A new scheme for the problem of centralized coded caching with non-uniform demands is proposed. The distinguishing feature of the proposed placement strategy is that it admits equal sub-packetization for all files while allowing the users to allocate more cache to the files which are more popular. This creates natural broadcasting opportunities in the delivery phase which are simultaneously helpful for the users who have requested files of different popularities. For the case of two files, we propose a new delivery strategy based on interference alignment which enables each user to decode his desired file following a two-layer peeling decoder. Furthermore, we extend the existing converse bounds for uniform demands under uncoded placement to the nonuniform case. To accomplish this, we construct N ! auxiliary users, corresponding to all permutations of the N files, each caching carefully selected sub-packets of the files. Each auxiliary user provides a different converse bound. The overall converse bound is the maximum of all these N ! bounds. We prove that our achievable delivery rate for the case of two files meets this converse, thereby establishing the optimal expected memory-rate trade-off for the case of K users and two files with arbitrary popularities under uncoded placement.

We propose a theoretical framework for the image source method that generalizes to arbitrary reflecting boundaries, e.g. boundaries that are curved or even with certain openings. Furthermore, it can seamlessly incorporate boundary absorption, source directivity, and nonspecular reflections. This framework is based on the notion of reflection paths that allows the introduction of the concepts of validity and visibility of virtual sources. These definitions facilitate the determination, for a given source and receiver location, of the distribution of virtual sources that explain the boundary effects of a wide range of reflecting surfaces. The structure of the set of virtual sources is then more general than just punctual virtual sources. Due to this more diverse configuration of image sources, we represent the room impulse response as an integral involving the temporal excitation signal against a measure determined by the source and receiver locations, and the original boundary. The latter smoothly enables, in an analytically tractable manner, the incorporation of more general boundary shapes as well as directivity of sources and boundary absorption while, at the same time, maintaining the conceptual benefits of the image source method.