——, “Group Radio Access Network, Evolved Universal Terrestrial Radio Access, Multiplexing and Channel Coding,” Release 8, Technical Specification (TS) 36.212, 2007.
  * [117] J. Taylor, “The condition of Gram matrices and related problems,” _Proc. the Royal Soc. of Edinburgh_ , vol. 80, no. 1-2, pp. 45–56, Nov. 1978\.
  * [118] S. Li, W. Yuan, C. Liu, Z. Wei, J. Yuan, B. Bai, and D. W. K. Ng, “A novel ISAC transmission framework based on spatially-spread orthogonal time frequency space modulation,” __to appear in_ IEEE J. Sel. Areas Commun._, 2022.
  * [119] S. Li, W. Yuan, J. Yuan, and C. Giuseppe, “On the potential of spatially-spread orthogonal time frequency space modulation for ISAC transmissions,” __to appear in_ Proc. - ICASSP IEEE Int. Conf. Acoust. Speech Signal Process._, pp. 1–5, 2022.
  * [120] C. Liu, W. Yuan, S. Li, X. Liu, D. W. K. Ng, and Y. Li, “Learning-based predictive beamforming for integrated sensing and communication in vehicular networks,” __to appear in_ IEEE J. Sel. Areas Commun._, 2022.
  * [121] S. Shrestha, M. D. Balachandran, M. Agarwal, L.-H. Zou, and K. Varahramyan, “A method to measure radar cross section parameters of antennas,” _IEEE Trans. Antennas Propag._ , vol. 56, no. 11, pp. 3494–3500, Nov. 2008.
  * [122] F. Liu, L. Zhou, C. Masouros, A. Li, W. Luo, and A. Petropulu, “Toward dual-functional radar-communication systems: Optimal waveform design,” _IEEE Trans. Signal Process._ , vol. 66, no. 16, pp. 4264–4279, Aug. 2018\.
  * [123] C. Liu, W. Yuan, S. Li, X. Liu, D. W. K. Ng, and Y. Li, “Predictive beamforming for integrated sensing and communication in vehicular networks: A deep learning approach,” in _Proc. IEEE Int. Conf. Commun._ , 2022, pp. 1–6.
  * [124] R. A. Horn and C. R. Johnson, _Matrix Analysis_. Cambridge university press, 2012.
  * [125] Y. Liao, M. Qiu, and J. Yuan, “Design and analysis of delayed bit-interleaved coded modulation with LDPC codes,” _IEEE Trans. Commun._ , vol. 69, no. 6, pp. 3556–3571, Mar. 2021.
  * [126] Y. Liao, L. Yang, J. Yuan, K. Huang, R. Leung, and J. Du, “LDPC code design for delayed bit-interleaved coded modulation,” in _IEEE Inf. Theory Workshop (ITW)_ , 2019, pp. 1–5.
  * [127] P. Kang, K. Cai, X. He, S. Li, and J. Yuan, “Generalized mutual information-maximizing quantized decoding of LDPC codes with layered scheduling,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 7, pp. 7258–7273, Jul. 2022.
  * [128] X. Wu, M. Qiu, and J. Yuan, “Partially information coupled bit-interleaved polar coded modulation,” _IEEE Trans. Commun._ , vol. 69, no. 10, pp. 6409–6423, Oct. 2021.
  * [129] J. Zhang, B. Bai, S. Li, M. Zhu, and H. Li, “Tail-biting globally-coupled LDPC codes,” _IEEE Trans. Commun._ , vol. 67, no. 12, pp. 8206–8219, Dec. 2019.
  * [130] S. Li, J. Zhang, B. Bai, X. Ma, and J. Yuan, “Self-superposition transmission: A novel method for enhancing performance of convolutional codes,” in _Int. Symp. Turbo Codes Iterative Inf. Process. (ISTC)_ , 2018, pp. 1–5.
  * [131] M. Qiu, X. Wu, A. Graell i Amat, and J. Yuan, “Analysis and design of partially information- and partially parity-coupled Turbo codes,” _IEEE Trans. Commun._ , vol. 69, no. 4, pp. 2107–2122, Apr. 2021.
  * [132] B. Liu, Y. Xie, and J. Yuan, “A deep learning assisted node-classified redundant decoding algorithm for BCH codes,” _IEEE Trans. Commun._ , vol. 68, no. 9, pp. 5338–5349, Sep. 2020.
  * [133] Z. Wei, L. Yang, D. W. K. Ng, J. Yuan, and L. Hanzo, “On the performance gain of NOMA over OMA in uplink communication systems,” _IEEE Trans. Commun._ , vol. 68, no. 1, pp. 536–568, Jan. 2020.
  * [134] Z. Wei, L. Zhao, J. Guo, D. W. K. Ng, and J. Yuan, “Multi-beam NOMA for hybrid mmWave systems,” _IEEE Trans. Commun._ , vol. 67, no. 2, pp. 1705–1719, Feb. 2019.
  * [135] S. Li, Z. Wei, W. Yuan, J. Yuan, B. Bai, and D. W. K. Ng, “On the achievable rates of uplink NOMA with asynchronized transmission,” in _IEEE Proc. Wireless Commun. Net. Conf._ , 2021, pp. 1–7.
  * [136] S. Hu, Z. Wei, Y. Cai, C. Liu, D. W. K. Ng, and J. Yuan, “Robust and secure sum-rate maximization for multiuser MISO downlink systems with self-sustainable IRS,” _IEEE Trans. Commun._ , vol. 69, no. 10, pp. 7032–7049, Oct. 2021.
  * [137] Y. Cai, Z. Wei, S. Hu, C. Liu, D. W. K. Ng, and J. Yuan, “Resource allocation and 3D trajectory design for power-efficient IRS-assisted UAV-NOMA communications,” __to appear in_ IEEE Trans. Wireless Commun._, pp. 1–1, 2022.
  * [138] C. Liu, W. Yuan, Z. Wei, X. Liu, and D. W. K. Ng, “Location-aware predictive beamforming for UAV communications: A deep learning approach,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 3, pp. 668–672, Mar. 2020.
  * [139] C. Liu, J. Wang, X. Liu, and Y.-C. Liang, “Maximum eigenvalue-based goodness-of-fit detection for spectrum sensing in cognitive radio,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 8, pp. 7747–7760, Aug. 2019.
  * [140] C. Liu, H. Li, J. Wang, and M. Jin, “Optimal eigenvalue weighting detection for multi-antenna cognitive radio networks,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 4, pp. 2083–2096, Apr. 2017.
  * [141] C. Liu, H. Li, and M. Jin, “Blind central-symmetry-based feature detection for spatial spectrum sensing,” _IEEE Trans. Veh. Technol._ , vol. 65, no. 12, pp. 10 147–10 152, Dec. 2016.
  * [142] C. Liu, M. Li, and M.-L. Jin, “Blind energy-based detection for spatial spectrum sensing,” _IEEE Wireless Commun. Lett._ , vol. 4, no. 1, pp. 98–101, Feb. 2015.
  * [143] Z. Sun, Z. Wei, N. Yang, and X. Zhou, “Two-tier communication for UAV-enabled massive IoT systems: Performance analysis and joint design of trajectory and resource allocation,” _IEEE J. Sel. Areas Commun._ , vol. 39, no. 4, pp. 1132–1146, Apr. 2021.
  * [144] Z. Wei, Y. Cai, Z. Sun, D. W. K. Ng, J. Yuan, M. Zhou, and L. Sun, “Sum-rate maximization for IRS-assisted UAV OFDMA communication systems,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 4, pp. 2530–2550, Apr. 2021\.
  * [145] Z. Wei, D. W. Kwan Ng, and J. Yuan, “NOMA for hybrid MmWave communication systems with beamwidth control,” _IEEE J. Sel. Top. Signal Process._ , vol. 13, no. 3, pp. 567–583, Jun. 2019.
  * [146] Z. Wei, X. Yu, D. W. K. Ng, and R. Schober, “Resource allocation for simultaneous wireless information and power transfer systems: A tutorial overview,” _Proc. IEEE_ , vol. 110, no. 1, pp. 127–149, Jan. 2022.
  * [147] M. Qiu, Y.-C. Huang, and J. Yuan, “Discrete signaling and treating interference as noise for the Gaussian interference channel,” _IEEE Trans. Inf. Theory_ , vol. 67, no. 11, pp. 7253–7284, Nov. 2021.
  * [148] ——, “Downlink non-orthogonal multiple access without SIC for block fading channels: An algebraic rotation approach,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 8, pp. 3903–3918, Aug. 2019.
  * [149] M. Qiu, Y.-C. Huang, J. Yuan, and C.-L. Wang, “Lattice-partition-based downlink non-orthogonal multiple access without SIC for slow fading channels,” _IEEE Trans. Commun._ , vol. 67, no. 2, pp. 1166–1181, Feb. 2019\.
  * [150] M. Qiu, Y.-C. Huang, S.-L. Shieh, and J. Yuan, “A lattice-partition framework of downlink non-orthogonal multiple access without SIC,” _IEEE Trans. Commun._ , vol. 66, no. 6, pp. 2532 – 2546, Jun. 2018.
  * [151] B. Liu, S. Li, Y. Xie, and J. Yuan, “A novel sum-product detection algorithm for faster-than-Nyquist signaling: A deep learning approach,” _IEEE Trans. Commun._ , vol. 69, no. 9, pp. 5975–5987, Sep. 2021.
  * [152] ——, “Deep learning assisted sum-product detection algorithm for faster-than-Nyquist signaling,” in _IEEE Inf. Theory Workshop (ITW)_ , 2019, pp. 1–5.
  * [153] Y. K. Enku, B. Bai, F. Wan, C. U. Guyo, I. N. Tiba, C. Zhang, and S. Li, “Two-dimensional convolutional neural network-based signal detection for OTFS systems,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 11, pp. 2514–2518, Nov. 2021.
  * [154] Y. K. Enku, B. Bai, S. Li, M. Liu, and I. N. Tiba, “Deep-learning based signal detection for MIMO-OTFS systems,” in _IEEE Int. Conf. Commun. Workshops (ICC Workshops)_ , 2022, pp. 1–5.