where       R=range (distance to echo)
                c = speed of electromagnetic radiation = 3 x 10⁸ m s^-1
                t = time since pulse was emitted
 

Maximum unambiguous range:

where    R_max = maximum unambiguous range
            c = 3 x 10⁸ m s^-1
                PRF = pulse repetition frequency [s^-1 ], i.e., number of pulses per second

The greater the frequency (more pulses per second), the shorter is the maximum unambiguous range.

Fundamental sampling theorem:  To measure a frequency, f_d, it is necessary to sample at a frequency of at least 2f_d .

The sampling rate is the PRF, so

Doppler theorem:

where    f = frequency shift
             = wavelength
            V = radial velocity, i.e. component of velocity toward of away from the radar

Maximum unambiguous Doppler velocity is

The "Doppler Dilemma:" Since the maximum unambiguous range is inversely related to the PRF while the maximum unambiguous Doppler velocity is directly related to the PRF, there is no single PRF that can maximize both at the same time.