Antenna arrays offer advantages in terms of spatial diversity, allowing for control over pattern specifications in space. The incorporation of frequency diversity in arrays presents an opportunity to manipulate beams in the Space-Time domain. Unlike conventional arrays, Frequency Diverse Arrays (FDA) with added frequency diversity exhibit time-variant and range-dependent patterns. These time variations impact both steering and auto-scanning applications. The array factor is influenced by the coherent interplay between frequency and spatial distributions of elements, thereby correlating the spatial and temporal behavior of the FDA's pattern. To address this space-frequency coherency, an adjoint spatial-frequency design algorithm is the most effective approach to controlling the array's spatial and temporal behavior. Given the complexity of the array factor formulations in FDAs, elements' frequency and spatial distribution have traditionally been designed separately. However, this study proposes an algorithm that simultaneously allocates the location and frequency of elements to achieve a desired pattern. Symmetrical FDA is initially designed using a straightforward formulation of the array factor obtained through symmetry, ensuring a stable and periodic scanning beam. Subsequently, two important design parameters and several crucial design criteria for scanning applications are suggested by analyzing the formulations. These parameters form the basis of a designing algorithm that enables the simultaneous design of element location and frequency in the space-frequency plan, thus meeting the temporal and spatial requirements of the pattern. To demonstrate the efficacy of the proposed algorithm, two different planar arrays are designed, and their results are compared with those of other planar configurations. This study lays the foundation for a novel approach to designing Frequency Diverse Arrays (FDAs), opening up new possibilities in array design.
