we can complete the calculation of gradient vectors $\frac{\partial
f(\bm{\theta})}{\partial\bm{\theta}}$ and $\frac{\partial
f_{c}(\bm{\theta})}{\partial\bm{\theta}}$. Due to the long expressions, we
present the final expressions of $\frac{\partial
f(\bm{\theta})}{\partial\bm{\theta}}$ and $\frac{\partial
f_{c}(\bm{\theta})}{\partial\bm{\theta}}$ in Appendix K. Finally, taking the
optimization of $f_{c}(\bm{\theta})$ for example, using the backtracking line
search, the pseudocode for performing the gradient ascent is presented in
Algorithm 1.

##  VII Numerical Results

This section aims to examine the properties and benefits after integrating RIS
into massive MIMO systems and characterize the impact of key system
parameters. We first consider a classic RIS application scenario where the RIS
is deployed in close proximity to some cell-edge users. In this case, the
direct links are relatively weak, and therefore it is expected that the RIS
could bring promising gains. Accordingly, assume that $K=8$ users are evenly
distributed on a semicircle centered at the RIS and of a radius $d_{UI}=20$ m.
The distance between the RIS and the BS is $d_{IB}=700$ m. The distance
between the user $k$ and the BS are obtained from geometric relationship,
i.e.,
$\left(d_{k}^{\mathrm{UB}}\right)^{2}=\left(d_{\mathrm{IB}}-d_{\mathrm{UI}}\cos\left(\frac{\pi}{9}k\right)\right)^{2}+\left(d_{\mathrm{UI}}\sin\left(\frac{\pi}{9}k\right)\right)^{2}$.
The path-loss exponent of the direct links is larger than the path-loss
exponent of the RIS-assisted links in order to characterize the more severe
signal attenuation due to the presence of blockages on the ground.
Accordingly, we set the distance-dependent path-loss factors equal to
$\alpha_{k}=10^{-3}d_{\mathrm{UI}}^{-2}$,
$\beta=10^{-3}d_{\mathrm{IB}}^{-2.5}$ and
$\gamma_{k}=10^{-3}\left(d_{k}^{\mathrm{UB}}\right)^{-4},\forall k$. The
number of symbols in each channel coherence time interval is
$\tau_{c}=196$[61, 62], and $\tau=K=8$ symbols are utilized for channel
estimation.  The power of noise is $\sigma^{2}=-104$ dBm (noise density of
$-174$ dBm/Hz with 10 MHz bandwidth). The other simulation parameters (unless
stated otherwise) are listed in Table IV.

TABLE IV: Simulation parameters. $\left({\varphi_{t}^{a},\varphi_{t}^{e}}\right)$ | $(4.17,0.09)$ | $\left({\phi_{r}^{a},\phi_{r}^{e}}\right)$ | $(6.28,4.21)$
---|---|---|---
$\left({\varphi_{1r}^{a},\varphi_{1r}^{e}}\right)$ | $(5.20,4.32)$ | $\left({\varphi_{2r}^{a},\varphi_{2r}^{e}}\right)$ | $(0.41,2.52)$
$\left({\varphi_{3r}^{a},\varphi_{3r}^{e}}\right)$ | $(3.84,1.78)$ | $\left({\varphi_{4r}^{a},\varphi_{4r}^{e}}\right)$ | $(1.35,4.15)$
$\left({\varphi_{5r}^{a},\varphi_{5r}^{e}}\right)$ | $(5.08,5.76)$ | $\left({\varphi_{6r}^{a},\varphi_{6r}^{e}}\right)$ | $(4.75,1.56)$
$\left({\varphi_{7r}^{a},\varphi_{7r}^{e}}\right)$ | $(4.74,5.36)$ | $\left({\varphi_{8r}^{a},\varphi_{8r}^{e}}\right)$ | $(0.09,1.40)$
BS antennas | $M=64$ | RIS elements | $N=64$
Transmit power |  $p=30$ dBm | Antenna spacing | $d_{bs}=\lambda/2$
Rician factors |  $\delta=1$, $\varepsilon_{k}=10,\forall k$ | Approximation factor | $\mu=100$

###  VII-A Spatial-independent Channels without EMI

We first validate our analytical results by assuming that the channels are
spatially independent and the EMI is neglected. This could help us gain a
preliminary but useful understanding for RIS-aided systems thanks to the easy-
understanding rate expressions and some explicit analytical insights in
Section IV. Specifically, the analytical results are obtained by using Theorem
2 and related corollaries. The Monte Carlo simulation results, which are
referred to as “Simulation” in the legends of the following figures, are
obtained from (35) by averaging over $10^{5}$ random channel realizations. The
phase shifts are designed by solving Problem (111) with respect to
$f(\bm{\theta})$.

####  VII-A1 Quality of LMMSE Channel Estimation

Figure 2: MSE and NMSE of user $1$ versus the number of RIS elements.

To begin with, we investigate the MSE and NMSE performance of the proposed
channel estimation scheme. The MSE and NMSE of the channel estimation
algorithm of the $k$-th user are characterized through the functions
$\mathrm{Tr}\left\\{\mathbf{MSE}_{k}\right\\}$ and $\mathrm{NMSE}_{k}$,
respectively. Without loss of generality, Fig. 2 illustrates the MSE and NMSE
of user $1$ versus the number of RIS elements $N$. In general Rician channels,
we observe that the MSE is an increasing function of $N$ while the NMSE is a
decreasing function of $N$, which is consistent with Corollaries 1, 2 and 3.
This is because the number of communication paths increases with $N$, but the
pilot length $\tau$ does not increase correspondingly, which increases the
estimation error. However, the intensity of the channel gains increases with
$N$, which, in turn, decreases the normalized errors. In purely LoS RIS-
assisted channels ($\delta=\varepsilon_{k}\to\infty$), the MSE and NMSE are,
on the other hand, independent of $N$. This is because LoS channels are
deterministic, and therefore do not introduce additional estimation errors.
Also, we see that the MSE tends to an upper bound but the NMSE tends to zero
when $N\to\infty$, which validates Corollary 1 and 2. Besides, by increasing
the length of the pilot signals from $8$ to $30$, we see that the NMSE
decreases. However, the NMSE that is obtained for $\tau=30$ can also be
obtained for $\tau=8$ but by using a larger value for $N$. This validates that
the increase of the RIS elements can play a similarly role as increasing
$\tau$. Finally, we see that the NMSE tends to a limit less than $1$ when the
transmit power is scaled proportionally to $p=100/N$, as $N\to\infty$. This
validates the correctness of (26).

####  VII-A2 Single-user Case

Next, we evaluate the ergodic achievable rate in the single-user scenario,
where only user $1$ is present.

Figure 3: Comparison of the two-timescale design
and instantaneous CSI-based design.

Figure 4:  Rate versus $N$ in a single-user system. The transmit power is
scaled as $p=E_{u}/N^{2}$ or $p=E_{u}/N$, where $E_{u}=20$ dB.

In Fig. 4, we compare the proposed two-timescale scheme with the conventional
instantaneous CSI-based scheme. The detailed implementation of the
instantaneous CSI-based scheme is presented in Appendix J. By assuming the
same rate loss factor (ideal but not achievable), it is seen that the
instantaneous CSI-based scheme outperforms the proposed two-timescale scheme,
especially when $N$ is large. This is because the LoS and NLoS channel
components are both exploited in the instantaneous CSI-based RIS design. By
contrast, the fast-fading NLoS channel information is not used in the proposed
statistical CSI-based RIS design. When considering the practical channel
estimation overhead, however, the proposed scheme outperforms the
instantaneous CSI-based scheme. This is because the instantaneous CSI-based
scheme requires a longer pilot length, which is proportional to $N$, even
though it can achieve a higher SNR. When $N$ is large, the instantaneous CSI-
based scheme needs a large number of time slots to transmit the pilot
sequence, and then only a few symbols are left for data transmission. As a
result of the high estimation overhead, the instantaneous CSI-based scheme
incurs in a rate loss, which leads to a severe decrease of the rate in the
large $N$ regime. Therefore, Fig. 4 validates the effectiveness of the
proposed two-timescale scheme.

In Fig. 4, we illustrate the power scaling law as a function of $N$ in a
single-user system. In agreement with Corollary 9, the rate converges to a
limit if we reduce the power proportionally to $1/N^{2}$ in Rician fading
channels. Also, the limit is maximized in LoS-only RIS-assisted channels
($\delta=\varepsilon_{k}\to\infty$). In NLoS-only RIS-assisted channels
($\delta=\varepsilon_{k}=0$), scaling the power proportionally to $1/N^{2}$
reduces the rate to zero. As proved in Corollary 10, in NLoS-only RIS-assisted
channels, the power can only be scaled proportionally to $1/N$ for maintaining
a non-zero rate. These observations highlight that LoS environments are
preferable for the deployment of RIS-aided single-user systems.

####  VII-A3 Multi-user Case

In the following, we evaluate the performance of RIS-aided systems in the
general multi-user scenario.

Figure 5: Minimum user rate versus the Rician factor $\delta$ or
$\varepsilon_{k},\forall k$.

Figure 6: Minimum user rate versus $M$.

Fig. 6 shows the impact of the Rician factors. It can be observed that the
achievable rate is a decreasing function of $\delta$ but an increasing
function of $\varepsilon_{k},\forall k$. This is because the rank of the LoS
channel $\overline{\bf H}_{2}$ between the RIS and the BS is $1$, while the
rank of the LoS channel $\overline{\mathbf{H}}_{1}$ between the users and the
RIS is not. When $\delta\to\infty$, the rank of the RIS-BS channel tends to
$1$, which leads to a rank-$1$ cascaded user-RIS-BS channel. As a result, the
RIS-assisted channel becomes rank-deficient, which cannot effectively sustain
the transmission of multiple users simultaneously.  It is known that the RIS
should be deployed either near the BS or near the users so that the product
pathloss effect is mitigated[68]. In addition, Fig. 6 provides some
suggestions with respect to spatial diversity when deploying the RIS. To make
$\varepsilon_{k}$ larger, it is better to install the RIS with a certain
height and increase the strength of the LoS components in RIS-user channels.
Besides, it is necessary to guarantee the high rank of the RIS-BS channel.
This condition holds for small $\delta$ under the considered Rician fading
model. Since small $\delta$ happens when the RIS is deployed far away from the
BS, it is still a good choice to place the RIS near the users after taking
into consideration the impact of spatial diversity. On the contrary, if the
RIS is deployed near the BS, $\delta$ could be large and the BS-RIS channel
could be rank-deficiency under the considered Rician fading model. In this
context, some additional operations are required to increase the rank, such as
introducing deterministic scatterers between the BS and the RIS or placing the
RIS very close to the BS so that the spherical wave model is valid[46].

In Fig. 6, we evaluate the rate as a function of the number of BS antennas.
This figure illustrates the impact of deploying an RIS in conventional massive
MIMO systems. It is observed that the deployment of an RIS effectively
improves the rate, and the improvement increases with the number of RIS
elements. It is worth nothing that this performance gain is obtained by using
a simple MRC receiver at the BS, and that the LMMSE channel estimator requires
the same amount of overhead as conventional massive MIMO systems. With the
help of an RIS, we can achieve the same rate as conventional massive MIMO
systems, but with a much smaller number of BS antennas. In particular, the
rate obtained by a $200$-antenna BS in conventional massive MIMO systems can
be obtained by a $100$-antenna BS in RIS-aided massive MIMO systems with
$N=64$ RIS elements. The number of BS antennas can be further decreased to
$M=64$ if the number of RIS elements is increased to $N=400$. Since the cost
and energy consumption of one RIS element is much lower than that of one BS
antenna, we conclude that the integration of RISs in conventional massive MIMO
systems is a promising and cost-effective solution for future wireless
communication systems.

Figure 7: Minimum user rate versus $M$ when $\delta=0$.
The transmit power is scaled as $p=E_{u}/\sqrt{M}$,
where $E_{u}=10$ dB.

Figure 8: Minimum user rate versus $N$ when $\delta=0$ or $\varepsilon_{k}=0$.
The transmit power is scaled as $p=E_{u}/N$, where $E_{u}=10$ dB.

In Fig. 8 and Fig. 8, finally, we investigate the power scaling law over the
purely NLoS RIS-BS channel ($\delta=0$) and the purely NLoS user-RIS channels
($\varepsilon_{k}=0,\forall k$). In Fig. 8, the transmit power is scaled
proportionally to $1/\sqrt{M}$ for the NLoS RIS-BS channel ($\delta=0$). In
agreement with Corollary 6, if $\delta=0$, the rate can be maintained to a
non-zero value when the power is scaled proportionally to $1/\sqrt{M}$ as
$M\to\infty$. Compared with conventional massive MIMO systems, the deployment
of an RIS effectively improves the asymptotic limit when $M\to\infty$, and the
rate gain could be further improved by increasing $N$.

In Fig. 8, the transmit power is scaled proportionally to $1/N$ over the
purely NLoS RIS-BS channel ($\delta=0$) or purely NLoS user-RIS channels
($\varepsilon_{k}=0,\forall k$). For $N\to\infty$, the rate maintains a non-
zero value, which is consistent with Corollaries 7 and 8. Besides, in
agreement with Corollary 7, the asymptotic limit for $\delta=0$ when
$N\to\infty$ can be significantly improved by increasing the number of BS
antennas from $M=64$ to $M=400$. This is because the RIS-BS channel has a high
rank if $\delta=0$, which decreases the spatial correlation among the users
and mitigates the multi-user interference. Furthermore, in agreement with
Corollary 8, the asymptotic limit for $\varepsilon_{k}=0,\forall k$ when
$N\to\infty$ only marginally increases when increasing $M$ from $64$ to $400$.
This observation confirms once again that it is beneficial to guarantee the
spatial diversity between the RIS and the BS.

###  VII-B Spatial-correlated Channels with EMI

The above simulation results have confirmed the gain of RIS in the case of
independent channels without EMI. After considering the spatial correlation
and EMI, this subsection will examine the performance of RIS in practical
cases. Meanwhile, we want to answer the question of whether the considered
independent case without EMI can serve as a high-accuracy approximation when
the element spacing is large and when the EMI is light. The strength of EMI is
characterized by the following ratio[58]

$\displaystyle\rho=\frac{\sigma_{e}^{2}}{\sigma^{2}},$ (124)

which defines the strength of EMI received at the RIS compared with the
thermal noise received at the BS.

Figure 9:  Achievable rate versus $N$ with different RIS element spacing
$d_{ris}$.

Figure 10:  The impact of EMI.

Fig. 10 illustrates the impact of RIS spatial correlation on the achievable
rate. Firstly, it can be seen that our results based on spatially independent
channels without EMI achieve a quite good approximation accuracy when the
element spacing is large ($d_{ris}=\lambda/2$) and when the EMI is light
($\rho=30$ dB). This means that the analytical insights drawn in Section IV
based on spatially independent channels and EMI-free assumptions, are still
meaningful for us to understand the properties of practical RIS-aided systems.
As element spacing decreases ($d_{ris}=\lambda/4$, $\lambda/8$), spatial
correlation becomes severe and it can be seen that there exists a degradation
for the achievable rate in the area of small $N$. This comes from the negative
impact of spatial correlation on the channel ranks and spatial diversity.
However, as $N$ increases, the rate with smaller $d_{ris}$ finally exceeds
that with larger $d_{ris}$. This is because the RIS has a greater ability to
customize the wireless channels in the presence of channel correlation, as
discussed in Theorem 3 and Remark 3. With larger $N$, the larger array and
passive beamforming gains provided by optimizing RIS outweigh the negative
rate loss from the decreased spatial diversity, which leads to an improved
achievable rate.

The impact of EMI is studied in Fig. 10. It can be found that when the power
of EMI is small ($\rho<60$ dB), the impact of EMI on the achievable rate is
negligible. This is because the users in RIS-aided systems share the common
RIS-BS link, which results in strong multi-user interference when using MRC.
As a result, when the EMI is light, its effects are overwhelmed by the effects
of the multi-user interference. As $\rho$ increases, the EMI becomes severer,
and it finally dominates the received signal at the RIS. In this context,
after reflected by the RIS, the EMI is stronger and then severely degrades the
received SINR at the BS, and the RIS-aided systems could even perform worse
than conventional massive MIMO systems.

Figure 11:  Achievable rate when the power is
scaled down proportionally to $p=10/N$.

Figure 12:  CDF of minimal user rate for RIS-aided and RIS-free systems,
$d_{ris}=\lambda/4$, $\rho=30$ dB, $p=20$ dBm.

Fig. 12 illustrates the variation of power scaling laws after considering
spatial correlation and EMI. Note that Corollary 11 proves that the power
scaling laws may not hold when $N\to\infty$ after considering EMI. However, if
EMI power is not very severe, we can see that the power scaling law still
holds when $N$ is large but not infinite. This is because even though the
power is reduced reversely proportional to $N$, when $N$ is not infinite, the
impact of multi-user interference is still stronger than that of EMI. Besides,
it again demonstrates that our analytical results are good approximations
under large element spacing ($d_{ris}=\lambda/2$) and small EMI. Furthermore,
with smaller element spacing ($d_{ris}=\lambda/4$), due to the larger gains
from optimizing RIS, the rate is improved with large $N$ as expected.

###  VII-C Other Use Cases

The above numerical results reveal an effective use case of the RIS when it is
deployed close to some cell-edge users. In this subsection, we would like to
examine the gain of using RIS in the more general scenarios. Specifically, in
practical systems, the users may distribute in a large region and the transmit
power could be small due to the use of large numbers of antennas. Therefore,
we consider that the transmit power is $p=20$ dBm and assume that the users
are randomly distributed in the $100$ m $\times$ $100$ m area from $(200\text{
m},0)$ to $(300\text{ m},100)$ [54]. The BS and RIS are deployed in $(0,0)$
and $(200\text{ m},0)$, respectively. Besides, we assume $d_{ris}=\lambda/4$
and $\rho=30$ dB for RIS element spacing and EMI.

In Fig. 12, we randomly generate the locations of users for $500$ times, and
then compare the RIS-aided and conventional massive MIMO systems. It can be
seen that in the considered scenarios, although the gain of using RIS still
exists, it is not as significant as that when the RIS is deployed very close
to the users. Meanwhile, if the RIS is not well optimized, the rate in RIS-
aided systems could be even worse than conventional massive MIMO systems. This
is due to the double pathloss effect in RIS-aided systems, and also because
the RIS-aided systems using MRC suffer from stronger multi-user interference
than conventional massive MIMO systems. When the direct link is strong enough,
the conventional massive MIMO systems can perform well, and the gain of RIS is
not obvious. Therefore, Fig. 12 indicates that when using MRC, it is more
suitable to deploy the RIS close to some cell-edge users whose direct links
are weak or even blocked. In that case, the RIS could bring additional high-
quality communication links for these users and therefore significantly
improve their rate performance.

##  VIII Conclusion

This paper investigated a two-timescale design for RIS-aided massive MIMO
systems by taking into account the channel estimation errors.  We first
considered the spatial-independent channel model and then extended it to the
spatial-correlated channel model in the presence of EMI. In both cases, we
obtained the LMMSE channel estimator for user-BS aggregated channels, employed
the MRC detectors, derived the UatF bound of the achievable rate, and designed
the phase shifts of the RIS based on gradient ascent methods. Besides, to gain
a better understanding of RIS-aided systems, we provided some explicit
insights under the simplified spatial-independent channel model. We
demonstrated that the transmit power can be reduced proportionally to $1/M$,
while maintaining a non-zero rate, as $M\to\infty$ over RIS-BS Rician
channels. If the RIS-BS channel is Rayleigh distributed, on the other hand, a
non-zero rate can be maintained when the power is scaled proportionally to
$1/\sqrt{M}$ as $M\to\infty$ or proportionally to $1/N$ as $N\to\infty$. Then,
by considering the practical spatial-correlated model and the EMI, we
demonstrated that the simplified model can be regarded as a high-quality
approximation when the element spacing of RIS is large and when the EMI is
light. Besides, it was shown that the presence of spatial correlation helps
the RIS play more roles in the improvement of the rate. We also found that it
is more suitable to place the RIS close to some cell-edge users so that the
gains of the RIS can be fully harvested.

##  Appendix A Some Useful Results

######  Lemma 7.

Consider a matrix ${\bf X}\in\mathbb{C}^{m\times n}$, $m,n\geq 1$, whose
entries are i.i.d. random variables with zero mean and ${v}_{x}$ variance.
Consider a deterministic matrix ${\bf W}\in\mathbb{C}^{n\times n}$. Then, we
have

$\displaystyle\mathbb{E}\left\\{\mathbf{X}\mathbf{W}\mathbf{X}^{H}\right\\}={v}_{{x}}\operatorname{Tr}\\{\mathbf{W}\\}\mathbf{I}_{m}.$
(125)

Proof: Consider the matrix $\mathbf{X}\mathbf{W}\mathbf{X}^{H}$. The
expectation of its $(i,j)$-th entry, where $i\neq j$, is given by

$\displaystyle\left[\mathbb{E}\left\\{\mathbf{X}\mathbf{W}\mathbf{X}^{H}\right\\}\right]_{ij}=\mathbb{E}\left\\{\sum_{l=1}^{n}\sum_{k=1}^{n}\mathbf{X}_{ik}\mathbf{W}_{kl}[\mathbf{X}^{H}]_{lj}\right\\}=\sum_{l=1}^{n}\sum_{k=1}^{n}\mathbb{E}\left\\{\mathbf{X}_{ik}\mathbf{X}_{jl}^{*}\right\\}\mathbf{W}_{kl}=0.$
(126)

Similarly, the expectation of its $(i,i)$-th entry is

$\displaystyle\left[\mathbb{E}\left\\{\mathbf{X}\mathbf{W}\mathbf{X}^{H}\right\\}\right]_{ii}=\sum_{l=1}^{n}\sum_{k=1}^{n}\mathbb{E}\left\\{\mathbf{X}_{ik}\mathbf{X}_{il}^{*}\right\\}\mathbf{W}_{kl}=\sum_{k=1}^{n}\mathbb{E}\left\\{\left|\mathbf{X}_{ik}\right|^{2}\right\\}\mathbf{W}_{kk}={v}_{{x}}\operatorname{Tr}\\{\mathbf{W}\\}.$
(127)

Therefore, the expectation of $\mathbf{X}\mathbf{W}\mathbf{X}^{H}$ is a
diagonal matrix and its diagonal entries are all equal to
${v}_{{x}}\operatorname{Tr}\\{\mathbf{W}\\}$. This completes the proof.
$\blacksquare$

By letting $m=1$ or $n=1$, corresponding results for random vectors can be
obtained.

######  Lemma 8.

Consider the deterministic matrices ${\bf W}\in\mathbb{C}^{N\times N}$ and
vectors ${\bf w}_{1},{\bf w}_{2}\in\mathbb{C}^{N\times 1}$, and ${\bf
w}_{3},{\bf w}_{4}\in\mathbb{C}^{M\times 1}$. Then, we have

$\displaystyle\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}\right\\}=\mathbb{E}\left\\{\operatorname{Re}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}\right\\}\right\\}=\mathbf{0},$
(128)
$\displaystyle\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{w}_{1}\tilde{\mathbf{h}}_{k}^{H}\mathbf{w}_{2}\right\\}=\mathbb{E}\left\\{\operatorname{Re}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{w}_{1}\tilde{\mathbf{h}}_{k}^{H}\mathbf{w}_{2}\right\\}\right\\}=0,$
(129)
$\displaystyle\mathbb{E}\left\\{\mathbf{w}_{3}^{H}\tilde{\mathbf{d}}_{k}\mathbf{w}_{4}^{H}\tilde{\mathbf{d}}_{k}\right\\}=\mathbb{E}\left\\{{\rm
Re}\left\\{\mathbf{w}_{3}^{H}\tilde{\mathbf{d}}_{k}\mathbf{w}_{4}^{H}\tilde{\mathbf{d}}_{k}\right\\}\right\\}=0.$
(130)

Proof: Let us consider a complex random variable $v=v_{r}+jv_{i}$ with
$v_{r},v_{i}\sim\mathcal{N}\left(0,1/2\right)$. Noting that for complex random
variables, different from the result that
$\mathbb{E}\left\\{\left|v\right|^{2}\right\\}=\mathbb{E}\left\\{v_{r}^{2}\right\\}+\mathbb{E}\left\\{v_{i}^{2}\right\\}=1$,
we have

$\displaystyle\mathbb{E}\left\\{v^{2}\right\\}=\mathbb{E}\left\\{v_{r}^{2}-v_{i}^{2}+2jv_{r}v_{i}\right\\}=\mathbb{E}\left\\{v_{r}^{2}\right\\}-\mathbb{E}\left\\{v_{i}^{2}\right\\}+2j\mathbb{E}\left\\{v_{r}\right\\}\mathbb{E}\left\\{v_{i}\right\\}=0,$
(131)
$\displaystyle\mathbb{E}\left\\{\operatorname{Re}\left\\{v^{2}\right\\}\right\\}=\mathbb{E}\left\\{v_{r}^{2}\right\\}-\mathbb{E}\left\\{v_{i}^{2}\right\\}=0.$
(132)

The entries of $\tilde{\mathbf{H}}_{2}$ are i.i.d., each having the same
distribution as $v$. Then, we have

$\displaystyle\mathbb{E}\left\\{\left[\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}\right]_{n1,n2}\right\\}=\mathbb{E}\left\\{\sum_{i=1}^{N}\sum_{m=1}^{N}\left[\tilde{\mathbf{H}}_{2}\right]_{n1,m}\mathbf{W}_{m,i}\left[\tilde{\mathbf{H}}_{2}\right]_{i,n2}\right\\}.$
(133)

For $(n1,m)\neq(i,n2)$ in (133), the expectation is zero, since the entries
are independent and zero-mean. For $(n1,m)=(i,n2)$ in (133), the expectation
is zero by using (131). Therefore, (128) is proved. Equations (129) and (130)
can be proved mutatis mutandis. $\blacksquare$

######  Lemma 9.

For deterministic matrices $\mathbf{C}\in\mathbb{C}^{M\times M}$ and
$\mathbf{W}\in\mathbb{C}^{N\times N}$, if $\mathbf{C}=\mathbf{C}^{H}$, there
is

$\displaystyle\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\right\\}=\operatorname{Tr}\\{\mathbf{W}\\}\operatorname{Tr}\left\\{\mathbf{C}^{2}\right\\}\mathbf{I}_{N}+|\operatorname{Tr}\\{\mathbf{C}\\}|^{2}\mathbf{W},$
(134)
$\displaystyle\mathbb{E}\left\\{\tilde{\mathbf{H}}_{c,2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{c,2}\mathbf{W}\tilde{\mathbf{H}}_{c,2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{c,2}\right\\}=\operatorname{Tr}\left\\{\mathbf{R}_{ris}\mathbf{W}\right\\}\operatorname{Tr}\left\\{\mathbf{C}^{2}\right\\}\mathbf{R}_{ris}+|\operatorname{Tr}\\{\mathbf{C}\\}|^{2}\mathbf{R}_{ris}\mathbf{WR}_{ris}.$
(135)

Proof: Define $\tilde{\mathbf{H}}_{2}=\left[{\bf J}_{1},\ldots,{\bf
J}_{N}\right]$, where ${\bf J}_{n}\in\mathbb{C}^{M\times 1}$, $1\leq n\leq N$,
are independent of each other, and ${\bf J}_{n}\sim\mathcal{CN}\left({\bf
0},{\bf I}_{M}\right)$. Denoting $[\mathbf{W}]_{m,n}={w}_{mn}$, then we have

$\displaystyle\begin{aligned}
\left[\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\right]_{i,j}=\sum_{h=1}^{N}\sum_{m=1}^{N}\mathbf{J}_{i}^{H}\mathbf{C}\mathbf{J}_{m}w_{mh}\mathbf{J}_{h}^{H}\mathbf{C}\mathbf{J}_{j}.\end{aligned}$
(136)

Note that
$\mathbb{E}\left\\{\left|\mathbf{J}_{i}^{H}\mathbf{C}\mathbf{J}_{i}\right|^{2}\right\\}=|\operatorname{Tr}(\mathbf{C})|^{2}+\operatorname{Tr}\left(\mathbf{C}^{2}\right)$
[74, (35)]. The expectation of the $i$-th diagonal term can be calculated as

$\displaystyle\begin{aligned}
&\mathbb{E}\left\\{\left[\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\right]_{i,i}\right\\}\\\
&=\mathbb{E}\left\\{\mathbf{J}_{i}^{H}\mathbf{C}\mathbf{J}_{i}w_{ii}\mathbf{J}_{i}^{H}\mathbf{C}\mathbf{J}_{i}\right\\}+\mathbb{E}\left\\{\sum_{m=1,m\neq
i}^{N}\mathbf{J}_{i}^{H}\mathbf{C}\mathbf{J}_{m}w_{mm}\mathbf{J}_{m}^{H}\mathbf{C}\mathbf{J}_{i}\right\\}\\\
&=w_{ii}\mathbb{E}\left\\{\left|\mathbf{J}_{i}^{H}\mathbf{C}\mathbf{J}_{i}\right|^{2}\right\\}+\mathbb{E}\left\\{\sum_{m=1,m\neq
i}^{N}w_{mm}\mathbf{J}_{i}^{H}\mathbf{C}\mathbb{E}\left\\{\mathbf{J}_{m}\mathbf{J}_{m}^{H}\right\\}\mathbf{C}\mathbf{J}_{i}\right\\}\\\
&=w_{ii}|\operatorname{Tr}(\mathbf{C})|^{2}+\operatorname{Tr}\\{\mathbf{W}\\}\operatorname{Tr}\left\\{\mathbf{C}^{2}\right\\}.\end{aligned}$
(137)

The expectation of the $(i,j)$-th non-diagonal term is given by

$\displaystyle\begin{aligned}
&\mathbb{E}\left\\{\left[\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}^{H}\mathbf{C}\tilde{\mathbf{H}}_{2}\right]_{i,j}\right\\}=\mathbb{E}\left\\{\mathbf{J}_{i}^{H}\mathbf{C}\mathbf{J}_{i}w_{ij}\mathbf{J}_{j}^{H}\mathbf{C}\mathbf{J}_{j}\right\\}\\\
&=w_{ij}\mathbb{E}\left\\{\mathbf{J}_{i}^{H}\mathbf{CJ}_{i}\right\\}\mathbb{E}\left\\{\mathbf{J}_{j}^{H}\mathbf{C}\mathbf{J}_{j}\right\\}=w_{ij}|\operatorname{Tr}\\{\mathbf{C}\\}|^{2}.\end{aligned}$
(138)

Combining (137) and (138) completes the proof of (134). Then, we can prove
(135) by using
$\tilde{\mathbf{H}}_{c,2}=\tilde{\mathbf{H}}_{2}\mathbf{R}_{ris}^{1/2}$.
$\blacksquare$

Applying Lemma 9, we can obtain some useful results as summarized in the
following Lemma.

######  Lemma 10.

For deterministic matrix $\mathbf{W}\in\mathbb{C}^{N\times N}$, we have

$\displaystyle\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}=e_{k1}^{2}M^{2}\mathbf{W}+e_{k3}M\operatorname{Tr}\\{\mathbf{W}\\}\mathbf{I}_{N},$
(139)
$\displaystyle\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\right\\}=M^{2}\mathbf{W}+M\operatorname{Tr}\\{\mathbf{W}\\}\mathbf{I}_{N},$
(140)
$\displaystyle\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{a}_{M}\mathbf{a}_{M}^{H}\tilde{\mathbf{H}}_{2}\mathbf{W}\tilde{\mathbf{H}}_{2}^{H}\mathbf{a}_{M}\mathbf{a}_{M}^{H}\tilde{\mathbf{H}}_{2}\right\\}=M^{2}\mathbf{W}+M^{2}\operatorname{Tr}\\{\mathbf{W}\\}\mathbf{I}_{N}.$
(141)

where $\mathbf{A}_{k}$ is defined in (18), and $e_{k1}$ and $e_{k3}$ are
defined in Lemma 2. For deterministic matrix $\mathbf{C}\in\mathbb{C}^{M\times
M}$ and random vector ${\bf u}\sim\mathcal{CN}\left({\bf 0},{\bf
I}_{M}\right)$, there are

$\displaystyle\mathbb{E}\left\\{\mathbf{u}\mathbf{u}^{H}\mathbf{C}\mathbf{u}\mathbf{u}^{H}\right\\}=\mathbf{C}+\operatorname{Tr}\\{\mathbf{C}\\}\mathbf{I}_{M},$
(142)
$\displaystyle\mathbb{E}\left\\{\left\|\mathbf{u}\right\|^{4}\right\\}=\operatorname{Tr}\left\\{\mathbb{E}\left\\{\mathbf{u}\mathbf{u}^{H}\mathbf{u}\mathbf{u}^{H}\right\\}\right\\}=M^{2}+M.$
(143)

## Appendix B

Recalling the definition of ${\bf q}_{k}$ in (II-A), where
$\tilde{\mathbf{H}}_{2}$, $\tilde{\mathbf{h}}_{k}$, $\tilde{\mathbf{d}}_{k}$,
and $\bf N$ are independent of each other and composed of zero-mean entries,
we have

$\displaystyle\mathbb{E}\left\\{\mathbf{y}_{p}^{k}\right\\}=\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}+\frac{1}{\sqrt{\tau
p}}\mathbb{E}\left\\{\mathbf{N}\right\\}{\bf
s}_{k}=\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}=\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{H}}_{2}{\bf\Phi}\overline{\mathbf{h}}_{k}.$
(144)

The covariance matrix between the unknown channel $\mathbf{q}_{k}$ and the
observation vector $\mathbf{y}_{p}^{k}$ can be written as

$\displaystyle\begin{array}[]{l}\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{y}_{p}^{k}\right\\}=\mathbb{E}\left\\{\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)\left(\mathbf{y}_{p}^{k}-\mathbb{E}\left\\{\mathbf{y}_{p}^{k}\right\\}\right)^{H}\right\\}\\\
=\mathbb{E}\left\\{\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)\left(\mathbf{q}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{Ns}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)^{H}\right\\}\\\
=\mathbb{E}\left\\{\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)^{H}\right\\}\\\
=\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\},\end{array}$
(149)

and

$\displaystyle\operatorname{Cov}\left\\{\mathbf{y}_{p}^{k},\mathbf{q}_{k}\right\\}=\left(\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{y}_{p}^{k}\right\\}\right)^{H}=\left(\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}\right)^{H}=\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}.$
(150)

Invoking the definition of $\mathbf{q}_{k}$, we obtain

$\displaystyle\begin{array}[]{l}\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}=\mathbb{E}\left\\{\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)^{H}\right\\}\\\
=\mathbb{E}\left\\{\begin{array}[]{l}\left(\sqrt{c_{k}\delta}\,\overline{\mathbf{H}}_{2}{\bf\Phi}\tilde{\mathbf{h}}_{k}+\sqrt{c_{k}\varepsilon_{k}}\,\tilde{\mathbf{H}}_{2}{\bf\Phi}\overline{\mathbf{h}}_{k}+\sqrt{c_{k}}\,\tilde{\mathbf{H}}_{2}{\bf\Phi}\tilde{\mathbf{h}}_{k}+\sqrt{\gamma_{k}}\tilde{\mathbf{d}}_{k}\right)\\\
\times\left(\sqrt{c_{k}\delta}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}+\sqrt{c_{k}\varepsilon_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}+\sqrt{c_{k}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}+\sqrt{\gamma_{k}}\tilde{\mathbf{d}}_{k}^{H}\right)\end{array}\right\\}\\\
=\mathbb{E}\left\\{c_{k}\delta\overline{\mathbf{H}}_{2}{\bf\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}{\bf\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}+c_{k}\varepsilon_{k}\tilde{\mathbf{H}}_{2}{\bf\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}{\bf\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}+c_{k}\tilde{\mathbf{H}}_{2}{\bf\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}{\bf\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}+\gamma_{k}\tilde{\mathbf{d}}_{k}\tilde{\mathbf{d}}_{k}^{H}\right\\}\\\
{{\mathop{=}\limits^{\left(b\right)}}}Nc_{k}\delta\mathbf{a}_{M}\mathbf{a}_{M}^{H}+\left(Nc_{k}\left(\varepsilon_{k}+1\right)+\gamma_{k}\right)\mathbf{I}_{M},\end{array}$
(157)

where ($b$) exploits Lemma 7 and the mutual independence of
$\tilde{\mathbf{H}}_{2}$ and $\tilde{\mathbf{h}}_{k}$.

Similarly, we have

$\displaystyle\begin{array}[]{l}\operatorname{Cov}\left\\{\mathbf{y}_{p}^{k},\mathbf{y}_{p}^{k}\right\\}=\mathbb{E}\left\\{\left(\mathbf{y}_{p}^{k}-\mathbb{E}\left\\{\mathbf{y}_{p}^{k}\right\\}\right)\left(\mathbf{y}_{p}^{k}-\mathbb{E}\left\\{\mathbf{y}_{p}^{k}\right\\}\right)^{H}\right\\}\\\
=\mathbb{E}\left\\{\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}+\frac{1}{\sqrt{\tau
p}}\mathbf{Ns}_{k}\right)\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}+\frac{1}{\sqrt{\tau
p}}\mathbf{Ns}_{k}\right)^{H}\right\\}\\\
=\mathbb{E}\left\\{\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)\left(\mathbf{q}_{k}-\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}\right)^{H}\right\\}+\frac{1}{\tau
p}\mathbb{E}\left\\{\mathbf{N}\mathbf{s}_{k}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\right\\}\\\
=\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}+\frac{\sigma^{2}}{\tau
p}\mathbf{I}_{M}.\end{array}$ (162)

Finally, by introducing the auxiliary variables of $a_{k1}=Nc_{k}\delta$ and
$a_{k2}=Nc_{k}\left(\varepsilon_{k}+1\right)+\gamma_{k}$, the proof is
completed.

## Appendix C

The LMMSE estimate of the channel ${\bf q}_{k}$ based on the observation
vector ${\bf y}^{k}_{p}$ can be written as[71, Chapter 12.5]

$\displaystyle\hat{\mathbf{q}}_{k}=\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}+\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{y}_{p}^{k}\right\\}\operatorname{Cov}^{-1}\left\\{\mathbf{y}_{p}^{k},\mathbf{y}_{p}^{k}\right\\}\left(\mathbf{y}_{p}^{k}-\mathbb{E}\left\\{\mathbf{y}_{p}^{k}\right\\}\right),$
(163)

where the mean and covariance matrices have been obtained in Lemma 1.

Let us compute
$\operatorname{Cov}^{-1}\left\\{\mathbf{y}_{p}^{k},\mathbf{y}_{p}^{k}\right\\}$.
Using the Woodbury matrix identity[71, Page 571], we have

$\displaystyle\operatorname{Cov}^{-1}\left\\{\mathbf{y}_{p}^{k},\mathbf{y}_{p}^{k}\right\\}$
$\displaystyle=\left(a_{k1}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)\mathbf{I}_{M}\right)^{-1}$
$\displaystyle=\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)^{-1}\mathbf{I}_{M}-\frac{a_{k1}\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)^{-2}}{1+Ma_{k1}\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)^{-1}}\mathbf{a}_{M}\mathbf{a}_{M}^{H}.$ (164)

As a result, we have

$\displaystyle\begin{aligned}
&\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{y}_{p}^{k}\right\\}\operatorname{Cov}^{-1}\left\\{\mathbf{y}_{p}^{k},\mathbf{y}_{p}^{k}\right\\}\\\
&=\left(a_{k1}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k2}\mathbf{I}_{M}\right)\left\\{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)^{-1}\mathbf{I}_{M}-\frac{a_{k1}\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)^{-2}}{1+Ma_{k1}\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)^{-1}}\mathbf{a}_{M}\mathbf{a}_{M}^{H}\right\\}\\\
&=\frac{a_{k1}\frac{\sigma^{2}}{\tau p}}{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)\left\\{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)+Ma_{k1}\right\\}}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+\frac{a_{k2}}{a_{k2}+\frac{\sigma^{2}}{\tau
p}}\mathbf{I}_{M}\\\ &\triangleq
a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k4}\mathbf{I}_{M}\triangleq\mathbf{A}_{k}=\mathbf{A}^{H}_{k}.\end{aligned}$
(165)

Since we have
$\mathbb{E}\left\\{\mathbf{q}_{k}\right\\}=\mathbb{E}\left\\{\mathbf{y}^{k}_{p}\right\\}=\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}$,
the LMMSE channel estimate in (163) is calculated as

$\displaystyle\begin{array}[]{l}\hat{\mathbf{q}}_{k}=\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{H}}_{2}{\bf\Phi}\overline{\mathbf{h}}_{k}+\mathbf{A}_{k}\left(\mathbf{y}_{p}^{k}-\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{H}}_{2}{\bf\Phi}\overline{\mathbf{h}}_{k}\right)\\\
\,\quad=\mathbf{A}_{k}\mathbf{y}_{p}^{k}+\left(\mathbf{I}_{M}-\mathbf{A}_{k}\right)\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{H}}_{2}{\bf\Phi}\overline{\mathbf{h}}_{k}\\\
\,\quad\triangleq\mathbf{A}_{k}\mathbf{y}_{p}^{k}+\mathbf{B}_{k}.\end{array}$
(169)

Additionally, we can expand the above linear expression and rewrite it as

$\displaystyle\begin{aligned}
\hat{\mathbf{q}}_{k}&=\mathbf{A}_{k}\left(\mathbf{q}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{Ns}_{k}\right)+\mathbf{B}_{k}\\\
&=\mathbf{A}_{k}\left(\sum\nolimits_{\omega=1}^{4}\mathbf{q}_{k}^{\omega}+\sqrt{\gamma_{k}}\tilde{\mathbf{d}}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{Ns}_{k}\right)+\left(\mathbf{I}_{M}-\mathbf{A}_{k}\right)\mathbf{q}_{k}^{1}\\\
&=\mathbf{q}_{k}^{1}+\sum\nolimits_{\omega=2}^{4}\mathbf{A}_{k}\mathbf{q}_{k}^{\omega}+\sqrt{\gamma_{k}}\mathbf{A}_{k}\tilde{\mathbf{d}}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{A}_{k}\mathbf{Ns}_{k}.\end{aligned}$ (170)

Then, by exploiting the property
$\mathbf{A}_{k}\overline{\mathbf{H}}_{2}=\left(a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k4}\mathbf{I}_{M}\right)\mathbf{a}_{M}\mathbf{a}_{N}^{H}=\left(Ma_{k3}+a_{k4}\right)\overline{\mathbf{H}}_{2}$,
we arrive at (17).

Based on the estimate $\hat{\mathbf{q}}_{k}$, we can obtain the estimation
error $\mathbf{e}_{k}=\mathbf{q}_{k}-\hat{\mathbf{q}}_{k}$. By direct
inspection, the mean of $\mathbf{e}_{k}$ is zero. Exploiting [71, Eq.
(12.21)], Lemma 1 and (165), the MSE matrix of the estimation error can be
calculated as

$\displaystyle\begin{array}[]{l}{\rm\bf{MSE}}_{k}=\mathbb{E}\left\\{\mathbf{e}_{k}\mathbf{e}_{k}^{H}\right\\}\\\
\qquad\,\quad=\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}-\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{y}_{p}^{k}\right\\}\operatorname{Cov}^{-1}\left\\{\mathbf{y}_{p}^{k},\mathbf{y}_{p}^{k}\right\\}\operatorname{Cov}\left\\{\mathbf{y}_{p}^{k},\mathbf{q}_{k}\right\\}\\\
\qquad\,\quad=\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}-\mathbf{A}_{k}\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}\\\
\qquad\,\quad=\left(\mathbf{I}_{M}-\mathbf{A}_{k}\right)\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}\\\
\qquad\,\quad=\left(\left(1-a_{k4}\right)\mathbf{I}_{M}-a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}\right)\left(a_{k1}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k2}\mathbf{I}_{M}\right)\\\
\qquad\,\quad=\left(a_{k1}\left(1-a_{k4}\right)-Ma_{k1}a_{k3}-a_{k2}a_{k3}\right)\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k2}\left(1-a_{k4}\right)\mathbf{I}_{M}\\\
\qquad\,\quad\triangleq
a_{k5}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k6}\mathbf{I}_{M},\end{array}$ (178)

where

$\displaystyle\begin{aligned}
a_{k5}&=a_{k1}\left(1-a_{k4}\right)-\left(Ma_{k1}+a_{k2}\right)a_{k3}\\\
&=a_{k1}\left(1-\frac{a_{k2}}{a_{k2}+\frac{\sigma^{2}}{\tau
p}}\right)-\left(Ma_{k1}+a_{k2}\right)\frac{a_{k1}\frac{\sigma^{2}}{\tau
p}}{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)\left\\{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)+Ma_{k1}\right\\}}\\\ &=\frac{a_{k1}\left(\frac{\sigma^{2}}{\tau
p}\right)^{2}}{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)\left(a_{k2}+\frac{\sigma^{2}}{\tau p}+Ma_{k1}\right)},\end{aligned}$
(179)

and

$\displaystyle
a_{k6}=a_{k2}\left(1-a_{k4}\right)=a_{k2}\left(1-\frac{a_{k2}}{a_{k2}+\frac{\sigma^{2}}{\tau
p}}\right)=\frac{a_{k2}\frac{\sigma^{2}}{\tau
p}}{a_{k2}+\frac{\sigma^{2}}{\tau p}}.$ (180)

Based on the MSE matrix, the NMSE of the estimation error can be expressed
as[53, Eq. (3.20)]

$\displaystyle\operatorname{NMSE}_{k}$
$\displaystyle=\frac{\operatorname{Tr}\left\\{{\rm\bf{MSE}}_{k}\right\\}}{\operatorname{Tr}\left\\{\operatorname{Cov}\left\\{\mathbf{q}_{k},\mathbf{q}_{k}\right\\}\right\\}}=\frac{M\left(a_{k5}+a_{k6}\right)}{M\left(a_{k1}+a_{k2}\right)}=\frac{a_{k5}+a_{k6}}{a_{k1}+a_{k2}}$
$\displaystyle=\frac{\frac{\sigma^{2}}{\tau
p}\left(Ma_{k1}a_{k2}+a_{k2}^{2}+\left(a_{k1}+a_{k2}\right)\frac{\sigma^{2}}{\tau
p}\right)}{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}+Ma_{k1}\right)\left(a_{k1}+a_{k2}\right)}.$ (181)

Hence, the proof is completed.

## Appendix D

Recall that
$\mathbf{A}_{k}=a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k4}\mathbf{I}_{M}$,
and $\overline{\mathbf{H}}_{2}=\mathbf{a}_{M}\mathbf{a}_{N}^{H}$. We can
readily obtain

$\displaystyle\begin{array}[]{l}\operatorname{Tr}\left\\{\mathbf{A}_{k}\right\\}=M\left(a_{k3}+a_{k4}\right)\triangleq
Me_{k1},\\\
\mathbf{A}_{k}\overline{\mathbf{H}}_{2}=a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}+a_{k4}\mathbf{I}_{M}\mathbf{a}_{M}\mathbf{a}_{N}^{H}=\left(Ma_{k3}+a_{k4}\right)\overline{\mathbf{H}}_{2}\triangleq
e_{k2}\overline{\mathbf{H}}_{2},\\\
\mathbf{A}_{k}\mathbf{A}_{k}=a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}\left(a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k4}\mathbf{I}_{M}\right)+a_{k4}\mathbf{I}_{M}\left(a_{k3}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k4}\mathbf{I}_{M}\right)\\\
\qquad\quad=Ma_{k3}^{2}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+2a_{k3}a_{k4}\mathbf{a}_{M}\mathbf{a}_{M}^{H}+a_{k4}^{2}\mathbf{I}_{M},\\\
\operatorname{Tr}\left\\{\mathbf{A}_{k}\mathbf{A}_{k}\right\\}=M\left(Ma_{k3}^{2}+2a_{k3}a_{k4}+a_{k4}^{2}\right)\triangleq
Me_{k3}.\end{array}$ (187)

By direct inspection of $e_{k1},e_{k2},e_{k3}$, we evince that they are
composed of non-negative terms. Therefore, we have $e_{k1},e_{k2},e_{k3}\geq
0$. Then, we aim to prove $e_{k1},e_{k2},e_{k3}\leq 1$. We first focus on the
parameter $e_{k2}$. Using the expressions of $a_{k3}$ and $a_{k4}$ in (20) and
(21), we can expand $e_{k2}$ as

$\displaystyle\begin{aligned}
e_{k2}&=Ma_{k3}+a_{k4}=\frac{Ma_{k1}\frac{\sigma^{2}}{\tau
p}}{\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}\right)\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}+Ma_{k1}\right)}+\frac{a_{k2}}{a_{k2}+\frac{\sigma^{2}}{\tau p}}\\\
&=\frac{a_{k2}\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}+Ma_{k1}\right)+Ma_{k1}\frac{\sigma^{2}}{\tau
p}}{a_{k2}\left(a_{k2}+\frac{\sigma^{2}}{\tau
p}+Ma_{k1}\right)+Ma_{k1}\frac{\sigma^{2}}{\tau p}+\frac{\sigma^{2}}{\tau
p}\left(a_{k2}+\frac{\sigma^{2}}{\tau p}\right)}.\end{aligned}$ (188)

It is clear that the numerator in (188) is smaller than the denominator.
Therefore, we proved that $e_{k2}\leq 1$. Then, we can directly obtain

$\displaystyle e_{k1}\leq e_{k2}\leq 1,$ (189) $\displaystyle e_{k3}\leq
e_{k2}^{2}\leq e_{k2}\leq 1.$ (190)

Finally, when $\tau p\to\infty$ or $N\to\infty$, we have $a_{k3}\to 0$ and
$a_{k4}\to 1$, which implies that $e_{k1}=e_{k2}=e_{k3}\to 1$. When $\tau p\to
0$, we have $a_{k3},a_{k4}\to 0$, which gives $e_{k1}=e_{k2}=e_{k3}\to 0$.
This completes the proof.

## Appendix E

###  E-A Signal Term and Noise Term

According to the orthogonality property of the LMMSE estimator, we have
$\mathbb{E}\left\\{\mathbf{e}_{k}\left(\mathbf{y}^{k}_{p}\right)^{H}\right\\}={\bf
0}$. Besides, since ${\bf e}_{k}$ has zero mean, we obtain
$\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{e}_{k}\right\\}=\mathbb{E}\left\\{\left(\mathbf{A}_{k}\mathbf{y}_{p}^{k}+\mathbf{B}_{k}\right)^{H}\mathbf{e}_{k}\right\\}=0$.
Therefore, we have

$\displaystyle\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}=\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\hat{\mathbf{q}}_{k}\right\\}+\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{e}_{k}\right\\}=\mathbb{E}\left\\{\left\|\hat{\mathbf{q}}_{k}\right\|^{2}\right\\}.$
(191)

Denote the signal term of (35) as
$\left|\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}\right|^{2}\triangleq
E_{k}^{\rm signal}\left(\mathbf{\Phi}\right)$, and denote the noise term of
(35) as
$\mathbb{E}\left\\{\left\|\hat{\mathbf{q}}_{k}\right\|^{2}\right\\}\triangleq
E_{k}^{\rm{noise}}\left(\mathbf{\Phi}\right)$. Clearly,
$\mathbb{E}\left\\{\left\|\hat{\mathbf{q}}_{k}\right\|^{2}\right\\}$ is a real
variable. Then, from (191), we obtain

$\displaystyle E_{k}^{\rm
signal}\left(\mathbf{\Phi}\right)=\left|\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}\right|^{2}=\left(\mathbb{E}\left\\{\left\|\hat{\mathbf{q}}_{k}\right\|^{2}\right\\}\right)^{2}=\left(E_{k}^{\rm{noise}}\left(\mathbf{\Phi}\right)\right)^{2}.$
(192)

Let us now derive $E_{k}^{\rm{noise}}\left(\mathbf{\Phi}\right)$. Recall the
expressions in (II-A) and (17). Since $\tilde{\bf H}_{2}$, $\tilde{\bf
h}_{k}$, $\tilde{\bf d}_{k}$ and ${\bf N}$ are independent of each other and
they all have a zero mean, we can derive the term
$\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}$ by
selecting the non-zero terms in the expansion as

$\displaystyle\begin{array}[]{l}E_{k}^{\rm{noise}}\left(\mathbf{\Phi}\right)=\mathbb{E}\left\\{\left\|\hat{\mathbf{q}}_{k}\right\|^{2}\right\\}=\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}\\\
=\mathbb{E}\left\\{\left(\sum_{\omega=1}^{4}\hat{\mathbf{q}}_{k}^{\omega}+\sqrt{\gamma_{k}}\mathbf{A}_{k}\tilde{\mathbf{d}}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{A}_{k}\mathbf{Ns}_{k}\right)^{H}\left(\sum_{\psi=1}^{4}\mathbf{q}_{k}^{\psi}+\sqrt{\gamma_{k}}\tilde{\mathbf{d}}_{k}\right)\right\\}\\\
=\sum_{\omega=1}^{4}\mathbb{E}\left\\{\left(\hat{\mathbf{q}}_{k}^{\omega}\right)^{H}\mathbf{q}_{k}^{\omega}\right\\}+\gamma_{k}\mathbb{E}\left\\{\tilde{\mathbf{d}}_{k}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{d}}_{k}\right\\}\\\
=c_{k}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+e_{k2}c_{k}\delta\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
\quad+c_{k}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+c_{k}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}+\gamma_{k}\mathrm{Tr}\left\\{\mathbf{A}_{k}^{H}\right\\}\\\
{{\mathop{=}\limits^{\left(c\right)}}}c_{k}\delta\varepsilon_{k}M\left|f_{k}(\mathbf{\Phi})\right|^{2}+c_{k}\delta
MNe_{k2}+c_{k}\varepsilon_{k}MNe_{k1}+c_{k}MNe_{k1}+\gamma_{k}Me_{k1}\\\
=M\left\\{\left|f_{k}(\mathbf{\Phi})\right|^{2}c_{k}\delta\varepsilon_{k}+Nc_{k}\delta
e_{k2}+\left(Nc_{k}\left(\varepsilon_{k}+1\right)+\gamma_{k}\right)e_{k1}\right\\},\end{array}$
(200)

where $(c)$ applies Lemma 7 and exploits the identities ${\rm Tr}\\{{\bf
A}_{k}\\}=Me_{k1}$, ${\bf\Phi}^{H}{\bf\Phi}={\bf I}_{N}$, and ${\rm
Tr}\left\\{\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\right\\}=MN$.
Substituting (200) into (192), we complete the calculation of $E_{k}^{\rm
signal}\left(\mathbf{\Phi}\right)$ and
$E_{k}^{\rm{noise}}\left(\mathbf{\Phi}\right)$.

We conclude this subsection by deriving some useful results that are obtained
by using a procedure similar to that used for obtaining (200). To be specific,
we aim to derive
$\mathbb{E}\left\\{{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}$,
$\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\hat{\underline{\mathbf{q}}}_{k}\right\\}$,
$\mathbb{E}\left\\{{\underline{\mathbf{q}}}_{k}^{H}{\bf A}_{k}{\bf
A}^{H}_{k}{\underline{\mathbf{q}}}_{k}\right\\}$,
$\mathbb{E}\left\\{{\underline{\mathbf{q}}}_{i}^{H}{\bf A}_{k}{\bf
A}^{H}_{k}{\underline{\mathbf{q}}}_{i}\right\\}$, and
$\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}{\underline{\mathbf{q}}}_{k}\right\\}$.

Firstly, when ${\bf A}_{k}={\bf I}_{M}$ and $\tau\to\infty$, the imperfect
estimate $\hat{\mathbf{q}}_{k}$ becomes the perfect estimate
${\mathbf{q}}_{k}$. Therefore, substituting ${\bf A}_{k}={\bf I}_{M}$ and
$\tau\to\infty$ into (200), we have

$\displaystyle\mathbb{E}\left\\{{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}=M\left\\{\left|f_{k}(\mathbf{\Phi})\right|^{2}c_{k}\delta\varepsilon_{k}+Nc_{k}\left(\delta+\varepsilon_{k}+1\right)+\gamma_{k}\right\\}.$
(201)

Secondly, by using the expression of $\hat{\underline{\mathbf{q}}}_{k}$ in
(17), we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\hat{\underline{\mathbf{q}}}_{k}\right\\}=\mathbb{E}\left\\{\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\left(\hat{\underline{\mathbf{q}}}_{k}^{\omega}\right)^{H}\hat{\underline{\mathbf{q}}}_{k}^{\psi}\right\\}=\sum_{\omega=1}^{4}\mathbb{E}\left\\{\left\|\hat{\underline{\mathbf{q}}}_{k}^{\omega}\right\|^{2}\right\\}\\\
{{\mathop{=}\limits^{\left(d\right)}}}M\left\\{\left|f_{k}(\mathbf{\Phi})\right|^{2}c_{k}\delta\varepsilon_{k}+Nc_{k}\delta
e_{k2}^{2}+Nc_{k}\left(\varepsilon_{k}+1\right)e_{k3}\right\\},\end{array}$
(204)

where $(d)$ follows by applying the identity
$\operatorname{Tr}\left\\{\mathbf{A}_{k}^{H}\mathbf{A}_{k}\right\\}=Me_{k3}$.

Thirdly, using $\mathbf{A}_{k}^{H}=\mathbf{A}_{k}$ and
$\mathbf{A}_{k}\overline{\mathbf{H}}_{2}=e_{k2}\overline{\mathbf{H}}_{2}$, we
have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\underline{\mathbf{q}}_{k}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}=\mathbb{E}\left\\{\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\left(\mathbf{A}_{k}\mathbf{q}_{k}^{\omega}\right)^{H}\left(\mathbf{A}_{k}\mathbf{q}_{k}^{\psi}\right)\right\\}\\\
=\left\|\sqrt{c_{k}\delta\varepsilon_{k}}\mathbf{A}_{k}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right\|^{2}+\sum_{\omega=2}^{4}\mathbb{E}\left\\{\left\|\hat{\mathbf{q}}_{k}^{\omega}\right\|^{2}\right\\}\\\
=M\left\\{\left|f_{k}(\mathbf{\Phi})\right|^{2}c_{k}\delta\varepsilon_{k}e_{k2}^{2}+Nc_{k}\delta
e_{k2}^{2}+Nc_{k}\left(\varepsilon_{k}+1\right)e_{k3}\right\\}.\end{array}$
(208)

Also, for $i\neq k$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\underline{\mathbf{q}}_{i}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right\\}=M\left\\{\left|f_{i}(\mathbf{\Phi})\right|^{2}c_{i}\delta\varepsilon_{i}e_{k2}^{2}+Nc_{i}\delta
e_{k2}^{2}+Nc_{i}\left(\varepsilon_{i}+1\right)e_{k3}\right\\}.\end{array}$
(210)

Finally, by substituting $\gamma_{k}=0$ into (200), we arrive at

$\displaystyle\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}{\underline{\mathbf{q}}}_{k}\right\\}=M\left\\{\left|f_{k}(\mathbf{\Phi})\right|^{2}c_{k}\delta\varepsilon_{k}+Nc_{k}\delta
e_{k2}+Nc_{k}\left(\varepsilon_{k}+1\right)e_{k1}\right\\}.$ (211)

###  E-B Interference Term

In this subsection, we derive the interference term of (35). The interference
term is denoted by
$\mathbb{E}\left\\{\left|\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{i}\right|^{2}\right\\}\triangleq
I_{ki}\left(\mathbf{\Phi}\right)$. First, it is worth noting that the
derivation of the interference term in the presence of imperfect CSI and
double-Rician channels in RIS-aided massive MIMO systems has two main
differences compared to conventional massive MIMO systems. Firstly, the
channel ${\bf q}_{k}$ and ${\bf q}_{i}$ are not independent, since different
users experience the same RIS-BS channel. This can be readily validated by
examining that
$\mathbb{E}\left\\{\mathbf{q}_{k}^{H}\mathbf{q}_{i}\right\\}\neq\mathbb{E}\left\\{\mathbf{q}_{k}^{H}\right\\}\mathbb{E}\left\\{\mathbf{q}_{i}\right\\}$.
Secondly, the LMMSE error ${\bf e}_{k}$ is uncorrelated with but dependent on
the estimate $\hat{\bf q}_{k}$, since the cascaded channel is not Gaussian
distributed. To tackle these two challenges, we derive the interference term
by decomposing it as

$\displaystyle I_{ki}\left(\mathbf{\Phi}\right)=$
$\displaystyle\mathbb{E}\left\\{\left|\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{i}\right|^{2}\right\\}=\mathbb{E}\left\\{\left|\left(\underline{\hat{\mathbf{q}}}_{k}+\mathbf{A}_{k}{\mathbf{d}}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{A}_{k}\mathbf{Ns}_{k}\right)^{H}\left(\underline{\mathbf{q}}_{i}+{\mathbf{d}}_{i}\right)\right|^{2}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\left|\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{i}+\underline{\hat{\mathbf{q}}}_{k}^{H}{\mathbf{d}}_{i}+{\mathbf{d}}_{k}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}+{\mathbf{d}}_{k}^{H}\mathbf{A}_{k}^{H}{\mathbf{d}}_{i}+\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}+\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}{\mathbf{d}}_{i}\right|^{2}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{\left|\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}+\mathbb{E}\left\\{\left|\underline{\hat{\mathbf{q}}}_{k}^{H}{\mathbf{d}}_{i}\right|^{2}\right\\}+\mathbb{E}\left\\{\left|{\mathbf{d}}_{k}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}+\mathbb{E}\left\\{\left|{\mathbf{d}}_{k}^{H}\mathbf{A}_{k}^{H}{\mathbf{d}}_{i}\right|^{2}\right\\}$
$\displaystyle+\frac{1}{\tau
p}\mathbb{E}\left\\{\left|\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}+\frac{1}{\tau
p}\mathbb{E}\left\\{\left|\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}{\mathbf{d}}_{i}\right|^{2}\right\\}.$
(212)

We aim to derive the six expectations in (E-B) one by one, but the first one
$\mathbb{E}\left\\{\left|\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}$
will be derived last. The second term in (E-B) is

$\displaystyle\mathbb{E}\left\\{\left|\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbf{d}_{i}\right|^{2}\right\\}=\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbf{d}_{i}\mathbf{d}_{i}^{H}\hat{\underline{\mathbf{q}}}_{k}\right\\}=\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbb{E}\left\\{\mathbf{d}_{i}\mathbf{d}_{i}^{H}\right\\}\hat{\underline{\mathbf{q}}}_{k}\right\\}=\gamma_{i}\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\hat{\underline{\mathbf{q}}}_{k}\right\\},$
(213)

where
$\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\hat{\underline{\mathbf{q}}}_{k}\right\\}$
is given in (204).

The third term in (E-B) is

$\displaystyle\mathbb{E}\left\\{\left|\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}=\mathbb{E}\left\\{\underline{\mathbf{q}}_{i}^{H}\mathbf{A}_{k}\mathbb{E}\left\\{\mathbf{d}_{k}\mathbf{d}_{k}^{H}\right\\}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right\\}=\gamma_{k}\mathbb{E}\left\\{\underline{\mathbf{q}}_{i}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right\\},$
(214)

where
$\mathbb{E}\left\\{\underline{\mathbf{q}}_{i}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right\\}$
is given in (210).

The fourth term in (E-B) is

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{i}\right|^{2}\right\\}=\mathbb{E}\left\\{\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbb{E}\left\\{\mathbf{d}_{i}\mathbf{d}_{i}^{H}\right\\}\mathbf{A}_{k}\mathbf{d}_{k}\right\\}\\\
=\gamma_{i}\mathbb{E}\left\\{\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{A}_{k}\mathbf{d}_{k}\right\\}=\gamma_{k}\gamma_{i}\operatorname{Tr}\left\\{\mathbf{A}_{k}^{H}\mathbf{A}_{k}\right\\}\\\
=\gamma_{k}\gamma_{i}Me_{k3}.\end{array}$ (218)

The fifth term in (E-B) is

$\displaystyle\frac{1}{{\tau
p}}\mathbb{E}\left\\{\left|\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}=\frac{1}{\tau
p}\mathbb{E}\left\\{\underline{\mathbf{q}}_{i}^{H}\mathbf{A}_{k}\mathbb{E}\left\\{\mathbf{N}\mathbf{s}_{k}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\right\\}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right\\}=\frac{\sigma^{2}}{\tau
p}\mathbb{E}\left\\{\underline{\mathbf{q}}_{i}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right\\},$
(219)

where
$\mathbb{E}\left\\{\underline{\mathbf{q}}_{i}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{i}\right\\}$
is given in (210).

The sixth term in (E-B) is

$\displaystyle\frac{1}{{\tau
p}}\mathbb{E}\left\\{\left|\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{i}\right|^{2}\right\\}=\frac{\sigma^{2}}{\tau
p}\gamma_{i}\operatorname{Tr}\left\\{\mathbf{A}_{k}\mathbf{A}_{k}^{H}\right\\}=\frac{\sigma^{2}}{\tau
p}\gamma_{i}Me_{k3}.$ (220)

Finally, we derive the first term
$\mathbb{E}\left\\{\left|\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}$,
which can be expanded as

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{i}\right|^{2}\right\\}=\mathbb{E}\left\\{\left|\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\left(\hat{\mathbf{q}}_{k}^{\omega}\right)^{H}\mathbf{q}_{i}^{\psi}\right|^{2}\right\\}\\\
=\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\mathbb{E}\left\\{\left|\left(\hat{\mathbf{q}}_{k}^{\omega}\right)^{H}\mathbf{q}_{i}^{\psi}\right|^{2}\right\\}+\sum_{\omega
1,\psi 1,\omega 2,\psi 2,\atop(\omega 1,\psi 1)\neq(\omega 2,\psi
2)}^{4}\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{\omega
1}\right)^{H}\mathbf{q}_{i}^{\psi
1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{\omega
2}\right)^{H}\mathbf{q}_{i}^{\psi 2}\right)^{H}\right\\},\end{array}$ (223)

where ${\bf q}^{1}_{k}$ \- ${\bf q}^{4}_{k}$ are defined in (II-A), and
$\hat{\bf q}^{1}_{k}$ \- $\hat{\bf q}^{4}_{k}$ are defined in (17).

Equation (223) can be derived by calculating the expectations of the $16$
modulus-square terms and the expectations of the other cross-terms. We first
calculate the former $16$ modulus-square terms in (223) one by one. The
derivation utilizes Lemma 7, Lemma 10, and the independence between
$\tilde{\bf H}_{2}$, $\tilde{\bf h}_{k}$, and $\tilde{\bf h}_{i}$.

Firstly, we consider the terms with $\omega=1$. When $\psi=1$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{i}\delta\varepsilon_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta^{2}\varepsilon_{k}\varepsilon_{i}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\delta^{2}\varepsilon_{k}\varepsilon_{i}M^{2}\left|f_{k}(\mathbf{\Phi})\right|^{2}\left|f_{i}(\mathbf{\Phi})\right|^{2}.\end{array}$
(227)

When $\psi=2$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{i}\delta}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta^{2}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\delta^{2}\varepsilon_{k}M^{2}N\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(231)

When $\psi=3$, using Lemma 7, we arrive at

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{i}\varepsilon_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}N\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}MN\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(236)

When $\psi=4$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\delta\varepsilon_{k}MN\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(241)

Secondly, we consider the terms with $\omega=2$. When $\psi=1$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{i}\delta\varepsilon_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta^{2}\varepsilon_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta^{2}\varepsilon_{i}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta^{2}\varepsilon_{i}\operatorname{Tr}\left\\{\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta^{2}\varepsilon_{i}M^{2}N\left|f_{i}(\mathbf{\Phi})\right|^{2}.\end{array}$
(247)

When $\psi=2$, we arrive at

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{i}\delta}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\bm{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta^{2}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta^{2}M^{2}N^{2}.\end{array}$ (252)

When $\psi=3$, we get

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{i}\varepsilon_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{i}N\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\bm{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{i}N\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{i}MN^{2}.\end{array}$ (258)

When $\psi=4$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\bm{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta
N\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta MN^{2}.\end{array}$ (264)

Thirdly, we consider the terms with $\omega=3$. When $\psi=1$, using
$\mathbf{A}_{k}\overline{\mathbf{H}}_{2}=e_{k2}\overline{\mathbf{H}}_{2}$, we
have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{i}\delta\varepsilon_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{e_{k2}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}e_{k2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}M\left|f_{i}(\mathbf{\Phi})\right|^{2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}MN\left|f_{i}(\mathbf{\Phi})\right|^{2}.\end{array}$
(270)

When $\psi=2$, we arrive at

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{i}\delta}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}\overline{\mathbf{H}}_{2}^{H}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{k}MN^{2}.\end{array}$ (275)

When $\psi=3$, using Lemma 10, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{i}\varepsilon_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\varepsilon_{k}\varepsilon_{i}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\varepsilon_{k}\varepsilon_{i}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\left(e_{k1}^{2}M^{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}+e_{k3}M\operatorname{Tr}\left\\{\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\right\\}\mathbf{I}_{N}\right)\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\varepsilon_{k}\varepsilon_{i}\left(e_{k1}^{2}M^{2}\overline{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\overline{\mathbf{h}}_{k}+e_{k3}M\operatorname{Tr}\left\\{\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\right\\}\overline{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{k}\right)\\\
=c_{k}c_{i}\varepsilon_{k}\varepsilon_{i}\left(e_{k1}^{2}M^{2}\left|\overline{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{i}\right|^{2}+e_{k3}MN^{2}\right).\end{array}$
(281)

When $\psi=4$, using Lemma 10 with ${\bf W}={\bf I}_{N}$, we get

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\varepsilon_{k}\left(e_{k1}^{2}M^{2}+e_{k3}MN\right)\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{I}_{N}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}c_{i}\varepsilon_{k}\left(e_{k1}^{2}M^{2}+e_{k3}MN\right)N.\end{array}$
(287)

Fourthly, we consider the terms with $\omega=4$. When $\psi=1$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{i}\delta\varepsilon_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta\varepsilon_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{i}M\left|f_{i}(\mathbf{\Phi})\right|^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\varepsilon_{i}MN\left|f_{i}(\mathbf{\Phi})\right|^{2}.\end{array}$
(293)

When $\psi=2$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{i}\delta}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\delta\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta
MN\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
=e_{k2}^{2}c_{k}c_{i}\delta MN^{2}.\end{array}$ (299)

When $\psi=3$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{i}\varepsilon_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\varepsilon_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}c_{i}\varepsilon_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\left(e_{k1}^{2}M^{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}+e_{k3}M\operatorname{Tr}\left\\{\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\right\\}\mathbf{I}_{N}\right)\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}c_{i}\varepsilon_{i}\mathbb{E}\left\\{e_{k1}^{2}M^{2}\tilde{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\tilde{\mathbf{h}}_{k}+e_{k3}MN\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}c_{i}\varepsilon_{i}\left(e_{k1}^{2}M^{2}N+e_{k3}MN^{2}\right).\end{array}$
(305)

When $\psi=4$, we get

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{i}}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right|^{2}\right\\}\\\
=c_{k}c_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}c_{i}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}c_{i}\mathbb{E}\left\\{\left(e_{k1}^{2}M^{2}+e_{k3}MN\right)\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}c_{i}\left(e_{k1}^{2}M^{2}N+e_{k3}MN^{2}\right).\end{array}$ (311)

The calculation of the expectations of the $16$ modulus-square terms in (223)
is completed. Then, we focus on the remaining cross-terms in (223). Even
though the total number of cross-terms is $16\times 15$, only a few terms are
non-zero. To help identify the non-zero cross-terms, we expand
$\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{i}$ as

$\displaystyle\begin{array}[]{l}\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{i}=\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\left(\hat{\mathbf{q}}_{k}^{\omega}\right)^{H}\mathbf{q}_{i}^{\psi}\\\
=\eta_{11}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{12}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}+\eta_{13}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{14}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\\\
+\eta_{21}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{22}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}+\eta_{23}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{24}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\\\
+\eta_{31}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{32}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\\\
+\eta_{33}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{34}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\\\
+\eta_{41}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{42}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\\\
+\eta_{43}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}+\eta_{44}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}.\end{array}$
(319)

For brevity, we use the notation $\eta_{11}$ \- $\eta_{44}$ to identify the
coefficients (path-loss factors and Rician factors) in front of the product of
vectors and matrices, since these coefficients are deterministic and do not
determine whether the expectations of the cross-terms are zero or not. Then,
we can calculate the cross-terms in (223) by calculating the expectation of
the product of one term in (319) with the conjugate transpose of another term
in (319). Therefore, by exploiting Lemma 8, the independence and the zero-mean
properties of $\tilde{\bf H}_{2}$, $\tilde{\bf h}_{k}$ and $\tilde{\bf
h}_{i}$, we find that only $8$ cross-terms have a non-zero expectation. They
can be combined as

$\displaystyle\begin{array}[]{l}\sum_{\omega 1,\psi 1,\omega 2,\psi
2,\atop(\omega 1,\psi 1)\neq(\omega 2,\psi
2)}^{4}\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{\omega
1}\right)^{H}\mathbf{q}_{i}^{\psi
1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{\omega
2}\right)^{H}\mathbf{q}_{i}^{\psi 2}\right)^{H}\right\\}\\\
=2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{i}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{i}^{3}\right)^{H}\right\\}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{i}^{2}\right)\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{i}^{4}\right)^{H}\right\\}\right\\}\\\
+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{i}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{i}^{3}\right)^{H}\right\\}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{i}^{2}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{i}^{4}\right)^{H}\right\\}\right\\}.\end{array}$
(323)

Then, we calculate these $4$ terms in (323) one by one. Using ${\rm
Tr}\left\\{{\bf A}_{k}\right\\}=Me_{k1}$, the first cross-term is derived as

$\displaystyle\begin{array}[]{l}2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{i}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{i}^{3}\right)^{H}\right\\}\right\\}\\\
=2\operatorname{Re}\left\\{\mathbb{E}\left\\{\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{i}\delta\varepsilon_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\bm{\Phi}\overline{\mathbf{h}}_{i}\left(\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{i}\varepsilon_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right)^{H}\right\\}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}\operatorname{Re}\left\\{\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}e_{k1}M\operatorname{Re}\left\\{\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\overline{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}e_{k1}M\operatorname{Re}\left\\{\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\overline{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{k}\varepsilon_{i}e_{k1}M^{2}\operatorname{Re}\left\\{f_{k}^{H}(\mathbf{\Phi})f_{i}(\mathbf{\Phi})\overline{\mathbf{h}}_{i}^{H}\overline{\mathbf{h}}_{k}\right\\}.\end{array}$
(330)

Note that the real operator ${\rm Re}\left\\{\cdot\right\\}$ will be omitted
for the calculation of the 2nd, 3rd, and 4th cross-terms, since the results
derived will show that they only have real parts. Then, the second cross-term
is

$\displaystyle\begin{array}[]{l}2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{i}^{2}\right)\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{i}^{4}\right)^{H}\right\\}\right\\}\\\
=2\mathbb{E}\left\\{\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{i}\delta}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\left(\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{i}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right)^{H}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=2c_{k}c_{i}\delta\varepsilon_{k}e_{k1}M\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=2c_{k}c_{i}\delta\varepsilon_{k}e_{k1}M^{2}\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(336)

The third cross-term is

$\displaystyle\begin{array}[]{l}2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{i}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{i}^{3}\right)^{H}\right\\}\right\\}\\\
=2\mathbb{E}\left\\{e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{i}\delta\varepsilon_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\left(\sqrt{c_{k}}\sqrt{c_{i}\varepsilon_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right)^{H}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{i}e_{k2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{i}e_{k1}e_{k2}M\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\overline{\mathbf{h}}_{i}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{i}e_{k1}e_{k2}M\operatorname{Tr}\left\\{\overline{\mathbf{h}}_{i}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{i}\right\\}\\\
=2c_{k}c_{i}\delta\varepsilon_{i}e_{k1}e_{k2}M^{2}\left|f_{i}(\mathbf{\Phi})\right|^{2}.\end{array}$
(343)

The fourth cross-term is

$\displaystyle\begin{array}[]{l}2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{i}^{2}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{i}^{4}\right)^{H}\right\\}\right\\}\\\
=2\mathbb{E}\left\\{e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{i}\delta}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\left(\sqrt{c_{k}}\sqrt{c_{i}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{i}\right)^{H}\right\\}\\\
=2c_{k}c_{i}\delta
e_{k2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{i}\tilde{\mathbf{h}}_{i}^{H}\right\\}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}c_{i}\delta
e_{k1}e_{k2}M\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}c_{i}\delta e_{k1}e_{k2}M^{2}N.\end{array}$ (349)

We have completed the calculation of the expectation of $8$ cross-terms.
Finally, the interference term ${I}_{ki}\left(\mathbf{\Phi}\right)$ is
obtained by combining (213) - (220), (227) - (311) and (330) - (349) to (E-B)
with some direct simplifications.

###  E-C Signal Leakage

In this subsection, we derive the signal leakage term of (35) as

$\displaystyle
E_{k}^{(\text{leakage})}\left(\mathbf{\Phi}\right)=\mathbb{E}\left\\{\left|\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right|^{2}\right\\}-\left|\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}\right|^{2},$
(350)

where $\mathbb{E}\left\\{\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right\\}$ is
given in (200). Therefore, we only need to derive the expectation
$\mathbb{E}\left\\{\left|\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right|^{2}\right\\}$.
By exploiting the zero-mean properties of ${\bf d}_{k}$ and $\bf N$, and
exploiting the independence between the cascaded channel, the direct channel,
and the noise, we can expand this term and remove the terms with zero
expectation as

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\hat{\mathbf{q}}_{k}^{H}\mathbf{q}_{k}\right|^{2}\right\\}=\mathbb{E}\left\\{\left|\left(\hat{\underline{\mathbf{q}}}_{k}^{H}+\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}+\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\right)\left(\underline{\mathbf{q}}_{k}+\mathbf{d}_{k}\right)\right|^{2}\right\\}\\\
=\mathbb{E}\left\\{\left|\hat{\underline{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}+\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbf{d}_{k}+\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}+\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}+\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right|^{2}\right\\}\\\
{{\mathop{=}\limits^{\left(d\right)}}}\mathbb{E}\left\\{\left|\hat{\underline{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\right|^{2}\right\\}+\mathbb{E}\left\\{\left|\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbf{d}_{k}\right|^{2}\right\\}+\mathbb{E}\left\\{\left|\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right|^{2}\right\\}+\mathbb{E}\left\\{\left|\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right|^{2}\right\\}\\\
+\mathbb{E}\left\\{\left|\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right|^{2}\right\\}+\mathbb{E}\left\\{\left|\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right|^{2}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\left(\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right)^{H}\right\\}\right\\},\end{array}$
(355)

where in $(d)$ the cross-term
$\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbf{d}_{k}\left(\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right)^{H}\right\\}$
is zero due to Lemma 8, and the cross-term
$\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbf{d}_{k}\left(\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right)^{H}\right\\}$
is zero because the odd-order central moments of a zero-mean Gaussian variable
are zero[75, Eq. (12)].

Next, we derive the $2$nd - $7$th terms in (355), but the first term in (355)
is calculated at the end. The second term in (355) is

$\displaystyle\mathbb{E}\left\\{\left|\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbf{d}_{k}\right|^{2}\right\\}=\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\mathbb{E}\left\\{\mathbf{d}_{k}\mathbf{d}_{k}^{H}\right\\}\hat{\underline{\mathbf{q}}}_{k}\right\\}=\gamma_{k}\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\hat{\underline{\mathbf{q}}}_{k}\right\\},$
(356)

where
$\mathbb{E}\left\\{\hat{\underline{\mathbf{q}}}_{k}^{H}\hat{\underline{\mathbf{q}}}_{k}\right\\}$
is given in (204).

The third term in (355) is

$\displaystyle\mathbb{E}\left\\{\left|\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right|^{2}\right\\}=\mathbb{E}\left\\{\underline{\mathbf{q}}_{k}^{H}\mathbf{A}_{k}\mathbb{E}\left\\{\mathbf{d}_{k}\mathbf{d}_{k}^{H}\right\\}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}=\gamma_{k}\mathbb{E}\left\\{\underline{\mathbf{q}}_{k}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\},$
(357)

where
$\mathbb{E}\left\\{\underline{\mathbf{q}}_{k}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}$
is given in (208).

Since
$\tilde{\mathbf{d}}_{k}\tilde{\mathbf{d}}_{k}^{H}\sim\mathcal{C}\mathcal{W}_{M}\left(\mathbf{I}_{M},1\right)$,
using the property of the Wishart distribution (142), the fourth term in (355)
can be obtained as

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right|^{2}\right\\}=\mathbb{E}\left\\{\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\mathbf{d}_{k}^{H}\mathbf{A}_{k}\mathbf{d}_{k}\right\\}\\\
=\operatorname{Tr}\left\\{\mathbb{E}\left\\{\mathbf{A}_{k}^{H}\mathbf{d}_{k}\mathbf{d}_{k}^{H}\mathbf{A}_{k}\mathbf{d}_{k}\mathbf{d}_{k}^{H}\right\\}\right\\}\\\
=\gamma_{k}^{2}\operatorname{Tr}\left\\{\mathbf{A}_{k}^{H}\left(\mathbf{A}_{k}+\operatorname{Tr}\left\\{\mathbf{A}_{k}\right\\}\mathbf{I}_{M}\right)\right\\}\\\
=\gamma_{k}^{2}\operatorname{Tr}\left\\{\mathbf{A}_{k}^{H}\mathbf{A}_{k}\right\\}+\gamma_{k}^{2}\left(\operatorname{Tr}\left\\{\mathbf{A}_{k}\right\\}\right)^{2}\\\
=\gamma_{k}^{2}Me_{k3}+\gamma_{k}^{2}M^{2}e_{k1}^{2}.\end{array}$ (363)

The fifth term in (355) is calculated as

$\displaystyle\mathbb{E}\left\\{\left|\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right|^{2}\right\\}=\frac{1}{\tau
p}\mathbb{E}\left\\{\underline{\mathbf{q}}_{k}^{H}\mathbf{A}_{k}\mathbb{E}\left\\{\mathbf{N}\mathbf{s}_{k}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\right\\}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}=\frac{\sigma^{2}}{\tau
p}\mathbb{E}\left\\{\underline{\mathbf{q}}_{k}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\},$
(364)

where
$\mathbb{E}\left\\{\underline{\mathbf{q}}_{k}^{H}\mathbf{A}_{k}\mathbf{A}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}$
is given in (208).

The sixth term in (355) is

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\frac{1}{\sqrt{\tau
p}}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right|^{2}\right\\}=\frac{1}{\tau
p}\mathbb{E}\left\\{\mathbf{d}_{k}^{H}\mathbf{A}_{k}\mathbb{E}\left\\{\mathbf{N}\mathbf{s}_{k}\mathbf{s}_{k}^{H}\mathbf{N}^{H}\right\\}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right\\}\\\
=\frac{\sigma^{2}}{\tau
p}\gamma_{k}\operatorname{Tr}\left\\{\mathbf{A}_{k}\mathbf{A}_{k}^{H}\right\\}=\frac{\sigma^{2}}{\tau
p}\gamma_{k}Me_{k3}.\end{array}$ (367)

The seventh term in (355) is

$\displaystyle\begin{array}[]{l}2\operatorname{Re}\left\\{\mathbb{E}\left\\{\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\left(\mathbf{d}_{k}^{H}\mathbf{A}_{k}^{H}\mathbf{d}_{k}\right)^{H}\right\\}\right\\}=2\operatorname{Re}\left\\{\mathbb{E}\left\\{\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}\mathbb{E}\left\\{\mathbf{d}_{k}^{H}\mathbf{A}_{k}\mathbf{d}_{k}\right\\}\right\\}\\\
=2\gamma_{k}Me_{k1}\operatorname{Re}\left\\{\mathbb{E}\left\\{\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}\right\\}=2\gamma_{k}Me_{k1}\mathbb{E}\left\\{\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\},\end{array}$
(370)

where
$\mathbb{E}\left\\{\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\right\\}$
is given in (211).

Finally, we derive the first term
$\mathbb{E}\left\\{\left|\hat{\underline{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\right|^{2}\right\\}$
in (355), which can be expanded as

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}\right|^{2}\right\\}=\mathbb{E}\left\\{\left|\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\left(\hat{\mathbf{q}}_{k}^{\omega}\right)^{H}\mathbf{q}_{k}^{\psi}\right|^{2}\right\\}\\\
=\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\mathbb{E}_{\bm{}}\left\\{\left|\left(\hat{\mathbf{q}}_{k}^{\omega}\right)^{H}\mathbf{q}_{k}^{\psi}\right|^{2}\right\\}+\sum_{\omega
1,\psi 1,\omega 2,\psi 2\atop(\omega 1,\psi 1)\neq(\omega 2,\psi
2)}^{4}\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{\omega
1}\right)^{H}\mathbf{q}_{k}^{\psi
1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{\omega
2}\right)^{H}\mathbf{q}_{k}^{\psi 2}\right)^{H}\right\\}.\end{array}$ (373)

In the following, we first calculate the $16$ modulus-square terms in (373),
and then calculate the remaining cross-terms.

Firstly, we consider the terms with $\omega=1$. When $\psi=1$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}^{2}\left|\left(\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\right)\mathbf{a}_{M}^{H}\mathbf{a}_{M}\left(\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right)\right|^{2}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}^{2}M^{2}\left|f_{k}(\mathbf{\Phi})\right|^{4}.\end{array}$
(377)

When $\psi=2$, we arrive at

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{k}\delta}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}M^{2}N\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}M^{2}N\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(384)

When $\psi=3$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{k}\varepsilon_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}^{2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}^{2}N\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}^{2}MN\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(389)

When $\psi=4$, we get

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}MN\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(393)

Secondly, we consider the terms with $\omega=2$. When $\psi=1$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{k}\delta\varepsilon_{k}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}e_{k2}^{2}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\right\\}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}e_{k2}^{2}\operatorname{Tr}\left\\{\mathbf{a}_{M}^{H}\mathbf{a}_{M}\left(\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\right)\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{a}_{N}\right\\}\\\
=c_{k}^{2}\delta^{2}\varepsilon_{k}e_{k2}^{2}M^{2}N\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(399)

Since
$\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\sim\mathcal{C}\mathcal{W}_{N}\left(\mathbf{I}_{N},1\right)$,
using (142), when $\psi=2$, we arrive at

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{k}\delta}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta^{2}e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta^{2}e_{k2}^{2}\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\right\\}\\\
=c_{k}^{2}\delta^{2}e_{k2}^{2}\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\left(\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}+\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\right\\}\mathbf{I}_{N}\right)\right\\}\\\
=c_{k}^{2}\delta^{2}e_{k2}^{2}\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}+MN\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\right\\}\\\
=c_{k}^{2}\delta^{2}e_{k2}^{2}\left(\operatorname{Tr}\left\\{\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\right\\}+\operatorname{Tr}\left\\{MN\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\right\\}\right)\\\
=2c_{k}^{2}\delta^{2}e_{k2}^{2}M^{2}N^{2}.\end{array}$ (407)

When $\psi=3$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{k}\varepsilon_{k}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}e_{k2}^{2}N\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}e_{k2}^{2}MN^{2}.\end{array}$ (412)

When $\psi=4$, we arrive at

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{k}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta
e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
{{\mathop{=}\limits^{\left(e\right)}}}c_{k}^{2}\delta
e_{k2}^{2}\mathbb{E}_{\tilde{\mathbf{h}}_{k}}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbb{E}_{\tilde{\mathbf{H}}_{2}}\left\\{\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\mid\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta
e_{k2}^{2}\mathbb{E}_{\tilde{\mathbf{h}}_{k}}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\operatorname{Tr}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\mid\tilde{\mathbf{h}}_{k}\right\\}\\\
{{\mathop{=}\limits^{\left(f\right)}}}c_{k}^{2}\delta
e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\left(\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right)\right\\}\\\
=c_{k}^{2}\delta
e_{k2}^{2}\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\right\\}\\\
{{\mathop{=}\limits^{\left(g\right)}}}c_{k}^{2}\delta
e_{k2}^{2}\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}(N+1)\mathbf{I}_{N}\right\\}\\\
=c_{k}^{2}\delta e_{k2}^{2}MN(N+1),\end{array}$ (421)

where $(e)$ utilizes the law of total expectation, which calculates the
conditional expectation of $\tilde{\mathbf{H}}_{2}$ given
$\tilde{\mathbf{h}}_{k}$, and then calculates the expectation of
$\tilde{\mathbf{h}}_{k}$. Since $\tilde{\mathbf{H}}_{2}$ is independent of
$\tilde{\mathbf{h}}_{k}$, the conditional expectation of
$\tilde{\mathbf{H}}_{2}$ given $\tilde{\mathbf{h}}_{k}$ is the same as its
unconditional expectation; $(f)$ comes from
$\operatorname{Tr}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}=\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}$
which is a scalar number and its place can be arbitrarily moved; and $(g)$
applies a special case of (142).

Thirdly, we consider the terms with $\omega=3$. When $\psi=1$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}^{2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}^{2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}^{2}e_{k2}^{2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}^{2}e_{k2}^{2}MN\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(427)

When $\psi=2$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{k}\delta}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\delta\varepsilon_{k}e_{k2}^{2}MN^{2}.\end{array}$ (432)

When $\psi=3$, using Lemma 10, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{k}\varepsilon_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\varepsilon_{k}^{2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\varepsilon_{k}^{2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\left(e_{k1}^{2}M^{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}+e_{k3}M\operatorname{Tr}\left\\{\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\right\\}\mathbf{I}_{N}\right)\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\varepsilon_{k}^{2}\left(e_{k1}^{2}M^{2}\overline{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{k}+e_{k3}MN\overline{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{k}\right)\\\
=c_{k}^{2}\varepsilon_{k}^{2}\left(e_{k1}^{2}M^{2}N^{2}+e_{k3}MN^{2}\right).\end{array}$
(438)

When $\psi=4$, using Lemma 10 with ${\bf W}={\bf I}_{N}$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}\varepsilon_{k}}\sqrt{c_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\varepsilon_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\varepsilon_{k}\left(e_{k1}^{2}M^{2}+e_{k3}MN\right)\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{I}_{N}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\\\
=c_{k}^{2}\varepsilon_{k}\left(e_{k1}^{2}M^{2}N+e_{k3}MN^{2}\right).\end{array}$
(444)

Fourthly, we consider the terms with $\omega=4$. When $\psi=1$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{k}\delta\varepsilon_{k}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}e_{k2}^{2}M\left|f_{k}(\mathbf{\Phi})\right|^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta\varepsilon_{k}e_{k2}^{2}MN\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(450)

When $\psi=2$, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{k}\delta}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\delta\mathbb{E}_{\tilde{\mathbf{h}}_{k}}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}_{\tilde{\mathbf{H}}_{2}}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\mid\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\delta
e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
{{\mathop{=}\limits^{\left(h\right)}}}c_{k}^{2}\delta
e_{k2}^{2}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\left(\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right)\right\\}\\\
=c_{k}^{2}\delta
e_{k2}^{2}\operatorname{Tr}\left\\{\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\right\\}\right\\}\\\
=c_{k}^{2}\delta
e_{k2}^{2}\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}+\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\right\\}\mathbf{I}_{N}\right\\}\\\
=c_{k}^{2}\delta
e_{k2}^{2}\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}+MN\mathbf{I}_{N}\right\\}\\\
=c_{k}^{2}\delta e_{k2}^{2}\left(MN+MN^{2}\right),\end{array}$ (459)

where $(h)$ comes from
$\operatorname{Tr}\left\\{\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\right\\}=\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}$,
which is a $1\times 1$ number and can be moved to the end of the equation.

When $\psi=3$, using Lemma 7 and Lemma 10, we have

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{k}\varepsilon_{k}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\varepsilon_{k}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\varepsilon_{k}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\left(e_{k1}^{2}M^{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}+e_{k3}M\operatorname{Tr}\left\\{\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\right\\}\mathbf{I}_{N}\right)\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\varepsilon_{k}\mathbb{E}\left\\{e_{k1}^{2}M^{2}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}+e_{k3}MN\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\varepsilon_{k}\mathbb{E}\left\\{e_{k1}^{2}M^{2}\tilde{\mathbf{h}}_{k}^{H}\overline{\mathbf{h}}_{k}\overline{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}+e_{k3}MN\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\varepsilon_{k}\left(e_{k1}^{2}M^{2}N+e_{k3}MN^{2}\right).\end{array}$
(466)

When $\psi=4$, we get

$\displaystyle\begin{array}[]{l}\mathbb{E}\left\\{\left|\sqrt{c_{k}}\sqrt{c_{k}}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right|^{2}\right\\}\\\
=c_{k}^{2}\mathbb{E}_{\tilde{\mathbf{h}}_{k}}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbb{E}_{\tilde{\mathbf{H}}_{2}}\left\\{\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}\tilde{\mathbf{H}}_{2}\right\\}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\mid\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\mathbb{E}_{\tilde{\mathbf{h}}_{k}}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\left(e_{k1}^{2}M^{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}+e_{k3}M\operatorname{Tr}\left\\{\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\right\\}\mathbf{I}_{N}\right)\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=c_{k}^{2}\mathbb{E}\left\\{e_{k1}^{2}M^{2}\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}+e_{k3}M\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\tilde{\mathbf{h}}_{k}^{H}\tilde{\mathbf{h}}_{k}\right\\}\\\
{{\mathop{=}\limits^{\left(i\right)}}}c_{k}^{2}\left\\{e_{k1}^{2}M^{2}N(N+1)+e_{k3}MN(N+1)\right\\},\end{array}$
(472)

where $(i)$ uses (143).

Herein, the calculation of the $16$ modulus-square terms are completed. Now,
we focus on the expectation of the remaining cross-terms. To better understand
the form of the cross-terms, we give the expansion of
$\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}$ as

$\displaystyle\begin{array}[]{l}\underline{\hat{\mathbf{q}}}_{k}^{H}\underline{\mathbf{q}}_{k}=\sum_{\omega=1}^{4}\sum_{\psi=1}^{4}\left(\hat{\mathbf{q}}_{k}^{\omega}\right)^{H}\mathbf{q}_{k}^{\psi}\\\
=\eta^{11}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{12}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}+\eta^{13}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{14}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\\\
+\eta^{21}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{22}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}+\eta^{23}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{24}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\\\
+\eta^{31}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{32}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\\\
+\eta^{33}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{34}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\\\
+\eta^{41}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{42}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\\\
+\eta^{43}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}+\eta^{44}\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\tilde{\mathbf{H}}_{2}^{H}\mathbf{A}_{k}^{H}\tilde{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}.\end{array}$
(480)

We use the notation $\eta^{11}$ \- $\eta^{44}$ to identify the coefficients in
front of the product of vectors and matrices. We can calculate the cross-terms
in (373) by calculating the expectation of the product of one term in (480)
with the conjugate transpose of another term in (480). There exist $16\times
15$ cross-terms, but only $20$ of them are non-zero. Using Lemma 8, the
independence and zero-mean properties of $\tilde{\bf H}_{2}$ and $\tilde{\bf
h}_{k}$, we can filter the $20$ non-zero cross-terms, and combine them into
the following $10$ terms:

$\displaystyle\begin{aligned} &\sum\nolimits_{\omega 1,\psi 1,\omega 2,\psi
2,\atop(\omega 1,\psi 1)\neq(\omega 2,\psi
2)}^{4}\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{\omega
1}\right)^{H}\mathbf{q}_{k}^{\psi
1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{\omega
2}\right)^{H}\mathbf{q}_{k}^{\psi 2}\right)^{H}\right\\}\\\
&=2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{k}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{k}^{2}\right)^{H}\right\\}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{k}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{k}^{3}\right)^{H}\right\\}\right\\}\\\
&+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{k}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{k}^{4}\right)^{H}\right\\}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{k}^{2}\right)\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{k}^{4}\right)^{H}\right\\}\right\\}\\\
&+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{1}\right)^{H}\mathbf{q}_{k}^{3}\right)\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{k}^{4}\right)^{H}\right\\}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{k}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{k}^{3}\right)^{H}\right\\}\right\\}\\\
&+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{k}^{2}\right)\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{k}^{3}\right)^{H}\right\\}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{2}\right)^{H}\mathbf{q}_{k}^{2}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{k}^{4}\right)^{H}\right\\}\right\\}\\\
&+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{k}^{1}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{k}^{2}\right)^{H}\right\\}\right\\}+2\operatorname{Re}\left\\{\mathbb{E}\left\\{\left(\left(\hat{\mathbf{q}}_{k}^{3}\right)^{H}\mathbf{q}_{k}^{3}\right)\left(\left(\hat{\mathbf{q}}_{k}^{4}\right)^{H}\mathbf{q}_{k}^{4}\right)^{H}\right\\}\right\\}.\end{aligned}$
(481)

Now, we derive these $10$ terms in sequence. Note that the real operator ${\rm
Re}\left\\{\cdot\right\\}$ is omitted, since the results show that they only
have real parts.

Let us begin with the calculation of the first term as follows

$\displaystyle\begin{array}[]{l}2\mathbb{E}\left\\{\sqrt{c_{k}\delta\varepsilon_{k}}\sqrt{c_{k}\delta\varepsilon_{k}}\,\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\left(e_{k2}\sqrt{c_{k}\delta}\sqrt{c_{k}\delta}\,\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right)^{H}\right\\}\\\
=2c_{k}^{2}\delta^{2}\varepsilon_{k}e_{k2}\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\mathbb{E}\left\\{\tilde{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\overline{\mathbf{H}}_{2}^{H}\overline{\mathbf{H}}_{2}\mathbf{\Phi}\tilde{\mathbf{h}}_{k}\right\\}\\\
=2c_{k}^{2}\delta^{2}\varepsilon_{k}e_{k2}\left(\overline{\mathbf{h}}_{k}^{H}\mathbf{\Phi}^{H}\mathbf{a}_{N}\right)\mathbf{a}_{M}^{H}\mathbf{a}_{M}\left(\mathbf{a}_{N}^{H}\mathbf{\Phi}\overline{\mathbf{h}}_{k}\right)\operatorname{Tr}\left\\{\mathbf{\Phi}^{H}\mathbf{a}_{N}\mathbf{a}_{M}^{H}\mathbf{a}_{M}\mathbf{a}_{N}^{H}\mathbf{\Phi}\right\\}\\\
=2c_{k}^{2}\delta^{2}\varepsilon_{k}e_{k2}M^{2}N\left|f_{k}(\mathbf{\Phi})\right|^{2}.\end{array}$
(486)

The second term is