Spectral Shape Due to Doppler Spread in Clarke's Model
- Gans developed a spectrum analysis for Clarke's model
- Let p (α) dα denote the fraction of the total incoming power within dα of the angle α, and let A denote the average received power with respect to an isotropic antenna.
- As N →∞, p (α) dα approaches a continuous, rather than a discrete, distribution. If G (α) is the azimuthal gain pattern of the mobile antenna as a function of the angle of arrival, the total received power can be expressed as
- ― (1)
- where AG (α)p (α)dα is the differential variation of received power with angle.
- If the scattered signal is a CW signal of frequency f_{c} then the instantaneous frequency of the received signal component arriving at an angle α is obtained using equation
― (2)
- Where f _{m} is the maximum Doppler shift. It should be noted that f(α) is an even function of α.
- If S (f) is the power spectrum of the received signal, the differential variation of received power with frequency is given by
- ― (3)
- Equating the differential variation of received power with frequency to the differential variation in received power with angle, we get ― (4)
- Differentiating equation (2),and rearranging the terms, we have
― (5)
- Using equation (2), α can be expressed as a function off as
― (6)
- This implies that
― (7)
- Substituting equation (5) and (7) into both sides of (4), the power spectral density S (f) can be expressed as
― (8)