Channel Estimation for Frequency Hopping

Frequency hopping systems have traditionally been applied in low-data rate applications where the received signal can be considered narrow-band. The multipath delays encountered in a wireless channel has considerably less a ect on a narrow-band signal. In such cases, delay estimation is not of much signi cance. However, delay estimation becomes increasingly important when the bandwidth of the system increases. The delay estimation problem for a high data rate frequency hopping system is addressed in this work. The multipath e ect on a narrow-band system is described by invoking what is commonly known as the at-fading approximation. This approximation leads to easier analysis and is employed whenever the bandwidth of the system is considerably less than the coherence bandwidth of the channel. However, the validity of the approximation is not clearly understood at higher bandwidths. An error bound is derived to describe the error involved in the at-fading approximation in terms of the bandwidth of the system and the channel coherence bandwidth. Based on this bound, it is demonstrated that the high data rate frequency hopping system indeed requires channel estimation. The channel estimation problem for a frequency hopping system is cast in the ESPRIT format. ESPRIT is a well known algorithm in array processing where the signal subspace is estimated based on the received signal samples and the rotational structure shared by the subspaces is used to estimate some signal parameters. It is shown that the received signal samples in a frequency hopping system share a similar rotational invariance structure. This structure is exploited for the estimation of channel delays. It is demonstrated that the received signal samples have more structure than what ESPRIT can exploit. Extensions of the ESPRIT algorithm to handle multiple invariances have been proposed but mostly at the cost of ease of computation that ESPRIT provides. An alternate algorithm that identi es a di erent constraint on the signal subspaces is demonstrated. This algorithm is based on the Cayley-Hamilton theorem for square matrices and is termed Signal Parameter Estimation based on the Cayley-Hamilton Constraint (SPECC). The SPECC algorithm is shown to yield better estimation of the delays with a computational cost comparable to that of the ESPRIT algorithm. The algorithm is shown to be robust and capable of providing good estimates even with inaccurate estimates of the signal subspace. The ESPRIT scheme requires the invariance structure to be shared among at least as many elements as the number of parameters estimated. It is shown that the SPECC algorithm requires no such constraint. A generalized version of the SPECC algorithm is presented and is shown to be applicable to all situations where ESPRIT is normally used.