## Radar Equation

### Radar Equation Theory

The point target radar range equation estimates the power at the input to the receiver for a target of a given radar cross section at a specified range. In this equation, the signal model is assumed to be deterministic. The equation for the power at the input to the receiver is

$${P}_{r}=\frac{{P}_{t}{G}_{t}{G}_{r}{\lambda}^{2}\sigma}{{(4\pi )}^{3}{R}_{t}^{2}{R}_{r}^{2}L},$$

where the terms in the equation are:

*P*— Received power in watts._{r}*P*— Peak transmit power in watts._{t}*G*— Transmitter gain._{t}*G*— Receiver gain._{r}*λ*— Radar operating frequency wavelength in meters.*σ*— Target's nonfluctuating radar cross section in square meters.*L*— General loss factor to account for both system and propagation loss.*R*— Range from the transmitter to the target._{t}*R*— Range from the receiver to the target. If the radar is monostatic, the transmitter and receiver ranges are identical._{r}

The equation for the power at the input to the receiver represents the signal term in the signal-to-noise ratio (SNR). To model the noise term, assume the thermal noise in the receiver has a white noise power spectral density (PSD) given by

$$P(f)=kT,$$

where *k* is the Boltzmann constant and
*T* is the effective noise temperature. The receiver acts as a
filter to shape the white noise PSD. Assume that the magnitude squared receiver
frequency response approximates a rectangular filter with bandwidth equal to the
reciprocal of the pulse duration, 1/*τ*. The total noise power at the output of the receiver is

$$N=\frac{kT{F}_{n}}{\tau},$$

where *F _{n}* is the receiver

*noise figure*.

The product of the effective noise temperature and the receiver noise factor is referred to as the *system temperature* and is denoted by *T _{s}*, so that

*T*=

_{s}*TF*.

_{n}Using the equation for the received signal power and the output noise power, the receiver output SNR is

$$\frac{{P}_{r}}{N}=\frac{{P}_{t}\tau \text{}\text{\hspace{0.05em}}{G}_{t}{G}_{r}{\lambda}^{2}\sigma}{{(4\pi )}^{3}k{T}_{s}{R}_{t}^{2}{R}_{r}^{2}L}.$$

Solving for the required peak transmit power,

$${P}_{t}=\frac{{P}_{r}{(4\pi )}^{3}k{T}_{s}{R}_{t}^{2}{R}_{r}^{2}L}{N\tau {G}_{t}{G}_{r}{\lambda}^{2}\sigma}.$$

### Plot Vertical Coverage Pattern Using Default Parameters

Set the frequency to 100 MHz, the antenna height to 10 m, and the free-space range to 200 km. The antenna pattern, surface roughness, antenna tilt angle, and field polarization assume their default values as specified in the `AntennaPattern`

, `SurfaceRoughness`

, `TiltAngle`

, and `Polarization`

properties.

Obtain an array of vertical coverage pattern values and angles.

freq = 100e6; ant_height = 10; rng_fs = 200; [vcp,vcpangles] = radarvcd(freq,rng_fs,ant_height);

To see the vertical coverage pattern, omit the output arguments.

radarvcd(freq,rng_fs,ant_height);

### Compute Peak Power Using Radar Equation Calculator App

The `radarEquationCalculator`

app lets you determine key radar characteristics such as detection range, required peak transmit power, and SNR. The App works for monostatic and bistatic radars.

**Open radarEquationCalculator App**

When you type `radarEquationCalculator`

from the command line or select the app from the **App Toolstrip**, an interactive window opens. The default window shows a calculation of target range from SNR, power, and other parameters. You can then select various options to compute different radar parameters.

radarEquationCalculator

**Compute Required Peak Transmit Power of Monostatic Radar**

As an example, use the app to compute the required peak transmit power for a monostatic radar to detect a large target at 100 km. The radar operates at 10 GHz with a 40 dB antenna gain. Set the probability of detection to 0.9 and the probability of false alarm to 0.0001.

From the

**Calculation Type**drop-down list, choose`Peak Transmit Power`

Set the

**Wavelength**to`3`

cmSpecify the

**Pulse Width**as`2`

microsecondsAssume total

**System Losses**of`5`

dBAssuming the target is a large airplane, set

**Target Radar Cross Section**value to`100`

${\mathrm{m}}^{2}$Choose

**Configuration**as`Monostatic`

Set the

**Gain**to be`40`

dBOpen the

**SNR**boxSpecify the

**Probability of Detections**as`0.9`

Specify the

**Probability of False Alarm**as`0.0001`

The app shows that the required peak transmit power is 0.2095 W.