Mathematical observation:
Assume a target at a distance R and has a radial velocity component of Vr. The roundtrip
distance to target is 2R. This is equivalent to 2R/λ wavelengths or (2R/λ)2π = 4πR/λ
radians. If the phase of the transmitted signal is φo, then the phase of the received signal
is
Alternatively, let the transmitted frequency be ft. The received signal can be represented
as sin(2 ( )) rec t R A = K πf t − t. The round-trip time tR is equal to 2R/c. With a radial
velocity of Vr the round-trip time is changing as R = R0 – Vrt. Thus, the received signal is
Thus, the received frequency changes by a factor 2ftVr/c = 2Vr/λ., which is the same as
before. The Nyquist criterion says that fmax = PRF/2, combining this with Equation (1),
results in
This is the maximum unambiguous velocity. Higher velocities cause velocity folding or
Velocity aliases.
This summarizes the Doppler dilemma: a large Rmax implies a small Vmax and vice versa
Doppler process recognizing and dealing with range and velocity aliases
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