Editing Continuous-wave radar (section)

===Modulated continuous-wave===
''Frequency-modulated continuous-wave radar'' (FM-CW) – also called continuous-wave frequency-modulated (CWFM) radar<ref>
Jim Lesurf.
[http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/RadCom/part15/page2.html "Improved forms of radar"].
accessdate=2014-01-24.
</ref>
– is a short-range measuring radar set capable of determining distance. This increases reliability by providing distance measurement along with speed measurement, which is essential when there is more than one source of reflection arriving at the radar antenna. This kind of radar is often used as "[[radar altimeter]]" to measure the exact height during the landing procedure of aircraft.<ref name=Radartutorial>{{Cite web|url=http://radartutorial.eu/02.basics/Frequency%20Modulated%20Continuous%20Wave%20Radar.en.html |work= Radartutorial |title= Frequency-Modulated Continuous-Wave Radar| access-date=2012-08-07}}</ref> It is also used as early-warning radar, [[wave radar]], and proximity sensors. Doppler shift is not always required for detection when FM is used. While early implementations, such as the APN-1 Radar Altimeter of the 1940s, were designed for short ranges, Over The Horizon Radars (OTHR) such as the Jindalee Operational Radar Network (JORN) are designed to survey intercontinental distances of some thousands of kilometres.

In this system the transmitted signal of a known stable frequency [[continuous wave]] varies up and down in frequency over a fixed period of time by a modulating signal. Frequency difference between the receive signal and the transmit signal increases with delay, and hence with distance. This smears out, or blurs, the Doppler signal. Echoes from a target are then mixed with the transmitted signal to produce a [[Beat (acoustics)|beat signal]] which will give the distance of the target after demodulation.

A variety of modulations are possible, the transmitter frequency can slew up and down as follows :
* [[Sine wave]], like air raid siren
* [[Sawtooth wave]], like the chirp from a bird
* [[Triangle wave]], like police siren in the United States
* [[Square wave (waveform)|Square wave]], like police siren in the United Kingdom

Range demodulation is limited to 1/4 wavelength of the transmit modulation. Instrumented range for 100&nbsp;Hz FM would be 500&nbsp;km. That limit depends upon the type of modulation and demodulation. The following generally applies.

:<math>\text{Instrumented Range} = F_r-F_t = \frac {\text{Speed of Light}}{(4 \times \text{Modulation Frequency})}</math>

The radar will report incorrect distance for reflections from distances beyond the instrumented range, such as from the moon. FMCW range measurements are only reliable to about 60% of the instrumented range, or about 300&nbsp;km for 100&nbsp;Hz FM.

====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>

====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.