  Antenna array technology enables directional transmission and reception of
wireless signals, for communications, localization, and sensing purposes. The
signal processing algorithms that underpin this technology began to be
developed several decades ago [1], but it is first with the ongoing deployment
of the fifth-generation (5G) wireless mobile networks that it becomes a
mainstream technology [2]. The number of antenna elements in the arrays of the
5G base stations and user devices can be measured at the order of 100 and 10,
respectively. As the networks shift towards using higher frequency bands, more
antennas fit into a given aperture.
  The 5G developments enhance the transmitter and receiver functionalities, but
the wireless channel propagation remains an uncontrollable system. This is
illustrated in Fig. 1(a) and its mathematical notation will be introduced
later. Transmitted signals with three different frequencies are shown to
illustrate the fact that attenuation can vary greatly across frequencies.
Looking beyond 5G, the advent of electromagnetic components that can shape how
they interact with wireless signals enables partial control of the propagation.
A reconfigurable intelligent surface (RIS) is a two-dimensional surface of
engineered material whose properties are reconfigurable rather than static [4].
  This article provides a tutorial on the fundamental properties of the RIS
technology from a signal processing perspective. It is meant as a complement to
recent surveys of electromagnetic and hardware aspects [4], [7], [11],
acoustics [12], communication theory [13], and localization [8]. We will
provide the formulas and derivations that are required to understand and
analyze RIS-aided systems using signal processing, and exemplify how they can
be utilized for improved communication, localization, and sensing.
