As explained in the literature, FM-CW ranging for a linear ramp waveform is given in the following set of equations: where is the radar frequency sweep amount and is the time to complete the frequency sweep. Then, , rearrange to a more useful: , where is the round trip time of the radar energy. It is then a trivial matter to calculate the physical one-way distance for an idealized typical case as: where is the speed of light in any transparent medium of refractive index n (n=1 in vacuum and 1.0003 for air). For practical reasons, receive samples are not processed for a brief period after the modulation ramp begins because incoming reflections will have modulation from the previous modulation cycle. This imposes a range limit and limits performance. Sinusoidal frequency modulation Sinusoidal FM is used when both range and velocity are required simultaneously for complex objects with multiple moving parts like turbine fan blades, helicopter blades, or propellers. This processing reduces the effect of complex spectra modulation produced by rotating parts that introduce errors into range measurement process. This technique also has the advantage that the receiver never needs to stop processing incoming signals because the modulation waveform is continuous with no impulse modulation. Sinusoidal FM is eliminated completely by the receiver for close in reflections because the transmit frequency will be the same as the frequency being reflected back into the receiver. The spectrum for more distant objects will contain more modulation. The amount of spectrum spreading caused by modulation riding on the receive signal is proportional to the distance to the reflecting object. The time domain formula for FM is: where (modulation index) A time delay is introduced in transit between the radar and the reflector. where time delay The detection process down converts the receive signal using the transmit signal. This eliminates the carrier. The Carson bandwidth rule can be seen in this equation, and that is a close approximation to identify the amount of spread placed on the receive spectrum: Receiver demodulation is used with FMCW similar to the receiver demodulation strategy used with pulse compression. This takes place before Doppler CFAR detection processing. A large modulation index is needed for practical reasons. Practical systems introduce reverse FM on the receive signal using digital signal processing before the fast Fourier transform process is used to produce the spectrum. This is repeated with several different demodulation