Introduction to Radar Basics - MATLAB Simulations for Radar Systems Design - page 12

where for simplicity the effects of the constant phase have been ignored in
Eq. (1.33). Therefore, the bandpass spectrum of the received signal is now cen-
tered at instead of . The difference between the two values corresponds
to the amount of Doppler shift incurred due to the target motion,

ψ 0

γω 0 ω 0

ω d = ω 0 – γω 0

(1.34)

is the Doppler frequency in radians per second. Substituting the value of ω d γ

in Eq. (1.34) and using yield 2π f = ω

2 v
c

2 v
λ

f d = ------ f 0 = ------ (1.35)

which is the same as Eq. (1.23). It can be shown that for a receding target the
Doppler shift is . This is illustrated in Fig. 1.9. f d = – 2 v λ ⁄

f d

f d

f 0

f 0

frequency frequency

closing target receding target
Figure 1.9. Spectra of received signal showing Doppler shift.

In both Eq. (1.35) and Eq. (1.23) the target radial velocity with respect to the
radar is equal to , but this is not always the case. In fact, the amount of Dop-
pler frequency depends on the target velocity component in the direction of the
radar (radial velocity). Fig. 1.10 shows three targets all having velocity : tar-
get 1 has zero Doppler shift; target 2 has maximum Doppler frequency as
defined in Eq. (1.35). The amount of Doppler frequency of target 3 is
, where

v

v

is the radial velocity; and is the total angle f d = 2 v cos θ ⁄ λ v cos θ θ

between the radar line of sight and the target.

Thus, a more general expression for that accounts for the total angle

f d

between the radar and the target is

2 v
λ

------

f d = cos θ

(1.36)

and for an opening target

– v
λ

---------

f d = cos θ

(1.37)

where . The angles and are, respectively, the eleva-

cos θ = cos θ e cos θ a θ e θ a

tion and azimuth angles; see Fig. 1.11 .