# 10.7 Doppler Radar

The Doppler effect is a shift in the frequency of a wave caused by the motion of the transmitting source, the reflecting object, or the receiving system. As illustrated in Fig. 10-15, a wave radiated by a stationary isotropic point source forms equally spaced concentric circles as a function of time travel from the source. In contrast, a wave radiated by a moving source is compressed in the direction of motion and is spread out in the opposite direction. Compressing a wave shortens its wavelength, which is equivalent to increasing its frequency. Conversely, spreading it out decreases its frequency. The change in frequency is called the Doppler frequency shift f d. That is, if f t is the frequency of the wave radiated by the moving source, the frequency f r of the wave that would be observed by a stationary receiver is

f r=f t+f d. (10.28)

# Figure 10-15 A wave radiated from a point source when (a) stationary and (b) moving. The wave is compressed in the direction of motion, spread out in the opposite direction, and unaffected in the direction normal to motion.

(a) Stationary source

(b) Moving source

Figure 10 15 Full Alternative Text

The magnitude and sign of f d depend on the direction of the velocity vector relative to the direction of the range vector connecting the source to the receiver.

Consider a source transmitting an electromagnetic wave with frequency f t (Fig. 10-16). At a distance R from the source, the electric field of the radiated wave is given by

E(R)=E 0 e j(ω t t−k R)=E 0 e j ϕ, (10.29)

where E 0 is the wave’s magnitude, ω t=2 π f t, and k=2 π/λ t with λ t being the wavelength of the transmitted wave. The magnitude depends on the distance R and the gain of the source antenna, but it is not of concern as far as the Doppler effect is concerned. The quantity

ϕ=ω t t−k R=2 π f t t−2 π λ t R (10.30)

is the phase of the radiated wave relative to its phase at R=0 and reference time t=0. If the source is moving toward the receiver, as in Fig. 10-16, or vice versa, at a radial velocity u r, then

R=R 0−u r t, (10.31)

where R 0 is the distance between the source and the receiver at t=0. Hence,

ϕ=2 π f t t−2 π λ t(R 0−u r t). (10.32)

# Figure 10-16 Transmitter with radial velocity u r approaching a stationary receiver.

This is the phase of the signal detected by the receiver. The frequency of a wave is defined as the time derivative of the phase ϕ divided by 2 π. Thus,

f r=1 2 π d ϕ d t=f t+u r λ t. (10.33)

Comparison of Eq. (10.33) with Eq. (10.28) leads to f d = u r/λ t. For radar, the Doppler shift happens twice: once for the wave from the radar to the target and again for the wave reflected by the target back to the radar. Hence, f d=2 u r/λ t. The dependence of f d on direction is given by the dot product of the velocity and range unit vectors, which leads to

f d=−2 u r λ t=−2 u λ t cos θ, (10.34)

where u r is the radial velocity component of u and θ is the angle between the range vector and the velocity vector (Fig. 10-17) with the direction of the range vector defined to be from the radar to the target. For a receding target (relative to the radar), 0≤θ≤90°, and for an approaching target, 90°≤θ≤180°.

# Figure 10-17 The Doppler frequency shift is negative for a receding target (0≤θ≤90°), as in (a), and positive for an approaching target (90°≤θ≤180°), as in (b).