M-ary orthogonal keying (MOK) systems under carrier frequency offset (CFO) are investigated. It is shown that spurious signals are introduced by the offset frequency components of spectrum after multiplication in correlation detection process, and some conditions on robust orthogonal signal sets are derived. Walsh function sets are found to be very weak against CFO, since they produce large spurious signals. As robust orthogonal signal sets against CFO, the rows of circulant Hadamard matrices are proposed and their error performanses are evaluated. The results show that they are good M-ary orthogonal signal sets in the presence of CFO.