Space-frequency equivalence in directional arrays, with special reference to superdirectivity and reciprocity

Previous proposals that a directional receiving array of given directional response can be made either from a number of receiving elements with a single narrow-band signal, or from two receiving elements with a signal comprising a number of separate narrow frequency bands, are examined from the points of view of the exactness of the equivalence and of the application of superdirectivity. It is concluded that no exact equivalence exists in practical terms, but that superdirectivity can certainly be obtained by the second method, as well as by the first. The application of the second method to a transmitting array is shown to be possible, and the law of reciprocity may be made to hold to a limited extent; but multi-frequency two-element transmitters and receivers cannot work together in the way in which multi-element single-frequency transmitters and receivers can.