By establishing a relationship between the classical ambiguity function and the sample cyclic cross correlation, it is shown that the former provides consistent estimates of delay and Doppler when the target echoes are observed in stationary noise. A novel fourth-order ambiguity function is proposed for estimating the delay and Doppler from random modulated Doppler-spread echoes. It is also shown that the conventional yields inconsistent estimates when the modulating process is zero-mean. Next, a method exploiting cyclostationarity of the signal is proposed for estimating the position and track of a generally maneuvering target or source. The received signal in this case can be modeled as a random amplitude polynomial phase process. Because of the statistical framework, the proposed approach also allows multiple signals, is theoretically insensitive to any stationary noise and provides a sequential estimation procedure.