som#1*#3#2[#2]#3#3

# Cache-Aided Communications in MISO Networks with Dynamic User Behavior

Milad Abolpour, , MohammadJavad Salehi, , and
Antti Tölli The authors are with the Centre for Wireless Communications,
University of Oulu, Finland. E-mail<EMAIL_ADDRESS>This
research has been supported by the Academy of Finland, 6G Flagship program
under Grant 346208, 343586 (CAMAIDE), and by the Finnish-American Research and
Innovation Accelerator (FARIA) program. Part of this work has been presented
at the IEEE International Symposium on Information Theory (ISIT) 2023, in
Taipei, Taiwan.

###### Abstract

Coded caching (CC) can substantially enhance network performance by leveraging
memory as an additional communication resource. However, the use of CC is
challenging in various practical applications due to dynamic user behavior.
The existing solutions, based on shared caching, cannot directly handle all
scenarios where users freely enter and depart the network at any time as they
are constrained by specific conditions on network parameters. This paper
proposes a universally applicable shared-caching scheme for dynamic setups
without any restriction on network parameters. The closed-form expressions for
the achievable degrees of freedom (DoF) are computed for the resulting
generalized scheme, and are shown to achieve the existing optimal bounds of
the shared-cache model. Furthermore, a successive-interference-cancellation-
free extension based on a fast iterative optimized beamformer design is
devised to optimize the use of excess spatial dimensions freed by cache-aided
interference cancellation. Extensive numerical experiments are carried out to
assess the performance of the proposed scheme. In particular, the results
demonstrate that while a dynamic setup may achieve a DoF substantially lower
than the optimal DoF of shared caching, our proposed scheme significantly
improves the performance at the finite signal-to-noise ratio compared to
unicasting, which only benefits from the local caching gain.

###### Index Terms:

coded caching; shared caching; dynamic networks; multi-antenna communications

##  I Introduction

The increasing volume and diversity of multimedia content, resulting from
emerging applications such as wireless immersive viewing, requires network
providers to develop their infrastructure to support data communications at
higher rates and with lower latency [1, 2, 3]. To facilitate the efficient
delivery of such multimedia content, coded caching (CC) has been proposed to
increase the data rates via leveraging the cache memory across the network as
an additional communication resource [4]. Incorporating CC into a single-
stream downlink network boosts the achievable rate by a multiplicative factor
proportional to the cumulative cache size in the entire network via
multicasting carefully designed codewords to different user groups. In a
network of $K$ users, each equipped with a cache memory large enough to store
a portion $\gamma$ of the entire library, CC can enable the so-called _global
caching gain_ of $K\gamma+1$. Interestingly, this new gain is also additive
with the spatial multiplexing gain, and hence, in a multi-input single-output
(MISO) setup with $L$ antennas and with the spatial multiplexing gain of
$\alpha\leq L$ at the transmitter, the delivery speed-up factor of CC is
increased to $K\gamma+\alpha$ [5, 6, 7, 8, 9, 10, 11], appealing given the
critical role of multi-antenna connectivity in next-generation networks [12].
This so-called combined degree-of-freedom (DoF) of $K\gamma+\alpha$ in MISO-CC
setups is even shown to be information-theoretically optimal for uncoded cache
placement and single-shot data delivery. Of course, a larger DoF is achievable
by using interference alignment techniques [13, 14, 15, 16]. However, such
schemes are complex and pose implementation challenges, contrasting our
primary goal of practical applicability in this paper.

Despite these promising gains, a practical implementation of CC is still
challenging due to various critical impediments. In this paper, we consider
one such bottleneck: the applicability of CC techniques in dynamic setups
where the users may freely enter or depart the network [17, 18, 19]. We use
the existing _shared-cache_ model to develop a generalized low-complexity
solution to enable the CC gain for every dynamic setup, coupled with an
iterative optimized beamformer design for enhanced performance over the entire
signal-to-noise ratio (SNR) range.

###  I-A Prior Art

The shared-cache model: One of the most critical and extensively investigated
challenges for the practical CC deployment is the complexity of the
subpacketization process [20, 21, 22, 23, 24, 25, 26]. That is, in a network
with $K$ users, each file should be split into many smaller parts, the number
of which grows exponentially with $K$. This process limits the real achievable
gain under practical constraints, such that in a network with the cache ratio
$\gamma=0.01$ and $K=1000$ users, enabling the caching gain of $K\gamma=10$
requires each file to be split into $\binom{1000}{10}>10^{23}$ mini-files,
which is not possible in practice [27]. While the subpacketization complexity
issue is inherent in single-antenna scenarios [28], in multi-antenna setups,
it is possible to address this challenge through the incorporation of signal-
level decoding strategies where the undesired terms are regenerated from the
local memory and removed before the received signal is retrieved at the
receiver [27].111The signal-level approach is not only beneficial in reducing
the subpacketization; it also enables a straightforward extension of CC
schemes to multi-input multi-output (MIMO) setups [29]. A systematic extension
of the signal-level scheme in [27] was later provided as the _shared-cache_
model [30], where there exist $P\leq K$ _caching profiles_ , and $\eta_{p}$
users are assigned to profile $p\in\left\\{1,\cdots,P\right\\}$. Even though
multiple users with a cache ratio of $\gamma$ can be assigned to the same
profile and cache exactly the same data, the scaling behavior of CC gains in
MISO setups can be close to the optimal case of a dedicated cache profile for
every user [31, 30]. More precisely, for small-antenna arrays with
$\alpha\leq\frac{K}{P}$ [30] and large-antenna arrays with
$\alpha>\frac{K}{P}$ [31], the shared-cache model enables the optimal DoF of
$\alpha\left(1+P\gamma\right)$ and $K\gamma+\alpha$, respectively.

CC in dynamic networks: In order to implement CC schemes in actual
applications, these schemes must be capable of effectively handling a dynamic
population of users departing and entering the network at any time. The
conventional CC schemes require the placement phase to be designed based on
the number of users known in advance, which is impossible in dynamic networks.
Interestingly, the shared-cache model developed for reducing the
subpacketization complexity can also be used to efficiently address this
issue, as the cache placement phase is built upon knowledge of the number of
profiles $P$, which can be chosen independently from the number of users $K$.
To this end, the authors in [17, 18, 19] applied the shared caching idea to
address the dynamic user behavior by utilizing the users’ cache ratio $\gamma$
to design the content placement phase. In this sense, for a given $\gamma$
value, they selected an appropriate number of caching profiles $P$ ensuring
that $P\gamma$ is an integer. However, unfortunately, the existing models are
not dynamic in the true sense, as they are applicable only under specific
constraints. For example, [30] is applicable only if the spatial multiplexing
gain is very small, i.e., $\alpha\leq\min_{p}\eta_{p}$, while [17] and [18]
can only support the large-antenna array networks with $\alpha>\hat{\eta}$,
given that $\alpha\geq K\gamma$ and $1\leq\hat{\eta}\leq\max_{p}\eta_{p}$.
Accordingly, no existing scheme in the literature can handle network
dynamicity properly in the following cases: I) small-antenna array:
$\alpha\leq\hat{\eta}$, with $\alpha\geq\min_{p}\eta_{p}$, and II) large-
antenna array: $\alpha>\hat{\eta}$, either with non-integer
$\nicefrac{{\alpha}}{{\hat{\eta}}}$ or with $\alpha<K\gamma$. Therefore, a
universal shared-cache setup that supports any user-to-profile association is
not yet available.

CC in finite-SNR: Initial multi-antenna CC schemes focused on improving the
DoF, using zero-forcing (ZF) precoders at the transmitter to completely null
out the inter-stream interference [7, 32]. Nevertheless, the DoF metric is an
appropriate performance indicator only at the high-SNR regime [33], and ZF
beamformers should also be replaced with optimized beamformers in finite-SNR
communications [34]. Accordingly, the work [6] focused on improving both
performance and reducing the complexity of the beamforming design for MISO-CC
schemes by presenting optimized beamformers that effectively controlled both
the spatial multiplexing gain and the size of the multiple-access channel that
is encountered during decoding. Similarly, the authors in [35] presented a CC
method that constrained the number of messages each user receives per time
slot to reduce the complexity without significantly compromising the
transmission rates.

Another possibility for reducing the beamforming complexity is to use signal-
level schemes that enable replacing the intricate multi-group multicast
beamforming design with a considerably simpler multi-user unicast solution
[36, 37]. This facilitates the deployment of CC schemes for very large
networks while maintaining an enhanced performance in the finite-SNR regime.
In addition, the signal-level decoding strategies allow for simplification of
the receiver as a successive interference cancellation (SIC) structure is not
required. In this regard, a sub-optimal beamforming design (ZF precoding) via
a SIC-free data delivery scheme was studied in [38].

In cache-aided MISO networks with dynamic user behavior, the achievable DoF is
adversely affected by non-uniform user distributions. However, this negative
impact can be mitigated by optimized beamforming techniques, which allow for
greater beamforming gain in the finite-SNR regime [33]. To this end, the early
results in [17] demonstrated how the joint application of the shared caching
model and optimized linear beamforming design can handle the impact of non-
uniform distribution of users and, at the same time, boost the finite-SNR
performance.

###  I-B Our Contributions

In this paper, a universal cache-assisted MISO system is designed to handle
dynamic setups with any network parameters by leveraging the shared caching
model, which also keeps the subpacketization overhead low. Moreover, we
present a novel SIC-free data delivery algorithm tailored for networks with
dynamic user behavior and propose a fast iterative beamforming design to
optimize its finite-SNR performance. In this regard, the contributions of this
paper are three-fold:

$\bullet$ Assuming the shared-cache model for content placement phase, two
transmission strategies are presented for the content delivery phase able to
support any user-to-profile association, including the regions not covered in
the existing literature, i.e.: i) uneven user association with
$\alpha>\hat{\eta}$ and non-integer $\frac{\alpha}{\hat{\eta}}$ (unlike [31]),
ii) $\min_{p}\eta_{p}\leq\alpha\leq\hat{\eta}$ (unlike [30]), and iii)
$\alpha>\hat{\eta}$ with $\alpha<K\gamma$ (unlike [18]).

$\bullet$ Closed-form expressions are obtained for the DoF, revealing the DoF
loss caused by non-uniformness in users’ distribution. Particularly, for the
uniform user associations, the proposed scheme achieves the optimal DoF not
only in the regions covered in the literature but also when
$\alpha\geq\hat{\eta}$ and $\nicefrac{{\alpha}}{{\hat{\eta}}}$ is non-integer.

$\bullet$ In contrast to the approaches outlined in [31, 30], which assume the
utilization of SIC structure for decoding requested files, a novel data
delivery algorithm is introduced, completely eliminating the need for SIC.
Consequently, optimized linear precoders are devised to maximize the symmetric
rate performance at the finite-SNR regime. In particular, it is shown that for
highly non-uniform user-to-profile associations, even though the DoF of our
scheme may fall substantially and tends to $\alpha$, i.e., the DoF of the
unicasting scheme, it still provides a noticeable performance improvement over
unicasting at finite SNR. Indeed, by employing our proposed scheme, users
benefit from the CC gain and can remove part of the interference to decode
their desired files by using their cache content. This cache-aided
interference removal frees up the spatial multiplexing gain and enables a
larger beamforming gain, resulting in improving the rate in power-limited
finite-SNR communications.

Notation: $\left[a:b\right]$ shows the set $\\{a,\cdots,b\\}$,
$[a]=\\{1,\cdots,a\\}$, $\left|{\mathcal{A}}\right|$ is the cardinality of the
set ${\mathcal{A}}$, and for $\Lambda\subseteq{\mathcal{A}}$,
${\mathcal{A}}_{\backslash\Lambda}$ represents ${\mathcal{A}}-\Lambda$.
$\left({\mathcal{A}}\|{\mathcal{A}}\right)_{n}$ denotes $n$ concatenations of
${\mathcal{A}}$ with itself,222We use generalized multiset definition, where
we allow the same elements to be repeated in the sets, e.g.,
$\left\\{a,a\right\\}$ cannot be reduced to $\left\\{a\right\\}$ and
${\mathcal{A}}\|{\mathcal{B}}$ is the concatenation of ${\mathcal{A}}$ and
${\mathcal{B}}$. Throughout the paper, bold lower-case letters show vectors.

##  II System Model

In this paper, we focus on a dynamic MISO network, where a base station (BS)
equipped with $L$ transmit antennas and the spatial multiplexing gain of
$\alpha\leq L$ serves several cache-enabled single-antenna users. The BS has
access to a library ${\mathcal{F}}$ with $N$ unit-sized files, and each user
is equipped with a large enough memory to store a portion $0<\gamma<1$ of the
entire library. Accordingly, for a given value of cache ratio $\gamma$, we
design the natural numbers $P$ and ${\bar{t}}$, such that ${\bar{t}}=P\gamma$.
In this dynamic setup, as illustrated in Fig. 1, users can move, enter and
depart the network at any time. Thus, the BS does not have any prior knowledge
about the number of available users during the transmission. When a user $k$
enters the network, it is assigned to a profile represented by
${\mathsf{p}}[k]\in\left[P\right]$, and its cache content is updated based on
a content placement algorithm.

Figure 1: A dynamic cache-aided network with $P=3$ profiles, where users
$\left\\{1,2,\cdots,7\right\\}$ are present and do not depart the network,
users $\left\\{8,9,10\right\\}$ and $\left\\{11,12,13,14\right\\}$ are
departing and entering the network, respectively.

In the placement phase, by following the same way as in [4], each file
$W^{n}\in{\mathcal{F}}$, $n\in[N]$, is split into $\binom{P}{{\bar{t}}}$
equal-sized mini-files $W^{n}_{{\mathcal{P}}}$ such that
$W^{n}\rightarrow\left\\{W^{n}_{{\mathcal{P}}}:{\mathcal{P}}\subseteq[P],\left|{\mathcal{P}}\right|={\bar{t}}\right\\}$.
The cache content associated with profile $p\in[P]$, represented by
${\mathcal{Z}}_{p}$, includes a portion $\gamma$ of each file $W^{n}$ as

${\mathcal{Z}}_{p}=\left\\{W^{n}_{{\mathcal{P}}}:{\mathcal{P}}\ni
p,{\mathcal{P}}\subseteq\left[P\right],\left|{\mathcal{P}}\right|={\bar{t}},\forall
n\in\left[N\right]\right\\}.$

Then, defining ${\mathcal{U}}_{p}$ as the set of users assigned to profile
$p$, i.e.,
${\mathcal{U}}_{p}=\left\\{k:{\mathsf{p}}\left[k\right]=p\right\\}$, each user
$k\in{\mathcal{U}}_{p}$ stores the cache content ${\mathcal{Z}}_{p}$ during
the placement phase.

The dynamic nature of the network causes a fluctuating user population
throughout time. During regular intervals, the network’s demanding users
reveal their required files from the library ${\mathcal{F}}$ to the BS. Then,
using a content delivery algorithm, the BS constructs and transmits a set of
codewords, enabling $K$ present users to retrieve their requested files.
Accordingly, during each transmission, the BS broadcasts a transmission vector
${\mathbf{x}}\in\mathbb{C}^{L\times 1}$ containing a superposition of the
precoded codewords such that each target user $k$ receives the signal

$y_{k}={\mathbf{h}}_{k}^{\rm H}{\mathbf{x}}+n_{k},$ (1)

where ${\mathbf{h}}_{k}\in\mathbb{C}^{L\times 1}$ is the channel gain of user
$k$ and $n_{k}$ is the zero-mean additive white Gaussian noise with variance
$N_{0}$.

Figure 2: System model for a dynamic coded caching setup, where $P=2$,
$\gamma=\frac{1}{2}$, ${\bar{t}}=1$, $\alpha=4$, $\hat{\eta}=4$, $Q=2$ and
$\beta=3$. During the placement phase, each user assigned to profiles $A$ and
$B$ stores the cache content associated with those profiles. For the delivery
phase, user $5$ is served via unicasting and other users are served via $3$
multicast transmissions.

##  III Resource Allocation and Data Delivery

In this section, we discuss the resource allocation and transmission
strategies during the content delivery phase. This phase commences once the
active users reveal their requested files and comprises two consecutive steps:
1) Coded caching (CC) data delivery; and 2) Unicast (UC) data delivery.

Let us define the number of users assigned to profile $p$ as the length of
profile $p$ denoted by $\eta_{p}$, where without loss of generality, it is
assumed that $\eta_{1}\geq\eta_{2}\geq\cdots\geq\eta_{P}$. By choosing a
delivery parameter $\hat{\eta}\leq\max_{p}\eta_{p}$,333Here, the delivery
parameter plays a similar role as the unifying length parameter in [18]. Both
delivery and unifying length parameters tune the DoF loss caused by the non-
uniformness in the user-to-profile association. the BS builds and transmits a
set of codewords to serve at most $\hat{\eta}$ users assigned to each profile
with a novel CC-based approach. In this regard, for every profile $p$
$\bullet$ if $\hat{\eta}<\eta_{p}$, we exclude $\eta_{p}-\hat{\eta}$ users,
and exempt BS to serve these users during the CC delivery step. Accordingly,
the excluded users are served in the UC delivery step.
$\bullet$ if $\hat{\eta}\geq\eta_{p}$, all $\eta_{p}$ users are served via the
CC delivery step.

Now, let us suppose that the set of users assigned to profile $p\in[P]$ and
served during the CC delivery step is denoted by ${\mathcal{V}}_{p}$ such that
$\left|{\mathcal{V}}_{p}\right|=\delta_{p}$,
${\mathcal{V}}_{p}=\left\\{v_{p,1},v_{p,2}\cdots,v_{p,\delta_{p}}\right\\}$,
and $v_{p,i}\in{\mathcal{U}}_{p}$ for $i\in[\delta_{p}]$. We note that
$\delta_{p}=\min\left(\hat{\eta},\eta_{p}\right)$, and clearly,
$\delta_{1}=\hat{\eta}$ and
$\delta_{1}\geq\delta_{2}\geq\cdots\geq\delta_{P}$.

In order to build the transmission vectors, as depicted in Fig. 2, the BS
selects a parameter $Q$, which represents the number of profiles served in
each transmission, and a parameter $\beta$, which expresses the number of
users chosen from each profile to serve in each transmission. The necessary
conditions for choosing any arbitrary values for $Q$ and $\beta$ are defined
in Remark 1.

###### Remark 1

In order to serve $Q$ profiles each with a maximum of $\beta$ users, the
network parameters should satisfy the constraints ${\bar{t}}+1\leq
Q\leq{\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\beta}}\right\rceil$ and
$\beta\leq\min\left(\alpha,\hat{\eta}\right)$.

###### Proof:

The proof is relegated to Appendix A. ∎

In the following, we present the system operation maximizing the DoF
performance separately for two regimes: 1) $\alpha\leq\hat{\eta}$ and 2)
$\alpha>\hat{\eta}$. For the CC delivery step, each of these cases operates
either with Strategy A (cf. Section III-A) or Strategy B (cf. Section III-B).
The chosen transmission strategy only depends on the parameters $\alpha$ and
$\hat{\eta}$, and it is independent of the content placement phase. In other
words, the server performs the placement phase only based on parameter
$\gamma$ without considering which transmission strategy the system will use
for the CC delivery step.

####  III-1 System operation for $\alpha\leq\hat{\eta}$

For the case of $\alpha\leq\hat{\eta}$, we set $\beta=\alpha$, and the only
option for $Q$ to maximize the DoF is $Q={\bar{t}}+1$. Here, Strategy A is
utilized to build the transmission vectors. Replacing the constraint
$\alpha\leq\hat{\eta}$ with $\alpha\leq\min_{p}\delta_{p}$ reduces our system
model to [30], while our proposed scheme also works for the scenarios with
$\min_{p}\delta_{p}<\alpha\leq\hat{\eta}$.

####  III-2 System operation for $\alpha>\hat{\eta}$

For this case, we set $\beta=\hat{\eta}$, and define
$\hat{\alpha}=\frac{\alpha}{\hat{\eta}}$. If $\hat{\alpha}$ is an integer,
then we follow Strategy A to build the transmission vectors. For non-integer
$\frac{\alpha}{\hat{\eta}}$, the server can serve users via Strategy A by
setting ${\bar{t}}+1\leq
Q\leq{\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$,
and via Strategy B by choosing
$Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil$. In
this case, if we assume that users are uniformly distributed among caching
profiles, and $\frac{\alpha}{\hat{\eta}}$ is an integer, the system
performance is simplified to [31], while our proposed scheme also covers the
uneven user-to-profile associations with non-integer
$\frac{\alpha}{\hat{\eta}}$.

###  III-A Transmission Strategy A

In this strategy, each mini-file $W_{{\mathcal{P}}}^{n}$ is split into $S_{\rm
A}=\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$ subpackets
$W_{{\mathcal{P}},q}^{n}$, where $q\in[S_{\rm A}]$ increases sequentially
after each transmission to ensure none of the subpackets is transmitted twice.
Next, we follow the so-called elevation process to serve $Q$ caching profiles
each with at most $\beta$ users.

#### Elevation process

In this process, the aim is to characterize the set of users that are served
in each transmission. Accordingly, we let
$\phi_{p}=\max\left(\beta,\delta_{p}\right)$, and use the so-called
transmission triple $\left(r,c,l\right)$, where $r\in\left[P-Q+1\right]$,
$c\in\left[\phi_{r}\right]$ and $l\in\left[\binom{P-r}{Q-1}\right]$. Also, it
is assumed that $\eta_{1}\geq\eta_{2}\geq\cdots\geq\eta_{P}$, which results in
$\phi_{1}\geq\phi_{2}\geq\cdots\geq\phi_{P}$. This process creates
${\mathcal{T}}_{r,c,l}$, the set of users that are served during the
transmission triple $\left(r,c,l\right)$. In this regard, first, for every
$p\in[P]$, we elevate the set ${\mathcal{V}}_{p}$ to the set
${\mathcal{R}}_{p}$ as follows.

${\mathcal{R}}_{p}={\mathcal{R}}_{p,1}\|\cdots\|{\mathcal{R}}_{p,\phi_{p}},$
(2)

where, for $j\in\left[\phi_{p}\right]$, ${\mathcal{R}}_{p,j}$ is defined as:

${\mathcal{R}}_{p,j}=\begin{cases}{\mathcal{V}}_{p}&\delta_{p}\leq{\beta}\\\
\left\\{v_{p,l}:l=\left(i+j-1\right)\%\delta_{p},1\leq
i\leq\beta\right\\}&\delta_{p}>\beta\end{cases}.$

Here, $\%$ sign demonstrates the mod operator, for which $c\%c=c$ and
$\left(d+c\right)\%c=d\%c$. Now, for $p\in[P]$, we define
${\mathcal{S}}_{p}={\mathcal{S}}_{p,1}\|\cdots\|{\mathcal{S}}_{p,\hat{\eta}}$,
while

${\mathcal{S}}_{p,j}=\begin{cases}{\mathcal{R}}_{p,j}&1\leq j\leq\phi_{p}\\\
\varnothing&\phi_{p}+1\leq j\leq\hat{\eta}\end{cases}.$ (3)

Then, for each $r\in[P-Q+1]$, we define the set ${\mathcal{M}}_{r}$ as:

$\displaystyle{\mathcal{M}}_{r}=\left\\{{\mathcal{D}}:{\mathcal{D}}\subseteq\left[r+1:P\right],\,\left|{\mathcal{D}}\right|=Q-1\right\\}.$
(4)

In the proceeding, we use ${\mathcal{M}}_{r}\left(l\right)$ to indicate the
$l$-th $\left(Q-1\right)$-tuple of ${\mathcal{M}}_{r}$. Finally, the set
${\mathcal{T}}_{r,c,l}$ for the transmission triple $\left(r,c,l\right)$ is
given by:

$\displaystyle{\mathcal{T}}_{r,c,l}\\!=\\!\left\\{{\mathcal{S}}_{r,c}\|{\mathcal{S}}_{b_{1},c}\|\\!\cdots\\!\|{\mathcal{S}}_{b_{Q-1},c}\\!:\\!b_{i}\\!\in\\!{\mathcal{M}}_{r}\left(l\right),\forall
i\\!\in\\![Q\\!-\\!1]\right\\}.$ (5)

Generally speaking, for the transmission triple $\left(r,c,l\right)$, if
$\delta_{r}=0$, the BS does not transmit any signal; otherwise, the BS serves
users assigned to
${\mathcal{T}}_{r,c,l}={\mathcal{S}}_{r,c}\|{\mathcal{S}}_{b_{1},c}\|\cdots\|{\mathcal{S}}_{b_{Q-1},c}$,
while $\left\\{b_{1},\cdots,b_{Q-1}\right\\}={\mathcal{M}}_{r}\left(l\right)$.
During the transmission triple $\left(r,c,l\right)$, by defining
${\mathcal{N}}=\left\\{r\right\\}\cup{\mathcal{M}}_{r}\left(l\right)$, the BS
constructs the transmission vector as follows.

$\displaystyle{\mathbf{x}}_{r,c,l}=\sum\limits_{\Lambda\subseteq{\mathcal{N}}:\left|\Lambda\right|={\bar{t}}}\,\,\sum\limits_{k\in{\mathcal{S}}_{p,c}:p\in{\mathcal{N}}_{\backslash\Lambda}}{\mathbf{w}}_{k}^{\Lambda}W_{\Lambda,q}^{k},$
(6)

where ${\mathbf{w}}_{k}^{\Lambda}\in\mathbb{C}^{L\times 1}$ is the precoder
that suppresses the interference of user $k$ at the set
${\mathcal{G}}_{k}^{\Lambda}=\left\\{j\in{\mathcal{S}}_{p,c}:\forall
p\in{\mathcal{N}}_{\backslash\Lambda},j\neq k\right\\}$. In Appendix B, it is
proven that all users served with Strategy A can decode their requested files
at the end of the CC delivery step. Accordinly, at the end of the transmission
triple $\left(r,c,l\right)$, user $k$ receives a signal as described in (1).
The following example shows the data delivery via Strategy A.

###### Example 1

For a cache-aided dynamic setup with $P=3$, $\gamma=\frac{1}{3}$,
${\bar{t}}=1$, $\alpha=6$, $\beta=3$ and
$Q={\bar{t}}+\nicefrac{{\alpha}}{{\beta}}=3$, let us assume that
${\mathcal{U}}_{1}=\left\\{1,2,3,4,5\right\\}$,
${\mathcal{U}}_{2}=\left\\{6,7,8,9\right\\}$ and
${\mathcal{U}}_{3}=\left\\{10,11,12\right\\}$ are the set of users assigned to
profiles $1,2$ and $3$, respectively. During the placement phase, each file
$W^{n}$ is split into $\binom{P}{{\bar{t}}}=3$ mini-files
$W_{{\mathcal{P}}}^{n}$, where ${\mathcal{P}}\in\left\\{1,2,3\right\\}$. For
$n\in\left[N\right]$, users assigned to profiles $1,2$ and $3$ store the mini-
files $W_{1}^{n}$, $W_{2}^{n}$ and $W_{3}^{n}$, respectively. For the delivery
phase, each mini-file $W_{{\mathcal{P}}}^{n}$ is further split into
$\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}=3$ subpackets
$W_{{\mathcal{P}},q}^{n}$, where $q\in\left[3\right]$.

Assuming, $\hat{\eta}\\!=\\!4$, we have
${\mathcal{V}}_{1}\\!=\\!\left\\{1,2,3,4\right\\}$,
${\mathcal{V}}_{2}=\left\\{6,7,8,9\right\\}$ and
${\mathcal{V}}_{3}=\left\\{10,11,12\right\\}$, while user $5$ will be served
during the UC delivery step.444Although it is mentioned in Section III to set
$\beta=\hat{\eta}$ to maximize the achievable DoF for the case of
$\alpha>\hat{\eta}$, here, we set $\beta<\hat{\eta}$ to give further insight
on the system operation with Strategy A. Then, we obtain
${\mathcal{R}}_{1}-{\mathcal{R}}_{3}$ as follows.

$\begin{array}[]{lll}&{\mathcal{R}}_{1}=\left(\left\\{1,2,3\right\\}\|\left\\{2,3,4\right\\}\|\left\\{3,4,1\right\\}\|\left\\{4,1,2\right\\}\right),\\\
&{\mathcal{R}}_{2}=\left(\left\\{6,7,8\right\\}\|\left\\{7,8,9\right\\}\|\left\\{8,9,6\right\\}\|\left\\{9,6,7\right\\}\right),\\\
&{\mathcal{R}}_{3}=\left(\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\right),\\\
\end{array}$

where ${\mathcal{R}}_{1,1}=\left\\{1,2,3\right\\}$ and so on. Accordingly,
using ${\mathcal{R}}_{1}-{\mathcal{R}}_{3}$,
${\mathcal{S}}_{1}-{\mathcal{S}}_{3}$ are given by:

$\begin{array}[]{lll}&{\mathcal{S}}_{1}=\left(\left\\{1,2,3\right\\}\|\left\\{2,3,4\right\\}\|\left\\{3,4,1\right\\}\|\left\\{4,1,2\right\\}\right),\\\
&{\mathcal{S}}_{2}=\left(\left\\{6,7,8\right\\}\|\left\\{7,8,9\right\\}\|\left\\{8,9,6\right\\}\|\left\\{9,6,7\right\\}\right),\\\
&{\mathcal{S}}_{3}=\left(\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\|\left\\{10,11,12\right\\}\|\varnothing\right).\end{array}$

In order to build the transmission vector for the transmission triple
$\left(1,1,1\right)$, first, we obtain
${\mathcal{M}}_{1}={\mathcal{M}}_{1}(1)=\left\\{2,3\right\\}$. Therefore, the
set of users that are served during the transmission triple $(1,1,1)$ is equal
to:

$\displaystyle{\mathcal{T}}_{1,1,1}$
$\displaystyle=\left({\mathcal{S}}_{1,1}\|{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}\right)=\left\\{1,2,3,6,7,8,10,11,12\right\\}.$

Finally, the transmission vector corresponding to the transmission triple
$\left(1,1,1\right)$ is constructed as follows.

$\displaystyle{\mathbf{x}}_{1,1,1}=$
$\displaystyle\sum\limits_{\Lambda\subseteq\left\\{1,2,3\right\\}:\left|\Lambda\right|=1}\,\,\sum\limits_{k\in{\mathcal{S}}_{p,c}:p\in\left\\{1,2,3\right\\}_{\backslash\Lambda}}{\mathbf{w}}_{k}^{\Lambda}W_{\Lambda,1}^{k}$
(7) $\displaystyle=$
$\displaystyle\sum\nolimits_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}{\mathbf{w}}_{k}^{1}W_{1,1}^{k}+\sum\nolimits_{k\in{\mathcal{S}}_{1,1}\|{\mathcal{S}}_{3,1}}{\mathbf{w}}_{k}^{2}W_{2,1}^{k}$
$\displaystyle+$
$\displaystyle\sum\nolimits_{k\in{\mathcal{S}}_{1,1}\|{\mathcal{S}}_{2,1}}{\mathbf{w}}_{k}^{3}W_{3,1}^{k}.$

Now, in order to show the decoding process, for any
$k,j\in{\mathcal{S}}_{p,c}$ with $p\in{\mathcal{N}}_{\backslash\Lambda}$, if
$j\neq k$, using ZF precoders, we have ${\mathbf{h}}_{k}^{\rm
H}{\mathbf{w}}_{j}^{\Lambda}=0$; otherwise ($j=k$), ${\mathbf{h}}_{k}^{\rm
H}{\mathbf{w}}_{k}^{\Lambda}\neq 0$. For example, ${\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{1}^{3}\neq 0$ and ${\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{2}^{3}=0$. Now, the received signal at user $1$ is given by:

$\begin{array}[]{c}y_{1}\\!=\\!\sum\limits_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}\\!{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{k}^{1}W_{1,1}^{k}\\!+\\!{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{1}^{2}W_{2,1}^{1}\\!+\\!{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{1}^{3}W_{3,1}^{1}+n_{1}.\end{array}$ (8)

User $1$ has all subpackets $W_{1,1}^{n}$, $\forall n\in[N]$ in its cache
memory. So, it can regenerate
$\sum\nolimits_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{k}^{1}W_{1,1}^{k}$555Similarly to [36], for each
$k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}$, the term
${\mathbf{h}}_{1}^{\rm H}{\mathbf{w}}_{k}^{1}W_{1,1}^{k}$ is estimated by
using downlink precoded pilots, leading user $1$ to regenerate
$\sum_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{k}^{1}W_{1,1}^{k}$. and subtract it from $y_{1}$, which
results in observing the superposition signal
$\tilde{y}_{1}={\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{1}^{2}W_{2,1}^{1}+{\mathbf{h}}_{1}^{\rm
H}{\mathbf{w}}_{1}^{3}W_{3,1}^{1}+n_{1}$. Here, we note that user $1$ needs to
implement SIC to decode the subpackets [6], [30].

###  III-B Transmission Strategy B

When $\alpha>\hat{\eta}$ and $\frac{\alpha}{\hat{\eta}}$ is not an integer, we
can serve users by setting $\beta=\hat{\eta}$, and
$Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil$.666For
$\alpha>\hat{\eta}$ and non-integer $\frac{\alpha}{\hat{\eta}}$, we can still
serve users with Strategy A. However, the achievable DoF is less than the one
with Strategy B. Here, first, we split each mini-file $W_{{\mathcal{P}}}^{n}$
into $S_{\rm
B}=\left(\hat{\eta}{\bar{t}}+\alpha\right)\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{Q-2}{Q-{\bar{t}}-2}$
subpackets $W_{{\mathcal{P}},q}^{n}$, where $q\in\left[S_{\rm B}\right]$.
Then, we use the elevation process to serve $Q$ profiles each with at most
$\beta$ users.

#### Elevation process

This process builds the set of users served in each transmission. First, for
$r\\!\in\\!\left[P\right]$, we define ${\mathcal{Y}}_{r}$ as:

$\displaystyle{\mathcal{Y}}_{r}=\begin{cases}{\mathcal{V}}_{r}&\delta_{r}=\hat{\eta}\\\
{\mathcal{V}}_{r}\|\left(f^{*}\|f^{*}\right)_{\hat{\eta}-\delta_{r}}&\mathrm{o.w.}\end{cases},$
(9)

where $f^{*}$ denotes the phantom (non-existent) users. Generally speaking, in
each transmission of Strategy B, we serve the users assigned to $Q$ profiles
such that we select at most $\hat{\eta}$ users from $Q-1$ profiles, and pick
$\theta=\alpha-\hat{\eta}\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$
users from another profile. Next, for $r\in\left[P\right]$ and
$m\in\left[\hat{\eta}\right]$, we consider the set ${\mathcal{E}}_{r}^{m}$ as:

$\displaystyle{\mathcal{E}}_{r}^{m}=\bigcup\nolimits_{i=0}^{\theta-1}{\mathcal{Y}}_{r}\left(\left(i+m\right)\%\hat{\eta}\right),$
(10)

where ${\mathcal{Y}}_{r}(i)$ is the $i$-th element of ${\mathcal{Y}}_{r}$.
Indeed, ${\mathcal{E}}_{r}^{m}$ shifts ${\mathcal{Y}}_{r}$ to the right for
$m$ times, and picks $\theta$ elements from it. Then, for each
$u\in{\mathcal{E}}_{r}^{m}$, we define the set ${\mathcal{K}}_{r}^{m,u}$ as
follows.

$\displaystyle{\mathcal{K}}_{r}^{m,u}=\left(u\|u\right)_{\nu_{1}}\|\left(f^{*}\|f^{*}\right)_{\nu_{2}-\nu_{1}},$
(11)

where $\nu_{1}=\binom{Q-2}{Q-{\bar{t}}-2}$ and
$\nu_{2}=\binom{Q-1}{Q-{\bar{t}}-1}$. Next, by defining
${\mathcal{K}}_{r}^{m,u}(s)$ as the $s$-th element of
${\mathcal{K}}_{r}^{m,u}$, for $s\in\left[\nu_{2}\right]$, it is assumed that
${\mathcal{K}}_{r,s}^{m,u}$ is the $s$ circular shifts of
${\mathcal{K}}_{r}^{m,u}$, such that:

$\displaystyle{\mathcal{K}}_{r,s}^{m,u}=\bigcup\nolimits_{i=0}^{\nu_{2}}{\mathcal{K}}_{r}^{m,u}\left(\left(i+s\right)\%\nu_{2}\right).$
(12)

Moreover, we assume that
$\overline{{\mathcal{P}}}_{r}=\left[P\right]_{\backslash r}$ for
$r\in\left[P\right]$, and $\bar{\delta}_{c}=\overline{{\mathcal{P}}}_{r}(c)$
is the $c$-th element of $\overline{{\mathcal{P}}}_{r}$. The caching profiles
in $\overline{{\mathcal{P}}}_{r}$ are sorted in descending order such that if
$i<j$, then
$\delta_{\overline{{\mathcal{P}}}_{r}(i)}\geq\delta_{\overline{{\mathcal{P}}}_{r}(j)}$.
Next, for a given $r$ and $c\in\left[P-Q+1\right]$, let
${\mathcal{I}}_{c}^{r}$ as:

$\small\begin{array}[]{c}{\mathcal{I}}_{c}^{r}\\!=\left\\{{\mathcal{D}}:\\!{\mathcal{D}}\\!\subseteq\\!\left\\{\overline{{\mathcal{P}}}_{r}(c\\!+\\!1),\cdots,\overline{{\mathcal{P}}}_{r}(P\\!-\\!1)\right\\},\left\lvert{\mathcal{D}}\right\rvert\\!=\\!Q\\!-\\!2\right\\}.\end{array}$
(13)

Furthermore, denote the $l$-th $\left(Q-2\right)$-tuple of
${\mathcal{I}}_{c}^{r}$ by ${\mathcal{I}}_{c}^{r}(l)$, where
$l\in\left[\binom{P-c-1}{Q-2}\right]$. For the delivery process with Strategy
B, we use the so-called transmission quintuple $\left(r,c,l,m,s\right)$, where
$r\in\left[P\right]$, $c\in\left[P-Q+1\right]$,
$l\in\left[\binom{P-c-1}{Q-2}\right]$, $m\in\left[\hat{\eta}\right]$ and
$s\in\left[\nu_{2}\right]$. In each transmission quintuple
$\left(r,c,l,m,s\right)$, users assigned to the caching profiles
${\mathcal{B}}=\left\\{\bar{\delta}_{c}\right\\}\cup{\mathcal{I}}_{c}^{r}(l)$,
and users in the set ${\mathcal{E}}_{r}^{m}$ are served. Suppose that
${\mathcal{C}}=\left\\{{\mathcal{D}}:{\mathcal{D}}\subseteq{\mathcal{B}},\left|{\mathcal{D}}\right|=\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor\right\\}$,
and ${\mathcal{C}}(n)$ is the $n$-th
$\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$-tuple of
${\mathcal{C}}$ with $n\in\left[\nu_{2}\right]$.

We define the function
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)$ such that
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=0$, if
${\mathcal{E}}_{r}^{m}=f^{*}$ and $\bar{\delta}_{c}=0$; otherwise,
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=1$. If
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=1$, after
eliminating the impacts of the phantom users $f^{*}$, the BS builds the
transmission vector for the transmission quintuple $\left(r,c,l,m,s\right)$ as
follows.

$\displaystyle{\mathbf{x}}_{r,c,l}^{m,s}=\sum\nolimits_{n=1}^{\nu_{2}}\,\,\sum\limits_{k\in{\mathcal{K}}_{r,s}^{m,u}(n)\cup{\mathcal{V}}_{p}:u\in{\mathcal{E}}_{r}^{m},p\in{\mathcal{C}}(n)}{\mathbf{w}}_{k}^{\Lambda}W_{\Lambda,q}^{k},$
(14)

where $\Lambda={\mathcal{B}}_{\backslash{\mathcal{C}}(n)}$ with
$\left|\Lambda\right|={\bar{t}}$, and
${\mathbf{w}}_{k}^{\Lambda}\in\mathbb{C}^{L\times 1}$ is the precoder that
suppresses the interference of user $k$ at the set
${\mathcal{H}}_{k}^{\Lambda}=\left\\{j\in{\mathcal{E}}_{r}^{m}\cup{\mathcal{V}}_{p}:p\in{\mathcal{C}}(n),j\neq
k,f^{*}\right\\}$. In Appendix B, we prove that all users can decode their
requested files with Strategy B. Accordingly, user $k$ receives the signal
shown in (1) at the end of the transmission quintuple $(r,c,l,m,s)$. The
following example studies the data delivery via Strategy B.

###### Example 2

With the given ${\mathcal{U}}_{1}-{\mathcal{U}}_{3}$ and
${\mathcal{V}}_{1}-{\mathcal{V}}_{3}$ in Example 1, suppose that $P=3$,
$\gamma=\frac{1}{3}$, ${\bar{t}}=1$, $\alpha=6$, $\beta=\hat{\eta}=4$,
$\theta=\alpha-\hat{\eta}\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor=2$
and $Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil=3$.
For the placement phase, similarly to Example 1, each file $W^{n}$ is split
into 3 mini-files, and users assigned to profiles $1,2$ and $3$, store the
mini-files $W_{1}^{n}$, $W_{2}^{n}$ and $W_{3}^{n}$, respectively. Then,
during the content delivery phase, each mini-file is split into
$\left(\hat{\eta}{\bar{t}}+\alpha\right)\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{Q-2}{Q-{\bar{t}}-2}=10$
subpackets. Now, for the transmission quintuple $(1,1,1,1,1)$, we have:

$\displaystyle{\mathcal{Y}}_{1}=\left\\{1,2,3,4\right\\},\,\,{\mathcal{Y}}_{2}=\left\\{6,7,8,9\right\\},\,\,{\mathcal{Y}}_{3}=\left\\{10,11,12,f^{*}\right\\}.$

At the next step, for $r=c=1$, it is found that
$\overline{{\mathcal{P}}}_{1}=\left\\{2,3\right\\}$,
$\overline{{\mathcal{P}}}_{1}(1)=2$, and
${\mathcal{I}}_{1}^{1}={\mathcal{I}}_{1}^{1}(1)=\left\\{3\right\\}$. During
this transmission, the users of the set
${\mathcal{E}}_{1}^{1}=\left\\{1,2\right\\}$, and the users assigned to the
profiles
${\mathcal{B}}=\overline{{\mathcal{P}}}_{1}(1)\cup{\mathcal{I}}_{1}^{1}(1)=\left\\{2,3\right\\}$
are served. Furthermore, ${\mathcal{K}}_{1,1}^{1,1}=\left\\{1,f^{*}\right\\}$
and ${\mathcal{K}}_{1,1}^{1,2}=\left\\{2,f^{*}\right\\}$, and, it is found
that ${\mathcal{C}}=\left\\{\left\\{2\right\\},\left\\{3\right\\}\right\\}$
such that ${\mathcal{C}}(1)=\left\\{2\right\\}$ and
${\mathcal{C}}(2)=\left\\{3\right\\}$.

Hence, after removing the phantom users $f^{*}$, the transmission vector for
the transmission quintuple $(1,1,1,1,1)$ takes the form of:

$\displaystyle{\mathbf{x}}_{1,1,1}^{1,1}$
$\displaystyle=\sum\limits_{n=1}^{2}\,\sum\limits_{k\in{\mathcal{K}}_{1,1}^{1,u}(n)\cup{\mathcal{V}}_{p}:u\in\left\\{1,2\right\\},p\in{\mathcal{C}}(n)}{\mathbf{w}}_{k}^{\Lambda}W_{\Lambda,1}^{k}$
$\displaystyle=\sum\limits_{k\in\left\\{1,2,6,7,8,9\right\\}}{\mathbf{w}}_{k}^{3}W_{3,1}^{k}+\sum\limits_{k\in\left\\{10,11,12\right\\}}{\mathbf{w}}_{k}^{2}W_{2,1}^{k},$

where ${\mathcal{H}}_{6}^{3}=\left\\{1,2,7,8,9\right\\}$,
${\mathcal{H}}_{10}^{2}=\left\\{1,2,11,12\right\\}$ and so on. Similarly to
Example 1, all users that are served via Strategy B are able to decode their
requested files.

###  III-C Unicast (UC) Data Delivery

In the UC delivery step, the BS transmits data to the users excluded from the
CC delivery step. Here, unlike the CC delivery step that benefits from the
global coded caching and spatial multiplexing gains, only local coded caching
and spatial multiplexing gains are available. Suppose that the BS serves
$K_{U}$ users during the UC delivery step such that
$K_{U}=\sum_{p=1}^{P}\max\left(0,\eta_{p}-\hat{\eta}\right)$. Each of the
requested files by these users is split into the same number of subpackets as
in the CC delivery step. Then, in order to transmit these missing subpackets,
we follow a greedy algorithm similar to [17], which comprises 3 processes: 1)
sort users based on the number of their missing subpackets in descending
order; 2) create a transmission vector to deliver one missing subpacket to
each of the first $\min\left(\alpha,K_{U}\right)$ users; 3) repeat processes 1
and 2 until all missing files are transmitted.

##  IV SIC-Free Data Delivery

As stated in Section III, during every transmission of the CC delivery step,
each user served via Strategy A and Strategy B receives
$\nu_{2}=\binom{Q-1}{Q-{\bar{t}}-1}$ and $\nu_{1}=\binom{Q-2}{Q-{\bar{t}}-2}$
subpackets of its requested file, respectively. Consequently, the retrieval of
subpackets may require the implementation of a SIC receiver at each user,
which is complex and undesired [33]. To address this issue, we propose a data
delivery algorithm that eliminates the SIC requirement. In the following, we
review this algorithm for both transmission strategies _A_ and _B_.

Strategy A. Let us denote the set of users served during the transmission
triple $\left(r,c,l\right)$, $r\in\left[P-Q+1\right]$,
$c\in\left[\phi_{r}\right]$, and $l\in[\binom{P-r}{Q-1}]$ with
${\mathcal{T}}_{r,c,l}$, and represent the set of subpackets received by user
$k\in{\mathcal{T}}_{r,c,l}$ in transmission triple $(r,c,l)$ with
${\mathcal{O}}_{r,c,l}^{k}$, i.e.,

$\displaystyle{\mathcal{O}}_{r,c,l}^{k}=\left\\{W^{k}_{\Lambda,q}:\Lambda\subseteq{\mathcal{N}},\left|\Lambda\right|={\bar{t}},{\mathsf{p}}\left[k\right]\in{\mathcal{N}}_{\backslash\Lambda}\right\\},$

and $|{\mathcal{O}}_{r,c,l}^{k}|=\nu_{2}$. Now, to eliminate the SIC
requirement, we split each transmission triple $(r,c,l)$ into $\nu_{2}$
transmissions, denoted by $(r,c,l)^{j}$, $j\in[\nu_{2}]$, such that in
transmission $(r,c,l)^{j}$, the BS transmits the superposition signal

$\begin{array}[]{c}{\mathbf{x}}_{r,c,l}^{j}=\sum\limits_{k\in{\mathcal{T}}_{r,c,l}}{\mathbf{w}}_{k}^{\Lambda^{\prime}}W^{k}_{\Lambda^{\prime},q},\end{array}$
(15)

where $W^{k}_{\Lambda^{\prime},q}$ is the $j$-th entry of
${\mathcal{O}}_{r,c,l}^{k}$, and
${\mathbf{w}}_{k}^{\Lambda^{\prime}}\in\mathbb{C}^{L\times 1}$ is the
beamforming vector suppressing the interference of subpacket
$W^{k}_{\Lambda^{\prime},q}$ at the set
${\mathcal{G}}_{k}^{\Lambda^{\prime}}$, defined in (6). Note that, following
(6) and (15), it is guaranteed that
${\mathbf{x}}_{r,c,l}=\sum_{j=1}^{\nu_{2}}{\mathbf{x}}_{r,c,l}^{j}$.

Strategy B. Let us denote the set of users served during the transmission
quintuple $\left(r,c,l,m,s\right)$, $r\in\left[P\right]$,
$c\in\left[P-Q+1\right]$, $l\in[\binom{P-c-1}{Q-2}]$,
$m\in\left[\hat{\eta}\right]$, $s\in\left[\nu_{2}\right]$, with
$\tilde{{\mathcal{T}}}_{r,c,l}^{m,s}$, and represent the set of subpackets
received by user $k\in\tilde{{\mathcal{T}}}_{r,c,l}^{m,s}$ in transmission
quintuple $(r,c,l,m,s)$ with $\tilde{{\mathcal{O}}}_{r,c,l}^{m,s,k}$, i.e.,

$\begin{array}[]{l}\tilde{{\mathcal{O}}}_{r,c,l}^{m,s,k}\\!=\\!\left\\{W^{k}_{\Lambda,q}\\!:\\!n\\!\in\\!\left[\nu_{2}\right],\Lambda\\!=\\!{\mathcal{B}}_{\backslash{\mathcal{C}}(\\!n\\!)},{\mathsf{p}}\left[k\right]\\!\in\\!{\mathcal{C}}(n)\\!\cup\\!\left\\{r\right\\}\right\\},\end{array}$

and $|\tilde{{\mathcal{O}}}_{r,c,l}^{m,s,k}|=\nu_{1}$. To eliminate the SIC
requirement, we split every transmission quintuple $(r,c,l,m,s)$ into
$\nu_{1}$ transmissions, denoted by $(r,c,l,m,s)^{j}$, $j\in[\nu_{1}]$, such
that in transmission $(r,c,l,m,s)^{j}$, the BS transmits the superposition
signal

$\begin{array}[]{c}{\mathbf{x}}_{r,c,l}^{m,s,j}=\sum\limits_{k\in\tilde{{\mathcal{T}}}_{r,c,l}^{m,s}}{\mathbf{w}}_{k}^{\tilde{\Lambda}}W^{k}_{\tilde{\Lambda},q},\end{array}$
(16)

where $W^{k}_{\tilde{\Lambda},q}$ denotes the $j$-th entry of
$\tilde{{\mathcal{O}}}_{r,c,l}^{m,s,k}$ and
${\mathbf{w}}_{k}^{\tilde{\Lambda}}\in\mathbb{C}^{L\times 1}$ is the precoder
suppressing the interference of subpacket $W^{k}_{\tilde{\Lambda},q}$ at every
user in ${\mathcal{H}}_{k}^{\tilde{\Lambda}}$, defined in (14). Clearly,
according to (14) and (16), we have
${\mathbf{x}}_{r,c,l}^{m,s}=\sum_{j=1}^{\nu_{1}}{\mathbf{x}}_{r,c,l}^{m,s,j}$.

###### Example 3

Let us focus on the cache-aided network considered in Example 1 and the
transmission vector ${\mathbf{x}}_{1,1,1}$ in (7). For this transmission
vector, the sets of subpackets received by users 1 and 2, for example, are
given as ${\mathcal{O}}_{1,1,1}^{1}=\\{W_{2,1}^{1},W_{3,1}^{1}\\}$ and
${\mathcal{O}}_{1,1,1}^{2}=\\{W_{2,1}^{2},W_{3,1}^{2}\\}$, respectively. So,
for SIC-free data delivery, ${\mathbf{x}}_{1,1,1}$ should be split into
$\nu_{2}=2$ transmission vectors

$\begin{array}[]{c}{\mathbf{x}}_{1,1,1}^{1}=\sum\limits_{k\in{\mathcal{S}}_{2,1}\|{\mathcal{S}}_{3,1}}{\mathbf{w}}_{k}^{1}W_{1,1}^{k}+\sum\limits_{k\in{\mathcal{S}}_{1,1}}{\mathbf{w}}_{k}^{2}W_{2,1}^{k},\end{array}$
$\begin{array}[]{c}{\mathbf{x}}_{1,1,1}^{2}=\sum\limits_{k\in{\mathcal{S}}_{3,1}}{\mathbf{w}}_{k}^{2}W_{2,1}^{k}+\sum\limits_{k\in{\mathcal{S}}_{1,1}\|{\mathcal{S}}_{2,1}}{\mathbf{w}}_{k}^{3}W_{3,1}^{k},\end{array}$

where, for example, ${\mathbf{w}}_{1}^{2}$ and ${\mathbf{w}}_{1}^{3}$
suppresses the interference at the sets of users
${\mathcal{G}}_{1}^{2}=\\{2,3,10,11,12\\}$ and
${\mathcal{G}}_{1}^{3}=\\{2,3,6,7,8\\}$, respectively.

##  V DoF Analysis

In this section, we use DoF as the metric of interest to measure performance.
Here, it is assumed that both strategies _A_ and _B_ employ the SIC-free data
delivery algorithm, and the DoF is defined as the average number of users
served concurrently during the delivery phase.777Note that utilizing the SIC-
free data delivery algorithm does not alter the DoF [6]. The assumption here
is only for the sake of simplicity. In CC and UC delivery steps, we denote the
total transmissions by $T_{M}$ and $T_{U}$, respectively, and the number of
transmitted subpackets by $J_{M}$ and $J_{U}$. Therefore, DoF is computed as
follows.

$\mathrm{DoF}=\frac{J_{M}+J_{U}}{T_{M}+T_{U}}.$ (17)

Furthermore, we suppose that $K_{M}$ and $K_{U}$ users are served during the
CC and UC delivery steps such that
$K_{M}=\sum_{p}\min\left(\hat{\eta},\eta_{p}\right)$ and
$K_{U}=\sum_{p}\max\left(0,\eta_{p}-\hat{\eta}\right)$. The next theorem
characterizes the DoF for the cache-aided networks operating with strategies A
and B during the CC delivery step.

###### Theorem 1

Consider a dynamic MISO network with the spatial multiplexing gain of
$\alpha$, cache ratio $\gamma$ and the delivery parameter $\hat{\eta}$. If the
system operates with Strategy A in the CC delivery step, the DoF is given by:

$\mathrm{DoF}=\begin{cases}\frac{K\binom{P-1}{Q-1}\beta}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}}&K_{U}=0\\\
\frac{K_{M}\binom{P-1}{Q-1}\beta\nu_{2}+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}\nu_{2}+\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil}&K_{U}\neq
0\end{cases},$ (18)

where $\beta^{\prime}=\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$ and
$D(\delta_{r})=\phi_{r}$ if $\delta_{r}\neq 0$; otherwise,
$D\left(\delta_{r}\right)=0$. If Strategy B is applied during the CC delivery
step, the DoF takes the form as follows.

$\mathrm{DoF}=\begin{cases}\frac{K\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}}{N_{M}}&K_{U}=0\\\
\frac{K_{M}\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}\nu_{1}+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{N_{M}\nu_{1}+N_{U}}&K_{U}\neq
0\end{cases},$ (19)

where
$\alpha^{\prime}=\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{1}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$,
$N_{U}=\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil$
and

$N_{M}=\sum_{r=1}^{P}\sum_{c=1}^{P-Q+1}\sum_{m=1}^{\hat{\eta}}\sum_{s=1}^{\nu_{2}}\binom{P-c-1}{Q-2}I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right).$
(20)

###### Proof:

The proof is relegated to Appendix B. ∎

###### Remark 2

Suppose that $K$ users are uniformly assigned to the caching profiles, i.e.,
$K=P\hat{\eta}$, and all users are served in the CC delivery step, i.e.,
$K_{M}=K$ and $K_{U}=0$. If $\alpha\leq\hat{\eta}$, our schemes achieves the
optimal DoF $\alpha\left(P\gamma+1\right)$ obtained in [30]. If
$\alpha>\hat{\eta}$, the achievable DoF of our scheme is simplified to the
optimal DoF $K\gamma+\alpha$ obtained in [31]. However, unlike the scheme
proposed in [31], our scheme supports the networks with non-integer
$\frac{\alpha}{\hat{\eta}}$ (cf. Appendix C).

Indeed, increasing non-uniformness in user distribution prevents the system
from achieving optimal DoF performance. For instance, for the case
$\alpha>\hat{\eta}$, suppose that all users are served with Strategy A during
the CC delivery step, such that $\beta=\hat{\eta}$ and
$Q\leq{\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$.
In this setup, we can set
$Q={\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$ to
boost the DoF performance. By defining $\eta_{\rm avg}=\frac{K}{P}$ and
assuming $\alpha$ is divisible by $\hat{\eta}$ and $\eta_{\rm avg}$, the DoF
loss (compared to uniform user distribution) is
$\alpha\left(1-\nicefrac{{\eta_{\rm avg}}}{{\hat{\eta}}}\right)$.

###### Remark 3

For the non-uniform user-to-profile association with $Q={\bar{t}}+1$ and
$\eta_{1}\leq\alpha$, the best possible DoF is achievable by setting
$\hat{\eta}=\max_{p}\eta_{p}=\eta_{1}$ (cf. Appendix D).

Generally speaking, for any network parameters, we should compare the
achievable DoF of our proposed scheme with $\alpha$ such that: i) if
$\mathrm{DoF}\geq\alpha$, we use the proposed CC scheme to simultaneously
benefit from global CC and spatial multiplexing gains; ii) if
$\mathrm{DoF}<\alpha$, we serve users via unicasting to not have any loss in
DoF.

###### Remark 4

In scenarios with non-uniform user-to-profile associations, some transmission
vectors may serve fewer than $\alpha$ users in the CC delivery step,
preventing the system from fully utilizing the available multiplexing gain. To
ensure that at least $\alpha$ users are served per transmission during the CC
delivery step, we can adopt the so-called efficient multicast method. In this
method, the BS disregards transmission vectors serving fewer than $\alpha$
users, and consequently, subpackets associated with these transmission vectors
are not transmitted during the CC delivery step. Instead, all these omitted
subpackets, in addition to the subpackets for $K_{U}$ users excluded from the
CC delivery step, are transmitted during the UC delivery step by following a
similar way as described in Section III-C. The derivation of a closed-form
expression for the DoF of our proposed scheme via the efficient multicast
method is beyond the scope of this paper, and we will explore it in our future
work. It’s worth noting that, without employing the efficient multicast
method, Strategy B outperforms Strategy A in terms of achievable DoF, as
outlined in Theorem 1. However, by using efficient multicast transmission,
Strategy A is not necessarily inferior to Strategy B in terms of DoF.

##  VI Beamforming Design

In this section, we design optimized beamformers to deliver data during the CC
and UC delivery steps. In this regard, let us assume that the system operates
with Strategy A via the SIC-free data delivery algorithm during the CC
delivery step, the set of users served in the transmission $(r,c,l)^{d}$ with
$d\in[\nu_{2}]$ is given by ${\mathcal{T}}_{r,c,l}$ (cf. Section IV), and the
aim is to design the beamformers for each transmission $(r,c,l)^{d}$. To this
end, by using (15) and (1), and following a similar way as in (8) to enable
the cache-aided interference removal, the achievable rate of user
$k\in{{\mathcal{T}}}_{r,c,l}$ is given by
$R_{k}=\log\left(1+\Gamma_{k}\right)$, where $\Gamma_{k}$ is the SINR at user
$k$ in this transmission, and is equal to:

$\displaystyle\Gamma_{k}=\frac{\left|{\mathbf{h}}_{k}^{\rm
H}{\mathbf{w}}_{k}^{\Lambda}\right|^{2}}{\sum\nolimits_{j\in{\mathcal{G}}_{k}^{\Lambda}}\left|{\mathbf{h}}_{k}^{\rm
H}{\mathbf{w}}_{j}^{\Lambda}\right|^{2}+N_{0}}.$ (21)

We note that if the system operates with Strategy B, by employing the SIC-free
data delivery algorithm, each transmission $(r,c,l,m,s)^{d}$ with
$d\in[\nu_{1}]$ serves the set of users $\tilde{{\mathcal{T}}}_{r,c,l}^{m,s}$.
Here, if ${\mathsf{p}}[k]=r$, ${\mathcal{G}}_{k}^{\Lambda}$ in (21) is
replaced by $\tilde{{\mathcal{T}}}_{r,c,l}^{m,s}-\\{k\\}$; otherwise, we
replace ${\mathcal{G}}_{k}^{\Lambda}$ with ${\mathcal{H}}_{k}^{\Lambda}$ (cf.
Section III-B). Similarly, for each transmission of the UC delivery step, by
considering $\hat{{\mathcal{T}}}$ as the set of served users, the interfering
set ${\mathcal{G}}_{k}^{\Lambda}$ in (21) is substituted with
$\hat{{\mathcal{T}}}-\\{k\\}$.

As stated in Section III-A, by using Strategy A, each mini-file
$W_{{\mathcal{P}}}^{n}$ is split into
$\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$ subpackets
$W_{{\mathcal{P}},q}^{n}$, where
$q\in[\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}]$. On the other hand, the size
of each mini-file is $\nicefrac{{1}}{{\binom{P}{{\bar{t}}}}}$ of the size of
the original file. Hence, user $k\in{{\mathcal{T}}}_{r,c,l}$ receives a
$\frac{1}{\mu_{k}}$ portion of its requested file at the end of the
transmission $(r,c,l)^{d}$, where
$\mu_{k}=\beta\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\binom{P}{{\bar{t}}}$. As a
result, during this transmission, the required time to deliver a subpacket to
user $k$, denoted by $T_{k}$, takes the form of
$T_{k}=\frac{1}{\mu_{k}R_{k}}$. Now, our aim is to design the beamformers to
minimize the delivery time or, equivalently, maximize the achievable rate for
the worst user served in the transmission $(r,c,l)^{d}$. Accordingly, given a
fixed transmit power $P_{\rm T}$, we should solve the following optimization
problem in order to obtain the optimized beamformers.

$\displaystyle\underset{{\mathbf{w}}_{k}^{\Lambda}:k\in{{\mathcal{T}}}_{r,c,l}}{\max\min}\quad{\mu_{k}R_{k}}$
(22)
$\displaystyle\quad\quad\mathrm{s.t.}\quad\sum\nolimits_{k\in{{\mathcal{T}}}_{r,c,l}}\left\|{\mathbf{w}}_{k}^{\Lambda}\right\|^{2}\leq
P_{\rm T}.$

$1$$2$$3$$4$$5$$6$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$A$$A$$B$$A$$A$$A$$B$$A$$B$AA$\hat{\eta}$DoF$Q=2$$Q=3$$Q=4$$Q=5$
(a) $\sigma=0$

$1$$2$$3$$4$$5$$6$$7$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$A$$A$$B$$A$$A$$A$$B$$A$$B$$A$$B$AA$\hat{\eta}$DoF$Q=2$$Q=3$$Q=4$$Q=5$
(b) $\sigma=0.63$

$1$$2$$3$$4$$5$$6$$7$$8$$9$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$$12$$13$$14$$A$$A$$B$$A$$A$$A$$B$$A$$B$$A$$B$$A$$A$AA$\hat{\eta}$DoF$Q=2$$Q=3$$Q=4$$Q=5$
(c) $\sigma=2.19$

Figure 3: The DoF versus $\hat{\eta}$ with different values of $Q$, $K=30$,
$\alpha=8$, $P=5$, $\gamma=0.2$ and ${\bar{t}}=1$ for: (a) $\sigma=0$, (b)
$\sigma=0.63$ and (c) $\sigma=2.19$. Here, _A_ and _B_ show that the system
operates with _Strategy A_ and _Strategy B_ , respectively.

Here, problem (22) can be efficiently solved as a max-min problem using
iterative methods based on Lagrangian duality, as demonstrated in various
studies including [36, 39]. In this sense, employing a Lagrangian duality-
based technique described in [36], we can iteratively find the optimum rate
$R_{\rm e}$ via bisection search, which is given by $R_{\rm e}=\mu_{k}R_{k}$.
For each fixed $R_{\rm e}$, the fixed point iteration method [40] is used to
obtain the dual variables $\lambda_{k}$ for $k\in{{\mathcal{T}}}_{r,c,l}$. In
this regard, at the first step, for $k\in{{\mathcal{T}}}_{r,c,l}$, we
initialize $\lambda_{k}\leftarrow\lambda_{k}[1]$. Then, until achieving the
desired level of convergence, we iteratively update the dual variables such
that for iteration $i+1$, the dual variable $\lambda_{k}$ is given by:

$\displaystyle\lambda_{k}[i+1]=\frac{\omega_{k}}{\omega_{k}+1}\left({\mathbf{h}}_{k}^{\rm
H}\boldsymbol{\Sigma}_{k,i}^{-1}{\mathbf{h}}_{k}\right)^{-1},$ (23)

where $\omega_{k}=2^{\nicefrac{{R_{\rm e}}}{{\mu_{k}}}}-1$, $\lambda_{k}[i]$
denotes the value of $\lambda_{k}$ during the $i$-th iteration, and
$\boldsymbol{\Sigma}_{k,i}={\mathbf{I}}_{L}+\sum_{j\in{\mathcal{G}}_{k}^{\Lambda}\cup\left\\{k\right\\}}\lambda_{j}[i]{\mathbf{h}}_{j}{\mathbf{h}}_{j}^{\rm
H}$.

After the convergence of (23), we obtain the normalized beamformer
$\tilde{{\mathbf{w}}}_{k}$. Accordingly, for $k\in{{\mathcal{T}}}_{r,c,l}$, as
per [36], $\tilde{{\mathbf{w}}}_{k}$ is given by
$\tilde{{\mathbf{w}}}_{k}=\nicefrac{{\tilde{\boldsymbol{\Sigma}}_{k}^{-1}{\mathbf{h}}_{k}}}{{\left\|\tilde{\boldsymbol{\Sigma}}_{k}^{-1}{\mathbf{h}}_{k}\right\|}}$,
where
$\tilde{\boldsymbol{\Sigma}}_{k}={\mathbf{I}}_{L}+\sum_{j\in{\mathcal{G}}_{k}^{\Lambda}\cup\left\\{k\right\\}}\lambda_{j}{\mathbf{h}}_{j}{\mathbf{h}}_{j}^{\rm
H}$. In order to find the power vector of the beamformers, denoted by
$\tilde{{\mathbf{p}}}$, we simply follow the same steps as in [36, eq. (26)].
Finally, during the transmission $(r,c,l)^{d}$, the beamforming vector
${\mathbf{w}}_{k}^{\Lambda}$ takes the form of
${\mathbf{w}}_{k}^{\Lambda}=\sqrt{\tilde{{\mathbf{p}}}(k)}\tilde{{\mathbf{w}}}_{k}$,
where $\tilde{{\mathbf{p}}}(k)$ is the $k$-th entry of $\tilde{{\mathbf{p}}}$.
Consequently, the transmission rate for the transmission $(r,c,l)^{d}$ during
the CC delivery step, denoted by $R_{r,c,l,d}^{\rm CC}$, is given by
$R_{r,c,l,d}^{\rm
CC}=\min_{k\in{\mathcal{T}}_{r,c,l}}\log\left(1+\Gamma_{k}\right)$. Similarly,
we can follow the same steps to obtain the transmission rates for each
transmission of Strategy B and UC delivery step. Finally, we evaluate the
system performance at finite-SNR regime based on the symmetric rate per user,
which is denoted by $R_{\rm sym}$. Assuming that the system operates with
Strategy A during the CC delivery step, $R_{\rm sym}$ is defined as follows.

$\begin{array}[]{l}R_{\rm
sym}\\!=\bigg{(}\\!\sum\limits_{r=1}^{P-Q+1}\sum\limits_{c=1}^{\phi_{r}}\sum\limits_{l=1}^{\binom{P-r}{Q-1}}\sum\limits_{d=1}^{\nu_{2}}\frac{1}{R_{r,c,l,d}^{\rm
CC}}\\!+\\!\sum\limits_{j=1}^{\tilde{N}_{\rm UC}}\frac{1}{R_{j}^{\rm
UC}}\\!\bigg{)}^{\\!-1}\\!,\end{array}$ (24)

where $\tilde{N}_{\rm UC}$ is the total number of transmissions during the UC
delivery step, and $R_{j}^{\rm UC}$ is the achievable rate associated with the
$j$-th transmission of the UC delivery step. Here, we note that if we use the
efficient multicast method during the CC delivery step, then $\tilde{N}_{\rm
UC}$ is not necessarily equal to $T_{U}$ in (17). Additionally, as mentioned
earlier, for the network setup operating with Strategy B in the CC delivery
step, we can follow the similar steps as in (21)-(23) to obtain the achievable
rate for each transmission in the CC delivery step. Then, we can simply
compute $R_{\rm sym}$ in (24) based on the total number of transmissions
during the CC delivery step via Strategy B.

##  VII Numerical Results

We use numerical simulations to analyze the performance of the proposed coded
caching scheme for dynamic networks in terms of both the achievable DoF and
the symmetric rate.

###  VII-A DoF Analysis

We consider a cache-aided MISO setup with ${\bar{t}}=1$, $P=5$, and
$\gamma=0.2$, in which $K=30$ users are present during the delivery phase.
Here, for each user-to-profile association, we compute the standard deviation
$\sigma$ as $\sigma^{2}=\frac{1}{P}\sum_{p=1}^{P}\left(\eta_{p}-\eta_{\rm
avg}\right)^{2}$, where $\eta_{\rm avg}=\nicefrac{{K}}{{P}}=6$.

Fig. 3 illustrates the achievable DoF for different $Q$ and $\sigma$ values
when $\alpha=8$. In the figure, we have also shown which transmission strategy
(i.e., _A_ or _B_) should be used for the selected parameters. From Fig. 3, we
see that even though choosing $\hat{\eta}=\max_{p}\eta_{p}$ is proven to
maximize the DoF only when $\alpha$ is large (i.e.
$\alpha\geq\max_{p}\eta_{p}$, cf. Remark 3), it is also a safe choice for
smaller $\alpha$ values. Moreover, for a given $\hat{\eta}$ value, selecting
the maximum possible $Q$, i.e.,
$Q={\bar{t}}+\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\rceil$, always maximizes
the achievable DoF. Moreover, _Strategy B_ enables our proposed scheme to be
truly universal and apply to any user-to-profile association. For instance,
although no existing scheme in the literature can achieve the optimal DoF of
$K\gamma+\alpha=14$ for the considered network with the uniform user-to-
profile association (as $\alpha>\hat{\eta}$ and
$\nicefrac{{\alpha}}{{\hat{\eta}}}$ is a non-integer, cf. Section III-B), our
proposed scheme can achieve this DoF with _Strategy B_ by setting
$\hat{\eta}=\eta_{\rm avg}=6$ and $Q=3$.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$2$$4$$6$$8$$10$$12$$14$Standard Deviation
($\sigma$)$\text{DoF}_{M}$Optimal: $\alpha=8$Proposed scheme: $\alpha=8$no-CC:
$\alpha=8$Optimal: $\alpha=5$Proposed scheme: $\alpha=5$no-CC: $\alpha=5$
Figure 4: The average of the maximum achievable DoF $\left({\rm
DoF}_{M}\right)$ versus the standard deviation $(\sigma)$ with $K=30$,
$\gamma=0.2$, $P=5$, ${\bar{t}}=1$ and $\alpha\in\left\\{5,8\right\\}$.

In Fig. 4, we analyze the effect of the non-uniformness in user-to-profile
associations (measured by $\sigma$) on the achievable DoF and compare the
results with two extreme cases: i) Optimal, where the users are uniformly
assigned to the caching profiles, and ii) no-CC, where the users only benefit
from the local CC gain, and $\min\left(\alpha,K\right)$ users are served per
transmission in a TDMA framework. The parameter $\mathrm{DoF}_{M}$ denotes the
average of the maximum achievable DoF over the user-to-profile associations,
and is calculated as follows: for any possible user-to-profile association for
the considered network, we find $\mathrm{DoF}_{\max}$ which is the maximum
achievable DoF obtained by a line search over $\hat{\eta}$ and $Q$ values, and
then, for the associations with the same $\sigma$ value, we set
$\mathrm{DoF}_{M}$ as the average of all the corresponding
$\mathrm{DoF}_{\max}$ values. As can be seen, although the placement phase of
our scheme was designed solely based on the knowledge of $\gamma$ (and not the
user count $K$), it can still boost the maximum achievable DoF by $10\%-70\%$
over the no-CC solution for moderate $\sigma$ values (e.g., $1\leq\sigma\leq
4.5$). This shows the strength of our algorithm in enabling a coded caching
gain for any given non-uniform user distribution.

$5$$10$$15$$20$$25$$30$$35$$40$$45$$50$$1$$2$$3$$4$$5$$6$$7$$8$SNR
[dB]Symmetric Rate [bits/s/Hz]_Strategy A_ : $\alpha=10$_Strategy B_ :
$\alpha=10$no-CC: $\alpha=10$_Strategy A_ : $\alpha=8$_Strategy B_ :
$\alpha=8$no-CC: $\alpha=8$ Figure 5: Average symmetric rate versus SNR with
random user-to-profile association, $K=30$, $L=10$, ${\bar{t}}=1$, $P=5$,
$\gamma=0.2$, $\hat{\eta}=\eta_{1}$ and $\alpha\in\\{8,10\\}$.

###  VII-B Symmetric Rate

Here, we study the symmetric rate performance of cache-aided setups at finite-
SNR regimes for the same network parameters considered in Section VII-A,
assuming the number of antennas at the transmitter is $L=10$. Fig. 5
demonstrates the average symmetric rate versus SNR for both transmission
strategies _A_ and _B_ , for $\alpha\in\\{8,10\\}$. The results are calculated
for 60 random user-to-profile associations. Upon observation, it is evident
that our proposed scheme yields a significant enhancement under both Strategy
A with
$Q={\bar{t}}+\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$ and
Strategy B with
$Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil$ in
comparison to the no-CC scheme. This enhanced performance is attributed to our
scheme’s utilization of both the global caching and the spatial multiplexing
gains.

As observed, when $\alpha=L=10$, Strategy B tends to surpass Strategy A at
high-SNR (e.g., 50dB). This is because Strategy A attains lower DoF compared
to Strategy B, i.e., it serves a smaller number of users per transmission in
comparison to Strategy B. However, maximizing the number of users served in
each transmission also causes Strategy B to keep the system fully loaded in
many transmission intervals, which is not an appropriate choice at finite SNR
as discussed in [6]. Therefore, as can be seen in Fig. 5, by setting
$\alpha=L$, the performance of Strategy B is inferior to Strategy A for
smaller SNR values. On the other hand, when $\alpha=L-2=8$, the system will
never be fully loaded even with _Strategy B_ , and hence, it outperforms
_Strategy A_ throughout the entire SNR range. Nevertheless, in both
comparisons, the difference between the two strategies is small, while
Strategy B poses a higher subpacketization complexity on the network.
Therefore, for real-world applications where the goal is to implement a cache-
aided dynamic setup with low complexity, Strategy A could be safely chosen to
serve users, albeit with a negligible loss in symmetric rate at finite-SNR
regimes.

$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$22$$24$$26$$28$$30$$32$$34$$36$$38$$40$$42$$44$$46$$48$$50$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$5.5$$6$$6.5$$7$$7.5$$8$SNR
[dB]Symmetric Rate [bits/s/Hz]_Strategy A_ : $\hat{\eta}=7$,
$\mathrm{DoF}=12$_Strategy B_ : $\hat{\eta}=7$, $\mathrm{DoF}=12.86$_Strategy
A_ : $\hat{\eta}=6$, $\mathrm{DoF}=8.57$_Strategy B_ : $\hat{\eta}=6$,
$\mathrm{DoF}=9.55$_Strategy A_ : $\hat{\eta}=3$, $\mathrm{DoF}=8.44$_Strategy
B_ : $\hat{\eta}=3$, $\mathrm{DoF}=9.26$no-CC: $\mathrm{DoF}=8$ Figure 6:
Symmetric rate versus SNR in a setup with $K=30$, $\sigma=0.63$, $\eta_{1}=7$,
$\alpha=L-2=8$, ${\bar{t}}=1$, $P=5$, $\gamma=0.2$ for different $\hat{\eta}$
values.
$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$22$$24$$26$$28$$30$$32$$34$$36$$38$$40$$42$$44$$46$$48$$50$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$4.5$$5$$5.5$$6$$6.5$$7$$7.5$$8$SNR
[dB]Symmetric Rate [bits/s/Hz]_Strategy A_ : $\hat{\eta}=9$,
$\mathrm{DoF}=11.43$_Strategy A_ : $\hat{\eta}=6$,
$\mathrm{DoF}=8.57$_Strategy B_ : $\hat{\eta}=6$, $\mathrm{DoF}=9.55$_Strategy
A_ : $\hat{\eta}=3$, $\mathrm{DoF}=8.44$_Strategy B_ : $\hat{\eta}=3$,
$\mathrm{DoF}=9.26$no-CC: $\mathrm{DoF}=8$ Figure 7: Symmetric rate versus SNR
in a network with $K=30$, $\sigma=2.19$, $\eta_{1}=9$, $\alpha=L-2=8$,
${\bar{t}}=1$, $P=5$, $\gamma=0.2$ for different $\hat{\eta}$ values.

Figs. 6 and 7 illustrate the symmetric rate of the considered network for
different $\hat{\eta}$ values, when $\alpha=8$. In Fig. 6, it is assumed that
$\sigma=0.63$ and $\eta_{1}=7$, while in Fig. 7, we have $\sigma=2.19$ and
$\eta_{1}=9$. As demonstrated, for both values of $\sigma$, the maximum
symmetric rate is attained with $\hat{\eta}=\eta_{1}=\max_{p}\eta_{p}$. This
is because setting $\hat{\eta}=\max_{p}\eta_{p}$ increases the number of users
served during the CC delivery step and maximizes the benefit from the
achievable global caching gain. Moreover, even though there is a notable
discrepancy in non-uniformness of the two cases, it is demonstrated that when
$\hat{\eta}=\max_{p}\eta_{p}$, the rate loss due to increased non-uniformness
in Fig. 7 becomes negligible in comparison to Fig. 6, particularly in the
finite-SNR regime. This underscores the effectiveness of our approach to
address the non-uniformness in the user-to-profile association by optimizing
the number of users served per transmission.

From Figs. 6 and 7, it can also be seen that when $\hat{\eta}$ is
significantly smaller than $\max_{p}\eta_{p}$, both the achievable DoF and the
symmetric rate become almost independent of the $\sigma$ value. For instance,
considering $\hat{\eta}=3$, Strategy A achieves exactly the same DoF and rate
values in both Fig. 6 and Fig. 7. This is because if $\hat{\eta}$ is small
enough, the user-to-profile association of the users served in the CC delivery
step becomes uniform, and the transmission vectors will be the same regardless
of the $\sigma$ value. Moreover, a small increase in $\hat{\eta}$ results in a
significant boost in the symmetric rate even though the DoF changes slightly.
For instance, when $\hat{\eta}=6$, Strategy A still delivers a modest boost in
DoF performance in comparison with the no-CC scheme ($7.13\%$). However, it
significantly elevates the symmetric rate over the no-CC scheme for both high-
and low-SNR scenarios ($18\%$ for 40dB and $27\%$ for 5dB). The reason for
this behavior is that increasing $\hat{\eta}$ improves the achievable global
caching gain, which means that fewer interference terms are suppressed by
beamformers, and hence, we can design more directed beams that are beneficial
in power-limited finite-SNR communications.

Table I shows how choosing the parameter $P$ affects the performance and
complexity of the proposed scheme. In this regard, we consider the same
network of $K=30$ users with $\gamma=0.2$ and $L=10$, assume uniform user-to-
profile association and $\alpha=L-1=9$, and compare the system performance
when $P\in\\{5,10\\}$ and either _Strategy A_ or _Strategy B_ is selected for
delivery phase (note that when $P=10$, _Strategy B_ is not applicable because
$\nicefrac{{\alpha}}{{\hat{\eta}}}$ is an integer). As observed, the maximum
achievable DoF and symmetric rate are almost independent of $P$. However,
setting $P=10$ significantly increases the subpacketization complexity and the
total number of transmissions compared to $P=5$. On the other hand, for $P=5$,
it is demonstrated that employing Strategy A achieves a higher symmetric rate
at finite-SNR scenarios, e.g., 20dB, compared to Strategy B with less
subpacketization complexity. Hence, in real-world scenarios with
$\alpha>\eta_{1}$ and a fixed $\gamma$ value, in order to lower the complexity
of the cache-aided setups and achieve a superior finite-SNR rate, it is better
to choose the smallest possible $P$, i.e., to find $P$ and ${\bar{t}}$ such
that $\mathrm{gcd}(P,{\bar{t}})=1$ and ${\bar{t}}=P\gamma$, then, use Strategy
A for the data delivery during the CC delivery step.

TABLE I: Impacts of $P$ on the System Performance | $P=5$ | $P=10$
---|---|---
| Strategy A | Strategy B | Strategy A
| $Q=2$ | $Q=3$ | $Q=5$
Maximum Achievable DoF | $12$ | $15$ | $15$
Subpacketization | $30$ | $225$ | $2835$
Number of Transmissions | $60$ | $360$ | $4536$
Rate at $20$dB (bits/s/Hz) | $2.5936$ | $2.2735$ | $2.2510$

##  VIII Conclusion

The dynamic nature of user behavior in practical applications poses a
challenge for implementing coded caching schemes, which can be partially
alleviated using a shared caching model. The conventional schemes in the
literature, however, lack true adaptability as they are limited by specific
network parameters. In this paper, we introduced a universally applicable
shared-caching scheme for dynamic setups with no restrictions on network
parameters. To this end, the placement phase was designed solely based on the
users’ cache ratio rather than the user count, and two transmission strategies
were proposed to facilitate the data delivery during the delivery phase.
Moreover, closed-form expressions were derived for the achievable DoF,
demonstrating that our scheme achieved the optimal bounds of the shared-cache
model. Additionally, we proposed a SIC-free data delivery algorithm and
designed fast iterative beamformers to optimize the finite-SNR performance.
Notably, it was shown that for uneven user distributions, although the
achievable DoF may fall from the optimal DoF of shared caching, our scheme
provided a significant performance improvement over unicasting at finite SNR
regimes.

##  Appendix A Proof of Remark 1

As stated earlier, each transmission vector is constructed to serve at most
$Q\beta$ users. Hence, each of these $Q\beta$ users is interfered by
$Q\beta-1$ interfering users. Since each user has stored a $\gamma$ portion of
the entire library, it can remove at most $\beta P\gamma=\beta{\bar{t}}$
interference terms using the stored cache content. Therefore, $\alpha$ must be
large enough to suppress the remaining $Q\beta-1-{\bar{t}}\beta$ interference
terms, i.e.,

$\alpha-1\geq Q\beta-1-{\bar{t}}\beta,$ (25)

which results in
$Q\leq{\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\beta}}\right\rceil$.
Furthermore, $Q\geq{\bar{t}}+1$ should hold to benefit from the maximum coded
caching gain.

Let us assume that profile $p$ is one of the $Q$ selected profiles for the
transmission. Therefore, $\min\left(\beta,\delta_{p}\right)$ users are served
from profile $p$ during this transmission. On the one hand, it is supposed
that the maximum number of users of each profile served in the CC delivery
step is $\hat{\eta}$, which results in having $\beta\leq\hat{\eta}$. On the
other hand, for each profile $p\in\left[P\right]$, we should have
$\min\left(\beta,\delta_{p}\right)\leq\alpha$ to be able to cancel out the
inter-stream interference between users assigned to the same profile. Hence,
according to the fact that $\delta_{1}=\hat{\eta}$, one can say
$\min\left(\beta,\delta_{1}\right)=\min\left(\beta,\hat{\eta}\right)=\beta\leq\alpha$,
which results in having the necessary condition
$\beta\leq\min\left(\alpha,\hat{\eta}\right)$ to select the parameter $\beta$.

##  Appendix B Proof of Theorem 1

Here, first, we present a lemma to simplify the calculations, then, prove the
theorem for strategies A and B in Sections B-A and B-B, respectively.

###### Lemma 1

Given $P$ and $Q$, we have
$\sum\limits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}=\binom{P}{Q}$.

###### Proof:

Assume a scenario at which we have a set
${\mathcal{X}}=\left\\{b_{1},\cdots,b_{P}\right\\}$ with
$\left|{\mathcal{X}}\right|=P$, and we want to select the subsets
$\bar{{\mathcal{X}}}\subseteq{\mathcal{X}}$ such that
$\left|\bar{{\mathcal{X}}}\right|=Q$. Here, our aim is to count the
possibilities for the subsets $\bar{{\mathcal{X}}}$. One way to count these
possibilities is to simply pick the subsets $\bar{{\mathcal{X}}}$ from
${\mathcal{X}}$, and then count them. Here, clearly, the number of
possibilities is equal to $\binom{P}{Q}$. Another way to choose
$\bar{{\mathcal{X}}}$ is to define $\left(P-Q+1\right)$ subsets
${\mathcal{L}}_{r}$ first, such that $r\in\left[P-Q+1\right]$,
${\mathcal{L}}_{r}=\left\\{b_{i}:r+1\leq i\leq P\right\\}$ and
$\left|{\mathcal{L}}_{r}\right|=P-r$. Then, the selection of
$\bar{{\mathcal{X}}}$ consists of $\left(P-Q+1\right)$ rounds. In round
$r\in\left[P-Q+1\right]$, we pick $b_{r}$ and choose $\left(Q-1\right)$
remaining elements from ${\mathcal{L}}_{r}$. In this regard, the number of
possibilities to choose $\bar{{\mathcal{X}}}$ in round $r$ is equal to
$\binom{P-r}{Q-1}$. By following this approach, we have
$\sum_{r=1}^{P-Q+1}\binom{P-r}{Q-1}$ possibilities to choose
$\bar{{\mathcal{X}}}$. Hence, it is sure that
$\sum_{r=1}^{P-Q+1}\binom{P-r}{Q-1}=\binom{P}{Q}$. ∎

###  B-A Proof of Theorem 1 for Strategy A

In order to prove the theorem, first, we compute the number of transmission
triples in Strategy A that serve each user during the CC delivery step.
Suppose that user $k$ with $p\left[k\right]=p$ is present in the CC delivery
step. As per (4) and (5), user $k$ is served in the transmission triple
$\left(r,c,l\right)$ such that $r\in\left[p\right]$. On the other hand,
according to (2) and (3), for each $r\in[p]$, there are $\beta$ values for $c$
that user $k$ is served in the transmission triple $\left(r,c,l\right)$. Now,
for any $r\in[p]$, let ${\mathcal{A}}_{k}^{(r)}$ be the set of values for $c$
at which user $k$ is served in the transmission triple $\left(r,c,l\right)$
such that $\left|{\mathcal{A}}_{k}^{(r)}\right|=\beta$. As mentioned in
Section III-A, for $r=p$ and $c\in{\mathcal{A}}_{k}^{(p)}$, there are
$\binom{P-p}{Q-1}$ values for $l$ at which user $k$ is served during the
transmission triple $\left(r,c,l\right)$. Furthermore, for
$r\in\left[p-1\right]$ and $c\in{\mathcal{A}}_{k}^{(r)}$, there exist
$\binom{P-r-1}{Q-2}$ transmission triples in which user $k$ receives the
subpackets. Hence, the number of transmission triples serving user $k$ is
given by:

$\begin{array}[]{c}\beta\left(\binom{P-p}{Q-1}+\sum\nolimits_{r=1}^{p-1}\binom{P-r-1}{Q-2}\right)=\beta\binom{P-1}{Q-1}.\end{array}$
(26)

As stated in Section IV, by using the SIC-free data delivery, each
transmission triple $(r,c,l)$ is split into $\nu_{2}$ transmissions
$(r,c,l)^{j}$ with $j\in[\nu_{2}]$, such that in each transmission
$(r,c,l)^{j}$, user $k\in{\mathcal{T}}_{r,c,l}$ receives only one subpacket of
its requested file. Therefore, using (26), the total number of subpackets
received by each user via _Strategy A_ is $\beta\binom{P-1}{Q-1}\nu_{2}$. Now,
according to that $K_{M}$ users are served via Strategy A, the total number of
transmitted subpackets in the CC delivery step is equal to:

$\begin{array}[]{c}J_{M}=K_{M}\beta\binom{P-1}{Q-1}\nu_{2}.\end{array}$ (27)

Next, we show that all users can decode their requested files with Strategy A.
User $k$ stores a $\gamma$ portion of the entire library in the placement
phase and needs to receive
$\left(1-\gamma\right)\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$
subpackets in the CC delivery step. As mentioned, user $k$ receives
$\beta\binom{P-1}{Q-1}\nu_{2}=\left(1-\gamma\right)\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$
subpackets via _Strategy A_ , demonstrating that user $k$ receives all its
missing subpackets in the CC delivery step.

Now, we calculate the total number of transmissions in the CC delivery step.
As per (5), if $\delta_{r}>0$, for any $r\in\left[P-Q+1\right]$ and
$c\in\left[\phi_{r}\right]$, users assigned to the set
${\mathcal{T}}_{r,c,l}={\mathcal{S}}_{r,c}\|{\mathcal{S}}_{b_{1},c}\|\cdots\|{\mathcal{S}}_{b_{Q-1},c}$
are served, where
$\left\\{b_{1},\cdots,b_{Q-1}\right\\}={\mathcal{M}}_{r}\left(l\right)$ and
$l\in\left[\binom{P-r}{Q-1}\right]$. So, assuming $\delta_{r}>0$, the total
number of transmission triples for any $r\in\left[P-Q+1\right]$ is equal to
$\phi_{r}\binom{P-r}{Q-1}$. Furthermore, as mentioned in Section IV, by
employing the SIC-free data delivery, each transmission triple $(r,c,l)$ is
divided into $\nu_{2}$ transmissions $(r,c,l)^{j}$ with $j\in[\nu_{2}]$. As a
result, the total number of transmissions of Strategy A is obtained as:

$\begin{array}[]{c}T_{M}=\sum\nolimits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}\nu_{2}.\end{array}$
(28)

In the next step, we compute the number of transmitted subpackets during the
UC delivery step. By following a similar way as in [18], each user served with
UC delivery step must receive
$\left(1-\gamma\right)\beta\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}$
subpackets. Since there are $K_{U}$ users in the UC delivery step, the number
of transmitted subpackets during the UC delivery step is

$\begin{array}[]{c}J_{U}=K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}.\end{array}$
(29)

Then, we obtain the total number of transmissions during the UC delivery step.
As stated earlier, during this step, we have to deliver
$K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}$ subpackets via
$T_{U}$ transmissions, such that in each transmission,
$\min\left(K_{U},\alpha\right)$ subpackets are transmitted. So, given
$K_{U}\neq 0$, the total number of transmissions to deliver these subpackets
is equal to:

$\begin{array}[]{c}T_{U}=\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil.\end{array}$
(30)

Now, for $K_{U}\neq 0$, by substituting (27)-(30) into (17), the DoF is
obtained as follows.

$\begin{array}[]{c}\mathrm{DoF}=\frac{K_{M}\binom{P-1}{Q-1}\beta\nu_{2}+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\sum\limits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}\nu_{2}+\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\beta^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil}.\end{array}$
(31)

Moreover, for $K_{U}=0$, we have $K_{M}=K$, and by applying (27) and (28) into
(17), the DoF takes the form as:

$\begin{array}[]{c}\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\beta}{\sum\nolimits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}}.\end{array}$
(32)

###  B-B Proof of Theorem 1 for Strategy B

In order to obtain the DoF for the dynamic MISO network with Strategy B
assisted by the SIC-free data delivery algorithm, first, we count the number
of transmitted subpackets during the CC delivery step. In this regard, assume
that user $k$ with $p\left[k\right]=p$ is present in the CC delivery step and
receives its subpackets with Strategy B. According to (10), (12) and (13),
when $c\in\left[P-Q+1\right]$, $l\in\left[\binom{P-c-1}{Q-2}\right]$,
$m\in\left[\theta\right]$ and $s\in\left[\nu_{2}\right]$ , user $k$ is served
during the transmission quintuples $\left(p,c,l,m,s\right)$. Hence, for $r=p$,
the number of transmission quintuples $\left(r,c,l,m,s\right)$ serving user
$k$ is given by:

$\begin{array}[]{c}\sum\nolimits_{c=1}^{P-Q+1}\binom{P-c-1}{Q-2}\theta\nu_{2},\end{array}$
(33)

which by using Lemma 1 is simplified to:

$\begin{array}[]{c}\sum\nolimits_{c=1}^{P-Q+1}\binom{P-c-1}{Q-2}\theta\nu_{2}=\theta\nu_{2}\binom{P-1}{Q-1}.\end{array}$
(34)

Furthermore, as per (10), (12) and (13), when $r\\!\neq\\!p$,
$c\\!\in\\!\left[P\\!-\\!Q\\!+\\!1\right]$,
$l\\!\in\\!\left[\binom{P-c-2}{Q-3}\right]$,
$m\\!\in\\!\left[\hat{\eta}\right]$ and $s\\!\in\\!\left[\nu_{2}\right]$, the
transmission quintuples $\left(r,c,l,m,s\right)$ serve user $k$. Therefore,
the number of transmission quintuples $\left(r,c,l,m,s\right)$ serving user
$k$ with $r\neq p$ is obtained as:

$\begin{array}[]{c}\sum\nolimits_{r=1}^{P-1}\sum\nolimits_{c=1}^{P-Q+1}\binom{P-c-2}{Q-3}\hat{\eta}\nu_{2},\end{array}$
(35)

which by using Lemma 1 takes the form as follows.

$\begin{array}[]{c}\sum\nolimits_{r=1}^{P-1}\sum\nolimits_{c=1}^{P-Q+1}\binom{P-c-2}{Q-3}\hat{\eta}\nu_{2}=\hat{\eta}\nu_{2}\left(P-1\right)\binom{P-2}{Q-2}.\end{array}$
(36)

As a result, by using (34) and (36), and according to that
$\theta=\alpha-\hat{\eta}\left\lfloor\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rfloor$,
the number of transmission quintuples serving user $k$ is equal to:

$\begin{array}[]{c}\theta\nu_{2}\binom{P-1}{Q-1}+\hat{\eta}\nu_{2}\left(P-1\right)\binom{P-2}{Q-2}=\binom{P-1}{Q-1}\nu_{2}\left(\hat{\eta}{\bar{t}}+\alpha\right).\end{array}$
(37)

Moreover, as discussed in Section IV, each transmission quintuple
$(r,c,l,m,s)$ is split into $\nu_{1}$ transmissions $(r,c,l,m,s)^{j}$ with
$j\in[\nu_{1}]$, where each transmission $(r,c,l,m,s)^{j}$ serves the user
$k\in\tilde{{\mathcal{T}}}_{r,c,l}^{m,s}$. Therefore, by using (37), the total
number of subpackets that each user $k\in\tilde{{\mathcal{T}}}_{r,c,l}^{m,s}$
receives during the CC delivery step is
$\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}\nu_{1}$. Now,
since $K_{M}$ users are served via Strategy B, the total number of transmitted
subpackets in the CC delivery step is given by:

$\begin{array}[]{c}J_{M}=K_{M}\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}\nu_{1}.\end{array}$
(38)

Next, we show that user $k$ receives all its subpackets during the CC delivery
step with Strategy B. Since a $\gamma$ portion of the entire library is stored
in the cache memory of user $k$, the total number of subpackets that must be
transmitted to user $k$ during the CC delivery step is
$\left(1-\gamma\right)\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{1}$.
As stated earlier, the total number of subpackets that user $k$ receives with
Strategy B is
$\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}\nu_{1}=\left(1-\gamma\right)\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{1}$,
showing that each user receives all its missing subpackets via Strategy B.

In the proceeding, we count the total number of transmissions with Strategy B.
As stated in Section III-B, if
$I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right)=1$, the server
delivers the subpackets to the users during the transmission quintuple
$\left(r,c,l,m,s\right)$; otherwise, it does not transmit any signal. In
addition, as per Section IV, each transmission quintuple $(r,c,l,m,s)$ is
divided into $\nu_{1}$ transmissions $(r,c,l,m,s)^{j}$ with $j\in[\nu_{1}]$.
Now, by following a similar approach to (28), since $j\in[\nu_{1}]$,
$r\in[P]$, $c\in\left[P-Q+1\right]$, $l\in\left[\binom{P-c-1}{Q-2}\right]$,
$m\in\left[\hat{\eta}\right]$ and $s\in\left[\nu_{2}\right]$, the total number
of transmissions in Strategy B is given by:

$\begin{array}[]{c}T_{M}=\sum\limits_{j=1}^{\nu_{1}}\sum\limits_{r=1}^{P}\sum\limits_{c=1}^{P-Q+1}\sum\limits_{m=1}^{\hat{\eta}}\sum\limits_{s=1}^{\nu_{2}}\binom{P-c-1}{Q-2}I^{+}\left(\bar{\delta}_{c},{\mathcal{E}}_{r}^{m}\right).\end{array}$
(39)

Now, in order to obtain $J_{U}$ and $T_{U}$, we follow the same way as in (29)
and (30). Accordingly, as stated in Section III-B, each file is split into
$\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\nu_{1}\left(\hat{\eta}{\bar{t}}+\alpha\right)$
subpackets. Therefore, according to the fact that the cache ratio at each user
is $\gamma$ and $K_{U}$ users are served in the UC delivery step, the total
number of transmitted subpackets during the UC delivery step is given by:

$\begin{array}[]{cl}J_{U}&=K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\binom{P-{\bar{t}}-1}{Q-{\bar{t}}-1}\nu_{1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\\\
&=K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}.\end{array}$
(40)

Furthermore, as mentioned in Section III-C, assuming $K_{U}\neq 0$, each
transmission of the UC delivery step serves $\min\left(K_{U},\alpha\right)$
users. Hence, the total number of transmissions in the UC delivery step is
obtained as follows.

$\begin{array}[]{l}T_{U}=\left\lceil\frac{K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{\min\left(K_{U},\alpha\right)}\right\rceil.\end{array}$
(41)

As a result, for $K_{U}\neq 0$, by applying (38)-(41) into (17), the DoF is
obtained as follows.

$\begin{array}[]{c}\mathrm{DoF}=\frac{K_{M}\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}\nu_{1}+K_{U}\left(1-\gamma\right)\binom{P}{{\bar{t}}}\alpha^{\prime}}{N_{M}\nu_{1}+N_{U}}.\end{array}$
(42)

In addition, when $K_{U}=0$ and $K_{M}=K$, by substituting (38) and (39) into
(17), the DoF is simplified to:

$\begin{array}[]{c}\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}}{N_{M}}.\end{array}$
(43)

##  Appendix C Proof of Remark 2

When $K$ users are uniformly assigned to $P$ caching profiles, i.e.,
$K=P\hat{\eta}$, and all users are served via CC delivery step, i.e.,
$K_{M}=K$ and $K_{U}=0$, we prove Remark 2 for three regimes: 1)
$\alpha\leq\hat{\eta}$, 2) $\alpha>\hat{\eta}$ with integer
$\frac{\alpha}{\hat{\eta}}$ and 3) $\alpha>\hat{\eta}$ with non-integer
$\nicefrac{{\alpha}}{{\hat{\eta}}}$.

1) $\alpha\leq\hat{\eta}$: In this regime, the system operates with Strategy A
during the CC delivery step, where $\beta=\alpha$ and $Q={\bar{t}}+1$. By
considering $\hat{\eta}=\eta_{r}$ for $\forall r\in[P]$, we know that
$D\left(\delta_{r}\right)=\hat{\eta}$. Next. we reformulate (18) as follows.

$\begin{array}[]{c}\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\alpha}{\hat{\eta}\sum\nolimits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}}.\end{array}$
(44)

Now, using Lemma 1 and setting $Q={\bar{t}}+1$, the DoF in (44) is simplified
to:

$\begin{array}[]{c}\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\alpha}{\hat{\eta}\binom{P}{Q}}=\frac{KQ\alpha}{P\hat{\eta}}=\alpha\left(P\gamma+1\right).\end{array}$
(45)

2) $\alpha>\hat{\eta}$ and integer $\nicefrac{{\alpha}}{{\hat{\eta}}}$: In
this regime, we know that the system uses Strategy A to deliver data during
the CC delivery step, where $\beta=\hat{\eta}$ and ${\bar{t}}+1\leq
Q\leq{\bar{t}}+\nicefrac{{\alpha}}{{\hat{\eta}}}$. Here, in order to maximize
the achievable DoF, we set $Q={\bar{t}}+\nicefrac{{\alpha}}{{\hat{\eta}}}$.
Then, similarly to the previous case, we consider $\hat{\eta}=\eta_{r}$ and
$D\left(\delta_{r}\right)=\hat{\eta}$ for all $r\in[P]$. So, the DoF in (18)
is given by:

$\begin{array}[]{c}\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\hat{\eta}}{\hat{\eta}\sum\nolimits_{r=1}^{P-Q+1}\binom{P-r}{Q-1}}.\end{array}$
(46)

Finally, by using Lemma 1 and (46), the maximum achievable DoF corresponding
to this regime is as follows.

$\begin{array}[]{c}\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}}{\binom{P}{Q}}=\frac{KQ}{P}=\hat{\eta}{\bar{t}}+\alpha=K\gamma+\alpha.\end{array}$
(47)

3) $\alpha>\hat{\eta}$ with non-integer $\nicefrac{{\alpha}}{{\hat{\eta}}}$:
In this regime, in order to maximize the achievable DoF, the server can
transmit data via Strategy B during the CC delivery step, such that
$\hat{\eta}=\eta_{r}$ for all $r\in[P]$, $\beta=\hat{\eta}$ and
$Q={\bar{t}}+\left\lceil\nicefrac{{\alpha}}{{\hat{\eta}}}\right\rceil$.
Accordingly, we have
$I^{+}\left(\bar{\delta_{c}},{\mathcal{E}}_{r}^{m}\right)=1$ for all
$r\in[P]$, $c\in\left[P-Q+1\right]$, $l\in\left[\binom{P-c-1}{Q-2}\right]$,
$m\in\left[\hat{\eta}\right]$ and $s\in\left[\nu_{2}\right]$. Hence, by using
Lemma 1 and (20), $N_{M}$ is simplified to:

$\begin{array}[]{c}N_{M}=P\hat{\eta}\binom{P-1}{Q-1}\nu_{2}.\end{array}$ (48)

Finally, by substituting (48) into (43) and using ${\bar{t}}=P\gamma$, the DoF
takes the form of:

$\begin{array}[]{c}\mathrm{DoF}=\frac{K\binom{P-1}{Q-1}\left(\hat{\eta}{\bar{t}}+\alpha\right)\nu_{2}}{P\hat{\eta}\binom{P-1}{Q-1}\nu_{2}}=\hat{\eta}{\bar{t}}+\alpha=K\gamma+\alpha.\end{array}$

##  Appendix D Proof of Remark 3

For $Q={\bar{t}}+1$ and $\eta_{1}\leq\alpha$, the system operates with
Strategy A during the CC delivery step, $\nu_{2}=1$ and
$\beta^{\prime}=\beta$, where $\beta=\alpha$, if $\alpha\leq\hat{\eta}$;
otherwise, $\beta=\hat{\eta}$. Assuming $\hat{\eta}<\eta_{1}$ (i.e. $K_{U}>0$)
and using (30), clearly, it is sure that
$\binom{P}{Q}+\frac{1}{\beta}T_{U}>\binom{P}{Q}$. So, by using Lemma 1, we
have:

$\begin{array}[]{c}\frac{1}{\beta}\sum\limits_{r=1}^{P-Q+1}\beta\binom{P-r}{Q-1}+\frac{1}{\beta}T_{U}>\frac{1}{\eta_{1}}\sum\limits_{r=1}^{P-Q+1}\eta_{1}\binom{P-r}{Q-1}.\end{array}$
(49)

Now, according to the fact that for any arbitrary parameters $w$, $x$, $y$ and
$z$, if $w+x\geq y$, then $\max\left(w,z\right)+x\geq y$, we rearrange (49) as
follows.

$\begin{array}[]{l}\frac{1}{\beta}\\!\sum\limits_{r=1}^{P-Q+1}\\!\max\left(\beta,\delta_{r}\right)\binom{P-r}{Q-1}\\!+\\!\frac{T_{U}}{\beta}\\!>\\!\frac{1}{\eta_{1}}\\!\sum\limits_{r=1}^{P-Q+1}\\!\eta_{1}\binom{P-r}{Q-1}.\end{array}$
(50)

We know that $D\left(\delta_{r}\right)=D\left(\eta_{r}\right)=0$ if
$\eta_{r}=0$; otherwise,
$D\left(\delta_{r}\right)=\max\left(\beta,\delta_{r}\right)$ and
$D\left(\eta_{r}\right)=\max\left(\eta_{r},\eta_{1}\right)=\eta_{1}$. Hence,
we reformulate (50) as follows.

$\begin{array}[]{c}\frac{K\binom{P-1}{Q-1}\beta}{\sum\nolimits_{r=1}^{P-Q+1}D\left(\delta_{r}\right)\binom{P-r}{Q-1}+T_{U}}<\frac{K\binom{P-1}{Q-1}\eta_{1}}{\sum\nolimits_{r=1}^{P-Q+1}D\left(\eta_{r}\right)\binom{P-r}{Q-1}}.\end{array}$
(51)

Furthermore, using $\beta^{\prime}=\beta$, $Q={\bar{t}}+1$, $\nu_{2}=1$, (27)
and (29), we have:

$\begin{array}[]{c}J_{M}+J_{U}=K\binom{P-1}{Q-1}\beta.\end{array}$ (52)

Using (52) and (18), and according to that $\nu_{2}=1$, the left side of (51)
demonstrates the DoF of a setup with $Q={\bar{t}}+1$ and $\hat{\eta}<\eta_{1}$
(i.e., $K_{U}\neq 0$), while the right side corresponds to the DoF with
$Q={\bar{t}}+1$ and $\hat{\eta}=\eta_{1}$ (i.e., $K_{U}=0$). Hence, for
$Q={\bar{t}}+1$ and $\eta_{1}\leq\alpha$, we should set $\hat{\eta}=\eta_{1}$
to maximize the achievable DoF.

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