DD domain symbols are assumed to be Gaussian distributed. We note that such a
case is not practically important but it can provide some interesting insights
for the analysis of the proposed algorithm. In particular, we have the
following Proposition.

Proposition 5-2 _(Detection Performance with Gaussian Constellation in DD
Domain)_ : For the case where the DD domain symbols are Gaussian distributed,
the DD domain detection cannot provide any error performance improvement,
i.e.,
$v_{z}^{{p},{\rm{DD}}}\left(l\right)=v_{z}^{{a},{\rm{DD}}}\left(l\right)$.

_Proof_ : With the Gaussian assumption, we have

$\displaystyle v_{z}^{p,{\rm{DD}}}\left(l\right)$
$\displaystyle=MSE\left(\eta\right)={\mathbb{E}}\left[{{{\left|{x-{\mathbb{E}}\left[{x|x+\xi}\right]}\right|}^{2}}}\right]$
$\displaystyle={\mathbb{E}}\left[{{{\left|\xi\right|}^{2}}}\right]=\frac{1}{\eta}=v_{z}^{{a},{\rm{DD}}}\left(l\right).$
(5.32)

This completes the proof of Proposition 5-2.$\hfill\blacksquare$

Proposition 5-2 suggests that, if the DD domain constellation is Gaussian
distributed, iteratively updating the extrinsic information between the TD
domain and DD domain will not introduce any error performance gain. In other
words, the error performance improvement is due to the non-Gaussian
constellation constraint in the DD domain. Intuitively speaking, the DD domain
detection can be viewed as a de-noising operation. If the constellation is
Gaussian distributed, the ML detection will give the detection output as
$x+\xi$ because ${\mathbb{E}}\left[{x|x+\xi}\right]=x+\xi$ always holds, thus
it cannot correct any error induced by the noise. On the other hand, when the
DD domain constellation is not Gaussian distributed, applying iterations
crossing time and DD domain can potentially improve the error performance.
This is because the TD domain estimation in module A assumes that the $\bf z$
is a Gaussian vector due to the spreading effect of ISFFT, which does not take
advantage of the DD domain constellation constraint. Therefore, by performing
DD domain symbol detection, the DD domain constellation constraint is
exploited by the proposed algorithm, which can lead to a potential error
performance improvement.

Fig. 5.2 shows the TD domain MSE performance of the proposed algorithm, where
$N=32$ and $M=64$, respectively. Without loss of generality, the TD domain
effective channel ${\bf{H}}_{\rm{TD}}^{{\rm{eff}}}$ is generated according to
(3.74), where $P=4$ and the channel coefficients are
$[-0.27+0.35i,0.17+0.01i,0.56-0.33i,0.31-0.56i]$, the delay indices are
$[0,9,4,7]$, and the Doppler indices are $[4.70,-2.26,1.23,-3.46]$,
respectively. In specific, we consider two constellation mappings at different
SNRs, including the quadrature phase shift keying (QPSK) at
${E_{s}}/{N_{0}}=12$ dB and 16-quadrature amplitude modulation (16-QAM) at
${E_{s}}/{N_{0}}=17$ dB and show state $v_{z}^{a,{\rm{T}}}\left({l}\right)$
and the corresponding MSE values in Fig. 5.2. As observed from the figure,
with an increased number of iterations, the MSE performance of the proposed
algorithm first decreases and then saturates at MSEs around $1.3\times
10^{-4}$ for the QPSK case and around $1.6\times 10^{-3}$ for the 16-QAM case.
Meanwhile, the derived state evolution shows a close match to the actual MSE
performance. This observation indicates that the derived state evolution is
consistent with the simulation results.

Figure 5.2: TD domain MSE performance for OTFS modulation with $P=4$, where
the frame contains $2048$ DD domain QPSK or 16-QAM symbols. In specific, the
SNR for the QPSK case is ${E_{s}}/{N_{0}}=12$ dB, while that for the 16-QAM
case is ${E_{s}}/{N_{0}}=17$ dB.

####  5.2.2 Analysis of Effective DD Domain SNR

In this subsection, we will investigate the converged error performance. In
particular, we are interested in the effective DD domain SNR $\eta_{\rm
DD}(l)$, because the DD domain detection is a simple component-wise operation,
where $\eta_{\rm DD}(l)$ determines the BER performance for a given
constellation. Let us define the second-order TD domain OTFS channel matrix
${\bf{G}}_{\rm{TD}}^{{\rm{eff}}}\buildrel\Delta\over{=}{\bf{H}}_{\rm{TD}}^{{\rm{eff}}}{\left({{\bf{H}}_{\rm{TD}}^{{\rm{eff}}}}\right)^{\rm{H}}}$.
Noticing that ${{\bf{G}}_{\rm{TD}}^{{\rm{eff}}}}$ is a Hermitian matrix by its
definition, there exists a unitary matrix $\bf U$ such that
${\bf{G}}_{\rm{TD}}^{{\rm{eff}}}={\bf{U\Lambda}}{{\bf{U}}^{\rm{H}}}$, where
${\bf{\Lambda}}$ is a diagonal matrix whose $(k,k)$-th entry is the $k$-th
eigenvalue ${\lambda_{k}}$ of ${\bf{G}}_{\rm{TD}}^{{\rm{eff}}}$. Thus, we can
rewrite the _a posteriori_ variance $v_{z}^{p,{\rm{T}}}\left(l\right)$ in
(5.25) according to the eigenvalues ${\lambda_{k}}$ as shown in the following
Lemma.

Lemma 5-1 _(TD domain a posteriori variance)_ : The TD domain _a posteriori_
variance $v_{z}^{p,{\rm{T}}}\left(l\right)$ in (5.25) can be simplified by

$\displaystyle
v_{z}^{p,{\rm{T}}}\left(l\right)=v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right)}}{{MN}}\sum\limits_{k=1}^{MN}{\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right){\lambda_{k}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right){\lambda_{k}}+{N_{0}}}}}.$
(5.33)

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

In order to obtain some important insights of the effective DD domain SNR
$\eta_{\rm DD}(l)$, we need to investigate the property of the eigenvalues
${\lambda_{k}}$. For the ease of further derivation, let us consider the
following assumption.

Assumption 5-2: The delay index associated to each resolvable path is
different to each other, i.e., ${l_{i}}\neq{l_{j}},\forall i\neq j,1\leq
i,j\leq P$.

It should be noted that the value of the delay index depends on the specific
reflectors corresponding to each resolvable path. Furthermore, in the case
where the maximum delay index $l_{\rm max}$ is much larger than $P$, it is
unlikely to have a channel realization where different paths share the same
delay index. However, later we will show that even without the above
assumption, our following derivation can still provide an accurate estimate
for the effective DD domain SNR $\eta_{\rm DD}(l)$. With Assumption 5-2 in
hand, we can derive the following Lemma for ${\bf G}_{\rm TD}^{\rm eff}$.

Lemma 5-2 _(Main diagonal entries of ${\bf G}_{\rm TD}^{\rm eff}$)_: Under
Assumption 5-2, the main diagonal entries of ${\bf G}_{\rm TD}^{\rm eff}$ are
of the same value ${\left\|{\bf{h}}\right\|^{2}}$, where
${\bf{h}}={\left[{{h_{1}},{h_{2}},...,{h_{P}}}\right]^{\rm{T}}}$ is the path
gain vector.

_Proof_ : The Lemma can be derived by noticing that
${{\bm{\Pi}}^{{l_{i}}}}{{\bm{\Delta}}^{{k_{i}}+{\kappa_{i}}}}{{\bm{\Delta}}^{-{k_{i}}-{\kappa_{i}}}}{{\bm{\Pi}}^{-{l_{i}}}}={{\bf{I}}_{MN}}$.$\hfill\blacksquare$

According to Lemma 5-1 and Lemma 5-2, we can now derive the lower-bound of the
DD domain _a priori_ variance $v_{z}^{a,{\rm{DD}}}\left(l\right)$. The
corresponding results are summarized in the following Theorem.

Theorem 5-2 _(Lower-bound of $v_{z}^{a,{\rm{DD}}}\left(l\right)$)_: Under
Assumption 2, the DD domain _a priori_ variance
$v_{z}^{a,{\rm{DD}}}\left(l\right)$ is lower-bounded by
$\frac{{N_{0}}}{{{{{\left\|{\bf{h}}\right\|}^{2}}}}}$, where the lower bound
becomes tighter if the TD domain _a priori_ variance
$v_{z}^{a,{\rm{T}}}\left(l\right)$ tends to zero and the lower bound is
achieved when $v_{z}^{a,{\rm{T}}}\left(l\right)=0$.

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

Immediately, we can derive the upper-bound of $\eta_{\rm DD}(l)$ based on
Theorem 5-2.

Corollary 5-1 _(Upper-bound of $\eta_{\rm DD}(l)$)_: Under Assumption 5-2, the
DD domain effective SNR ${\eta_{\rm{DD}}}\left({l}\right)$ is upper-bounded by
$\frac{{{{\left\|{\bf{h}}\right\|}^{2}}}}{{{N_{0}}}}$, and the upper-bound
becomes tighter with more iterations.

_Proof_ : The proof can be derived from (5.23) and Theorem
5-2.$\hfill\blacksquare$

It is interesting to see from Theorem 5-2 and Corollary 5-1 that the effective
DD domain SNR of the proposed algorithm can theoretically approach the maximum
receiver SNR for a given fading channel [8], with a sufficient number of
iterations. Equivalently, this observation indicates that the proposed
algorithm can approach the error performance of MLSE theoretically given a
sufficient number of iterations. This is because the applied DD domain
detection in (5.12) is derived from the ML detection rule, which is optimal in
the sense of minimizing the symbol error rate given the effective DD domain
SNR. Therefore, as the effective DD domain SNR approaches its maximum with an
increased number of iterations, the proposed algorithm gradually approaches
the MLSE performance given the channel conditions. It should be noted that the
MLSE can provide the optimal ML error performance, but usually requires a
prohibitively high complexity [8]. Therefore, the proposed algorithm can be
viewed as a type of reduced-complexity detection algorithm that can
potentially approach the optimal error performance. On the other hand, we note
that the above analysis is based on Assumption 5-2. For the case where
different resolvable paths share the same delay index, we will demonstrate
that the effective DD domain SNR also follows Corollary 5-1 by numerical
simulations.

Figure 5.3: Effective DD domain SNRs for OTFS modulation with different number
of iterations, where the frame contains $2048$ DD domain QPSK or 16-QAM
symbols. In specific, the SNR for the QPSK case is ${E_{s}}/{N_{0}}=14$ dB,
while that for the 16-QAM case is ${E_{s}}/{N_{0}}=17$ dB. The considered
wireless channel contains $P=4$ paths with different delay indices for each
path.

Fig. 5.3 shows the effective DD domain SNRs with respect to number of
iterations. Without loss of generality, the TD domain effective channel
${\bf{H}}_{\rm{TD}}^{{\rm{eff}}}$ is generated according to (3.74), where
$P=4$. Specifically, the channel coefficients are given by
$[-0.04-0.31i,0.40-0.11i,-0.43+0.18i,0.59+0.21i]$, the delay indices are given
by $[0,5,2,8]$, and the Doppler indices are given by
$[-3.08,3.45,-3.94,-0.72]$, respectively. Similarly, we consider both QPSK and
16-QAM constellations with ${E_{s}}/{N_{0}}=14$ dB and ${E_{s}}/{N_{0}}=17$
dB, respectively. Meanwhile, we also plot the SNR derived based on the state
evolution and the corresponding SNR upper bound, i.e.,
$\frac{{{{\left\|{\bf{h}}\right\|}^{2}}}}{{{N_{0}}}}$. As observed from the
figure, with an increased number of iterations, the effective DD domain SNR
increases. In specific, the derived SNR based on the state evolution shows a
close match to the actual SNR performance based on the simulation. More
importantly, the derived SNR upper bound agrees with the simulation results
and state evolution, and the bound becomes tighter as the number of iteration
increases, which is consistent with the above analysis.

Figure 5.4: Effective DD domain SNRs for OTFS modulation with different number
of iterations, where the frame contains $2048$ DD domain QPSK or 16-QAM
symbols. In specific, the SNR for the QPSK case is ${E_{s}}/{N_{0}}=14$ dB,
while that for the 16-QAM case is ${E_{s}}/{N_{0}}=17$ dB. The considered
wireless channel contains $P=4$ paths and the first two paths share the same
delay index.

To have a fair comparison, we show the effective DD domain SNRs with respect
to the number of iterations with a specific channel realization in Fig. 5.4,
where different resolvable paths share the same delay index. In specific, the
channel contains $P=4$ paths and the channel coefficients are
$[-0.21+0.27i,0.17+0.01i,0.22-0.66i,0.31-0.46i]$, the delay indices are
$[0,0,4,7]$, and the Doppler indices are $[-3.28,4.45,-1.94,-0.23]$,
respectively. We observe that both QPSK case and 16-QAM case have similar SNR
performance to the previous figure. This observation indicates that the above
analysis is also valid when different resolvable paths share the same delay
index.

####  5.2.3 Analysis of Detection Complexity

We first consider the computational complexity of module A. It is obvious that
the most computation relates to the matrix inverse in (5.2). Generally, the
computation complexity order of the matrix inversion is ${\cal
O}\left({{{\left({MN}\right)}^{3}}}\right)$. However, this complexity can be
further reduced by considering the sparse and banded structure of the TD
domain effective channel ${\bf{H}}_{\rm{TD}}^{{\rm{eff}}}$. For example, the
banded structure enables an LU decomposition-based method to calculate the
matrix inverse with a reduced complexity [96]. Furthermore, with some
appropriate approximations, some algorithms with an even lower complexity can
also be used for the calculation, such as the algorithm proposed in [97]. Due
to the space limitation, we leave the details of reduced-complexity TD domain
L-MMSE detection designs for our future work. As for module B, it is obvious
that the computational complexity is of order ${\cal O}\left({{MN}}\right)$
because of the component-wise operation. On the other hand, the computational
complexity for the domain transformation can be low by considering the special
structure of the corresponding kernels. In particular, the Kronecker products
${\bf{F}}_{N}\otimes{{\bf{I}}_{M}}$ and
${\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}$ in (5.7) and (5.16) can be
efficiently calculated based on FFT and IFFT and the corresponding computation
complexity will be ${\cal O}\left({MN\log N}\right)$. Therefore, we can
calculate the total detection complexity of the proposed algorithm per
iteration, which is given by ${\cal O}\left({{{\left({MN}\right)}^{3}}+2MN\log
N+MN}\right)$. It should be noted that the detection complexity does not
increase in the presence of the fractional Doppler indices. In comparison of
the proposed algorithm, the detection complexity of the optimal MLSE detection
is exponential to the number of non-zero entries per row/column of the
corresponding channel matrix, which can be as high as ${\cal O}\left({{\cal
X}{{\left({\mathbb{A}}\right)}^{MN}}}\right)$ when the fractional Doppler
shift exists. Therefore, the proposed algorithm can significantly reduce the
detection complexity compared to the optimal MLSE detection, especially in the
presence of fractional Doppler shifts.

###  5.3 Numerical Results

In this section, we will evaluate the BER performance of the proposed
algorithm by assuming perfect CSI. We consider the average BER performance
with a sufficient number of realizations of the channel. Specifically, the
corresponding channel matrix is generated according to (3.74). The channel
coefficients are randomly generated based on a uniform power delay profile,
and the delay and Doppler indices are randomly generated within the range of
$[0,l_{\rm max}]$ and $[-k_{\rm max},k_{\rm max}]$, where $l_{\rm max}=10$ and
$k_{\rm max}=5$, unless otherwise specified. We note that, as mentioned
before, the delay index can only be an integer number, while the Doppler index
can be a fractional number [25]. Without loss of generality, we consider the
QPSK modulated OTFS system with different numbers of paths, where $M=64$ and
$N=32$, respectively, unless otherwise specified. Specifically, we set the
subcarrier spacing as $15$ kHz, and the total bandwidth of the transmission is
$960$ kHz, unless otherwise specified. We also provide other detection methods
for comparison that include the MMSE detection based on the DD domain
effective channel ${\bf H}_{\rm DD}^{\rm eff}$, DD domain detection based on
the SPA [42], and the DD domain message passing algorithm in [25]. The
considered SPA detection is derived based on the graphical model corresponding
to the DD domain effective channel, whose computational complexity can be
${\cal O}\left({{\cal X}{{\left({\mathbb{A}}\right)}^{MN}}}\right)$ in the
case of complex fractional Doppler shifts [42]. In particular, the considered
SPA detection can theoretically approach the error performance of the optimal
MLSE detection and achieve the same performance when the graphical model does
not contain any cycle [42]. However, since the DD domain SPA detection
requires a very high detection complexity in the fractional Doppler case, we
only consider the integer Doppler case for simplicity.

Figure 5.5: BER performance for OTFS modulation with $P=4$, where the error
performance of the proposed algorithm is compared with that of DD domain MRC
and MMSE detections, as well as the SPA detection with integer Doppler shifts.

Fig. 5.5 shows the BER performance with $P=4$. As observed from the figure,
the proposed algorithm with only one iteration outperforms the maximum ratio
combining (MRC) scheme, where the MRC scheme is implemented by multiplexing
the Hermitian conjugate of the effective DD domain channel matrix with the DD
domain received symbol vector. Meanwhile, the error performance of the
proposed algorithm with one iteration is almost the same as that with the MMSE
detection. This observation indicates that applying the same detection method
for OTFS modulation over different domains can result in the similar error
performance. Furthermore, with the increased number of iterations, the
proposed algorithm outperforms the MMSE detection and almost achieves the
performance of SPA detection with only integer Doppler indices. In specific,
with BER below $1\times 10^{-4}$, the proposed algorithm with $5$ iterations
has only a marginal performance gap (around $0.2$ dB) compared to the SPA
detection. This observation clearly substantiates our theoretical analysis in
the previous section.

Figure 5.6: BER performance for OTFS modulation with fractional Doppler
shifts, where $P=10$.

Fig. 5.6 shows the BER performance with $P=10$. Under such a complex channel
condition, the DD domain effective channel ${\bf H}_{\rm DD}^{\rm eff}$ can be
very dense and conventional detection methods based on ${\bf H}_{\rm DD}^{\rm
eff}$ may not have a good error performance. As shown in the figure, the
proposed algorithm with one iteration performs almost the same with the DD
domain MMSE detection, which is consistent with the observation in the
previous figure. Meanwhile, the DD domain message passing algorithm [25] also
shows a similar error performance to both the DD domain MMSE detection and the
proposed algorithm with one iteration. However, with the increased number of
iterations, the proposed algorithm significantly outperforms the MMSE
detection and message passing algorithm. Specifically, at BER $\approx
1\times{10^{-3}}$, with only $2$ iterations, the error performance of the
proposed algorithm shows an around $3.2$ dB gain compared to that of the MMSE
detection and the gain to the message passing algorithm is even more.
Furthermore, with $5$ iterations, the SNR gain to the MMSE detection
performance is increased to around $4.1$ dB. This observation shows the
advantage of the proposed algorithm over conventional detection algorithms,
which also agrees with our previous theoretical analysis.

Figure 5.7: SNR gains vs. number of iterations, where both QPSK and 16-QAM
signals are considered with $P=4$ and $P=6$.

In Fig. 5.7, we provide the simulation results of the SNR gains of the
proposed algorithm relative to the MMSE detection in the presence of
fractional Doppler shifts, at BER $\approx{10^{-3}}$. We set $M=32$, $N=16$,
$l_{\rm max}=3$, and $k_{\rm max}=6$, where both QPSK and 16-QAM signals are
considered. As shown in the figure, the SNR gains increase with iterations
until saturating at their maximum values. We observe that $5$ iterations are
usually sufficient for acquiring the maximum SNR gains. We also notice that
the SNR gains increase with more paths for both QPSK and 16-QAM. Furthermore,
we observe that the proposed algorithm enjoys a larger SNR gain with a larger
constellation size, which is consistent with our discussions in Section 5.2.2.
This is because the error performance gap between the MMSE detection and ML
detection increases with an increased size of constellations, and the proposed
algorithm can substantially reduce this performance loss with more iterations.

###  5.4 Summary of the Chapter

In this chapter, we proposed a novel cross domain iterative detection for OTFS
modulation. We derived the state evolution for the proposed algorithm and
investigated its detection performance including both the MMSE performance and
the DD domain effective SNR. In particular, we show that the proposed
algorithm can approach the error performance of MLSE detection even in the
presence of complex fractional Doppler shifts, but only requires a much lower
detection complexity. Our analytical results are explicitly verified by
simulation results, where a significant performance improvement can be
observed compared to the conventional detection algorithms for channels with
fractional Doppler shifts. The cross domain signal processing may be a new
research direction for OTFS modulation and general multi-carrier modulation
schemes. Our future work will focus on the extension of the proposed algorithm
to general wireless channels. An interesting starting point for the extension
could be the application of the eigenvalue decomposition for the underlying
channel matrix, where two unitary matrices are comprised by left and right
eigenvectors. The details of this extension will be studied in the future.

###  5.5 Appendices of the Chapter

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

####  5.5.1 Justification of Problem Formulation (5.8)

The problem formulation (5.8) is based on the following relationship between
the extrinsic mean of $\bf z$ and the exact value of $\bf z$, i.e.,

$\displaystyle{\bf{m}}_{\bf{z}}^{e,{\rm{T}}}={\bf{z}}+{{\bf{\hat{w}}}},$
(5.34)

where ${{\bf{\hat{w}}}}$ is a white Gaussian noise sample vector with zero
mean and a diagonal covariance matrix
${\bf{C}}_{\bf{z}}^{a,{\rm{DD}}}={\bf{C}}_{\bf{z}}^{e,{\rm{T}}}$ standing for
the inaccuracy of the MMSE estimation in the TD domain. The rational behind
(5.34) is that unitary transformation
${{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}$ can suppress the residual
correlations due to the spreading effect of ${\bf{F}}_{N}^{\rm{H}}$ [89].

We note that the considered problem formulation is not the only way for DD
domain detection. A more straightforward way is to first convert the extrinsic
information ${\bf{m}}_{\bf{z}}^{e,{\rm{T}}}$ of the TD domain signal $\bf z$
to DD domain and then compare it with the DD domain symbol constellation. In
specific, we can directly apply the transformation
${{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}$ onto (5.34) and obtain the
corresponding DD domain symbol estimates, i.e.,

$\displaystyle{\bf{m}}_{\bf{x}}^{a,{\rm{DD}}}=\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){\bf{m}}_{\bf{z}}^{e,{\rm{T}}}={\bf{x}}+\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){{\bf{\hat{w}}}},$
(5.35)

where the term
$\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){{\bf{\hat{w}}}}$ is the
equivalent noise in the DD domain and its covariance matrix is denoted by
${\bf{C}}_{\bf{x}}^{a,{\rm{DD}}}$. Recalling (3.73), we have

$\displaystyle{\bf{C}}_{\bf{x}}^{a,{\rm{DD}}}$
$\displaystyle=\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){\bf{C}}_{\bf{z}}^{a,{\rm{DD}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right).$
(5.36)

It should be noted that in the asymptotical regime, i.e., $MN$ tends to
infinity, the diagonal entries of the diagonal matrix
${\bf{C}}_{\bf{z}}^{a,{\rm{DD}}}$ tend to be of the same value owing to the
law of large numbers, i.e.,
${\bf{C}}_{\bf{z}}^{a,{\rm{DD}}}\propto{{\bf{I}}_{MN}}$. In this case, we have
${\bf{C}}_{\bf{x}}^{a,{\rm{DD}}}={\bf{C}}_{\bf{z}}^{a,{\rm{DD}}}$ and the DD
domain symbol detection can be done in a straightforward symbol-by-symbol
fashion. However, in practice, the diagonal entries of
${\bf{C}}_{\bf{z}}^{a,{\rm{DD}}}$ may not be of the same value due to the
specific noise values. In this case, ${\bf{C}}_{\bf{x}}^{a,{\rm{DD}}}$ can be
non-diagonal and dense. As mentioned before, a common approach for this issue
is to discard the non-diagonal entries, which is simple but may introduce
performance loss. In contrast to the above solution, we can consider the DD
domain detection problem formulation in (5.8). As can be noticed from Section
5.1.3, the proposed DD domain detection algorithm based on (5.8) bypasses the
problem that the diagonal entries of ${\bf{C}}_{\bf{z}}^{a,{\rm{DD}}}$ are of
different values and can still be efficiently carried out in a symbol-by-
symbol fashion.

####  5.5.2 Proof of Proposition 5-1

According to the definition of extrinsic information, we can derive the
extrinsic mean $m_{\bf{x}}^{e,{\rm{DD}}}\left[k\right]$ by excluding the
contribution of $x[k]$ in (5.14). By noticing that
$m_{\bf{x}}^{p,{\rm{DD}}}\left[k\right]={\mathbb{E}}\left[{x\left[k\right]|{\bf{m}}_{\bf{z}}^{e,{\rm{T}}}}\right]={\mathbb{E}}\left[{x\left[k\right]|{\bf{m}}_{\bf{x}}^{a,{\rm{DD}}}}\right]$,
we have

$\displaystyle
m_{\bf{x}}^{e,{\rm{DD}}}\left[k\right]={\mathbb{E}}\left[{x\left[k\right]|{\bf{m}}_{\bf{x}}^{a,{\rm{DD}}}/m_{\bf{x}}^{a,{\rm{DD}}}\left[k\right]}\right],$
(5.37)

where
${{\bf{m}}_{\bf{x}}^{a,{\rm{DD}}}/m_{\bf{x}}^{a,{\rm{DD}}}\left[k\right]}$
denotes ${\bf m}_{\bf{x}}^{e,{\rm{DD}}}$ excluding the entry of
${m_{\bf{x}}^{a,{\rm{DD}}}\left[k\right]}$. Furthermore, it can be noticed
that the _a posteriori_ probability of $x[k]$ only relates to
${m_{\bf{x}}^{a,{\rm{DD}}}\left[k\right]}$ instead of other entries in ${\bf
m}_{\bf{x}}^{e,{\rm{DD}}}$. Therefore, it can be shown that
$m_{\bf{x}}^{e,{\rm{DD}}}\left[k\right]=0$ [89]. Similarly, it can be shown
that DD domain detection cannot provide any extrinsic information in terms of
the covariance matrix either, due to the component-wise operation [89]. This
completes the proof of Proposition 1.$\hfill\blacksquare$

####  5.5.3 Proof of Lemma 5-2

According to (5.25), we have

$\displaystyle v_{z}^{{p},{\rm{T}}}\left(l\right)=$ $\displaystyle
v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right)}\right)}^{2}}}}{{MN}}{\rm{Tr}}\left({{{\left({{\bf{H}}_{\rm{TD}}^{{\rm{eff}}}}\right)}^{\rm{H}}}{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right){\bf{H}}_{\rm{TD}}^{{\rm{eff}}}{{\left({{\bf{H}}_{\rm{TD}}^{{\rm{eff}}}}\right)}^{\rm{H}}}+{N_{0}}{{\bf{I}}_{MN}}}\right)}^{-1}}{\bf{H}}_{\rm{TD}}^{{\rm{eff}}}}\right)$
$\displaystyle=$ $\displaystyle
v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right)}\right)}^{2}}}}{{MN}}{\rm{Tr}}\left({{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right){\bf{G}}_{\rm{TD}}^{{\rm{eff}}}+{N_{0}}{{\bf{I}}_{MN}}}\right)}^{-1}}{\bf{G}}_{\rm{TD}}^{{\rm{eff}}}}\right).$
(5.38)

Furthermore, by considering
${\bf{G}}_{\rm{TD}}^{{\rm{eff}}}={\bf{U\Lambda}}{{\bf{U}}^{\rm{H}}}$, (5.38)
can be further simplified by (5.39), as shown at the top of next page.

$\displaystyle v_{z}^{{p},{\rm{T}}}\left(l\right)=$ $\displaystyle
v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right)}\right)}^{2}}}}{{MN}}{\rm{Tr}}\left({{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right){\bf{U\Lambda}}{{\bf{U}}^{\rm{H}}}+{N_{0}}{{\bf{I}}_{MN}}}\right)}^{-1}}{\bf{U\Lambda}}{{\bf{U}}^{\rm{H}}}}\right)$
$\displaystyle=$ $\displaystyle
v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right)}\right)}^{2}}}}{{MN}}{\rm{Tr}}\left({{\bf{U}}{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right){\bf{\Lambda}}+{N_{0}}{{\bf{I}}_{MN}}}\right)}^{-1}}{\bf{\Lambda}}{{\bf{U}}^{\rm{H}}}}\right)$
$\displaystyle=$ $\displaystyle
v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right)}}{{MN}}\sum\limits_{k=1}^{MN}{\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right){\lambda_{k}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right){\lambda_{k}}+{N_{0}}}}}.$
(5.39)

This completes the proof of Lemma 5-2.$\hfill\blacksquare$

####  5.5.4 Proof of Theorem 5-2

The proof is based on the application of Jensen’s inequality. Let us consider
the following function
$f\left(\lambda\right)={\frac{{v\lambda}}{{v\lambda+{N_{0}}}}}$, whose second-
order derivative with respect to $\lambda$ is of the form
${f^{{}^{\prime\prime}}}\left(\lambda\right)=-\frac{{2{v^{2}}{N_{0}}}}{{{{\left({v\lambda+{N_{0}}}\right)}^{3}}}}$.
It can be shown that with $v$ and $N_{0}$ strictly above zero, the above
function is a concave function. Furthermore, with $v$ close to zero,
${f^{{}^{\prime\prime}}}\left(\lambda\right)$ tends to be a zero, which
indicates that $f\left(\lambda\right)$ tends to be a linear function.
Therefore, according to Jensen’s inequality, it is obvious that

$\displaystyle\frac{1}{{MN}}\sum\limits_{k=1}^{MN}{f\left({{\lambda_{k}}}\right)}\leq
f\left({\frac{1}{{MN}}\sum\limits_{k=1}^{MN}{{\lambda_{k}}}}\right)=\frac{{\frac{v}{{MN}}\sum\limits_{k=1}^{MN}{{\lambda_{k}}}}}{{\frac{v}{{MN}}\sum\limits_{k=1}^{MN}{{\lambda_{k}}}+{N_{0}}}},$
(5.40)

where the bound becomes tighter with decreasing $v$ and the equality is
achieved when $v=0$. Therefore, we have

$\displaystyle v_{z}^{p,{\rm{T}}}\left(l\right)$
$\displaystyle=v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right)}}{{MN}}\sum\limits_{k=1}^{MN}{\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right){\lambda_{k}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right){\lambda_{k}}+{N_{0}}}}}$
$\displaystyle\geq
v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right)}\right)}^{2}}\frac{1}{{MN}}\sum\limits_{k=1}^{MN}{{\lambda_{k}}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right)\frac{1}{{MN}}\sum\limits_{k=1}^{MN}{{\lambda_{k}}}+{N_{0}}}},$
(5.41)

where the lower bound becomes tighter with the decrease of
$v_{z}^{{a},{\rm{T}}}\left(l\right)$ and the equality is achieved when
$v_{z}^{{a},{\rm{T}}}\left(l\right)$ becomes zero. Notice that
$\sum\limits_{k=1}^{MN}{{\lambda_{k}}}={\rm{Tr}}\left({{\bf{G}}_{\rm{TD}}^{{\rm{eff}}}}\right)$.
Thus, by considering Lemma 5-2, (5.41) becomes

$\displaystyle v_{z}^{{p},{\rm{T}}}\left(l\right)\geq
v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right)}\right)}^{2}}{{\left\|{\bf{h}}\right\|}^{2}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right){{\left\|{\bf{h}}\right\|}^{2}}+{N_{0}}}}.$
(5.42)

By substituting (5.42) into (5.24), after some manipulations, we arrive at

$\displaystyle v_{z}^{{a},{\rm{DD}}}\left(l\right)$
$\displaystyle=\frac{1}{{\frac{1}{{v_{z}^{{p},{\rm{T}}}\left(l\right)}}-\frac{1}{{v_{z}^{{a},{\rm{T}}}\left(l\right)}}}}$
$\displaystyle\geq\frac{1}{{\frac{1}{{v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right){{\left\|{\bf{h}}\right\|}^{2}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right){{\left\|{\bf{h}}\right\|}^{2}}+{N_{0}}}}}}-\frac{1}{{v_{z}^{{a},{\rm{T}}}\left(l\right)}}}}$
$\displaystyle=\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right)-\frac{{{{\left({v_{z}^{{a},{\rm{T}}}\left(l\right)}\right)}^{2}}{{\left\|{\bf{h}}\right\|}^{2}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right){{\left\|{\bf{h}}\right\|}^{2}}+{N_{0}}}}}}{{\frac{{v_{z}^{{a},{\rm{T}}}\left(l\right){{\left\|{\bf{h}}\right\|}^{2}}}}{{v_{z}^{{a},{\rm{T}}}\left(l\right){{\left\|{\bf{h}}\right\|}^{2}}+{N_{0}}}}}}=\frac{{{N_{0}}}}{{{{\left\|{\bf{h}}\right\|}^{2}}}}.$
(5.43)

This completes the proof of Theorem 5-2.$\hfill\blacksquare$

##  Chapter 6 Performance Analysis of Coded OTFS Systems over High-Mobility
Channels

In this chapter, we aim to analyze the error performance of coded OTFS systems
over general Rayleigh-fading WSSUS channels [48, 49]. To this end, we start
from the study of conditional PEP [98, 99] for a given channel realization. In
order to obtain an accurate performance analysis, we consider two cases of the
OTFS transmission depending on the number of independent resolvable paths of
the channel and derive the corresponding conditional PEPs. Since the exact
unconditional PEP is generally intractable [100], we resort to the application
of some proper bounding techniques to study the conditional PEP and derive the
unconditional performance upper bounds. Based on the unconditional performance
bound, the impact of channel coding parameters on the performance of OTFS
modulation is unveiled. In particular, we find that the squared Euclidean
distance between a pair of codewords is the key parameter that determines the
coding gain for coded OTFS systems, given the number of independent resolvable
paths. Therefore, the code design criterion is formulated to optimize the
coding gain by maximizing the minimum squared Euclidean distance between all
codeword pairs. The main contributions of this chapter can be summarized as
follows.

  * •

We investigate the conditional PEP of OTFS systems for a given channel
realization by studying the pairwise Euclidean distance between OTFS
codewords. Based on the conditional PEP, we derive the unconditional
performance upper bounds for OTFS systems, according to the number of
independent resolvable paths. We also show a few important properties of the
codeword difference matrix. Furthermore, according to the derived bounds, we
define the coding gain and diversity gain of OTFS systems. More importantly,
we show that the coding gain of OTFS systems depends on the squared Euclidean
distance and the number of independent resolvable paths.

  * •

According to the derived performance bounds, we show that there is a
fundamental trade-off between the diversity gain and the coding gain for OTFS
systems. In particular, the diversity gain of OTFS systems improves with the
number of independent resolvable paths, while the coding gain declines.

  * •

Based on the derived performance bounds, we propose our code design criterion
to optimize the coding gain, which is to maximize the minimum squared
Euclidean distance among all codeword pairs. In other words, traditional good
codes with a large minimum Euclidean distance can be directly applied to OTFS
systems.

  * •

We demonstrate a significant performance improvement achieved by the coded
OTFS modulation over the coded OFDM modulation over high-mobility channels by
numerical simulations. We also provide numerical results of coded OTFS systems
over high-mobility channels with various channel codes, such as classical
convolutional codes and state-of-art low-density parity-check (LDPC) codes.
Our performance analysis and code design are explicitly verified by these
results.

###  6.1 Error Performance Analysis

Without loss of generality, let us focus on the rectangular pulse shaped
point-to-point OTFS transmissions with fractional delay and fractional
Doppler. For the ease of derivation, let us write the DD domain system model
based on (3.81) by

$\displaystyle{\bf{y}}_{\rm
DD}={\bf{\Phi}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{x}}_{\rm
DD}\right){\bf{h}}+{\bf{w}},$ (6.1)

where ${{\bm{\omega}}_{\tau}}$ and ${{\bm{\omega}}_{\nu}}$ denote the vectors
of delay indices and Doppler indices, respectively, i.e.,
${{\bm{\omega}}_{\tau}}=\left[{{l_{1}}+{\iota_{1}},{l_{2}}+{\iota_{2}},\ldots,{l_{P}}+{\iota_{P}}}\right]^{\rm{T}}$,
${{\bm{\omega}}_{\nu}}=\left[{{k_{1}}+{\kappa_{1}},{k_{2}}+{\kappa_{2}},\ldots,{k_{P}}+{\kappa_{P}}}\right]^{\rm{T}}$,
respectively. In specific, we refer to
${\bf{\Phi}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{x}}_{\rm
DD}\right)$ as the _equivalent codeword matrix_ and it is a concatenated
matrix of size $MN\times P$ constructed by the column vector
${{\bf{\Xi}}_{i}}{\bf{x}}_{\rm DD}$ , i.e.,

${\bf{\Phi}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{x}_{\rm
DD}}\right)=\left[{{{\bf{\Xi}}_{1}}{\bf{x}_{\rm
DD}}\quad{{\bf{\Xi}}_{2}}{\bf{x}}_{\rm
DD}\quad\cdots\quad{{\bf{\Xi}}_{P}}{\bf{x}}_{\rm DD}}\right],$ (6.2)

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

${{\bf{\Xi}}_{i}}\buildrel\Delta\over{=}\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){{\bm{\Pi}}^{{l_{i}}+{\iota_{i}}}}{{\bm{\Delta}}^{{k_{i}}+{\kappa_{i}}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right),1\leq
i\leq P.$ (6.3)

In (6.1), $\bf{h}$ is the channel coefficient vector of size $P\times 1$,
i.e., ${\bf{h}}={\left[{{h_{1}},{h_{2}},...,{h_{P}}}\right]^{\rm{T}}}$, where
the elements in ${\bf{h}}$ are assumed to be independent and identically
distributed complex Gaussian random variables. Besides, we assume a uniform
power delay and Doppler profile of the channel such that the channel
coefficient $h_{i}$ has mean $\mu$ and variance $1/(2P)$ per real dimension
for $1\leq i\leq P$ and is independent from the delay and Doppler indices
[66]. In particular, we note that if $\mu=0$, $\left|{{h_{i}}}\right|$ follows
the Rayleigh distribution, which will be considered as a special case in our
error performance analysis and code design.

In order to investigate the theoretical error performance of the coded OTFS
systems, we assume that ideal CSI is available at the receiver, including
$\bf{h}$, ${{\bm{\omega}}_{\tau}}$, and ${{\bm{\omega}}_{\nu}}$111Some
references regarding the channel estimation for OTFS modulation can be found
in [33]. Meanwhile, recent advances of machine learning based methods can also
potentially improve the channel estimation and detection performance of OTFS
modulation [101, 102, 103, 104].. We note that matrix
${\bf{\Phi}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{x}}\right)$
depends on ${{\bm{\omega}}_{\tau}}$, ${{\bm{\omega}}_{\nu}}$, and the
transmitted symbol vector ${\bf{x}}_{\rm DD}$. Therefore, for a given channel
realization, we define the _conditional Euclidean distance_
${d_{\bf{h},{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}^{2}\left({{\bf{x}},{\bf{x^{\prime}}}}\right)}$
between a pair of codewords ${\bf{x}}$ and ${\bf{x^{\prime}}}$
(${\bf{x}}\neq{\bf{x^{\prime}}}$) as

$\displaystyle
d_{\bf{h},{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}^{2}\left({{\bf{x}},{\bf{x^{\prime}}}}\right)=$
$\displaystyle
d_{\bf{h},{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}^{2}\left({\bf{e}}\right)$
$\displaystyle\buildrel\Delta\over{=}$
$\displaystyle{\left\|{{{\bf{\Phi}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{e}}\right)}{\bf{h}}}\right\|^{2}}={{\bf{h}}^{\rm{H}}}{\bf{\Omega}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{e}}\right){\bf{h}},$
(6.4)

where ${\bf{e}}={\bf{x}}-{\bf{x^{\prime}}}$ is the corresponding codeword
difference (error) sequence and
${\bf{\Omega}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{e}}\right)={\left({{\bf{\Phi}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{e}}\right)}\right)^{\rm{H}}}\left({\bf{\Phi}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{e}}\right)\right)$
is referred to as the _codeword difference matrix_. Without loss of generality
and for notational simplicity, we henceforth drop the subscript of
${\bf{\Omega}}_{{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}\left({\bf{e}}\right)$
and we now have

${\bf{\Omega}}\left({\bf{e}}\right)\\!=\\!\\!\left[{\begin{array}[]{*{20}{c}}{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{1}^{\rm{H}}{{\bf{\Xi}}_{1}}{\bf{e}}}&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{1}^{\rm{H}}{{\bf{\Xi}}_{2}}{\bf{e}}}&\cdots&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{1}^{\rm{H}}{{\bf{\Xi}}_{P}}{\bf{e}}}\\\
{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{2}^{\rm{H}}{{\bf{\Xi}}_{1}}{\bf{e}}}&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{2}^{\rm{H}}{{\bf{\Xi}}_{2}}{\bf{e}}}&{}\hfil&\vdots\\\
\vdots&{}\hfil&\ddots&\vdots\\\
{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{P}^{\rm{H}}{{\bf{\Xi}}_{1}}{\bf{e}}}&\cdots&\cdots&{{{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{P}^{\rm{H}}{{\bf{\Xi}}_{P}}{\bf{e}}}\end{array}}\right].$
(6.5)

The conditional PEP is upper-bounded by [98, 99]

$\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{\bf{h}},{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)\leq\exp\left({-\frac{{{E_{s}}}}{{4{N_{0}}}}{d_{\bf{h},{{\bm{\omega}}_{\tau}},{{\bm{\omega}}_{\nu}}}^{2}\left({{\bf{x}},{\bf{x^{\prime}}}}\right)}}\right),$
(6.6)

where $E_{s}$ is the average symbol energy. Note that the codeword difference
matrix ${\bf{\Omega}}\left({\bf{e}}\right)$ is positive semidefinite Hermitian
with a rank $r$, where $r\leq P$. Let us denote by
$\left\\{{{{\bf{v}}_{1}},{{\bf{v}}_{2}},...,{{\bf{v}}_{P}}}\right\\}$ the
eigenvectors of ${\bf{\Omega}}\left({\bf{e}}\right)$ and
$\left\\{{{\lambda_{1}},{\lambda_{2}},...,{\lambda_{P}}}\right\\}$ the
corresponding nonnegative real eigenvalues sorted in the descending order,
where ${{\lambda_{i}}}>0$ for $1\leq i\leq r$ and ${{\lambda_{i}}}=0$ for
$r+1\leq i\leq P$. Thus, (6.6) can be further expanded as [98, 99]

$\Pr\left({{\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{\bf{h}},{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}}\right)\leq\exp\left({-\frac{{{E_{s}}}}{{4{N_{0}}}}\sum\limits_{i=1}^{r}{{\lambda_{i}}{{\left|{{{\tilde{h}}_{i}}}\right|}^{2}}}}\right),$
(6.7)

where ${{\tilde{h}}_{i}}={\bf{h}}^{\rm T}\cdot{{\bf{v}}_{i}}$, for $1\leq
i\leq r$. It can be shown that
$\left\\{{{{\tilde{h}}_{1}},{{\tilde{h}}_{2}},...,{{\tilde{h}}_{r}}}\right\\}$
are independent complex Gaussian random variables with mean
${\mu_{{{\tilde{h}}_{i}}}}=(\mathbb{E}\left[{\bf{h}}\right])^{\rm
T}\cdot{{\bf{v}}_{i}}$ and variance $1/(2P)$ per real dimension. Thus, it is
obvious that ${\small|{{{\tilde{h}}_{i}}}\small|}$ follows the Rician
distribution with a Rician factor
${K_{i}}={\left|{{\mu_{{{\tilde{h}}_{i}}}}}\right|^{2}}$ [98], and its PDF is
given by

$\displaystyle
p\left({\left|{{{\tilde{h}}_{i}}}\right|}\right)=2P\left|{{{\tilde{h}}_{i}}}\right|\exp\left({-P{{\left|{{{\tilde{h}}_{i}}}\right|}^{2}}-P{K_{i}}}\right){I_{0}}\left({2P\left|{{{\tilde{h}}_{i}}}\right|\sqrt{{K_{i}}}}\right),$
(6.8)

where ${I_{0}}(\cdot)$ denotes the Bessel function of the first kind. In the
following, we will target on the analysis of the unconditional PEP. To this
end, we aim to calculate the average of (6.7) over the channel distribution
according to (6.8). More specifically, we will discuss two important cases
depending on the number of independent resolvable paths $P$.

Remark 6-1: It has been defined in the previous works [9, 10, 11, 14] that the
rank of ${\bf{\Omega}}\left({\bf{e}}\right)$ is the diversity gain of the OTFS
modulation. Specifically, it has been shown in [11] that the diversity gain of
uncoded OTFS modulation systems can be one but the full diversity can be
obtained by suitable precoding schemes. Furthermore, [9] has shown that the
full diversity can be achieved almost surely for the case of $P=2$ when the
frame size is sufficiently large, even for uncoded OTFS modulation systems.

####  6.1.1 Error Performance Analysis for Coded OTFS systems

Notice that $\bf{h}$, ${{\bm{\omega}}_{\tau}}$, and ${{{\bm{\omega}}_{\nu}}}$
are independent from each other with the assumptions of WSSUS channel and the
uniform power delay profile [7]. Therefore, the unconditional PEP can be
derived by firstly averaging (6.7) over ${|{{{\tilde{h}}_{i}}}|}$ term by term
which results in

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)\leq\prod\limits_{i=1}^{r}{\frac{1}{{1+\frac{{{E_{s}}}}{{4{N_{0}}}}\cdot\frac{{{\lambda_{i}}}}{{P}}}}\exp\left({-\frac{{{K_{i}}\frac{{{E_{s}}}}{{4{N_{0}}}}\cdot\frac{{{\lambda_{i}}}}{{P}}}}{{1+\frac{{{E_{s}}}}{{4{N_{0}}}}\cdot\frac{{{\lambda_{i}}}}{{P}}}}}\right)}.$
(6.9)

Furthermore, we consider a special case where ${{K_{i}}}=0$ and
${\small|{{{\tilde{h}}_{i}}}\small|}$ follows the Rayleigh distribution, i.e.,
${\small|{{{h}_{i}}}\small|}$ also follows the Rayleigh distribution. In the
case of Rayleigh fading, (6.9) can be further simplified as

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)\leq{\left({\prod\limits_{i=1}^{r}{{\lambda_{i}}/P}}\right)^{-1}}{\left({\frac{{{E_{s}}}}{{4{N_{0}}}}}\right)^{-r}}=\frac{1}{{\prod\limits_{i=1}^{r}{{\lambda_{i}}}}}{\left({\frac{{{E_{s}}}}{{4{N_{0}}P}}}\right)^{-r}},$
(6.10)

It should be noted that (6.10) is consistent with the analysis in [10]. On the
other hand, the PEP in (6.10) depends on the delay and Doppler indices
${\bm{\omega}}_{\tau}$ and ${\bm{\omega}}_{\nu}$. In order to derive the
unconditional PEP, we need to find the statistical distribution for the term
$\prod\nolimits_{i=1}^{r}{{\lambda_{i}}}$ regarding the delay and Doppler
indices. Unfortunately, such a task is generally intractable [100] and is
normally handled by applying the Monte Carlo method without providing any
important insight. Instead of resorting to the Monte Carlo method, we apply
proper bounding techniques to evaluate the value of
$\prod\nolimits_{i=1}^{r}{{\lambda_{i}}}$ in order to obtain some general
results about the unconditional PEP. Specifically, we have the following
property.

Property 6-1 _(Gram matrix[105])_: Let
${\bar{\bf{u}}_{i}}\buildrel\Delta\over{=}{{\bf{\Xi}}_{i}}{\bf{e}}$, for
$1\leq i\leq P$, where ${\bf{\Xi}}_{i}$ is given by (6.3). Then,
$\left\\{{{\bar{\bf{u}}_{1}},{\bar{\bf{u}}_{2}},\ldots{\bar{\bf{u}}_{P}}}\right\\}$
form a list of vectors chosen from the $P$-dimensional complex inner-product
subspace $\mathbb{H}^{P}$. Thus, the codeword difference matrix
${\bf{\Omega}}\left({\bf{e}}\right)$ is positive semidefinite Hermitian and it
is a _Gram matrix_ corresponding to the vectors
$\left\\{{{\bar{\bf{{u}}}_{1}},{\bar{\bf{{u}}}_{2}},\ldots{\bar{\bf{{u}}}_{P}}}\right\\}$.

Based on the property of ${\bf{\Omega}}\left({\bf{e}}\right)$, we now
introduce four important lemmas, which will be served as the building blocks
for our error performance analysis for coded OTFS systems.

Lemma 6-1 _(Main diagonal elements of ${\bf{\Omega}}\left({\bf{e}}\right)$)_:
The main diagonal elements of the codeword difference matrix
${\bf{\Omega}}\left({\bf{e}}\right)$ are of the same value
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$, where
$d_{\rm{E}}^{2}\left({\bf{e}}\right)={{\bf{e}}^{\rm{H}}}{\bf{e}}$ is the
squared Euclidean distance for a pair of codewords ${\bf{x}}$ and
${\bf{x^{\prime}}}$ corresponding to the error sequence $\bf{e}$.

_Proof_ : By considering (6.2), the $i$-th diagonal element of
${\bf{\Omega}}\left({\bf{e}}\right)$ is given by
${{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i}^{\rm{H}}{{\bf{\Xi}}_{i}}{\bf{e}}$, and it
is equal to the inner product of ${{{{\bf{\bar{u}}}}_{i}}}$. Furthermore, we
have

$\displaystyle{\bf{\Xi}}_{i}^{\rm{H}}{{\bf{\Xi}}_{i}}=$
$\displaystyle\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){\left({{{\bm{\Delta}}^{{k_{i}}+{\kappa_{i}}}}}\right)^{\rm{H}}}{\left({{{\bm{\Pi}}^{{l_{i}}+{\iota_{i}}}}}\right)^{\rm{H}}}{{\bm{\Pi}}^{{l_{i}}+{\iota_{i}}}}{{\bm{\Delta}}^{{k_{i}}+{\kappa_{i}}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right)$
$\displaystyle=$
$\displaystyle\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){{\bm{\Delta}}^{-{k_{i}}-{\kappa_{i}}}}{{\bm{\Pi}}^{-{l_{i}}-{\iota_{i}}}}{{\bm{\Pi}}^{{l_{i}}+{\iota_{i}}}}{{\bm{\Delta}}^{{k_{i}}+{\kappa_{i}}}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right)$
(6.11) $\displaystyle=$ $\displaystyle{{\bf{I}}_{MN}},$ (6.12)

where (6.11) is due to the properties of ${\bm{\Pi}}$ and $\bm{\Delta}$,
respectively, and (6.12) is due to the property of the Kronecker product.
Therefore, the term
${{\bf{e}}^{\rm{H}}}{\bf{\Xi}}_{i}^{\rm{H}}{{\bf{\Xi}}_{i}}{\bf{e}}$ is
further simplified as ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$. This completes
the proof of Lemma 6-1. $\blacksquare$

Lemma 6-2 _(Trace of ${\bf{\Omega}}\left({\bf{e}}\right)$)_: The trace of the
codeword difference matrix ${\bf{\Omega}}\left({\bf{e}}\right)$ is
${Pd_{\rm{E}}^{2}\left({\bf{e}}\right)}$. Equivalently, the summation of the
first $r$ eigenvalues satisfies
$\sum\nolimits_{i=1}^{r}{{\lambda_{i}}}=Pd_{\rm{E}}^{2}\left({\bf{e}}\right)$.

_Proof_ : This lemma is a straightforward extension of Lemma 6-2 by noticing
that the ${\lambda_{i}}=0$ for $r+1\leq i\leq P$. $\blacksquare$

Lemma 6-3 _(Lower bound on $\sum\nolimits_{i=1}^{r}{1/{\lambda_{i}}}$)_: The
summation of first $r$ eigenvalue inverses of
${\bf{\Omega}}\left({\bf{e}}\right)$ is lower-bounded by

$\sum\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}\geq\frac{{{r^{2}}}}{{Pd_{\rm{E}}^{2}\left({\bf{e}}\right)}},$
(6.13)

where the equality holds if ${\bf{\Omega}}\left({\bf{e}}\right)$ is a diagonal
matrix, i.e.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.

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

Lemma 6-4 _(Lower bound on $\sum\nolimits_{i=1}^{r}{\lambda_{i}^{2}}$)_: The
summation of the first $r$ squared eigenvalues of
${{{{{\bf{\Omega}}\left({\bf{e}}\right)}}}}$ is lower-bounded by

$\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}\geq\frac{{{P^{2}}}}{r}{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{2}},$
(6.14)

where the equality holds if ${\bf{\Omega}}\left({\bf{e}}\right)$ is a diagonal
matrix, i.e.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.

_Proof_ : Notice that the first $r$ eigenvalues of
${{\bf{\Omega}}\left({\bf{e}}\right)}$ are all positive. Therefore, we apply
the Cauchy-Schwarz inequality and the following holds

$\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}\geq\frac{1}{r}{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)^{2}}=\frac{{{P^{2}}}}{r}{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{2}},$
(6.15)

where the equality is achieved when the eigenvalues of
${\bf{\Omega}}\left({\bf{e}}\right)$ are of the same value, e.g.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.
This completes the proof of Lemma 6-4. $\blacksquare$

The above lemmas show some important properties of the codeword difference
matrix ${\bf{\Omega}}\left({\bf{e}}\right)$. Based on these properties of
${\bf{\Omega}}\left({\bf{e}}\right)$, we can now consider the following lower
bounds of the eigenvalue product.

Theorem 6-1 _(Lower bound on $\prod\nolimits_{i=1}^{r}{{\lambda_{i}}}$)_: The
product of the first $r$ eigenvalues of ${\bf{\Omega}}\left({\bf{e}}\right)$
is lower-bounded by

$\prod\limits_{i=1}^{r}{{\lambda_{i}}}\geq{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}}\exp\left({r-d_{\rm{E}}^{2}\left({\bf{e}}\right)\sum\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}}\right),$
(6.16)

where the equality holds if ${\bf{\Omega}}\left({\bf{e}}\right)$ is a diagonal
matrix, i.e.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.

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

It should be noted that (6.16) still depends on the channel parameters
${\bm{\omega}}_{\tau}$ and ${\bm{\omega}}_{\nu}$. To obtain an unconditional
lower bound, we apply an approximation to the lower bound in (6.16), which is
summarized in the following Theorem.

Theorem 6-2 _(Approximated lower bound on
$\prod\nolimits_{i=1}^{r}{{\lambda_{i}}}$)_: The product of the first $r$
eigenvalues of ${\bf{\Omega}}\left({\bf{e}}\right)$ can be approximately
lower-bounded by

$\prod\limits_{i=1}^{r}{{\lambda_{i}}}\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}},$
(6.17)

where the approximation is exact if ${\bf{\Omega}}\left({\bf{e}}\right)$ is a
diagonal matrix, i.e.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.

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

Based on Theorem 6-2, it is not hard to verify that the eigenvalue product can
be approximated by the term
${\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}}$, regardless of the
specific distributions of the delay and Doppler indices. Meanwhile, the above
approximation is quite insightful in the sense that it only relates to the
rank of the codeword difference matrix ${\bf{\Omega}}\left({\bf{e}}\right)$
(this is actually the diversity gain as we will introduce later) and the
squared Euclidean distance $d_{\rm{E}}^{2}\left({\bf{e}}\right)$, which does
not depend on the exact value of the delay and Doppler indices. In particular,
the approximation becomes exact if
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$,
which can be interpreted as the projections of the error sequence $\bf{e}$
onto each independent resolvable path, i.e., ${\bar{\bf{u}}_{i}}$, are
orthogonal to each other. According to Theorem 6-2, we approximate the
unconditional PEP by

$\Pr\left({{{\bf{x}},{\bf{x^{\prime}}}}}\right)\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{\left({\frac{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{P}}\right)^{-r}}{\left({\frac{{{E_{s}}}}{{4{N_{0}}}}}\right)^{-r}}.$
(6.18)

Based on (6.18), we note that the unconditional PEP for OTFS modulation only
depends on ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$, the rank of
${\bf{\Omega}}\left({\bf{e}}\right)$, and number of independent resolvable
paths $P$, and is independent from the specific distribution of delay and
Doppler indices. Our derived result is a generalized framework of the
unconditional PEP for small $P$ that is suitable for various OTFS transmission
cases, including the case where $r<P$ and the delay and Doppler indices are
not uniformly distributed. In particular, the term $P$ in the denominator can
be interpreted as the energy averaging with respect to the number of
independent paths, while the term
${\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{-r}}$ indicates the
potential improvement of the error performance introduced by channel coding.
Furthermore, according to (6.18), we refer to the power of the SNR as the
diversity gain, which dominates the exponential behaviour of the error
performance for OTFS systems against the average SNR. On the other hand, the
term
${{d_{\rm{E}}^{2}\left({\bf{e}}\right)}\mathord{\left/{\vphantom{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}r}}\right.\kern-1.2pt}P}$
is referred to as the coding gain, which characterizes the approximate
improvement of coded OTFS systems over the uncoded counterpart with the same
diversity gain, i.e., the same exponent $-r$ [98]. Considering the diversity
property of OTFS modulation discussed in the previous remarks, it is
interesting to see from (6.18) that there exists a fundamental trade-off
between the diversity gain and the coding gain which is formally stated in the
following.

Corollary 6-1 _(Trade-off between diversity and coding gain)_ : For a given
channel code, the diversity gain of OTFS systems improves with the number of
independent resolvable paths $P$, while the coding gain declines.

Based on Corollary 6-1, we note that when $P$ (the rank of
${\bf{\Omega}}\left({\bf{e}}\right)$) is small, the diversity gain is small.
In this case, the squared Euclidean distance between codewords is crucial for
OTFS systems as an optimized code can greatly improve the error performance.
On the other hand, when $P$ (the rank of ${\bf{\Omega}}\left({\bf{e}}\right)$)
is large, there is a large diversity gain. In this case, it is expected that
the code design can only offer a limited error performance improvement.
However, it should also be noted that the coding gain always improves with the
increase of $d_{\rm{E}}^{2}\left({\bf{e}}\right)$, regardless of the value of
the diversity gain according to (6.18). Therefore, a preliminary guideline for
the code design for the OTFS systems is to maximize the minimum value of
$d_{\rm{E}}^{2}\left({\bf{e}}\right)$ among all pairs of codewords of the
code.

To verify the accuracy of the derived unconditional PEP bound, we numerically
compare the coding gain and the derived bound corresponding to (6.18) and
(6.10). In particular, recalling (6.10), after some manipulations, we obtain

$\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)\leq{\left({\frac{{{{\left({\prod\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{\frac{{\rm{1}}}{r}}}}}{P}}\right)^{-r}}{\left({\frac{{{E_{s}}}}{{4{N_{0}}}}}\right)^{-r}}.$
(6.19)

Hence, we refer to the term
${{{{\left({\prod\nolimits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{\frac{{\rm{1}}}{r}}}}\mathord{\left/{\vphantom{{{{\left({\prod\nolimits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{\frac{{\rm{1}}}{r}}}}P}}\right.\kern-1.2pt}P}$
as the _conditional coding gain_ of the OTFS systems for given channel
parameters ${\bm{\omega}}_{\tau}$ and ${\bm{\omega}}_{\nu}$, as the
eigenvalues are related to ${{{\bm{\omega}}_{\tau}}}$ and
${{{\bm{\omega}}_{\nu}}}$. Based on the conditional coding gain, we can also
obtain the _average coding gain_ with respect to various channel parameters
${{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}$ and error sequences
$\bf{e}$ by means of Monte Carlo simulation. On the other hand, from the
unconditional PEP upper bound (6.18), we call the function
$f\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)={{d_{\rm{E}}^{2}\left({\bf{e}}\right)}\mathord{\left/{\vphantom{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}P}}\right.\kern-1.2pt}P}$
the _coding gain bound_ of the OTFS systems.

Figure 6.1: Average coding gain for error sequences with $P=2$ and
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ in terms of various delay and Doppler
indices, comparing with the coding gain bound.

Now let us compare the average coding gain and the coding gain obtained from
the performance bound via simulations. We numerically average the conditional
coding gains subjected to all error sequences with
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ and channel parameters
${{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}$ to obtain the average
coding gain. Without loss of generality, we consider BPSK signals and error
sequences with ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ for OTFS systems with
integer delay and Doppler shifts.

Fig. 6.1 shows the comparison between the average coding gain and the
corresponding coding gain bound in decibels for $P=2$ with various maximum
delay and Doppler indices, where the OTFS system with $N=5$ and $M=2$ is
considered. As shown in the figure, different values of maximum delay and
Doppler indices do not have a strong impact on the average coding gain.
Meanwhile, it can be observed in the figure that the average coding gain
improves with the increase of the squared Euclidean distance
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$. Furthermore, the derived coding gain
bound shows a close match with the overall average coding gain, especially
when ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ is small.

Figure 6.2: Average coding gain for error sequences with
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ in terms of different numbers of
independent resolvable paths, comparing with the coding gain bounds (in dashed
lines), where the maximum delay and Doppler indices are set to be
$l_{\rm{max}}=2$ and $k_{\rm{max}}=4$, respectively.

Fig. 6.2 illustrates the average coding gains and the corresponding coding
gain bounds (in dashed lines) in decibels with respect to different values of
$P$, where the OTFS system with $N=5$ and $M=2$ is considered. As shown in the
figure, given ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$, the average coding gain
decreases with the increase of the number of paths $P$, which is consistent
with Corollary 6-1. Similar to the previous figure, the coding gain bounds
match well with the general behaviour of average coding gains, especially when
$P$ is small, which verifies the correctness of our derivation. On the other
hand, we notice that the derived coding gain bound slightly diverge from the
average coding gains, when $P$ is large. To verify this observation, we in the
following consider the case with a large number of paths and a larger OTFS
frame.

Figure 6.3: Average coding gain for error sequences with
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ in terms of different numbers of
independent resolvable paths, comparing with the coding gain bounds (in dashed
lines), where the maximum delay and Doppler indices are set to be
$l_{\rm{max}}=2$ and $k_{\rm{max}}=2$, respectively.

We consider an OTFS system with $M=32$ and $N=16$ in Fig. 6.3, 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
coding gains and the corresponding coding gain bounds with given error
sequences222Without loss of generality, the error sequences are of the form
${\bf e}=[2,0,-2,2,0,-2,...,0...0]^{\rm T}$.. Similar to the previous figure,
we observe that the derived coding gain bounds diverge from the average coding
gains for a large number of paths and the gap between them becomes wider with
the increase of $P$. This observation motivates us to derive a more suitable
approximation for the coding gain for the case of a large $P$, the details of
which will be given in the following subsection.

####  6.1.2 Error Performance Analysis for Large Values of $P$

When there is a large number of independent resolvable paths of the channel,
i.e., the value of $P$ is large, the unconditional PEP can be more accurately
bounded by considering the strong law of large number [106, 99]. In specific,
the term $\sum\nolimits_{i=1}^{r}{{\lambda_{i}}{{|{{{\tilde{h}}_{i}}}|}^{2}}}$
in (6.7) approaches a Gaussian random variable, due to the central limit
theorem [107] and the fact that OTFS modulation can achieve the full diversity
with proper designs using precoding techniques [11].

Notice that ${{\tilde{h}}_{i}}$ follows the complex Gaussian distribution,
with mean ${\mu_{{{\tilde{h}}_{i}}}}=(\mathbb{E}\left[{\bf{h}}\right])^{\rm
H}\cdot{{\bf{v}}_{i}}$ and variance $1/(2P)$ per real dimension. Therefore,
for the ease of derivation, we normalize the variance and rewrite (6.7) as

$\Pr\left({{\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{\bf{h}},{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}}\right)\leq\exp\left({-\frac{{{E_{s}}}}{{4{N_{0}}}}\sum\limits_{i=1}^{r}{\frac{{{\lambda_{i}}}}{P}{{\left|{{{\bar{h}}_{i}}}\right|}^{2}}}}\right),$
(6.20)

where ${\bar{h}_{i}}=\sqrt{P}{\tilde{h}_{i}}$. Note that
$\left\\{{{{\bar{h}}_{1}},{{\bar{h}}_{2}},...,{{\bar{h}}_{r}}}\right\\}$ are
independent complex Gaussian random variables with mean
${\mu_{{{\bar{h}}_{i}}}}={\sqrt{P}}{\mu_{{{\tilde{h}}_{i}}}}$ and variance
$1/2$ per real dimension. Furthermore, it is obvious that
$\left|{{{\bar{h}}_{i}}}\right|$ follows the Rician distribution with a Rician
factor ${{\bar{K}}_{i}}={\left|{{\mu_{{{\bar{h}}_{i}}}}}\right|^{2}}$ and a
unit variance. Thus, it can be shown that
${{{\left|{{{\bar{h}}_{i}}}\right|}^{2}}}$ follows a noncentral chi-squared
distribution with one DoF and noncentrality parameter $S={{\bar{K}}_{i}}$,
whose mean and variance are given by [92]

$\displaystyle{\mu_{{{\left|{{{\bar{h}}_{i}}}\right|}^{2}}}}$
$\displaystyle=1+{{\bar{K}}_{i}},$ (6.21)
$\displaystyle\sigma_{{{\left|{{{\bar{h}}_{i}}}\right|}^{2}}}^{2}$
$\displaystyle=2+4{{\bar{K}}_{i}}.$ (6.22)

Next, we derive the unconditional PEP by means of Gaussian approximation. To
start with, let
$\psi=\sum\limits_{i=1}^{r}{{\lambda_{i}}{{\left|{{{\bar{h}}_{i}}}\right|}^{2}}}$.
According to (6.21) and (6.22), we approximate $\psi$ as a Gaussian random
variable, whose mean is
${\mu_{\psi}}=\sum\limits_{i=1}^{r}{{\lambda_{i}}\left({1+{{{\bar{K}}_{i}}}}\right)}$
and variance is
$\sigma_{\psi}^{2}=\sum\limits_{i=1}^{r}{\lambda_{i}^{2}\left({2+4{{\bar{K}}_{i}}}\right)}$.
Thus, according to the Gaussian distribution of $\psi$, the conditional PEP in
(6.7) is upper-bounded by

$\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)\leq\int\limits_{0}^{+\infty}{\exp\left({-\frac{{{E_{s}}}}{{4{N_{0}}P}}\psi}\right)p}\left(\psi\right)d\psi.$
(6.23)

Considering

$\displaystyle\int\limits_{0}^{+\infty}{\exp\left({-\gamma\psi}\right)p}\left(\psi\right)d\psi=\exp\left({\frac{1}{2}{\gamma^{2}}\sigma_{\psi}^{2}-\gamma{\mu_{\psi}}}\right)Q\left({\frac{{\gamma\sigma_{\psi}^{2}-{\mu_{\psi}}}}{{{\sigma_{\psi}}}}}\right),\gamma>0,$
(6.24)

we obtain

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)\leq\exp\left({\frac{1}{2}{{\left({\frac{{{E_{s}}}}{{4{N_{0}}}}}\right)}^{2}}\cdot\frac{{\sigma_{\psi}^{2}}}{{{P^{2}}}}-\frac{{{E_{s}}}}{{4{N_{0}}}}\cdot\frac{{{\mu_{\psi}}}}{P}}\right)Q\left({\frac{{{E_{s}}}}{{4{N_{0}}}}\cdot\frac{{{\sigma_{\psi}}}}{P}-\frac{{{\mu_{\psi}}}}{{{\sigma_{\psi}}}}}\right).$
(6.25)

Similar to the previous subsection, we consider the special case of Rayleigh
fading.

In the case of Rayleigh fading, i.e., $|{{{\bar{h}}_{i}}}|$ and $|{{h_{i}}}|$
follow the Rayleigh distribution, we have
${\mu_{\psi}}=\sum\limits_{i=1}^{r}{{\lambda_{i}}}$ and
$\sigma_{\psi}^{2}=2\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}$. Therefore, the
right hand side of (6.25) is given by

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)\leq$
$\displaystyle\exp\left({{{\left({\frac{{{E_{s}}}}{{4{N_{0}}}}}\right)}^{2}}\sum\limits_{i=1}^{r}{\frac{{\lambda_{i}^{2}}}{{{P^{2}}}}}-\frac{{{E_{s}}}}{{4{N_{0}}}}\sum\limits_{i=1}^{r}{\frac{{{\lambda_{i}}}}{P}}}\right)$
$\displaystyle
Q\left({\frac{{{E_{s}}}}{{4{N_{0}}P}}\sqrt{2\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}-\frac{{\sum\nolimits_{i=1}^{r}{{\lambda_{i}}}}}{{\sqrt{2\sum\nolimits_{i=1}^{r}{\lambda_{i}^{2}}}}}}\right).$
(6.26)

Furthermore, we consider the Chernoff bound of the Q-function [106]

$Q\left(\gamma\right)\leq\exp\left({-\frac{1}{2}{\gamma^{2}}}\right),\gamma>0,$
(6.27)

and (6.26) can be further upper-bounded by

$\displaystyle\Pr\left({\left.{{\bf{x}},{\bf{x^{\prime}}}}\right|{{{\bm{\omega}}_{\tau}}},{{{\bm{\omega}}_{\nu}}}}\right)$
$\displaystyle\leq$
$\displaystyle\exp\left({{{\left({\frac{{{E_{s}}}}{{4{N_{0}}}}}\right)}^{2}}\sum\limits_{i=1}^{r}{\frac{{\lambda_{i}^{2}}}{{{P^{2}}}}}-\frac{{{E_{s}}}}{{4{N_{0}}}}\sum\limits_{i=1}^{r}{\frac{{{\lambda_{i}}}}{P}}}\right)\exp\left({-{{\left({\frac{{{E_{s}}}}{{4{N_{0}}}}}\right)}^{2}}\sum\limits_{i=1}^{r}{\frac{{\lambda_{i}^{2}}}{{{P^{2}}}}}-\frac{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}}{{4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}}+\frac{{{E_{s}}}}{{4{N_{0}}}}\sum\limits_{i=1}^{r}{\frac{{{\lambda_{i}}}}{P}}}\right)$
$\displaystyle=$
$\displaystyle\exp\left({-\frac{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}}{{4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}}}\right),$
(6.28)

when

$\frac{{{E_{s}}}}{{4{N_{0}}}}\geq\frac{P{\sum\nolimits_{i=1}^{r}{{\lambda_{i}}}}}{{2\sum\nolimits_{i=1}^{r}{\lambda_{i}^{2}}}}.$
(6.29)

Based on (6.28), the unconditional PEP can be approximately upper-bounded as
shown in the following Theorem.

Theorem 6-3 (_Unconditional PEP upper bound for large $P$_): For a large value
of $P$ and a reasonably high SNR, i.e.,
$\frac{{{E_{s}}}}{{4{N_{0}}}}\geq\frac{r}{2{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}$,
the unconditional PEP of OTFS systems can be approximately upper-bounded by

$\Pr\left({{\bf{x}},{\bf{x^{\prime}}}}\right)\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}\exp\left({-\frac{{{E_{s}}}}{{16{N_{0}}}}d_{\rm{E}}^{2}}\left({\bf{e}}\right)\right).$
(6.30)

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

It should be noted that, for $r\geq 4$, the approximation in Theorem 3 is
sufficiently accurate owing to the strong law of the large number [99, 106].
On the other hand, the value of $r$ is usually smaller than the squared
Euclidean distance $d_{\rm{E}}^{2}\left({\bf{e}}\right)$ for practical
wireless transmissions with good channel codes333For example, a popular
industry-standard rate-$1/2$ convolutional code with code memory of $6$ has a
minimum squared Euclidean distance $d_{\rm{E}}^{2}\left({\bf{e}}\right)=40$
[108].. Therefore, our SNR assumption is reasonable. Compared with Corollary
6-1, it is not surprising that the unconditional PEP only depends on the
squared Euclidean distance $d_{\rm{E}}^{2}\left({\bf{e}}\right)$, regardless
of the delay and Doppler indices. Furthermore, we note that (6.30) is of the
similar form of the error performance for AWGN channels [109]. This is because
the impact of fading is mitigated by a large number of diversity branches and
consequently, in other words, the channel with a large number of diversity
paths approaches an AWGN model [109].

Notice that the upper bound in Theorem 6-3 is based on the PEP analysis, which
does not directly indicate the average error performance of the coded system
[110]. But it can be used to approximate the coding gain for large $P$. Based
on the results from Theorem 6-3, we apply the commonly adopted coding gain
approximation for AWGN channels [108] to evaluate the coding gain of OTFS
systems with a large $P$. In specific, we have

${\rm{Coding}}\>{\rm{gain}}\simeq
10{\log_{10}}\left({\frac{{d_{{\rm{c}},\min}^{2}}}{{d_{{\rm{u}},\min}^{2}}}}\right){\rm{dB}},$
(6.31)

where ${d_{{\rm{c}},\min}^{2}}$ and ${d_{{\rm{u}},\min}^{2}}$ are the minimum
squared Euclidean distances for coded and uncoded OTFS systems, respectively.

In the previous subsections, we have derived the error performance analysis of
coded OTFS systems, which is valid for general underspread WSSUS channels
including both the integer and fractional delay and Doppler cases. In the
following, we will discuss the design of channel codes according to our error
performance analysis.

####  6.1.3 Code Design Issues

According to the derived analysis, the rule-of-thumb channel code design
criterion is discussed in this section. Without loss of generality, we only
consider the Rayleigh fading channel in the following. We can see from the
previous analysis that the code design criterion for the coded OTFS system is
to maximize the minimum squared Euclidean distance
$d_{\rm{E}}^{2}\left({\bf{e}}\right)$.

Proposition 6-1 _(The squared Euclidean distance criterion)_ : The channel
code should be designed to maximize the minimum squared Euclidean distance
among all pairs of possible codewords.

We note that even with the designed code, the error performance of coded OTFS
systems may still vary with different channel parameters e.g.,
${\bm{\omega}}_{\tau}$ and ${\bm{\omega}}_{\nu}$. This detrimental effect due
to channel realizations is widely observed in the system designs for fading
channels, such as in [100, 111]. In specific, with different channel
parameters, the value of the conditional coding gain can be different even
with the same error sequence, which may potentially jeopardize the overall
error performance of OTFS systems. In order to obtain a more robust
performance, it is desirable to apply an interleaver to permute the coded
symbols before sending to the constellation mapper or the OTFS modulator in
the DD domain. As pointed out in [111], such an interleaver can “whiten” the
transmitted symbols from the information theoretic point of view and the
detrimental effect on the error performance due to the channel parameters can
thus be alleviated.

To examine our performance analysis of the coded OTFS systems, we perform
numerical simulations for OTFS systems over high-mobility channels, the
results of which will be shown in the next section.

###  6.2 Numerical Results

In this section, the error performance of the coded OTFS system with various
channel codes is evaluated via numerical simulations by considering perfect
CSI at the receiver side. We consider the BPSK signal for the OTFS system,
where the data sequence is firstly encoded and interleaved, and then BPSK
mapped. Without loss of generality, we consider the SPA [84, 41] for OTFS
detection, where the details can be found in [42]. In order to verify the
accuracy of the analytical results, we consider four different convolutional
codes (with trellis termination) with different minimum squared Euclidean
distance $d_{\rm{E}}^{2}{\left({\bf{e}}\right)}$ among all possible codeword
pairs. The details of the code parameters are given in Table 6.1, including
the generator matrix and the memory length. In particular, we also show the
smallest value of squared Euclidean distance
$d_{\rm{E}}^{2}\left({\bf{e}}\right)$ among all possible codeword pairs for
each code. In specific, we consider a coded OTFS system with $N=8$ and $M=16$
and correspondingly the codeword length for all considered simulations is
$128$ bits unless otherwise specified. The channel decoder adopts the
logarithm domain Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm [112]. Furthermore,
we consider the Rayleigh fading case. If not otherwise specified, we only
consider the integer delay and Doppler case and set the maximum delay index as
$l_{\max}=3$ and the maximum Doppler index as $k_{\max}=5$, which is
corresponding to a relative speed around $250$ km/h with $4$ GHz carrier
frequency and $1.5$ kHz sub-carrier spacing. For each channel realization, we
randomly select the delay and Doppler indices according to the uniform
distribution, such that we have $-{k_{\max}}\leq{k_{i}}\leq{k_{\max}}$ and
$0\leq{l_{i}}\leq{l_{\max}}$.

Table 6.1: Code Parameters Code structure |  Generator matrix |  Memory length |  Minimum $d_{\rm{E}}^{2}\left({\bf{e}}\right)$
---|---|---|---
A |  $\left[1+D,D\right]$ |  1 |  12
B |  $\left[1+{D^{2}},1+D+{D^{2}}\right]$ |  2 |  20
C |  $\left[{1+{D^{2}}+{D^{5}},1+D+{D^{2}}+{D^{3}}+{D^{4}}+{D^{5}}}\right]$ |  5 |  32
D |  $\left[{1\\!+\\!D\\!+\\!{D^{2}}\\!+\\!{D^{5}}\\!+\\!{D^{6}},1\\!+\\!{D^{2}}\\!+\\!{D^{3}}\\!+\\!{D^{4}}\\!+\\!{D^{6}}}\right]$ |  6 |  40
Figure 6.4: FER performance for OTFS modulation with different codes for
$P=2$, where the relative user equipment (UE) speed is $250$ km/h.

The frame-error-rate (FER) performance of the OTFS systems with $P=2$ is shown
in Fig. 6.4. It can be observed that the slope of the FER curve for the
uncoded OTFS system is slightly worse than that for coded OTFS systems. This
indicates that the uncoded OTFS system with $P=2$ does not guarantee the full
diversity for all possible channel realizations, which is consistent with the
analysis in [9]. More importantly, this observation also shows that the
application of channel coding can improve the diversity gain of OTFS systems
in the case where the OTFS modulation fails to achieve the full diversity.
Moreover, we observe that employing the channel code with a larger minimum
squared Euclidean distance $d_{\rm{E}}^{2}\left({\bf{e}}\right)$ indeed leads
to a larger coding gain. In specific, we observe from the figure that for FER
$\approx{10^{-2}}$, the required SNRs for codes A, B, C and D, are $15.28$ dB,
$13.64$ dB, $12.65$ dB, and $12.54$ dB, respectively. Compared to uncoded OTFS
systems, these four coded OTFS systems achieve coding gains roughly $2.99$ dB,
$4.63$ dB, $5.62$ dB, and $5.73$ dB, respectively. This observation clearly
substantiates the proposed performance analysis and code design criterion in
Proposition 6-1.

Figure 6.5: FER performance for OTFS modulation with different codes for
$P=8$, where the relative UE speed is $250$ km/h.

Fig. 6.5 shows the FER performance of the OTFS systems with $P=8$. We observe
that the channel code with a larger minimum squared Euclidean distance
$d_{\rm{E}}^{2}\left({\bf{e}}\right)$ enjoys a larger coding gain compared
with the uncoded OTFS system, which is consistent with the analysis of Theorem
6-3. Furthermore, we can calculate the approximated coding gains according to
(6.31) for the adopted channel codes, which are $1.83$ dB, $4.12$ dB, $6.37$
dB, and $7.42$ dB for code A to D, respectively, while we also obtain the
actual coding gains based on the simulation results that are $1.75$ dB, $3.52$
dB, $5.13$ dB, and $5.42$ dB for code A to D, respectively. These observations
verify the accuracy of our analysis for large $P$ in Theorem 6-3 and the
coding gain approximation. Together with the observations from Fig. 6.4, we
can also conclude that our proposed code design criterion in Proposition 6-1
is universal for general OTFS systems regardless of the channel parameters and
the number of paths $P$.

Figure 6.6: FER performance of code D for OTFS modulation with $P=3$ and
$P=8$, comparing with that of uncoded OTFS systems, where the relative UE
speed is $250$ km/h.

Fig. 6.6 presents the trade-off between the diversity and coding gain. In
particular, we consider the FER performance of code D with $P=3$ and $P=8$,
comparing with that of the corresponding uncoded OTFS systems. As shown in
figure, at FER $\approx{10^{-3}}$, the coded OTFS system with $P=3$ exhibits
around $5.7$ dB coding gain compared to that of the uncoded OTFS system with
the same FER, while only around $5.0$ dB coding gain is obtained for the coded
OTFS system with $P=8$. This observation matches the prediction in Corollary
1, which indicates that the coding gain reduces with the increase of $P$,
given the same channel code.

Figure 6.7: FER performance of code A for QPSK/16-QAM mapped OTFS systems with
different number of paths and the relative UE speed of $250$ km/h.

Fig. 6.7 presents the trade-off between the diversity and coding gain with
practical mappings, namely, QPSK and 16-QAM mappings. In particular, we
consider the FER performance of code A with $P=2$ and $P=4$, comparing with
that of the corresponding uncoded OTFS systems. Similar to the previous
figure, the trade-off between the diversity and coding gain is clearly
demonstrated by the simulation results.

The FER performance of the OTFS modulation with code D and different number of
paths $P$ is illustrated in Fig. 6.7. It can be observed from the figure that
given the same code, the error performance of the coded OTFS systems improves
with the increase of number of distinguishable paths. Furthermore, it is
obvious that with the same code, a larger value of $P$ corresponds to a larger
diversity advantage as indicated in the figure, which is consistent with our
analysis.

Figure 6.8: FER performance of code A for OTFS modulation with $P=3$ and
$P=8$, compared to an OFDM system, where the relative UE speed is $250$ km/h.

We compare the FER performance of code A with $P=3$ and $P=8$, and that of the
corresponding OFDM systems in Fig. 6.8. As observed from the figure, the OTFS
system enjoy better error performance than that of the OFDM system with the
same code, for both $P=3$ and $P=8$. Furthermore, the FER curve of the OFDM
system shares almost the same slope as that of the OTFS system, for $P=3$. As
for $P=8$, the achieved diversity gain of the OTFS system is clearly higher
than that of the OFDM system. Note that the diversity gain of coded OFDM
systems is determined by the smaller value of the minimum symbol-wise Hamming
distance of the code ${\delta_{\rm{H}}}$ and the number of paths
$P$444According to [113], we have
$r_{\rm{OFDM}}={\rm{min}}({\delta_{\rm{H}}},P)$, where $r_{\rm{OFDM}}$ is the
achievable diversity gain of a coded OFDM system. In specific, we have
$r_{\rm{OFDM}}=3$ for both $P=3$ and $P=8$ with Code A. [113], while OTFS
systems can obtain the full diversity almost surely regardless of the
employment of channel codes. Therefore, this observation clearly shows the
advantage of the OTFS systems over the OFDM systems.

Figure 6.9: FER performance of code A for OTFS modulation and OFDM modulation
with $P=4$, where the values of the relative UE speed are $150$ km/h, $250$
km/h, and $300$ km/h, respectively.

The FER performance of both OTFS systems and OFDM systems with various values
of the relative UE speed is compared in Fig. 6.9. We consider $l_{\max}=3$ and
$k_{\max}=3,5,6$, which corresponds to the cases where the relative UE speeds
are $150$ km/h, $250$ km/h, and $300$ km/h, respectively. We apply the near-
optimal ML detection [42] for both the OTFS and OFDM systems to have a fair
comparison. We observe that the error performance of both OTFS and OFDM
systems with ML detection does not change much with different relative UE
speeds, which is consistent with the observations in [25]. Note that, the ML
detection is not practically feasible for conventional OFDM systems due to the
high detection complexity. Therefore, frequency domain equalization is usually
deployed for OFDM systems [114]. In this case, the error performance of OFDM
systems will degrade dramatically with the increase of the speed [114] due to
the severe ICI induced by the Doppler spread. Furthermore, the FER performance
of the OTFS systems outperform that of the OFDM systems, including both coded
and uncoded cases with various values of the speed. Similar to the previous
figure, we also observe that the achieved diversity gain of the OTFS system is
higher than that of the OFDM system, which agrees with our analysis.

Table 6.2: Code Parameters for Fig. 6.10 Code |  Data length |  Codeword length |  Code rate
---|---|---|---
Convolutional Code D |  250 |  512 |  0.488
5G LDPC |  256 |  512 |  0.5
LTE Turbo |  250 |  512 |  0.488
Figure 6.10: FER performance of 5G LDPC code and 3GPP Turbo code for OTFS
modulation with $P=4$ and a relative UE speed of $250$ km/h, where the
codeword length is $512$ bits.

We present the FER results of coded OTFS systems with modern codes in Fig.
6.10. We consider an LDPC code from the 5G communication standard [115]
(referred to as the 5G LDPC code) and the Turbo code from the 3GPP long term
evolution (LTE) standard [116] (referred to as the LTE Turbo code), where code
parameters are given in Table 6.2 and we have $N=16$ and $M=32$ for OTFS
modulation. As a benchmark, the FER performance of the convolutional code D is
also given in Fig. 6.10. We observe that, the LTE Turbo code achieves the best
error performance compared to the 5G LDPC code and the convolutional code D
with $P=4$, although they all share the same diversity gain. More
specifically, at FER $\approx{10^{-3}}$, 5G LDPC code and LTE Turbo code show
around $0.5$ dB and $0.7$ dB SNR gain compared to the convolutional code D.
Note that a similar observation of the error performance can be observed over
the AWGN channel, where the LTE Turbo code has the best performance while the
convolutional code D has the worst performance. Therefore, this observation
indicate that codes optimized for AWGN channels can also achieve a good error
performance in the OTFS systems, which is consistent with our analysis.

Remark 2: Based on the above simulation results, we summarize that the slopes
of FER curves for both uncoded and coded OTFS systems are generally different.
This is because the diversity of uncoded OTFS systems is determined only by
channel parameters while the diversity of coded OTFS systems is determined by
both the channel and code parameters. In particular, uncoded OTFS systems
cannot guarantee the exploitation of the full diversity in all possible
channel coefficients and transmitted symbols realizations, while coded OTFS
systems can achieve the full diversity with a careful code design. Note that
how to design practical codes to guarantee full diversity is an interesting
research work and one initial idea is to ensure the _codeword difference
matrix_ in (6.5) to be a diagonal matrix for any pair of codewords. However,
we will explore the details on the code design in our future work due to the
space limitation.

###  6.3 Summary of the Chapter

In this chapter, we studied the performance analysis of coded OTFS systems
over high-mobility channels. We first derived the conditional PEP for a given
channel realization and then obtained the unconditional PEP by leveraging
proper bounding techniques. We discussed two cases of the OTFS transmission
according to the number of independent resolvable paths of the channel, where
we showed that the coding improvement of OTFS systems depends on the squared
Euclidean distance between a pair of codewords. More importantly, we
demonstrated the fundamental trade-off between the diversity gain and the
coding gain for OTFS systems. Furthermore, we proposed a code design criterion
based on the derived unconditional bound. The analysis and the code design
criterion are verified by numerical simulations with various channel codes.
Our future work will focus on the modern code design for OTFS systems by
considering analytical tools such as the density evolution and the extrinsic
information transfer (EXIT) chart.

###  6.4 Appendices of the Chapter

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

####  6.4.1 Proof of Lemma 6-3

Notice that, ${{\bf{\Omega}}\left({\bf{e}}\right)}$ is positive semidefinite
Hermitian. Hence, the eigenvalues $\left\\{{{\lambda_{i}}}\right\\}$ of
${{\bf{\Omega}}\left({\bf{e}}\right)}$ for $1\leq i\leq r$ are all positive.
Considering the arithmetic mean and geometric mean (AM-GM) inequality, we
obtain

$\sum\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}\geq
r{\left({\prod\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}}\right)^{\frac{1}{r}}}=\frac{r}{{{{\left({\prod\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{\frac{1}{r}}}}}.$
(6.32)

Then, we apply the Cauchy-Schwarz inequality to the denominator of the right
hand side of (6.32), which yields

$\frac{r}{{{{\left({\prod\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{\frac{1}{r}}}}}\geq\frac{{{r^{2}}}}{{\sum\limits_{i=1}^{r}{{\lambda_{i}}}}}=\frac{r^{2}}{P{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}.$
(6.33)

It is obvious that the equality only holds when the eigenvalues
$\left\\{{{\lambda_{i}}}\right\\}$ are of the same value, e.g., the codeword
difference matrix ${{\bf{\Omega}}\left({\bf{e}}\right)}$ is a diagonal matrix,
i.e.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.
This completes the proof of Lemma 6-3. $\blacksquare$

####  6.4.2 Proof of Theorem 6-1

The product of the first $r$ eigenvalues of
${{\bf{\Omega}}\left({\bf{e}}\right)}$ can be written as

$\prod\limits_{i=1}^{r}{{\lambda_{i}}}=\exp\left({\ln\left({\prod\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}\right)=\exp\left({\sum\limits_{i=1}^{r}{\ln\left({{\lambda_{i}}}\right)}}\right).$
(6.34)

Let us consider the inequality $\ln\left(\gamma\right)\geq
1-\frac{1}{\gamma},\gamma\in\left({0,+\infty}\right)$, where the equality only
holds when $\gamma=1$. Therefore, we have

$\displaystyle\prod\limits_{i=1}^{r}{{\lambda_{i}}}$
$\displaystyle=\exp\left({\sum\limits_{i=1}^{r}{\ln\left({{\lambda_{i}}}\right)}}\right)$
$\displaystyle=\exp\left({r\ln\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)+\sum\limits_{i=1}^{r}{\ln\left({\frac{{{\lambda_{i}}}}{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}}\right)}}\right)$
$\displaystyle\geq\exp\left({r\ln\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)+\sum\limits_{i=1}^{r}{\left({1-\frac{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{{{\lambda_{i}}}}}\right)}}\right)$
(6.35)
$\displaystyle={\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}}\exp\left({r-d_{\rm{E}}^{2}\left({\bf{e}}\right)\sum\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}}\right).$
(6.36)

It is obvious that the equality holds in (6.35) only if the first $r$
eigenvalues $\\{\lambda_{i}\\}$ of ${{\bf{\Omega}}\left({\bf{e}}\right)}$
equal to ${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$. Notice that the eigenvalues
$\\{\lambda_{i}\\}$ of ${{\bf{\Omega}}\left({\bf{e}}\right)}$ equal to the
main diagonal elements when ${{\bf{\Omega}}\left({\bf{e}}\right)}$ is a
diagonal matrix, i.e.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}\left({\bf{e}}\right),\ldots,d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right\\}$,
in which case we have
$\det\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)={\left({d_{\rm{E}}^{2}}\left({\bf{e}}\right)\right)^{P}}$.
This completes the proof of Theorem 6-1. $\blacksquare$

####  6.4.3 Proof of Theorem 6-2

Following Theorem 6-1, we note that value of the term
${r-d_{\rm{E}}^{2}\left({\bf{e}}\right)\sum\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}}$
is smaller than 1 according to Lemma 6-3. Notice that the value of
$f(x)=\exp(-x)$ decays relatively slowly with the increase of $x$, if $x<1$.
This fact motives us to consider the approximation based on Lemma 6-3 as
considered in [106, 99], such as

$\displaystyle\prod\limits_{i=1}^{r}{{\lambda_{i}}}$
$\displaystyle\geq{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}}\exp\left({r-d_{\rm{E}}^{2}\left({\bf{e}}\right)\sum\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}}\right)$
$\displaystyle\mathbin{\lower
1.29167pt\hbox{$\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}}\exp\left({r-d_{\rm{E}}^{2}\left({\bf{e}}\right){{\left({\sum\limits_{i=1}^{r}{\frac{1}{{{\lambda_{i}}}}}}\right)}_{\min}}}\right)$
$\displaystyle={\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}}\exp\left({r-d_{\rm{E}}^{2}\left({\bf{e}}\right)\frac{{{r^{2}}}}{{Pd_{\rm{E}}^{2}\left({\bf{e}}\right)}}}\right)$
$\displaystyle={\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}}\exp\left({r-\frac{{{r^{2}}}}{P}}\right),$
$\displaystyle\geq{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)^{r}},$
(6.37)

where the equality holds if ${\bf{\Omega}}\left({\bf{e}}\right)$ is a diagonal
matrix, i.e.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.
Mathematically, the above approximation may be loose if
${\bf{\Omega}}\left({\bf{e}}\right)$ is ill-conditioned. Therefore, we justify
the accuracy of our approximation as follows. A commonly adopted approach in
testifying if a matrix is in ill-condition is the $\bf{P}$-condition number
[117]. In specific, we consider the lower bound on the $\bf{P}$-condition
number of a Gram matrix [117]. The ${\bf{P}}$-condition number of
${\bf{\Omega}}\left({\bf{e}}\right)$ is defined by

${\bf{P}}\left({\bf{\Omega}}\left({\bf{e}}\right)\right)\buildrel\Delta\over{=}{\rm
Radius}\left({\bf{\Omega}}\left({\bf{e}}\right)\right){\rm
Radius}\left({\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)^{-1}}\right),$
(6.38)

where ${\rm Radius}\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)$ is the
_spectral radius_ of ${{\bf{\Omega}}\left({\bf{e}}\right)}$, i.e., ${\rm
Radius}\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)$ equals to the
largest eigenvalues of ${{\bf{\Omega}}\left({\bf{e}}\right)}$. In particular,
the matrix ${{\bf{\Omega}}\left({\bf{e}}\right)}$ is said to be ill-
conditioned if ${\bf{P}}\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)$ is
large and is to be well-conditioned if
${\bf{P}}\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)$ is small.
According to [117], we have
${\bf{P}}\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)\geq{{{{\left({{{\left\|{{{\bf{u}}_{i}}}\right\|}^{2}}}\right)}_{\max}}}\mathord{\left/{\vphantom{{{{\left({{{\left\|{{{\bf{u}}_{i}}}\right\|}^{2}}}\right)}_{\max}}}{{{\left({{{\left\|{{{\bf{u}}_{j}}}\right\|}^{2}}}\right)}_{\min}}}}}\right.\kern-1.2pt}{{{\left({{{\left\|{{{\bf{u}}_{j}}}\right\|}^{2}}}\right)}_{\min}}}}$
for $1\leq i,j\leq P$, which yields

${\bf{P}}\left({{\bf{\Omega}}\left({\bf{e}}\right)}\right)\geq{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}\mathord{\left/{\vphantom{{d_{\rm{E}}^{2}\left({\bf{e}}\right)}{d_{\rm{E}}^{2}\left({\bf{e}}\right)=1}}}\right.\kern-1.2pt}{d_{\rm{E}}^{2}\left({\bf{e}}\right)=1}}.$
(6.39)

We can see that the $\bf{P}$-condition number of
${{\bf{\Omega}}\left({\bf{e}}\right)}$ always greater than or equal to $1$,
which indicates that ${{\bf{\Omega}}\left({\bf{e}}\right)}$ is generally well-
conditioned. This completes the proof of Theorem 6-2. $\blacksquare$

####  6.4.4 Proof of Theorem 6-3

Recalling (6.29), we note that (6.28) only holds if

$\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}\geq\frac{{P\sum\nolimits_{i=1}^{r}{{\lambda_{i}}}}}{{{{{E_{s}}}\mathord{\left/{\vphantom{{{E_{s}}}{\left({4{N_{0}}}\right)}}}\right.\kern-1.2pt}{\left({4{N_{0}}}\right)}}}}=\frac{{{P^{2}}d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{{{{{E_{s}}}\mathord{\left/{\vphantom{{{E_{s}}}{\left({4{N_{0}}}\right)}}}\right.\kern-1.2pt}{\left({4{N_{0}}}\right)}}}}.$
(6.40)

Therefore, we consider the approximation of (6.28) as follows

$\displaystyle\Pr\left({{\bf{x}},{\bf{x^{\prime}}}}\right)$
$\displaystyle\leq\exp\left({-{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}\mathord{\left/{\vphantom{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}{\left({4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}\right)}}}\right.\kern-1.2pt}{\left({4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}\right)}}}\right)$
$\displaystyle\mathbin{\lower
1.29167pt\hbox{$\buildrel<\over{\smash{\scriptstyle\sim}\vphantom{{}_{x}}}$}}\exp\left({-{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}\mathord{\left/{\vphantom{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}{{{\left({4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}\right)}_{\min}}}}}\right.\kern-1.2pt}{{{\left({4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}\right)}_{\min}}}}}\right)$
$\displaystyle=\exp\left({-{{{{\left({Pd_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)}^{2}}}\mathord{\left/{\vphantom{{{{\left({Pd_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)}^{2}}}{\left({4\frac{{{P^{2}}d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{{{{{E_{s}}}\mathord{\left/{\vphantom{{{E_{s}}}{\left({4{N_{0}}}\right)}}}\right.\kern-1.2pt}{\left({4{N_{0}}}\right)}}}}}\right)}}}\right.\kern-1.2pt}{\left({\frac{{4{P^{2}}d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{{{{{E_{s}}}\mathord{\left/{\vphantom{{{E_{s}}}{\left({4{N_{0}}}\right)}}}\right.\kern-1.2pt}{\left({4{N_{0}}}\right)}}}}}\right)}}}\right)$
$\displaystyle=\exp\left({-\frac{{{E_{s}}}}{{16{N_{0}}}}d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right).$
(6.41)

The above approximation is reasonable because the value of
$\exp\left({-{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}\mathord{\left/{\vphantom{{{{\left({\sum\limits_{i=1}^{r}{{\lambda_{i}}}}\right)}^{2}}}{\left({4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}\right)}}}\right.\kern-1.2pt}{\left({4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}\right)}}}\right)$
changes relatively slowly with the increase of
${4\sum\limits_{i=1}^{r}{\lambda_{i}^{2}}}$ [106, 99]. On the other hand, the
justification of the SNR assumption of (6.29) is necessary. According to Lemma
6-4, we have

$\frac{{P\sum\nolimits_{i=1}^{r}{{\lambda_{i}}}}}{2{\sum\nolimits_{i=1}^{r}{\lambda_{i}^{2}}}}\leq\frac{{{P^{2}}d_{\rm{E}}^{2}\left({\bf{e}}\right)}}{{\frac{{{2P^{2}}}}{r}{{\left({d_{\rm{E}}^{2}\left({\bf{e}}\right)}\right)}^{2}}}}=\frac{r}{{2d_{\rm{E}}^{2}\left({\bf{e}}\right)}},$
(6.42)

where the equality only holds when the eigenvalues of
${\bf{\Omega}}\left({\bf{e}}\right)$ share the same value, e.g.,
${\bf{\Omega}}\left({\bf{e}}\right)=\textrm{diag}\left\\{{d_{\rm{E}}^{2}{\left({\bf{e}}\right)},\ldots,d_{\rm{E}}^{2}}{\left({\bf{e}}\right)}\right\\}$.
Therefore, we can see that the term
$\frac{P{\sum\nolimits_{i=1}^{r}{{\lambda_{i}}}}}{2{\sum\nolimits_{i=1}^{r}{\lambda_{i}^{2}}}}$
is upper-bounded by $\frac{r}{2{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}$. Hence,
the assumption of SNR of (6.29) can be further restricted as
$\frac{{{E_{s}}}}{{4{N_{0}}}}\geq\frac{r}{2{d_{\rm{E}}^{2}\left({\bf{e}}\right)}}$.
This completes the proof of Theorem 6-3. $\blacksquare$

##  Chapter 7 OTFS for Integrated Sensing and Communications

In this chapter, we focus on the design of ISAC by using OTFS waveforms [118,
119]. Although the applications of OTFS modulation in ISAC transmissions have
shown promising performances as mentioned in the literature review, they often
rely on sophisticated beamforming schemes [61, 63, 120] that are designed
according to the CSI at transmitter (CSIT). However, it should be noted that
for some practical scenarios, such as co-located radar and communication
antennas, channel fading coefficients for communication cannot be directly
obtained from radar sensing. In specific, the strengths of channel fading
coefficients for communication usually depend on the path loss and channel
scatterers [7], while the echo strengths for radar sensing also depend on the
effective area of the radar receiving antenna and the radar cross section
(RCS) [121]. Therefore, there is generally a mismatch of the reflection
strengths between radar sensing and communication. In other words, the path
with the strongest echo power for radar sensing may not be the strongest path
for communication. Consequently, if the beams for communication steer towards
the strongest path indicated by radar sensing, the communication performance
may degrade dramatically.

Considering the potential mismatch between radar sensing and communication, we
propose a novel ISAC transmission framework based on spatially-spread OTFS
(SS-OTFS) modulation in this chapter. To facilitate the ISAC design, we
introduce the concept of SS-OTFS modulation to further exploit the delay-
Doppler-angular (DDA) domain channel characteristics. Compared to conventional
MIMO-OTFS modulation, SS-OTFS applies the so-called “spatial spreading” and
“spatial de-spreading” modules at the transmitter and receiver, respectively.
The key novelty of applying those modules is the discretization of the angular
domain, which results in simple and insightful input-output relationships for
both radar sensing and communication. The most interesting feature of those
relationships is that each antenna (pair) corresponds to a specific angle
(pair) according to the _angular resolution_. As such, it is possible to
greatly mitigate the multi-path effect, which enables efficient system designs
that are based only on estimates of the delays, Dopplers, angles-of-departure
(AoDs), and radar reflection coefficients from radar sensing, without the _a
priori_ knowledge of the fading coefficients of the communication channels. It
is worth emphasizing that most of the existing ISAC work in the literature
assume that the communication fading coefficients are available at the BS for
system designs [59, 122]. In contrast, the proposed framework does not require
the _a priori_ knowledge of communication fading coefficients. In particular,
we propose a practical precoding scheme and a power allocation strategy to
improve the communication performance based only on the estimated AoDs, and
delay and Doppler shifts from radar sensing at the previous time instant. The
main contributions of this chapter can be summarized as follows.

  * •

We derive both the communication and radar models for SS-OTFS-enabled ISAC
transmission. In particular, we show that the interference from spatial
multiplexing can be approximately eliminated by spatial spreading and de-
spreading with a sufficiently large number of antennas, which results in
simple and insightful effective channel matrices for both radar sensing and
communication.

  * •

Based on the radar sensing model, we develop simple beam tracking and AoA
estimation algorithms by exploiting the special structure of the effective
radar sensing matrix. We show that the transmitted beam width can be easily
controlled by the power allocation among the antennas, which is independent
from the precoding design. Furthermore, the AoA estimation can be
straightforwardly implemented by checking the received power among different
antennas, which is due to the discretization of the angular domain.
Furthermore, we introduce the power allocation for radar sensing, which is
designed to maximize the minimum power of the received echoes.

  * •

Based on the derived communication model, we analytically unveil the impacts
of precoding matrices and power allocation on the PEP. In particular, we show
that the PEP is minimized when the _codeword difference matrix_ has a diagonal
structure and the geometric mean of the allocated power associated to
corresponding paths is maximized. Based on this finding, we develop our
precoding design by introducing _virtual_ delay and Doppler indices to shape
the codeword difference matrix, while we show that the equal power allocation
can maximize the geometric mean.

  * •

We notice that radar sensing and communication require different power
allocations. Therefore, we discuss the radar sensing and communication
performances with respect to different power allocations. Based on our
discussions and simulation results, we show that the power allocation should
be designed leaning towards radar sensing in practical scenarios. Meanwhile,
the effectiveness of the proposed ISAC framework has also been verified by our
simulation results.

For the sake of clarity, the related notations in this chapter are listed in
Table 7.1, Table 7.2, and Table 7.3, corresponding to the ISAC transmitter,
ISAC channels, and ISAC receiver, respectively.

Table 7.1: List of Notations for ISAC Transmitter $K$ | Number of UEs
---|---
$N_{\rm BS}$ | Number of antennas at BS
$M$ | Number of sub-carriers/delay bins
$N$ | Number of time slots/Doppler bins
${\Delta f}$ | Sub-carrier spacing
$T$ | Time slot duration
${\theta_{{\rm{range}}}}$ | Beamwidth
${N_{{\rm{range}}}}$ | Number of antennas to support the beamwidth
${{\mathbb{U}}_{i,p}}$ |  Antenna set corresponding to the $p$-th path of the $i$-th UE
$\bf X$ and $\bf x$ | Matrix and vector of the DD domain symbols
$\bf V$ and $\bf v$ | Matrix and vector of the time-delay (TD) domain symbols
${\bf d}_{n_{\rm t}}$ |  Symbol vector of the $n_{\rm t}$-th antenna after precoding
${\bf d}$ | Concatenated symbol vector after precoding
${\bf z}_{n_{\rm t}}$ |  Symbol vector of the $n_{\rm t}$-th antenna after power allocation
${\bf z}$ | Concatenated symbol vector after power allocation
${\bf W}_{n_{\rm t}}$ |  Precoding matrix associated to the $n_{\rm t}$-th antenna
${\bf W}$ | Concatenated precoding matrix
${\bf W}_{i}^{\rm com}$ |  Set of precoding matrices for the $i$-th UE
${\alpha}_{n_{\rm t}}$ |  Allocated power for the $n_{\rm t}$-th antenna
${\beta}_{i}$ |  Power assigned to the $i$-th UE per antenna
${{\tilde{\alpha}}_{i}}$ |  Total power for the $i$-th UE
${\alpha}_{\rm total}$ | Total power budget for transmission
${\bm{\alpha}}$ | Power allocation matrix
${\bm{\alpha}}_{i}^{\rm com}$ |  Set of allocated power values for the $i$-th UE
$\bf S$ and $\bf s$ | Matrix and vector of the TDS domain symbols
Table 7.2: List of Notations for ISAC Channels $P$ | Number of independent resolvable paths
---|---
$N_{0}$ and ${\tilde{N}}_{0}$ | Noise power for communication and radar
$h_{i,p}$ and ${\tilde{h}}_{i,p}$ |  Communication and radar fading coefficients for the $p$-th path of the $i$-th UE
${\bf{h}}_{i}^{{\rm{eff}}}$ |  Set of communication fading coefficients for $i$-th UE
${{\varphi_{i,p}}}$ |  AoD for the $p$-th path of the $i$-th UE
$\tau_{i,p}$ and ${\tilde{\tau}}_{i,p}$ |  Communication and radar delay shifts for $p$-th path of $i$-th UE
$l_{i,p}$ and ${\tilde{l}}_{i,p}$ |  Communication and radar delay indices for $p$-th path of $i$-th UE
$\nu_{i,p}$ and ${\tilde{\nu}}_{i,p}$ |  Communication and radar Doppler shifts for $p$-th path of $i$-th UE
$k_{i,p}$ and ${\tilde{k}}_{i,p}$ |  Communication and radar Doppler indices for $p$-th path of $i$-th UE
$\kappa_{i,p}$ |  Fractional communication Doppler index for $p$-th path of $i$-th UE
${\tilde{\kappa}}_{i,p}$ |  Fractional radar Doppler index for $p$-th path of $i$-th UE
${\dot{l}}_{p}$ and ${\dot{k}}_{p}$ |  Virtual delay and Doppler indices for the $p$-th path
${\bm{\omega}}_{i}^{\tau}$ and ${\bm{\omega}}_{i}^{\nu}$ |  Set of communication delay and Doppler indices for the $i$-th UE
${\bf{a}}\left({{\varphi_{i,p}}}\right)$ | Steering vector and steering matrix
${\bf{A}}\left({{\varphi_{i,p}}}\right)$ | Steering matrix
${\bm{\Pi}}$ | Permutation matrix (forward cyclic shift) for characterizing delay influence
${\bm{\Delta}}$ | Diagonal phase shift matrix for characterizing Doppler influence
${\bf{H}}_{{n_{\rm{t}}},i}^{{\rm{TDS}}}$ |  TDS domain communication channel between $n_{\rm t}$-th antenna and $i$-th UE
${\bf{\tilde{H}}}_{{n_{\rm{t}}},i,{n_{\rm{r}}}}^{{\rm{TDS}}}$ |  TDS domain radar channel for $(n_{\rm t},n_{\rm r})$-th antenna pair and $i$-th UE
${\bf{H}}_{i,p}^{{\rm{TD}}}$ and ${\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}$ |  TD domain communication and radar channels for $p$-th path and $i$-th UE
${\bf{h}}_{i,p}^{{\rm{A}}}$ |  Angular domain communication channel for the $p$-th path and the $i$-th UE
${\bf{\tilde{H}}}_{i,p}^{{\rm{A}}}$ |  Angular domain radar channel for the $p$-th path and the $i$-th UE
${\bf{\tilde{H}}}_{{\rm{Radar}}}^{{\rm{TDA}}}$ | TDA domain effective radar sensing matrix
${\bf{H}}_{i}^{{\rm{DD}}}$ | Effective DD domain communication channel matrix
Table 7.3: List of Notations for ISAC Receivers ${\bf r}_{i}$ |  TD domain received symbol vector of the $i$-th UE for communication
---|---
${\bf\tilde{r}}_{n_{\rm r}}$ |  TDS domain received symbol vector of the ${n_{\rm r}}$-th antenna
${\bf r}$ | Concatenated vector for TD domain received symbols for communication
${\bf\tilde{r}}$ | Concatenated vector for TDS domain received symbols for radar
${\bf q}_{i}$ |  TD domain communication noise vector of the $i$-th UE
${\bf\tilde{q}}_{n_{\rm r}}$ |  TDS domain radar noise vector for the ${n_{\rm r}}$-th antenna
${\bf q}$ and ${\bf\tilde{q}}$ | Concatenated TD(S) domain noise vector for communication and radar
${\bm{\eta}}$ and ${\bm{\tilde{\eta}}}$ | DD domain and TDA domain noise vector for communication and radar
${\bf\tilde{z}}$ | TDA domain radar received symbol vector
${\bf y}_{i}$ |  DD domain communication received symbol vector for the $i$-th UE
${\bf{\Phi}}_{i}^{\omega_{i}^{\tau},\omega_{i}^{{}_{\nu}},{\bf{W}}_{i}^{{\rm{com}}}}\left({\bf{x}}\right)$ | Equivalent codeword matrix
${\bf{\Omega}}_{i}\left({\bf{e}}\right)$ | Codeword difference matrix
${\bf{\tilde{\Omega}}}_{i}\left({\bf{e}}\right)$ | Weighted codeword difference matrix
${d_{{\bf{h}}_{i}^{{\rm{eff}}},{{\bm{\omega}}_{\tau}},\\!{{\bm{\omega}}_{\nu}},\\!{{\bm{\alpha}}_{i}^{{\rm{com}}}},\\!{\bf{W}}_{i}^{{\rm{com}}}}^{2}\\!\left({{\bf{x}},\\!{\bf{x^{\prime}}}}\right)}$ |  Conditional Euclidean distance between $\bf x$ and $\bf x^{\prime}$
${d_{\rm{E}}^{2}\left({\bf{e}}\right)}$ |  Euclidean distance of the error sequence $\bf e$

###  7.1 System Model

Without loss of generality, let us consider an ISAC system in a mobile
network, where one BS broadcasts a common message to $K$ randomly distributed
UEs within the service coverage and senses the radar-related information of
the UEs based on the received echoes. In particular, we consider a multiple-
input single-output (MISO) case, where the BS is equipped $N_{\rm BS}$
antennas while each UE has only one antenna. We assume that the system
operates in an open area as shown in Fig. 7.1, where there are $P$ independent
resolvable paths between the BS and each UE111The existence of resolvable
paths indicates that the transmitted electromagnetic wave can be successfully
reflected/received by the sensing target/communication receiver, which is
crucial for both radar sensing and communication functionalities..

Figure 7.1: A brief diagram of the considered ISAC transmission.

####  7.1.1 Transmitter Structure

Without loss of generality, let us consider the SS-OTFS-enabled ISAC
transmitter structure as shown in Fig. 7.2.

Figure 7.2: The block diagram of the SS-OTFS-enabled ISAC transmitter, where
the two arrows represent different beam directions.

Let ${\bf{X}}\in{{\mathbb{A}}^{M\times N}}$ be the DD domain transmitted
symbol matrix (broadcast information message) of size $M\times N$, where $M$
denotes the number of orthogonal sub-carriers and $N$ denotes the number of
time slots, respectively. Let ${\Delta f}$ and $T$ be the sub-carrier spacing
and the time slot duration, respectively. According to the principle of OTFS
modulation [6], the TD domain transmitted symbol matrix
${\bf{V}}\in{{\mathbb{A}}^{M\times N}}$ of size $M\times N$ can be obtained by
performing the ISFFT and IFFT to ${\bf{X}}$. For simplicity, we consider the
vector form representation of OTFS transmissions according to [26]. Let
${\bf{x}}\buildrel\Delta\over{=}{\rm{vec}}\left({\bf{X}}\right)$ and
${\bf{v}}\buildrel\Delta\over{=}{\rm{vec}}\left({\bf{V}}\right)$ be the DD
domain and the TD domain transmitted symbol vectors of length $MN$,
respectively. According to [26, 48], we have

$\displaystyle{\bf{v}}\buildrel\Delta\over{=}{\rm{vec}}\left({\bf{V}}\right)={\rm{vec}}\left({{\bf{F}}_{M}^{\rm{H}}{{\bf{F}}_{M}}{\bf{XF}}_{N}^{\rm{H}}}\right)=\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right){\bf{x}}.$
(7.1)

After obtaining the TD domain broadcast message ${\bf{v}}$, the BS multiplexes
the message onto each antenna with respect to the precoding matrices222It is
worth noticing that the conventional precoding matrix for narrow band
multiple-input systems is of size $N_{\rm BS}\times N_{\rm BS}$. However, the
transmitted signal on each antenna is generally a wideband signal in the
considered system. Therefore, we propose to apply precoding to each antenna’s
transmitted signal in order to combat the multi-path interference, while apply
spatial spreading to combat the interference after spatial multiplexing.
$\left\\{{{{\bf{W}}_{1}},{{\bf{W}}_{2}},...,{{\bf{W}}_{{N_{{\rm{BS}}}}}}}\right\\}$
and we have

$\displaystyle{{\bf{d}}_{n_{\rm t}}}={{\bf{W}}_{n_{\rm
t}}}{{\bf{v}}},\quad\forall{n_{\rm{t}}}\in\left\\{{1,2,...,{N_{{\rm{BS}}}}}\right\\},$
(7.2)

where the precoding matrix ${{\bf{W}}_{n_{\rm t}}}$ is of size $MN\times MN$
and it has a normalized energy with respect to the message length, i.e.,
${\left\|{{{\bf{W}}_{{n_{\rm{t}}}}}}\right\|_{\rm{F}}}=MN$, for
$1\leq{n_{\rm{t}}}\leq{N_{{\rm{BS}}}}$. In particular, we restrict ourselves
to only consider the symbol-wise precoding, where each row/column of
${{\bf{W}}_{n_{\rm t}}}$ only has one non-zero element and
${{\bf{W}}_{{n_{\rm{t}}}}}{\bf{W}}_{{n_{\rm{t}}}}^{\rm{H}}={{\bf{I}}_{MN}}$,
for $1\leq{n_{\rm{t}}}\leq{N_{{\rm{BS}}}}$. Based on (7.2), we allocate power
to each antenna’s signal, such that the transmitted symbol vector ${\bf
z}_{n_{\rm t}}$ after power allocation333For the ease of presentation, we
henceforth use the term energy and power interchangeably, without raising
ambiguities. for the $n_{\rm t}$-th antenna is given by

$\displaystyle{{\bf{z}}_{n_{\rm
t}}}=\sqrt{{\alpha_{{n_{\rm{t}}}}}}{{\bf{d}}_{n_{\rm t}}},$ (7.3)

where $\alpha_{n_{\rm t}}$, for $1\leq{n_{\rm{t}}}\leq{N_{{\rm{BS}}}}$, is the
allocated power for the ${n_{\rm{t}}}$-th antenna, and
$\sum\nolimits_{{N_{\rm{t}}}=1}^{{N_{{\rm{BS}}}}}{{\alpha_{{n_{\rm{t}}}}}}={\alpha_{{\rm{total}}}}$
with ${\alpha_{{\rm{total}}}}$ being the total transmit power budget. Let us
define the transmitted symbol vector before and after power allocation by
${\bf{d}}\buildrel\Delta\over{=}{\left[{{\bf{d}}_{1}^{\rm{H}},{\bf{d}}_{2}^{\rm{H}},...,{\bf{d}}_{{N_{{\rm{BS}}}}}^{\rm{H}}}\right]^{\rm{H}}}$
and
${\bf{z}}\buildrel\Delta\over{=}{\left[{{\bf{z}}_{1}^{\rm{H}},{\bf{z}}_{2}^{\rm{H}},...,{\bf{z}}_{{N_{{\rm{BS}}}}}^{\rm{H}}}\right]^{\rm{H}}}$,
respectively. Then, it can be shown that

$\displaystyle{\bf{z}}=\left({{\bm{\alpha}}\otimes{{\bf{I}}_{MN}}}\right){\bf{d}},$
(7.4)

where ${\bm{\alpha}}\buildrel\Delta\over{=}{\rm
diag}\left\\{{\sqrt{\alpha_{1}},\sqrt{\alpha_{2}},...,\sqrt{\alpha_{{N_{{\rm{BS}}}}}}}\right\\}$
is the diagonal power allocation matrix of size $N_{\rm BS}\times N_{\rm BS}$.
Let us rearrange the transmitted symbol vectors for different antennas into a
matrix $\bf Z$ of size $MN\times N_{\rm BS}$ based on
${\bf{z}}={\rm{vec}}\left({\bf{Z}}\right)$. Then, we apply $N_{\rm BS}$-point
IFFT, i.e., ${\bf F}_{N_{\rm BS}}^{\rm H}$, to the symbols among different
antennas for spatial spreading, yielding

$\displaystyle{\bf{S}}={\bf{ZF}}_{{N_{{\rm{BS}}}}}^{\rm{H}},$ (7.5)

where ${\bf{S}}$ is the time-delay-spatial (TDS) domain transmitted symbol
matrix of size $MN\times N_{\rm BS}$. Assuming that a rectangular pulse is
applied as the transmitter shaping pulse, it can be shown that the TDS domain
transmitted signal for the $n_{\rm t}$-th antenna can be fully characterized
by the $n_{\rm t}$-th column of ${\bf{S}}$ [26]. Denote by ${\bf s}_{n_{\rm
t}}$ the $n_{\rm t}$-th column of ${\bf{S}}$, and we have
${\bf{s}}={\rm{vec}}\left({\bf{S}}\right)$, where
${\bf{s}}\buildrel\Delta\over{=}{\left[{{\bf{s}}_{1}^{\rm{H}},{\bf{s}}_{2}^{\rm{H}},...,{\bf{s}}_{{N_{{\rm{BS}}}}}^{\rm{H}}}\right]^{\rm{H}}}$.
By combining (7.1), (7.4), and (7.5), and considering the property of
Kronecker product, we have

$\displaystyle{\bf{s}}$
$\displaystyle=\left({{\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}\otimes{{\bf{I}}_{MN}}}\right){\bf{z}}=\left({{\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}\otimes{{\bf{I}}_{MN}}}\right)\left({{\bm{\alpha}}\otimes{{\bf{I}}_{MN}}}\right){\bf{Wv}}$
$\displaystyle=\left({{\left({{\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}{\bm{\alpha}}}\right)}\otimes{{\bf{I}}_{MN}}}\right){\bf{W}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right){\bf{x}},$
(7.6)

where
${\bf{W}}\buildrel\Delta\over{=}{\left[{{\bf{W}}_{1}^{\rm{H}},{\bf{W}}_{2}^{\rm{H}},...,{\bf{W}}_{{N_{{\rm{BS}}}}}^{\rm{H}}}\right]^{\rm{H}}}$
is the concatenated precoding matrix of size $N_{\rm BS}MN\times MN$. For a
better understanding, we provide a diagram in Fig. 7.3, characterizing the
domain transformations for the transmitter step by step.

Figure 7.3: The domain transformations the SS-OTFS enabled ISAC transmitter.

As shown in Fig. 7.3, the OTFS modulation transforms the broadcast message
from the DD domain to the TF domain and then to the TD domain. The precoding
and power allocation are performed at the time-delay-angular (TDA) domain
after repeating the signals onto each antenna. After that, the IFFT converts
the signals from the TDA domain to the TDS domain for signal transmission.

####  7.1.2 Communication Model

Without loss of generality, let us assume that there is $P$ independent
resolvable paths between the BS and each UE444In practice, the number of paths
$P$ could be estimated by the DD domain channel estimation algorithms. Some
related discussions could be found in [52, 39].. According to the far-field
assumption [8] and the DD domain channel characteristics [7], the
communication channel with respect to the antenna index $n_{\rm t}$, for
$1\leq n_{\rm t}\leq N_{\rm BS}$, and the UE index $i$, for $1\leq i\leq K$,
can be modeled by

$\displaystyle
h\left({{n_{\rm{t}}},i,\tau,\nu}\right)=\sum\limits_{p=1}^{P}{{h_{i,p}}\exp\left({j\pi\left({{n_{\rm{t}}}-1}\right)\sin{\varphi_{i,p}}}\right)}\delta\left({\tau-{\tau_{i,p}}}\right)\delta\left({\nu-{\nu_{i,p}}}\right),$
(7.7)

where we assume that the distance between the adjacent antennas is half of the
signal wavelength. In (7.7), $h_{i,p}\in{\mathbb{C}}$, ${{\varphi_{i,p}}}$,
$\tau_{i,p}$, and $\nu_{i,p}$ are the communication fading coefficient, AoD,
delay shift, and Doppler shift corresponding to the $p$-th path of the $i$-th
UE, respectively. We assume that the communication fading coefficient
$h_{i,p}$ follows the uniform power delay and power Doppler profiles, such
that the $h_{i,p}$ has zero mean and variance $1/(2P)$ per real dimension for
$1\leq p\leq P$, and is independent from the delay and Doppler indices [66].

Assuming that a rectangular pulse is applied as the matched-filtering pulse
for each UE, then with a reduced CP structure (introduced in Chapter 3) [26],
the TDS domain channel response based on (7.7) can be equivalently represented
by its matrix form [26], i.e.,

$\displaystyle{\bf{H}}_{{n_{\rm{t}}},i}^{{\rm{TDS}}}\buildrel\Delta\over{=}$
$\displaystyle\sum\limits_{p=1}^{P}{{h_{i,p}}}\exp\left({j\pi\left({{n_{\rm{t}}}-1}\right)\sin{\varphi_{i,p}}}\right){{\bf{\Pi}}^{{l_{i,p}}}}{{\bf{\Delta}}^{{k_{i}}+{\kappa_{i,p}}}},\forall
i,p.$ (7.8)

Specifically, we denote by $l_{i,p}$ and $k_{i,p}$ the indices of delay and
Doppler associated with the $p$-th path of the $i$-th UE, respectively, where
we have

${\tau_{i,p}}=\frac{{l_{i,p}}}{{M\Delta f}},\quad{\rm
and}\quad{\nu_{i,p}}=\frac{{{k_{i,p}}+{\kappa_{i,p}}}}{{NT}},$ (7.9)

respectively. Similar to the descriptions in Chapter 3, we have $0\leq
l_{i,p}\leq M-1$, and $0\leq k_{i,p}\leq N-1$, respectively [26]. Note that
the term
$-{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}\leq{\kappa_{i,p}}\leq{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$
denotes the fractional Doppler which corresponds to the fractional shift from
the nearest Doppler grid [25]. On the other hand, since the typical value of
the sampling time ${1\mathord{\left/{\vphantom{1{M\Delta
f}}}\right.\kern-1.2pt}{M\Delta f}}$ in the delay domain is usually
sufficiently small, the impact of fractional delays in typical wide-band
systems can be neglected [8]. With (7.8), the TD domain555Since the UE only
has one antenna, the spatial/angular domain features at the receiver side,
e.g., receiver steering vector, are disappeared. Therefore, we use the term TD
instead of TDS/TDA for the relevant descriptions for the communication
receiver in sequel.  received symbol vector for the $i$-th UE is written by

${{\bf{r}}_{i}}=\sum\limits_{{n_{\rm{t}}}=1}^{{N_{{\rm{BS}}}}}{{\bf{H}}_{{n_{\rm{t}}},i}^{{\rm{TDS}}}{{\bf{s}}_{{n_{\rm{t}}}}}+{{\bf{q}}_{i}}},$
(7.10)

where ${{\bf{q}}_{i}}$ denotes the AWGN samples with one-sided power spectral
density (PSD) $N_{0}$. Equivalently, by separating the angular features in
(7.8), (7.10) can be rearranged as

$\displaystyle{{\bf{r}}_{i}}=\sum\limits_{p=1}^{P}\left({{{\bf{a}}^{\rm{T}}}\left({{\varphi_{i,p}}}\right)\otimes{\bf{H}}_{i,p}^{{\rm{TD}}}}\right){\bf{s}}+{{\bf{q}}_{i}},$
(7.11)

where ${\bf{a}}\left({{\varphi_{i,p}}}\right)$ is the transmit steering vector
given by666For the sake of a better clarification of the advantages of the
proposed scheme, we consider the energy normalization of the underlying
channel in the following contents of this chapter. In practice, the
normalization term in (7.12) should not exist, yielding a multiplexing gain
related to the number of antennas $N_{\rm BS}$.

$\displaystyle{\bf{a}}\left({{\varphi_{i,p}}}\right)\buildrel\Delta\over{=}\frac{1}{{\sqrt{{N_{{\rm{BS}}}}}}}{\left[{1,\exp\left({j\pi\sin{\varphi_{i,p}}}\right),...,\exp\left({j\pi\left({{N_{{\rm{BS}}}}-1}\right)\sin{\varphi_{i,p}}}\right)}\right]^{\rm{T}}},$
(7.12)

and ${{\bf{H}}_{i,p}^{{\rm{TD}}}}$ is defined as the TD domain equivalent
communication channel matrix for the $p$-th path of the $i$-th UE, given by

$\displaystyle{\bf{H}}_{i,p}^{{\rm{TD}}}\buildrel\Delta\over{=}{{h_{i,p}}}{{\bf{\Pi}}^{{l_{i,p}}}}{{\bf{\Delta}}^{{k_{i,p}}+{\kappa_{i,p}}}}.$
(7.13)

Then, according to the connections between the TD domain to the DD domain
[26], the DD domain received signal for the $i$-th UE is given by

$\displaystyle{{\bf{y}}_{i}}=\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right)\sum\limits_{p=1}^{P}{\left({{{\bf{a}}^{\rm{T}}}\left({{\varphi_{i,p}}}\right)\otimes{\bf{H}}_{i,p}^{{\rm{TD}}}}\right)}{\bf{s}}+{{\bm{\eta}}_{i}},$
(7.14)

where
${{\bm{\eta}}_{i}}\buildrel\Delta\over{=}\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right){{\bf{q}}_{i}}$
is the equivalent AWGN noise vector in the DD domain. Finally, by substituting
(7.6) into (7.14), we obtain

$\displaystyle{{\bf{y}}_{i}}=$
$\displaystyle\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right)\sum\limits_{p=1}^{P}{\left({{{\bf{a}}^{\rm{T}}}\left({{\varphi_{i,p}}}\right)\otimes{\bf{H}}_{i,p}^{{\rm{TD}}}}\right)}\left({{\left({{\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}{\bm{\alpha}}}\right)}\otimes{{\bf{I}}_{MN}}}\right){\bf{W}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right){\bf{x}}+{{\bm{\eta}}_{i}}$
$\displaystyle=$
$\displaystyle\left({{{\bf{F}}_{N}}\otimes{{\bf{I}}_{M}}}\right)\sum\limits_{p=1}^{P}{\left({\left({{{\bf{a}}^{\rm{T}}}\left({{\varphi_{i,p}}}\right){\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}{\bm{\alpha}}}\right)\otimes{\bf{H}}_{i,p}^{{\rm{TD}}}}\right)}{\bf{W}}\left({{\bf{F}}_{N}^{\rm{H}}\otimes{{\bf{I}}_{M}}}\right){\bf{x}}+{{\bm{\eta}}_{i}}.$
(7.15)

####  7.1.3 Radar Model

Similar to the communication model, we consider the radar channel response
with respect to the transmit antenna index $n_{\rm t}$, for $1\leq n_{\rm
t}\leq N_{\rm BS}$, the UE index $i$, for $1\leq i\leq K$, the receive antenna
index $n_{\rm r}$, for $1\leq n_{\rm r}\leq N_{\rm BS}$, which is modeled by

$\displaystyle\tilde{h}\left({{n_{\rm{t}}},i,{n_{\rm{r}}},\tilde{\tau},\tilde{\nu}}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{p=1}^{P}{{{\tilde{h}}_{i,p}}\exp\left({j\pi\left({{n_{\rm{t}}}-1}\right)\sin{\varphi_{i,p}}}\right)}\exp\left({j\pi\left({{n_{\rm{r}}}-1}\right)\sin{\varphi_{i,p}}}\right)\\!\delta\left({\tau\\!-\\!{{\tilde{\tau}}_{i,p}}}\right)\\!\delta\left({\nu\\!-\\!{{\tilde{\nu}}_{i,p}}}\right),$
(7.16)

where we assume that the receive and transmit antennas are co-located such
that AoDs and AoAs are of the same values [59]. In (7.16),
${\tilde{h}}_{i,p}\in{\mathbb{C}}$, ${\tilde{\tau}}_{i,p}$, and
${\tilde{\nu}}_{i,p}$ are the radar reflection coefficient, round-trip delay
shift, and round-trip Doppler shift associated to the $p$-th path of the
$i$-th UE, respectively, where the round-trip delay and Doppler shifts satisfy
${{{\tilde{\tau}}_{i,p}}}=2{{{\tau}_{i,p}}}$ and
${{{\tilde{\nu}}_{i,p}}}=2{{{\nu}_{i,p}}}$ [59, 61]. In practice, the radar
reflection coefficient ${\tilde{h}}_{i,p}$ relates to the distance between the
$i$-th UE and the BS with respect to the $p$-th path, the effective area of
the radar receiving antenna, the RCS, the wavelength, and the transmit and
receive antenna gains [121]. Similar to the communication model, we assume
that a rectangular pulse is applied as the filtering pulse at the receiver,
then with the reduced CP structure [26], the TDS domain equivalent radar
sensing matrix is given by [26]

$\displaystyle{\bf{\tilde{H}}}_{{n_{\rm{t}}},i,{n_{\rm{r}}}}^{{\rm{TDS}}}\buildrel\Delta\over{=}$
$\displaystyle\sum\limits_{p=1}^{P}{{{\tilde{h}}_{i,p}}}\exp\left({j\pi\left({{n_{\rm{t}}}-1}\right)\sin{\varphi_{i,p}}}\right)\exp\left({j\pi\left({{n_{\rm{r}}}-1}\right)\sin{\varphi_{i,p}}}\right){{\bf{\Pi}}^{{{\tilde{l}}_{i,p}}}}{{\bf{\Delta}}^{{{\tilde{k}}_{i}}\\!+\\!{{\tilde{\kappa}}_{i,p}}}},$
(7.17)

where ${{{\tilde{l}}_{i,p}}}=2{{{l}_{i,p}}}$ and
${{\tilde{k}}_{i}}+{{\tilde{\kappa}}_{i,p}}=2\left({{k_{i}}+{\kappa_{i,p}}}\right)$.
Let ${{\tilde{N}}_{0}}$ denote the one-sided PSD for the radar noise, which
takes into account of both the AWGN noise power from the channel and the
interference power from the transmit signals after interference cancellation
[59]. Then, similar to (7.10), the radar received TDS domain symbol vector for
the $n_{\rm r}$-th antenna is written by

${{{\bf{\tilde{r}}}}_{{n_{\rm{r}}}}}=\sum\limits_{i=1}^{K}{\sum\limits_{{n_{\rm{t}}}=1}^{{N_{{\rm{BS}}}}}{{\bf{\tilde{H}}}_{{n_{\rm{t}}},i,{n_{\rm{r}}}}^{{\rm{TDS}}}{{\bf{s}}_{{n_{\rm{t}}}}}+{{{\bf{\tilde{q}}}}_{n_{\rm
r}}}}},$ (7.18)

where ${{{\bf{\tilde{q}}}}_{n_{\rm r}}}$ denotes the TDS domain radar AWGN
vector. By stacking the TDS domain received symbols from each antenna, we have

$\displaystyle{\bf{\tilde{r}}}$
$\displaystyle={\left[{{\bf{\tilde{r}}}_{1}^{\rm{H}},{\bf{\tilde{r}}}_{2}^{\rm{H}},...,{\bf{\tilde{r}}}_{{N_{{\rm{BS}}}}}^{\rm{H}}}\right]^{\rm{H}}}=\sum\limits_{i=1}^{K}{\sum\limits_{p=1}^{P}{\left({{\bf{A}}\left({{\varphi_{i,p}}}\right)\otimes{\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}}\right)}}{\bf{s}}+{\bf{\tilde{q}}},$
(7.19)

where
${\bf{\tilde{q}}}\buildrel\Delta\over{=}{\left[{{\bf{\tilde{q}}}_{1}^{\rm{H}},{\bf{\tilde{q}}}_{2}^{\rm{H}},...,{\bf{\tilde{q}}}_{{N_{{\rm{BS}}}}}^{\rm{H}}}\right]^{\rm{H}}}$
is the equivalent TDS domain radar noise,

$\displaystyle{\bf{A}}\left({{\varphi_{i,p}}}\right)\buildrel\Delta\over{=}{\bf{a}}\left({{\varphi_{i,p}}}\right){{\bf{a}}^{\rm{T}}}\left({{\varphi_{i,p}}}\right),\forall
i,p,$ (7.20)

is the steering matrix associated with the $p$-th path of the $i$-th UE, and

$\displaystyle{\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}\buildrel\Delta\over{=}{{{\tilde{h}}_{i,p}}}{{\bf{\Pi}}^{{{\tilde{l}}_{i,p}}}}{{\bf{\Delta}}^{{{\tilde{k}}_{i}}+{{\tilde{\kappa}}_{i,p}}}},$
(7.21)

is the TD domain equivalent radar channel matrix for the $p$-th path of the
$i$-th UE, respectively. Without loss of generality, we consider the radar
sensing in the TDA domain. Therefore, we apply the spatial de-spreading to the
TDS domain radar received symbol vector ${\bf{\tilde{r}}}$, yielding

$\displaystyle{\bf{\tilde{z}}}=$
$\displaystyle\left({{{\bf{F}}_{{N_{{\rm{BS}}}}}}\otimes{{\bf{I}}_{MN}}}\right){\bf{\tilde{r}}}$
$\displaystyle=$
$\displaystyle\left({{{\bf{F}}_{{N_{{\rm{BS}}}}}}\otimes{{\bf{I}}_{MN}}}\right)\sum\limits_{i=1}^{K}{\sum\limits_{p=1}^{P}\left({{\bf{A}}\left({{\varphi_{i,p}}}\right)\otimes{\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}}\right)}{\left({{\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}\otimes{{\bf{I}}_{MN}}}\right){\bf{z}}}+{\bm{\tilde{\eta}}}$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{K}{\sum\limits_{p=1}^{P}{\left({\left({{{\bf{F}}_{{N_{{\rm{BS}}}}}}{\bf{A}}\left({{\varphi_{i,p}}}\right){\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}{\bm{\alpha}}}\right)\otimes{\bf{\tilde{H}}}_{i,p}^{{\rm{TD}}}}\right){\bf{d}}}+{\bm{\tilde{\eta}}}},$
(7.22)

where ${\bm{\tilde{\eta}}}$ is the TDA domain equivalent radar AWGN vector
with one-sided PSD ${{\tilde{N}}_{0}}$.

####  7.1.4 Model Simplifications with Spatial Spreading and De-spreading

To characterize the effect of spatial spreading and de-spreading, we are
interested in the structure of the equivalent angular domain channel
vector/matrix for both communication and radar. For the $p$-th path of the
$i$-th UE, let us define the equivalent angular domain channel vector for the
communication channel by

$\displaystyle{\bf{h}}_{i,p}^{\rm{A}}\buildrel\Delta\over{=}{{\bf{a}}^{\rm{T}}}\left({{\varphi_{i,p}}}\right){\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}{\bm{\alpha}},$
(7.23)

and the equivalent angular domain channel matrix for the radar channel by

$\displaystyle{\bf{\tilde{H}}}_{i,p}^{\rm{A}}\buildrel\Delta\over{=}{{\bf{F}}_{{N_{{\rm{BS}}}}}}{\bf{A}}\left({{\varphi_{i,p}}}\right){\bf{F}}_{{N_{{\rm{BS}}}}}^{\rm{H}}{\bm{\alpha}},$
(7.24)

respectively. Based on (7.23) and (7.24), we can derive the elements in
${\bf{h}}_{i,p}^{\rm{A}}$ and ${\bf{\tilde{H}}}_{i,p}^{\rm{A}}$ after some
manipulations. The corresponding results are summarized in Lemma 7-1.

Lemma 7-1 (_Angular Domain Channel_): Given $1\leq k\leq N_{\rm BS}$, and
$1\leq l\leq N_{\rm BS}$, the elements in the equivalent angular domain
communication and radar channels can be expressed by

$\displaystyle
h_{i,p}^{\rm{A}}\left[l\right]=\frac{{\sqrt{{\alpha_{l}}}}}{{{N_{{\rm{BS}}}}}}\sum\limits_{{n_{\rm{t}}}=0}^{{N_{{\rm{BS}}}}-1}{\exp\left({j{n_{\rm{t}}}\pi\left({\sin{\varphi_{i,p}}+\frac{{2\left({l-1}\right)}}{{{N_{{\rm{BS}}}}}}}\right)}\right)},$
(7.25)

and

$\displaystyle\tilde{H}_{i,p}^{\rm{A}}\left[{k,l}\right]=$
$\displaystyle\frac{{\sqrt{{\alpha_{l}}}}}{{{{\left({{N_{{\rm{BS}}}}}\right)}^{2}}}}\left({\sum\limits_{{n_{\rm{r}}}=0}^{{N_{{\rm{BS}}}}-1}{\exp\left({j\pi{n_{\rm{r}}}\left({\sin{\varphi_{i,p}}-\frac{{2\left({k-1}\right)}}{{{N_{{\rm{BS}}}}}}}\right)}\right)}}\right)$
$\displaystyle\quad\quad\quad\\!\\!\quad\left({\sum\limits_{{n_{\rm{t}}}=0}^{{N_{{\rm{BS}}}}-1}{\exp\left({j\pi{n_{\rm{t}}}\left({\sin{\varphi_{i,p}}+\frac{{2\left({l-1}\right)}}{{{N_{{\rm{BS}}}}}}}\right)}\right)}}\right),$
(7.26)

respectively.

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

Based on (7.25) and (7.26), we notice that if the value of
$\sin{\varphi_{i,p}}$ is integer multiples of $2/N_{\rm BS}$, the values of
${{h}}_{i,p}^{\rm{A}}\left[l\right]$ and
${{\tilde{H}}}_{i,p}^{\rm{A}}\left[k,l\right]$ will be either
$\sqrt{\alpha_{l}}$ or zero. Therefore, let us define the angular resolution