Editing Continuous-wave radar (section)

====Sinusoidal frequency modulation====
[[File:Amfm3-en-de.gif|thumb|right|200px|Sinusoidal FM modulation identifies range by measuring the amount of spectrum spread produced by propagation delay (AM is not used with FMCW).|alt=Animation of audio, AM and FM signals]]

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 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:

:<math> y(t) = \cos \left\{ 2 \pi [ f_{c} + \Beta \cos \left( 2 \pi f_{m} t \right) ] t \right\}\,</math>

::where <math>\Beta = \frac{f_{\Delta}}{f_{m}}</math> (modulation index)

A time delay is introduced in transit between the radar and the reflector.

:<math> y(t) = \cos \left\{ 2 \pi [ f_{c} + \Beta \cos \left( 2 \pi f_{m} (t + \delta t) \right) ] (t + \delta t) \right\}\,</math>

::where <math>\delta t =</math> time delay

The detection process down converts the receive signal using the transmit signal. This eliminates the carrier.

:<math> y(t) = \cos \left\{ 2 \pi [ f_{c} + \Beta \cos \left( 2 \pi f_{m} (t + \delta t) \right) ] (t + \delta t) \right\}\;\cos \left\{ 2 \pi [ f_{c} + \Beta \cos \left( 2 \pi f_{m} t \right) ] t \right\}\,</math>

:<math> y(t) \approx \cos \left\{ -4 t \pi \Beta \sin ( 2 \pi f_{m} (2t + \delta t) \sin ( \pi f_{m} \delta t) + 2 \delta t \pi \Beta \cos (2 \pi f_{m} ( t + \delta t) ) \right\}\,</math>

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:

:<math>\text{Modulation Spectrum Spread} \approx 2 (\Beta + 1 ) f_m \sin (\delta t ) </math>

:<math>\text{Range} = 0.5 C / \delta t </math>

Receiver demodulation is used with FMCW similar to the receiver demodulation strategy used with pulse compression. This takes place before [[Pulse-Doppler signal processing#Detection|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 values. Range is found by identifying the receive spectrum where width is minimum.

Practical systems also process receive samples for several cycles of the FM in order to reduce the influence of sampling artifacts.