Mathematical observation:

Assume a target at a distance *R* and has a radial velocity component of *V**r*. The roundtrip

distance to target is 2*R*. This is equivalent to 2*R*/λ wavelengths or (2*R*/λ)2π = 4π*R*/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 *f**t**.* The received signal can be represented

as sin(2 ( )) rec *t R* *A* = *K* π*f t* − *t*. The round-trip time *t**R* is equal to 2*R*/*c*. With a radial

velocity of *V**r* the round-trip time is changing as *R* = *R*0 – *V**r**t.* Thus, the received signal is

Thus, the received frequency changes by a factor 2*f**t**V**r**/c = 2V**r/*λ*.,* which is the same as

before. The Nyquist criterion says that *f*max = 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 *R*max implies a small *V*max and vice versa

Doppler process recognizing and dealing with range and velocity aliases

- On examining horizontal and vertical shapes of object. For example range-aliased incoming objects become skinny close-by objects. Incoming objects suspicious 3m – 150m wide, 130m – 6.2 km height.
- Then we examine reflectivity in conjunction with other factors,
- Changed PRF–real echoes will not change position, but aliases will. This is not always an option.
- Velocity folding causes a change in sign, which is relatively easy to spot if the Folding is within a larger region.

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