Radar
In the common case where the transmitter and the receiver are at the same location, Rt = Rr and the term Rt² Rr² can be replaced by R4, where R is the range.
This yields:

  
    
      
        
          P
          
            r
          
        
        =
        
          
            
              
                P
                
                  t
                
              
              
                G
                
                  t
                
              
              
                A
                
                  r
                
              
              σ
              
                F
                
                  4
                
              
            
            
              
                
                  (
                  4
                  π
                  )
                
                
                  2
                
              
              
                R
                
                  4
                
              
            
          
        
        .
      
    
    {\displaystyle P_{r}={{P_{t}G_{t}A_{r}\sigma F^{4}} \over {{(4\pi )}^{2}R^{4}}}.}
  

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, 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 and shadowing and depends on the details of the environment. In a real-world situation, pathloss effects are also  considered.

Doppler effect: 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 
  
    
      
        
          F
          
            D
          
        
      
    
    {\displaystyle F_{D}}
  
 is Doppler frequency, 
  
    
      
        
          F
          
            T
          
        
      
    
    {\displaystyle F_{T}}
  
 is transmit frequency, 
  
    
      
        
          V
          
            R
          
        
      
    
    {\displaystyle V_{R}}
  
 is radial velocity, and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is the speed of light:

  
    
      
        
          F
          
            D
          
        
        =
        2
        ×
        
          F
          
            T
          
        
        ×
        
          (
          
            
              
                V
                
                  R
                
              
              C
            
          
          )
        
      
    
    {\displaystyle F_{D}=2\times F_{T}\times \left({\frac {V_{R}}{C}}\right)}
  
.
Passive radar is applicable to electronic countermeasures and radio astronomy as follows:

  
    
      
        
          F
          
            D
          
        
        =
        
          F
          
            T
          
        
        ×
        
          (
          
            
              
                V
                
                  R
                
              
              C
            
          
          )
        
      
    
    {\displaystyle F_{D}=F_{T}\times \left({\frac {V_{R}}{C}}\right)}
  
.
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 (
  
    
      
        
          F
          
            T
          
        
      
    
    {\displaystyle F_{T}}
  
) is pulsed, using a pulse repeat frequency of 
  
    
      
        
          F
          
            R
          
        
      
    
    {\displaystyle F_{R}}
  
, the resulting frequency spectrum will contain harmonic frequencies above and below 
  
    
      
        
          F
          
            T
          
        
      
    
    {\displaystyle F_{T}}
  
 with a distance of 
  
    
      
        
          F
          
            R
          
        
      
    
    {\displaystyle F_{R}}
  
. As a result, the Doppler measurement is only non-ambiguous if the Doppler frequency shift is less than half of 
  
    
      
        
          F
          
            R
          
        
      
    
    {\displaystyle F_{R}}
  
, called the Nyquist frequency, since the returned frequency otherwise cannot be distinguished from shifting of a harmonic frequency above or below, thus requiring:

  
    
      
        
          |
        
        
          F
          
            D
          
        
        
          |
        
        <
        
          
            
              F
              
                R
              
            
            2
          
        
      
    
    {\displaystyle |F_{D}|<{\frac {F_{R}}{2}}}
  

Or when substituting with 
  
    
      
        
          F
          
            D
          
        
      
    
    {\displaystyle F_{D}}
  
:

  
    
      
        
          |
        
        
          V
          
            R
          
        
        
          |
        
        <
        
          
            
              
                F
                
                  R
                
              
              ×
              
                
                  C
                  
                    F
                    
                      T
                    
                  
                
              
            
            4
          
        
      
    
    {\displaystyle |V_{R}|<{\frac {F_{R}\times {\frac {C}{F_{T}}}}{4}}}
  

As an example, a Doppler weather radar with a pulse rate of 2 kHz and transmit frequency of 1 GHz can reliably measure weather speed up to at most 150 m/s (340 mph), thus cannot reliably determine radial velocity of aircraft moving 1,000 m/s (2,200 mph).

Polarization: In all electromagnetic radiation, the electric field is perpendicular to the direction of propagation, and the electric field direction is the polarization of the wave. For a transmitted radar signal, the polarization can be controlled to yield different effects. Radars use horizontal, vertical, linear, and circular polarization to detect different types of reflections. For example, circular polarization is used to minimize the interference caused by rain. Linear polarization returns usually indicate metal surfaces. Random polarization returns usually indicate a fractal surface, such as rocks or soil, and are used by navigation radars.

Limiting factors: 

Beam path and range: A radar beam follows a linear path in vacuum but follows a somewhat curved path in atmosphere due to variation in the refractive index of air, which is called the radar horizon. Even when the beam is emitted parallel to the ground, the beam rises above the ground as the curvature of the Earth sinks below the horizon. Furthermore, the signal is attenuated by the medium the beam crosses, and the beam disperses.
The maximum range of conventional radar can be limited by a number of factors:

Line of sight, which depends on the height above the ground. Without a direct line of sight, the path of the beam is blocked.
The maximum non-ambiguous range, which is determined by the pulse repetition frequency. The maximum non-ambiguous range is the distance the pulse can travel to and return from before the next pulse is emitted.
Radar sensitivity and the power of the return signal as computed in the radar equation. This component includes factors such as the environmental conditions and the size (or radar cross section) of the target.

Noise: Signal noise is an internal source of random variations in the signal, which is generated by all electronic components.
Reflected signals decline rapidly as distance increases, so noise introduces a radar range limitation. The noise floor and signal-to-noise ratio are two different measures of performance that affect range performance. Reflectors that are too far away produce too little signal to exceed the noise floor and cannot be detected. Detection requires a signal that exceeds the noise floor by at least the signal-to-noise ratio.
Noise typically appears as random variations superimposed on the desired echo signal received in the radar receiver. The lower the power of the desired signal, the more difficult it is to discern it from the noise. The noise figure is a measure of the noise produced by a receiver compared to an ideal receiver, and this needs to be minimized.
Shot noise is produced by electrons in transit across a discontinuity, which occurs in all detectors. Shot noise is the dominant source in most receivers. There will also be flicker noise caused by electron transit through amplification devices, which is reduced using heterodyne amplification. Another reason for heterodyne processing is that for fixed fractional bandwidth, the instantaneous bandwidth increases linearly in frequency. This allows improved range resolution. The one notable exception to heterodyne (downconversion) radar systems is ultra-wideband radar. Here a single cycle, or transient wave, is used similar to UWB communications, see List of UWB channels.
Noise is also generated by external sources, most importantly the natural thermal radiation of the background surrounding the target of interest. In modern radar systems, the internal noise is typically about equal to or lower than the external noise. An exception is if the radar is aimed upwards at clear sky, where the scene is so "cold" that it generates very little thermal noise. The thermal noise is given by kB T B, where T is temperature, B is bandwidth (post matched filter) and kB is the Boltzmann constant.