7.6 Multipath Doppler Channel

7.6.1 Constant Relative Radial Speeds – Discrete Scatterers

In the case of multiple reflections on point scatterers, propagation can be described by the multipath Doppler channel (Izzo and Napolitano 2002b), (Napolitano 2003), (Izzo and Napolitano 2005), that is a LTV system with impulse-response function

(7.249)

where K is the number of the channel paths.

By double Fourier transforming the right-hand side of (7.249), one obtains the transmission function (see (1.44) and (6.68))

(7.250)

from which, accounting for (4.77a), it follows that the multipath Doppler channel is a FOT deterministic LTV system with Ω = {1, …, K},

(7.251)

(7.252)

Equivalently, accounting for (4.78), we get

(7.253)

(7.254)

The output y(t) corresponding to the input signal x(t) is obtained by substituting (7.249) into (1.41):

(7.255)

7.6.2 Continuous Scatterer

In the case of distributed moving reflector, assuming that each volume element (ξ, ξ + dξ) has constant relative radial speed with respect to transmitter and receiver, the input/output relationship for the propagation channel is

(7.256)

which corresponds to a LTV system with impulse-response function

(7.257)

The system with impulse-response function (7.249) is deterministic in the FOT probability sense (Section 6.3.8) (Izzo and Napolitano 2002a,b) whereas the system (7.257) is random in the FOT probability sense. Similar models are considered in (Middleton 1967; Sadowsky and Kafedziski 1998).

If the narrow-band condition (7.219) can be assumed to be verified for the speed of each volume element, then the input/output relation can be expressed in terms of the (narrow-band) spreading function SF(τ, ν) (Bello 1963):

(7.258)

Analogously, if the narrow-band condition is not satisfied, the input/output relation can be expressed in terms of the wide-band spreading function WSF(τ, s) (Weiss 1994):

(7.259)

If a frequency shift ν = (s − 1)f_c is present in the propagation model as in (7.129) and (7.156), then (7.259) modifies into

(7.260)