Editing Continuous-wave radar (section)

====Sawtooth frequency modulation====
[[File:Fmcw prinziple.png|thumb|upright=1.5|Ranging with an FM-CW radar system: if the error caused by a possible Doppler frequency <math>f_D</math> can be ignored and the transmitter's power is linearly frequency modulated, then the time delay (<math>\Delta t</math>) is proportional to the difference of the transmitted and the received signal (<math>\Delta f</math>) at any time.]]

Sawtooth modulation is the most used in FM-CW radars where range is desired for objects that lack rotating parts. Range information is mixed with the Doppler velocity using this technique. Modulation can be turned off on alternate scans to identify velocity using unmodulated carrier frequency shift. This allows range and velocity to be found with one radar set. Triangle wave modulation can be used to achieve the same goal.

As shown in the figure the received waveform (green) is simply a delayed replica of the transmitted waveform (red). The transmitted frequency is used to down-convert the receive signal to [[baseband]], and the amount of frequency shift between the transmit signal and the reflected signal increases with time delay (distance). The time delay is thus a measure of the range; a small frequency spread is produced by nearby reflections, a larger frequency spread corresponds with more time delay and a longer range.

With the advent of modern electronics, [[digital signal processing]] is used for most detection processing. The beat signals are passed through an [[analog-to-digital converter]], and digital processing is performed on the result. As explained in the literature, FM-CW ranging for a linear ramp waveform is given in the following set of equations:<ref name=Radartutorial/>

::<math>k = \frac {\Delta{f_{radar}}} {\Delta{t_{radar}}}</math>

:::where <math>\Delta{f_{radar}}</math> is the radar frequency sweep amount and <math>\Delta{t_{radar}}</math> is the time to complete the frequency sweep.

Then, <math>\Delta{f_{echo}} = t_rk</math>, rearrange to a more useful:

::<math>t_r = \frac {\Delta{f_{echo}}} {k}</math>, where <math>t_r</math> 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:

::<math>\text{dist}_{oneway} = \frac {c' t_r}{2}</math>

:::where <math>c'=c/n</math> 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.

::<math>\text{Range Limit} = 0.5 \ c' \ t_{radar} </math>