Following the recent proposed nonlinear frequency division multiplexing (NFDM) scheme [1] for integrable communication channels, we have designed a class of signals called spatially periodic signals that maintain their shape periodically during transmission. This class of signals is of interest because the encoding and decoding processes are much simpler. This paper establishes properties of spatially periodic signals, leading to a better understanding of how spatially periodic signals should be designed and the tradeoff between various system parameters including signal energy and minimum period.