Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

7.3 Constant Relative Radial Speed

Let ξ(t) be the projection of the vector P _R(t) − P _T(t − D(t)) along the propagation direction i_k (oriented from P _T(t − D(t)) to P _R(t)), that is

(7.77)

The delay D(t) experienced by the wavefront transmitted at time t − D(t) in P _T(t − D(t)) and received at time t in P _R(t) depends on the distance ||P_R(t) − P_T(t − D(t))||. Accordingly with (7.11) we have

(7.78)

As observed in Section 7.1.3, |ξ(t)| is the time-varying distance responsible of the Doppler effect, viz, the time-varying delay D(t), and should not be confused with

(7.79)

Let us assume that the relative radial speed between P _T(t − D(t)) and P _R(t) is constant within the observation interval. Thus, the projection of the vector P _R(t) − P _T(t − D(t)) along the propagation direction i_k is a linear function of t. That is,

(7.80)

Furthermore, let us assume that, within the observation interval, RX does not collapse on TX. Thus, ξ(t) does not change sign and it is always (t, D(t)) > 0. Under ...