# SNR ratio in radar for stationary targets

Is there a mathematical model which provides relationship between the Signal to Noise Ratio in radar measurements and velocity/range of targets. i.e.

1. How SNR varies with the speed of targets?
2. How SNR varies with speed of the radar?
3. How SNR varies range of target?

Is the SNR very low when the target and the radar, both, are stationary?

Without a lot more specific description of the system and environment it's really not possible to provide a meaningful answer to the first two questions. To a 1st order, the signal to noise ratio (SNR) does not vary with the relative range rate of the radar and target, but other effects, such as Doppler shifting, can impact the resulting SNR after processing, but that depends on the specifics of the processing. There are also effects, such as interference from clutter that can degrade performance that are velocity dependent, but those are usually considered a degradation in signal to interference plus noise rather than a reduction of SNR.

For the third question, the following relationship, known as the "radar range equation" applies (ref: The Radar Equation):

$$SNR = \frac{S}{N} = \frac{P_tG^2\lambda^2\sigma}{(4\pi)^3R^4kT_SB_nL}$$

where $$S$$ is the signal power, $$N$$ is the noise power, $$P_t$$ is the transmitter power, $$G$$ is the antenna gain (assuming transmit and receive gain are the same), $$\lambda$$ is the wavelength, $$\sigma$$ is the radar cross section of the target, $$R$$ is the distance (range) between the radar and target, $$k$$ is the Boltzmann constant, $$T_S$$ is the system noise temperature, $$B_n$$ is the noise bandwidth, and $$L$$ is any additional system losses.

As can be seen from this expression, the SNR will vary inversely with the 4th power of the range to the target and will be independent of the velocity of either the target or radar.

• While I understand the inverse relation between range of target and power, let us take just the clutter for example when the target is stationary. Will not the SINR be bad enough to make it difficult to distinguish between stationary target and clutter? Jul 30 at 17:33
• @user146290 This is one of the classic problems in radar. However you mentioned SNR, which in this sense is strictly the signal against thermal noise. If you want to know more about a signal against noise and clutter, that's a different scenario that takes more consideration. Jul 30 at 22:11
• How SNR varies range of target?

For a typical target and radar, by 1/(range ** 4) (see @GrapefruitisAwesome's answer).

This is because in a typical radar, a target looks like a point, so there's a 1/(range ** 2) loss from the transmit antenna to the target, which is multiplied by a 1/(range ** 2) loss from the target's return to the receiving antenna.

• How SNR varies with the speed of targets? How SNR varies with speed of the radar? Is the SNR very low when the target and the radar, both, are stationary?

For a raw radar signal, none of these matter unless the relative motion of the target and radar set are so great that the signal goes out of the bandwidth of the receiver because of doppler. Speeds would have to be ridiculously high for this to be an issue, however.

There are radar sets, such as police radar and automotive driving-aid radar that sense range and velocity, and filter out any returns that are not moving with respect to the radar set, or are not moving with respect to the ground. This is, however, not a function of the raw radar signal -- it's a design feature of the radar set that is intentionally introduced, because such returns are considered to be clutter.

• With reference to the motion w.r.t. ground, doesn’t it mean that when the target is stationary, there will be no doppler shift and will have the same signature as roadside clutter? Jul 30 at 17:23
• Yes. And there have been instances of ADAS (automatic driver-assist system) systems driving cars into parked cars. Jul 30 at 19:34