Editing Radar (section)

===Radar range equation===
The power ''P<sub>r</sub>'' returning to the receiving antenna is given by the equation:

:<math>P_{r}=\frac{P_{t}G_{t}A_{r}\sigma F^{4}}{{(4\pi )}^{2}R_{t}^{2}R_{r}^{2}}</math>

where
* ''P''<sub>t</sub> = transmitter power
* ''G''<sub>t</sub> = [[Antenna gain|gain]] of the transmitting antenna
* ''A''<sub>r</sub> = [[effective aperture]] (area) of the receiving antenna; this can also be expressed as <math>{G_r\lambda^2}\over{4\pi}</math>, where
:* <math>\lambda</math> = transmitted wavelength
:* ''G''<sub>r</sub> = gain of receiving antenna<ref>{{cite book |last=Stimson |first=George |url=https://archive.org/details/introductiontoai0000stim/page/98 |title=Introduction to Airborne Radar |date=1998 |publisher=SciTech Publishing Inc. |isbn=978-1-891121-01-2 |page=98 |url-access=registration}}</ref>
* ''σ'' = [[radar cross section]], or scattering coefficient, of the target
* ''F'' = pattern propagation factor
* ''R''<sub>t</sub> = distance from the transmitter to the target
* ''R''<sub>r</sub> = distance from the target to the receiver.

In the common case where the transmitter and the receiver are at the same location, ''R''<sub>t</sub> = ''R''<sub>r</sub> and the term ''R''<sub>t</sub>² ''R''<sub>r</sub>² can be replaced by ''R''<sup>4</sup>, where ''R'' is the range.
This yields:
:<math>P_r = {{P_t G_t  A_r \sigma F^4}\over{{(4\pi)}^2 R^4}}.</math>

This shows that the received power declines as the fourth power of the range, which means that the received power from distant targets is relatively very small.

Additional filtering and pulse integration modifies the radar equation slightly for [[Pulse-Doppler radar#Performance|pulse-Doppler radar performance]], which can be used to increase detection range and reduce transmit power.

The equation above with ''F'' = 1 is a simplification for transmission in a vacuum without interference. The propagation factor accounts for the effects of [[Multipath propagation|multipath]] and shadowing and depends on the details of the environment. In a real-world situation, [[pathloss]] effects are also  considered.

===Doppler effect===
{{Main|Doppler radar|Pulse-Doppler radar}}
[[File:Doppler effect diagrammatic.svg|thumb|upright=1.2|Change of [[wavelength]] caused by motion of the source]]
Frequency shift is caused by motion that changes the number of wavelengths between the reflector and the radar. This can degrade or enhance radar performance depending upon how it affects the detection process. As an example, [[moving target indication]] can interact with Doppler to produce signal cancellation at certain radial velocities, which degrades performance.

Sea-based radar systems, [[semi-active radar homing]], [[active radar homing]], [[weather radar]], military aircraft, and [[radar astronomy]] rely on the Doppler effect to enhance performance. This produces information about target velocity during the detection process. This also allows small objects to be detected in an environment containing much larger nearby slow moving objects.

Doppler shift depends upon whether the radar configuration is active or passive. Active radar transmits a signal that is reflected back to the receiver. Passive radar depends upon the object sending a signal to the receiver.

The Doppler frequency shift for active radar is as follows, where <math>F_D</math> is Doppler frequency, <math>F_T</math> is transmit frequency, <math>V_R</math> is radial velocity, and <math>C</math> is the speed of light:<ref>{{cite news|title=Exploration: The Doppler Effect|author=M. Castelaz|publisher=Pisgah Astronomical Research Institute}}</ref>

:<math>F_D =  2 \times F_T \times \left (\frac {V_R}{C} \right)</math>.

Passive radar is applicable to [[electronic countermeasures]] and [[radio astronomy]] as follows:

:<math>F_D = F_T \times \left (\frac {V_R}{C} \right)</math>.

Only the radial component of the velocity is relevant. When the reflector is moving at right angle to the radar beam, it has no relative velocity. Objects moving parallel to the radar beam produce the maximum Doppler frequency shift.

When the transmit frequency (<math>F_T</math>) is pulsed, using a pulse repeat frequency of <math>F_R</math>, the resulting frequency spectrum will contain harmonic frequencies above and below <math>F_T</math> with a distance of <math>F_R</math>. As a result, the Doppler measurement is only non-ambiguous if the Doppler frequency shift is less than half of <math>F_R</math>, called the [[Nyquist frequency]], since the returned frequency otherwise cannot be distinguished from shifting of a harmonic frequency above or below, thus requiring:

:<math>|F_D| < \frac {F_R}{2}</math>

Or when substituting with <math>F_D</math>:

:<math>|V_R| < \frac {F_R \times \frac {C}{F_T}}{4}</math>

As an example, a Doppler weather radar with a pulse rate of 2&nbsp;kHz and transmit frequency of 1&nbsp;GHz can reliably measure weather speed up to at most {{convert|150|m/s|mph|abbr=on}}, thus cannot reliably determine radial velocity of aircraft moving {{convert|1000|m/s|mph|abbr=on}}.