Cache-Aided Communications in MISO Networks with Dynamic User Behavior

—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.


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]- [3].To facilitate the efficient delivery of such multimedia content, coded caching (CC) has been proposed to increase the data rates and reduce the communication load during peak traffic times [4].A large amount of memory is available in modern network devices.The CC models provide the opportunity to employ this distributed memory as a supplementary communication resource, enabling them to improve the scalability, latency, and costeffectiveness in various real-world applications such as largescale video-on-demand (VoD) [5], extended reality (XR) [2], [3], and coded distributed computing [6].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 γ of the entire library, CC can enable the so-called global caching gain of Kγ + 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 α ≤ L at the transmitter, the delivery speed-up factor of CC is increased to Kγ + α [7]- [13], appealing given the critical role of multiantenna connectivity in next-generation networks [14].This so-called combined degree-of-freedom (DoF) of Kγ + α 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 [15]- [18].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 [19]- [21].We use the existing shared-cache model to develop a generalized lowcomplexity 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.

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 [22]- [28].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 γ = 0.01 and K = 1000 users, enabling the caching gain of Kγ = 10 requires each file to be split into > 10 23 mini-files, which is not possible in practice [29].While the subpacketization complexity issue is inherent in single-antenna scenarios [30], 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 [29]. 1 A systematic extension of the signal-level scheme in [29] was later provided as the shared-cache model [32], where there exist P ≤ K caching profiles, and η p users are assigned to profile p ∈ {1, • • • , P }.Even though multiple users with a cache ratio of γ 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 [32], [33].More precisely, for small-antenna arrays with α ≤ K P [32] and large-antenna arrays with α > K P [33], the shared-cache model enables the optimal DoF of α(1 + P γ) and Kγ + α, 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.The authors in [34], [35] have applied the maximum distance separable (MDS) codes in the content placement phase to manage the networks with dynamic conditions and have demonstrated that using MDS coding not only handles the dynamicity of networks but also can guarantee the privacy of users.However, implementation of MDS coding in the placement phase requires having a very large finite field.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 [19]- [21] applied the shared caching idea to address the dynamic user behavior by utilizing the users' cache ratio γ to design the content placement phase.In this sense, for a given γ value, they selected an appropriate number of caching profiles P ensuring that P γ 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, [32] is applicable only if the spatial multiplexing gain is very small, i.e., α ≤ min p η p , while [19] and [20] can only support the large-antenna array networks with α > η, given that α ≥ Kγ and 1 ≤ η ≤ max p η p .Accordingly, no existing scheme in the literature can handle network dynamicity properly in the following cases: I) smallantenna array: α ≤ η, with α ≥ min p η p , and II) large-antenna array: α > η, either with non-integer α /η or with α < Kγ.Therefore, a universal shared-cache setup that supports any user-to-profile association is not yet available.The primary objective of this paper is to design a transmission strategy that not only supports the regions already covered but also supports the regions so far omitted in the literature.Moreover, this strategy should minimize the DoF loss in the dynamic networks, which is caused by the non-uniform association of users to the caching profiles.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 [9], [36].Nevertheless, the DoF metric is an appropriate performance indicator only at the high-SNR regime [37], and ZF beamformers should also be replaced with optimized beamformers in finite-SNR communications [38].Accordingly, the work [8] 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 [39] 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 [40], [41].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 [42].
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 [37].To this end, the early results in [19] 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.

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: • 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 α > η and non-integer α η (unlike [33]), ii) min p η p ≤ α ≤ η (unlike [32]), and iii) α > η with α < Kγ (unlike [20]).We note that the proposed shared caching model can be applied to any baseline CC scheme, as long as the constraints of the baseline scheme are satisfied.For instance, for networks with the number of files N exceeding the number of users K, applying the proposed shared caching model to the introduced YMA scheme in [43] can outperform the baseline scheme [4], as YMA removes the linearly dependent multicast messages that may occur when multiple users request the same file.
• 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 α ≥ η and α /η is non-integer.
• In contrast to the approaches outlined in [32], [33], 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 α, 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:

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 α ≤ L serves several cacheenabled single-antenna users.The BS has access to a library F with N unit-sized files, and each user is equipped with a large enough memory to store a portion 0 < γ = M N < 1 of the entire library, where M is the number of files that can be stored in cache memory of each user.Accordingly, for a given value of cache ratio γ, we design the natural numbers P and t, such that t = P γ.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.In the placement phase, by following the same way as in [4], each file 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 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 x ∈ C L×1 containing a superposition of the precoded codewords such that each target user k receives the signal where h k ∈ C L×1 is the channel gain of user k and n k is the zero-mean additive white Gaussian noise with variance N 0 .

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 η p , where without loss of generality, it is assumed that η 1 ≥ η 2 ≥ • • • ≥ η P .By choosing a delivery parameter η ≤ max p η p , 3 the BS builds and transmits a set of codewords to serve at most η users assigned to each profile with a novel CC-based approach.In this regard, for every profile p • if η < η p , we exclude η p − η users, and exempt BS to serve these users during the CC delivery step.Accordingly, the excluded users are served in the UC delivery step.
• if η ≥ η p , all η p users are served via the CC delivery step.Now, let us suppose that the set of users assigned to profile p ∈ [P ] and served during the CC delivery step is denoted by We note that δ p = min(η, η p ), and clearly, δ 1 = η and 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 β, 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 β are defined in Remark 1.
Remark 1: In order to serve Q profiles each with a maximum of β users, the network parameters should satisfy the constraints t + 1 ≤ Q ≤ t + ⌈ α /β⌉ and β ≤ min(α, η).
Proof: The proof is relegated to Appendix A. It is worth mentioning that the key parameters of this work are classified into two categories: 'system' and 'design' parameters.The 'system' parameters rely on the physical characteristics and structure of the network.Moreover, the 'design' parameters are used to construct the transmission strategies and manage the resource allocation.Table I illustrates the 'system' and 'design' parameters, while the letters "S" and "D" represent the categories 'system' and 'design', respectively.We note that the 'system' and 'design' parameters are not completely independent, and some correlation exists between them.For instance, α is limited by L.
In the following, we present the system operation maximizing the DoF performance separately for two regimes: 1) α ≤ η and 2) α > η.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 α and η, and it is independent of the content placement phase.In other words, the server performs the placement phase only based on parameter γ without considering which transmission strategy the system will use for the CC delivery step.
1) System operation for α ≤ η: For the case of α ≤ η, we set β = α, and the only option for Q to maximize the DoF is Q = t+1.Here, Strategy A is utilized to build the transmission vectors.Replacing the constraint α ≤ η with α ≤ min p δ p reduces our system model to [32], while our proposed scheme also works for the scenarios with min p δ p < α ≤ η.
2) System operation for α > η: For this case, we set β = η, and define α = α η .If α is an integer, then we follow Strategy A to build the transmission vectors.For non-integer α η , the server can serve users via Strategy A by setting t + 1 ≤ Q ≤ t + ⌊ α /η⌋, and via Strategy B by choosing Q = t + ⌈ α /η⌉.In this case, if we assume that users are uniformly distributed among caching profiles, and α η is an integer, the system performance is simplified to [33], while our proposed scheme also covers the uneven user-to-profile associations with non-integer α η .

A. Transmission Strategy A
In this strategy, each mini-file , where q ∈ [S 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 β users.
Elevation process: In this process, the aim is to characterize the set of users that are served in each transmission.To this end, we utilize the sliding-window approach.Recall that the set of users assigned to profile p and served during the CC delivery step is denoted by V p with |V p | = δ p .This slidingwindow approach extends the set V p to η windows, such that each of the first φ p = max(β, δ p ) windows contains min(δ p , β) users from the set V p circularly shifted to the right, and the other η − φ p windows are the empty windows without any users.Accordingly, during each transmission, in order to serve the users assigned to profile p, the BS selects one of these η windows and serves users associated with that window.Here, we use the so-called transmission triple A = (r, c, l), where r In each transmission triple A = (r, c, l), Q profiles are selected, such that the first selected profile is the caching profile r, and the remaining Q − 1 profiles are chosen from the set [r + 1 : P ], and hence, there exist P −r Q−1 possibilities to choose these Q−1 profiles.The l-th possibility (out of P −r Q−1 possibilities) is represented by index l in the transmission triple A = (r, c, l).Generally speaking, the indices r and l determine the Q caching profiles that are selected in the transmission triple A = (r, c, l).The elevation process creates T A , the set of users that are served during the transmission triple A, which is characterized by concatenating the c-th window of each of the Q selected profiles.As mentioned earlier, it is assumed that In other words, if the c-th window of profile p is empty, then the c-th window of profile p + 1 is also empty.Hence, for  2, 1) and A 3 = (1, 3, 1).
fixed values of r and l, the total number of transmission tiples serving Q selected profiles is equal to φ r , as each of these Q profiles has only φ r non-empty windows.Following the elevation process, the BS constructs the transmission vector x A to serve users in the set T A during the transmission triple A.
In the proceeding, we present the mathematical representation of the elevation process.In this regard, first, for every p ∈ [P ], we elevate the set V p to the set R p as follows.
where, for j ∈ [φ p ], R p,j is defined as: Otherwise, R p,j is obtained by making a circular shift on V p and picking β elements from that.Now, for p ∈ [P ], we define As per (3), for each p ∈ [P ], S p consists of η sets (windows) S p,j , such that for the first φ p windows, i.e., j ≤ [φ p ], we have S p,j = R p,j , while the remaining η −φ r windows are the empty windows.Indeed, (3) equalizes the number of available windows associated with each profile to η.Then, for each r ∈ [P − Q + 1], we define the set M r as: In the proceeding, we use M r (l) to indicate the l-th (Q − 1)tuple of M r .Finally, the set T A for the transmission triple A = (r, c, l) is given by: ) In Fig. 3, an illustrative example for the elevation process to create the set of served users for a network with In this network, it is assumed that α = 2, η = 3, and therefore, we have During the transmission triple A = (r, c, l), by defining N = {r} ∪ M r (l), the BS constructs the transmission vector as follows.
where w Λ k ∈ C L×1 is the precoder that suppresses the interference of user k at the set G Λ k = j ∈ S p,c : ∀p ∈ N \Λ , j = k .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.Accordingly, at the end of the transmission triple A, user k receives a signal as described in (1).The following example shows the data delivery via Strategy A.
where R 1,1 = {1, 2, 3} and so on.Accordingly, using R 1 − R 3 , S 1 − S 3 are given by: In order to build the transmission vector for the transmission triple Therefore, the set of users that are served during the transmission triple A • = (1, 1, 1) is equal to: Finally, the transmission vector corresponding to the transmission triple A • = (1, 1, 1) is constructed as follows.
Now, in order to show the decoding process, for any k, j ∈ S p,c with p ∈ N \Λ , if j = k, using ZF precoders, we have . Now, the received signal at user 1 is given by: User 1 has all subpackets 5 and subtract it from y 1 , which results in observing the superposition signal Here, we note that user 1 needs to implement SIC to decode the subpackets [8], [32].

B. Transmission Strategy B
When α > η and α η is not an integer, we can serve users by setting β = η, and Q = t + ⌈ α /η⌉. 6Here, first, we split each mini-file , where q ∈ [S B ].Then, we use the elevation process to serve Q profiles each with at most β users.
Elevation process: This process builds the set of users served in each transmission.Similarly to Strategy A, this process uses the sliding-window approach to create the set of users served in each transmission.In this regard, the set of users assigned to profile r ∈ [P ] and served in the CC delivery step, described as V r , is extended to the set Y r , such that |Y r | = η.To this end, if |V r | = δ r < η, then, η − δ r phantom (non-existing) users are added to the set V r ; otherwise, Y r = V r .In other words, for r ∈ [P ], we define Y r as: where f * denotes the phantom users.Generally speaking, in each transmission of Strategy B, we serve the users assigned to Q profiles such that we select at most η users from Q − 1 = t + ⌊ α /η⌋ profiles, and pick θ = α − η⌊ α /η⌋ users from another profile, which results in serving ηt + α users 5 Similarly to [40], for each k ∈ S 2,1 S 3,1 , the term h H 1 w 1 k W k 1,1 is estimated by using downlink precoded pilots, leading user 1 to regenerate 6 For α > η and non-integer α η , we can still serve users with Strategy A. However, the achievable DoF is less than the one with Strategy B. per transmission.Here, we note that if K users are uniformly assigned to P caching profiles, i.e., η = K P , the number of served users per transmission is obtained as ηt + α = Kγ + α, which coincides with the optimal DoF in MISO CC schemes as described in [7].In this process, first, we select a profile r ∈ [P ], and then, pick θ users from Y r .To this end, for r ∈ [P ] and m ∈ [η], we consider the set E m r as: where Y r (i) is the i-th element of Y r .Indeed, E m r shifts Y r to the right for m times, and picks θ elements from it.This circular shift on Y r to create E m r is used to make the user selection independent of the users' indices.
Moreover, we assume that P r = [P ] \r for r ∈ [P ], and δc = P r (c) is the c-th element of P r .The caching profiles in P r are sorted in descending order such that if i < j, then δ Pr(i) ≥ δ Pr (j) .Next, for a given r and c ∈ [P − Q + 1], let I r c as: Furthermore, denote the l-th , the BS serves the users assigned to the set E m r ∪{Y p : p ∈ { δc }∪I r c (l)}.In order to create a transmission vector to serve these users, first, all subsets of { δc } ∪ I r c (l) with the cardinality of Q − t − 1 = ⌊ α /η⌋ is aggregated in the set B as follows r is assigned to ν 1 tuples of B to ensure that each user receives ν 1 subpackets per transmission.To this end, for any u ∈ E m r , we define the set K m,u r as follows As observed, ν 1 elements of K m,u r are equal to u, and the remaining ν 2 − ν 1 elements are the phantom users f * .Next, by considering K m,u r (n) as the n-th element of K m,u r , in order to create the transmission vectors, the element K m,u r (n) should be merged with B(n) for n ∈ [ν 2 ] and u ∈ E m r .However, in order to make this transmission strategy independent of user indexing, we add another round of circular shift on K m,u r , such that for any s ∈ [ν 2 ], the set represents s circular shifts of K m,u r to the right.Moreover, For the delivery process with Strategy B, we use the socalled transmission quintuple B = (r, c, l, m, s), where r ∈ x where is the precoder that suppresses the interference of user k at the set 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 B.
The following example studies the data delivery via Strategy B.
Example 2: With the given 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 n 1 , W n 2 and W n 3 , respectively.Then, during the content delivery phase, user k ∈ [K] requests the file W k , and each mini-file is split into (η t + α) 10 subpackets.Now, for the transmission quintuple B • = (1, 1, 1, 1, 1), we have: At the next step, for r = c = 1, it is found that P 1 = {2, 3}, P 1 (1) = 2, and I 1 1 = I 1 1 (1) = {3}.During this transmission, the users of the set E 1 1 = {1, 2}, and the users assigned to the set Here, we note that we use the generalized multiset definition, where the same element can appear multiple times in the set.Hence, after removing the phantom users f * , the transmission vector for the transmission quintuple B • = (1, 1, 1, 1, 1) takes the form of: where H 3 6 = {1, 2, 7, 8, 9}, H 2 10 = {1, 2, 11, 12} and so on.Similarly to Example 1, all users that are served via Strategy B are able to decode their requested files.

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 = P p=1 max(0, η p − η).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 [19], 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(α, K U ) 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 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 [37].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 A = (r, c, l), r

and represent the set of subpackets received by user k ∈ T A in transmission triple
. Now, to eliminate the SIC requirement, we split each transmission triple A into ν 2 transmissions, denoted by A j , j ∈ [ν 2 ], such that in transmission A j , the BS transmits the superposition signal where W k Λ ′ ,q is the j-th entry of O k A , and w Λ ′ k ∈ C L×1 is the beamforming vector suppressing the interference of subpacket W k Λ ′ ,q at the set G Λ ′ k , defined in (6).Note that, following ( 6) and ( 14), it is guaranteed that x A = ν2 j=1 x j A .Strategy B. Let us denote the set of users served during the transmission quintuple B = (r, c, l, m, s), r with TB , and represent the set of subpackets received by user k ∈ TB in transmission quintuple B = (r, c, l, m, s) with Õk B , i.e., To eliminate the SIC requirement, we split every transmission quintuple B into ν 1 transmissions, denoted by B j , j ∈ [ν 1 ], such that in transmission B j , the BS transmits the superposition signal where W k Λ,q denotes the j-th entry of Õk B and w Λ k ∈ C L×1 is the precoder suppressing the interference of subpacket W k Λ,q at every user in H Λ k , defined in (13).Clearly, according to ( 13) and ( 15), we have x B = ν1 j=1 x j B .Example 3: Let us focus on the cache-aided network considered in Example 1 and the transmission vector x A• in (7).For this transmission vector, the sets of subpackets received by users 1 and 2, for example, are given as 3,1 }, respectively.So, for SIC-free data delivery, x A• should be split into ν 2 = 2 transmission vectors where, for example, w 2 1 and w 3 1 suppresses the interference at the sets of users G 2 1 = {2, 3, 10, 11, 12} and G 3 1 = {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. 7In 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.
Furthermore, we suppose that K M and K U users are served during the CC and UC delivery steps such that K M = p min(η, η p ) and K U = p max(0, η p − η).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 α, cache ratio γ and the delivery parameter η.If the system operates with Strategy A in the CC delivery step, the DoF is given by: where β ′ = β P − t−1 Q− t−1 and D(δ r ) = φ r if δ r = 0; otherwise, D(δ r ) = 0.If Strategy B is applied during the CC delivery step, the DoF takes the form as follows.
where α ′ = (η t + α)ν 1 and The proof is relegated to Appendix B. Corollary 1: To give further insight into Theorem 1, assume that the length of all caching profiles is non-zero, i.e., η p = 0 for p ∈ [P ], all users are served during the CC delivery step, i.e., η p = δ p for p ∈ [P ], and η = η 1 = max p η p .If the system operates with Strategy A and α ≥ η, then β = min(α, η) = η and the DoF in ( 17) is simplified to DoF = KQ /P .Similarly, when the system operates with Strategy B in the CC delivery step, the DoF in ( 18) is rearranged as DoF = K(η t+α) /P η.
Remark 2: Suppose that K users are uniformly assigned to the caching profiles, i.e., K = P η, and all users are served in the CC delivery step, i.e., K M = K and K U = 0.If α ≤ η, our schemes achieves the optimal DoF α(P γ + 1) obtained in [32].If α > η, the achievable DoF of our scheme is simplified to the optimal DoF Kγ + α obtained in [33].However, unlike the scheme proposed in [33], our scheme supports the networks with non-integer α η (cf.Appendix C).Indeed, increasing non-uniformness in user distribution prevents the system from achieving optimal DoF performance.For instance, for the case α > η, suppose that all users are served with Strategy A during the CC delivery step, such that β = η and Q ≤ t + ⌊ α /η⌋.In this setup, we can set Q = t + ⌊ α /η⌋ to boost the DoF performance.By defining η avg = K P and assuming α is divisible by η and η avg , the DoF loss (compared to uniform user distribution) is α(1 − ηavg /η).
Remark 3: For the non-uniform user-to-profile association with Q = t+1 and η 1 ≤ α, the best possible DoF is achievable by setting η = max p η p = η 1 (cf.Appendix D).
Generally speaking, for any network parameters, we should compare the achievable DoF of our proposed scheme with α such that: i) if DoF ≥ α, we use the proposed CC scheme to simultaneously benefit from global CC and spatial multiplexing gains; ii) if DoF < α, 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 α users in the CC delivery step, preventing the system from fully utilizing the available multiplexing gain.To ensure that at least α 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 α 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, and without loss of generality, suppose that Strategy A is comprised of N A transmissions.Obviously, if the length of all caching profiles is non-zero, i.e., η p = 0 for all p ∈ [P ], we have where |Π| = N A and Π(n) represents the n-th element of Π. Hereafter, for national simplicity, the aim is to design the beamformers for the transmission n ∈ [N A ], for which Π(n) = A d = (r, c, l) d and the set of users served in this transmission is given by T A (cf. Section IV).To this end, by using ( 14) and ( 1), and enabling the cache-aided interference removal, the achievable rate of user k ∈ T A is given by R k = log(1 + Γ k ), where Γ k is the SINR at user k in this transmission, and is equal to: We note that if the system operates with Strategy B, by employing the SIC-free data delivery algorithm, each transmission 20) is replaced by TB − {k}; otherwise, we replace G Λ k with H Λ k (cf.Section III-B).Similarly, for each transmission of the UC delivery step, by considering T as the set of served users, the interfering set G Λ k in ( 20) is substituted with T − {k}.
As stated in Section III-A, by using Strategy A, each minifile . On the other hand, the size of each minifile is 1 /( P t ) of the size of the original file.Hence, user k ∈ T A receives a 1 µ k portion of its requested file at the end of the transmission n ∈ 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 = 1 µ 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 n ∈ [N A ]. Accordingly, given a fixed transmit power P T , we should solve the following optimization problem in order to obtain the optimized beamformers.max min Here, problem ( 21) can be efficiently solved as a max-min problem using iterative methods based on Lagrangian duality, as demonstrated in various studies including [40], [44].In this sense, employing a Lagrangian duality-based technique described in [40], we can iteratively find the optimum rate R e via bisection search, which is given by R e = µ k R k .For each fixed R e , the fixed point iteration method [45] is used to obtain the dual variables λ k for k ∈ T A .In this regard, at the first step, for k ∈ T A , we initialize λ k ← λ 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 λ k is given by: where denotes the value of λ k during the i-th iteration, and After the convergence of ( 22), we obtain the normalized beamformer wk .Accordingly, for k ∈ T A , as per [40], wk is given by wk = In order to find the power vector of the beamformers, denoted by p, we simply follow the same steps as in [40, eq. (26)].Finally, during the transmission n ∈ [N A ], the beamforming vector w Λ k takes the form of w Λ k = p(k) wk , where p(k) is the k-th entry of p. Consequently, the transmission rate for the transmission n ∈ [N A ] during the CC delivery step, denoted by R A n , is given by R A n = min k∈T A log(1 + Γ k ).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 sym .Assuming that the system operates with Strategy A during the CC delivery step, R sym is defined as follows.
where ÑUC is the total number of transmissions during the UC delivery step, and R UC j 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 ÑUC is not necessarily equal to T U in (16).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 ( 20)-( 22) to obtain the achievable rate for each transmission in the CC delivery step.Then, we can simply compute R sym in (23) based on the total number of transmissions during the CC delivery step via Strategy B. Here, we note that any beamforming design can be applied to the proposed data delivery scheme, and the finite-SNR performance can be measured via various metrics such as data delivery time [46].

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.

A. DoF Analysis
We consider a cache-aided MISO setup with t = 1, P = 5, and γ = 0.2, in which K = 30 users are present during the delivery phase.Here, for each user-toprofile association, we compute the standard deviation σ as 2 , where η avg = K /P = 6.Fig. 4 illustrates the achievable DoF for different Q and σ values when α = 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. 4, we see that even though choosing η = max p η p is proven to maximize the DoF only when α is large (i.e.α ≥ max p η p , cf.Remark 3), it is also a safe choice for smaller α values.Moreover, for a given η value, selecting the maximum possible Q, i.e., Q = t + ⌈ α /η⌉, 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 Kγ + α = 14 for the considered network with the uniform (a) σ = 0     user-to-profile association (as α > η and α /η is a non-integer, cf.Section III-B), our proposed scheme can achieve this DoF with Strategy B by setting η = η avg = 6 and Q = 3.In Fig. 5, we analyze the effect of the non-uniformness in user-to-profile associations (measured by σ) 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(α, K) users are served per transmission in a TDMA framework.The parameter 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 DoF max which is the maximum achievable DoF obtained by a line search over η and Q values, and then, for the associations with the same σ value, we set DoF M as the average of all the corresponding DoF max values.As can be seen, although the placement phase of our scheme was designed solely based on the knowledge of γ (and not the user count K), it can still boost the maximum achievable DoF by 10% − 70% over the no-CC solution for moderate σ values (e.g., 1 ≤ σ ≤ 4.5).This shows the strength of our algorithm in enabling a coded caching gain for any given non-uniform user distribution.
In Table II, for different user-to-profile associations and α values, the achievable DoF of our proposed scheme is compared with the schemes [20], [32], [33] and no-CC.Here,  "-" shows that the scheme is not applicable to the region.Moreover, it is assumed that K = 30, γ = 0.2, t = 1, P = 5 and the length of caching profiles is represented as (η 1 , η 2 , η 3 , η 4 , η 5 ).For the association (6, 6, 6, 6, 6) and α = 8, our scheme achieves the optimal DoF of 14 as same as the methods introduced in [20], [33].However, the scheme [32] is not applicable to these network parameters, as the constraint α ≤ min p η p is not satisfied.Since the scheme [33] only supports the networks with the uniform distribution of users, only the introduced method in [20] and our proposed scheme can manage the data delivery in a setup with the association (9, 8, 6, 5, 2) and α = 8 and achieve the DoF of 11.43.

B. Symmetric Rate
Here, we study the symmetric rate performance of cacheaided 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.Moreover, the block-fading (quasi-static) i.i.d.Rayleigh fading channels h k ∼ CN (0, I) with k ∈ [K] are considered in simulations.Fig. 6 demonstrates the average symmetric rate versus SNR for both transmission strategies A and B, for α ∈ {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 = t+⌊ α /η⌋ and Strategy B with Q = t+⌈ α /η⌉ 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 α = 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 [8].Therefore, as can be seen in Fig. 6, by setting α = L, the performance of Strategy B is inferior to Strategy A for smaller SNR values.On the other hand, when α = 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.
Figs. 7 and 8 illustrate the symmetric rate of the considered network for different η values, when α = 8.In Fig. 7, it is assumed that σ = 0.63 and η 1 = 7, while in Fig. 8, we have σ = 2.19 and η 1 = 9.As demonstrated, for both values of σ, the maximum symmetric rate is attained with η = η 1 = max p η p .This is because setting η = max p η 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 η = max p η p , the rate loss due to increased nonuniformness in Fig. 8 becomes negligible in comparison to Fig. 7, particularly in the finite-SNR regime.This underscores the effectiveness of our approach to address the nonuniformness in the user-to-profile association by optimizing the number of users served per transmission.
From Figs. 7 and 8, it can also be seen that when η is significantly smaller than max p η p , both the achievable DoF and the symmetric rate become almost independent of the σ value.For instance, considering η = 3, Strategy A achieves exactly the same DoF and rate values in both Fig. 7 and Fig. 8.This is because if η 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 σ value.Moreover, a small increase in η results in a significant boost in the symmetric rate even though the DoF changes slightly.For instance, when η = 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 η 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 III and Table IV show how choosing the parameter P affects the performance and complexity of the proposed scheme for large (α > η) and small antenna arrays (α ≤ η), respectively.In Table III, we consider the same network of K = 30 users with γ = 0.2 and L = 10, assume uniform user-to-profile association and α = L − 1 = 9, and compare the system performance when P ∈ {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 α /η is an integer).Here, we note that setting P ∈ {5, 10} satisfies the largeantenna constraint α > η, as for the uniform user distributions η = K /P .As observed, the maximum achievable DoF  In Table IV, a network of K = 30 users, γ = 0.2, L = 10, α = 2, and a uniform association of users to profiles is considered for P ∈ {5, 10, 15}.We note that for P ∈ {5, 10, 15}, the small-antenna array constraint α ≤ η is satisfied, the system can only operate with Strategy A, and it functions similarly to the proposed schemes in [32] and [41] during the CC delivery step.As per Theorem 1 and [32], the optimal achievable DoF for the region α ≤ η is α(P γ + 1).On the other hand, the optimal DoF for a network of K users with the spatial multiplexing gain of α is given by Kγ + α.In other words, the achievable DoF α(P γ + 1) is upper bounded by Kγ + α for the region α ≤ η.Therefore, as observed from Table IV, as long as the constraint P ≤ ⌈ K /α⌉ hold, an increase in P enhances the achievable DoF.Assuming K /α ∈ N, setting P = K /α maximizes the achievable rate, and the DoF takes the form of α( K /αγ + 1) = Kγ + α with the cost of increased subpacketization complexity.
Fig. 9 compares the achievable symmetric of our proposed scheme with no-CC in a network with K = 50, γ = 0.08, P = 25, t = 2, L = 50, SNR = 20dB, and different α values, where blue and black curves demonstrate the symmetric of our proposed scheme and no-CC, respectively, and the red curve illustrates the performance improvement of our proposed scheme over no-CC.As observed, increasing α does not monotonically increase the symmetric rate.For instance, increasing α from 2 to 40 improves the symmetric rate, while an increment in α from 40 to 50 reduces the achievable rate, due to that for mid-range SNR values (e.g., 20dB), we should serve fewer users at a time to achieve a higher symmetric rate [8], revealing that for large-antenna arrays, setting α = L does not maximize the achievable rate, and we have to optimize α with respect to SNR and the other network parameters.Moreover, when α ≤ 45, the gap between our proposed scheme and the no-CC scheme decreases with α, such that for α = 2 the performance improvement over the no-CC scheme is more than 150%, while for α = 45, the improvement is about 7%.On the other hand, increasing α from 46 to 50 increases the performance improvement to 18%, due to that for these values of α, the achievable DoF is equal to min(Kγ + α, K) = K, and consequently, we get some beamforming gain over the no-CC scheme.

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 Qβ users.Hence, each of these Qβ users is interfered by Qβ − 1 interfering users.Since each user has stored a γ portion of the entire library, it can remove at most βP γ = β t interference terms using the stored cache content.Therefore, α must be large enough to suppress the remaining Qβ − 1 − tβ interference terms, i.e., α − 1 ≥ Qβ − 1 − tβ, (24) which results in Q ≤ t+⌈ α /β⌉.Furthermore, Q ≥ 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(β, δ p ) 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 η, which results in having β ≤ η.On the other hand, for each profile p ∈ [P ], we should have min(β, δ p ) ≤ α to be able to cancel out the inter-stream interference between users assigned to the same profile.Hence, according to the fact that δ 1 = η, one can say min(β, δ 1 ) = min(β, η) = β ≤ α, which results in having the necessary condition β ≤ min(α, η) to select the parameter β.

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.Here, our aim is to count the possibilities for the subsets X .One way to count these possibilities is to simply pick the subsets X from X , and then count them.Here, clearly, the number of possibilities is equal to P Q .Another way to choose X is to define (P − Q + 1) subsets L r first, such that r ∈ [P − Q + 1], L r = {b i : r + 1 ≤ i ≤ P } and |L r | = P − r.Then, the selection of X consists of (P − Q + 1) rounds.In round r ∈ [P − Q + 1], we pick b r and choose (Q − 1) remaining elements from L r .In this regard, the number of possibilities to choose X in round r is equal to P −r Q−1 .By following this approach, we have

A. Proof of Theorem 1 for Strategy A
In order to prove the theorem, we compute the total number of transmitted subpackets during the content delivery phase and divide it by the total number of transmissions in the CC and UC delivery steps.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[k] = p is present in the CC delivery step.As per (4) and ( 5 k , there exist 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: As stated in Section IV, by using the SIC-free data delivery, each transmission triple A is split into ν 2 = Q−1 Q− t−1 transmissions A j with j ∈ [ν 2 ], such that in each transmission A j , user k ∈ T A receives only one subpacket of its requested file.Therefore, using (25), the total number of subpackets received by each user via Strategy A is β P −1 Q−1 ν 2 .In order to give further insight into (25), assume that user k is assigned to profile p.To count the transmissions serving user k, the number of possibilities to choose Q profiles with profile p being selected is equal to P −1 Q−1 .Now, we serve η times (assume that the length of each profile is greater than β) each of these Q selected profiles (including profile p), such that in each transmission triple, the indices of users assigned to profiles are circularly shifted.Therefore, for each set of Q selected profiles, user k is served in β transmission triples due to the fact that in each transmission triple, β users from each profile are selected, and the indices of selected users are shifted in each transmission triple.However, each of these transmission triples contains ν 2 subpackets for user k.Therefore, we split each transmission triple to ν 2 transmissions to ensure that user k receives only one of its desired subpackets per transmission.As a result, the total number of transmissions serving user k and, equivalently, the total number of received subpackets by user k is computed as β P −1 Q−1 ν 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: Next, we show that all users can decode their requested files with Strategy A. User k stores a γ portion of the entire library in the placement phase and needs to receive .So, assuming δ r > 0, the total number of transmission triples for any r ∈ [P − Q + 1] is equal to φ r P −r Q−1 .We recall that φ r = max(β, δ r ).Furthermore, as mentioned in Section IV, by employing the SIC-free data delivery, each transmission triple A is divided into ν 2 transmissions A j with j ∈ [ν 2 ].As a result, the total number of transmissions of Strategy A is obtained as: 2 and A B is the concatenation of A and B. Throughout the paper, bold lower-case letters show vectors.v H is the conjugatetranspose (Hermitian) of the vector v.The letter A represents the triple (r, c, l) for some r, c, l ∈ N, and B expresses the quintuple (r, c, l, m, s) for some r, c, l, m, s ∈ N. f * represents the phantom (non-existent) users.% sign demonstrates the mod operator, for which c%c = c and (d + c)%c = d%c.Furthermore, for integers a and b, a b = a! b!(b−a)! .• denotes the norm operator.Finally, ⌊y⌋ and ⌈y⌉ show the floor and ceiling of y ∈ R.

Fig. 2 .
Fig. 2. System model for a dynamic coded caching setup, where P = 2, γ = 1 2 , t = 1, α = 4, η = 4, Q = 2 and β = 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.

( 1 ,
2, 1) and A 3 = (1, 3, 1), for which the BS creates the set of served users T A1 , T A2 and T A3 , respectively.Generally speaking, for the transmission triple A = (r, c, l), if δ r = 0, the BS does not transmit any signal; otherwise, the BS serves users assigned to T A = S r,c S b1,c • • • S bQ−1,c , which represents the concatenation of the c-th window of profile r and the profiles in the set M r

= 1 ,
and s ∈ [ν 2 ].In each transmission quintuple B = (r, c, l, m, s), for any n ∈ [ν 2 ], we create the set C(n) = K m,u r,s (n) ∪ B(n), and define the function I + δc , E m r such that I + δc , E m r = 0, if E m r = f * and δc = 0; otherwise, I + δc , E m r = 1.If I +δc , E m r after eliminating the impacts of the phantom users f * , the BS builds the transmission vector for the transmission quintuple B = (r, c, l, m, s) as follows. 63

Lemma 1 :
Given P and Q, we have Assume a scenario at which we have a set X = {b 1 , • • • , b P } with |X | = P , and we want to select the subsets X ⊆ X such that | X | = Q.
), user k is served in the transmission triple A = (r, c, l) such that r ∈ [p].On the other hand, according to (2) and (3), for each r ∈ [p], there are β = min(α, η) values for c that user k is served in the transmission triple A = (r, c, l).Now, for any r ∈ [p], let A (r) k be the set of values for c at which user k is served in the transmission triple A = (r, c, l) such that |A (r) k | = β.As mentioned in Section III-A, for r = p and c ∈ A (p) k , there are P −p Q−1 values for l at which user k is served during the transmission triple A. Furthermore, for r ∈ [p − 1] and c ∈ A (r)

( 1 −
γ)β P t P − t−1 Q− t−1 subpackets in the CC delivery step.As mentioned, user k receives β P −1 Q−1 ν 2 = (1 − γ)β P t P − t−1 Q− t−1subpackets 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 δ r = min(η r , η) > 0, for any r ∈ [P − Q + 1] and c ∈ [φ r ], users assigned to the set T A = S r,c S b1,c • • • S bQ−1,c are served, where {b 1 , • • • , b Q−1 } = M r (l) and l ∈ [ P −r Q−1 ] is split into P P such that W n → {W n P : P ⊆ [P ], |P| = t}.The cache content associated with profile p ∈ [P ], represented by Z p , includes a portion γ of each file W n as Z p = {W n P : P ∋ p, P ⊆ [P ], |P| = t, ∀n ∈ [N ]}.Then, defining U p as the set of users assigned to profile p, i.e., U p = {k : p[k] = p}, each user k ∈ U p stores the cache content Z p during the placement phase.

TABLE III IMPACTS
OF P ON THE PERFORMANCE FOR LARGE-ANTENNA ARRAYS 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 α > η 1 and a fixed γ 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 t such that gcd(P, t) = 1 and t = P γ, then, use Strategy A for the data delivery during the CC delivery step.

TABLE IV IMPACTS
OF P ON THE PERFORMANCE FOR SMALL-ANTENNA ARRAYS