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:

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

where the terms in the equation are:

• Pr — Received power in watts.

• Pt — Peak transmit power in watts.

• Gt — Transmitter gain.

• 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.

• Rt — Range from the transmitter to the target.

• Rr — Range from the receiver to the target. If the radar is monostatic, the transmitter and receiver ranges are identical.

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

`$P\left(f\right)=kT$`

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 Fn 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 Ts, so that Ts = TFn .

Using the equation for the received signal power and the output noise power, the receiver output SNR is:

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

Solving for the required peak transmit power:

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

The preceding equations are implemented in the Phased Array System Toolbox™ by the functions: `radareqpow`, `radareqrng`, and `radareqsnr`. These functions and the equations on which they are based are valuable tools in radar system design and analysis.

This example shows how to compute the required peak transmit power using the radar equation. You implement a noncoherent detector with a monostatic radar operating at 5 GHz. Based on the noncoherent integration of 10 1μs pulses, you want to achieve a detection probability of 0.9 with a maximum false-alarm probability of $1{0}^{-6}$ for a target with a nonfluctuating radar cross section (RCS) of $1{m}^{2}$ at 30 km. The transmitter gain is 30 dB. Determine the required SNR at the receiver and use the radar equation to calculate the required peak transmit power.

Use Albersheim's equation to determine the required SNR for the specified detection and false-alarm probabilities

```Pd = 0.9; Pfa = 1e-6; NumPulses = 10; SNR = albersheim(Pd,Pfa,10)```
```SNR = 4.9904 ```

The required SNR is approximately 5 dB. Use the function `radareqpow` to determine the required peak transmit power in watts.

```tgtrng = 30e3; fc = 5e9; c = physconst('Lightspeed'); lambda = c/fc; RCS = 1; pulsedur = 1e-6; G = 30; Pt = radareqpow(lambda,tgtrng,SNR,pulsedur,'rcs',RCS,'gain',G)```
```Pt = 5.6485e+03 ```

The required peak power is approximately 5.6 kW.

### Maximum Detectable Range for a Monostatic Radar

Assume that the minimum detectable SNR at the receiver of a monostatic radar operating at 1 GHz is 13 dB. Use the radar equation to determine the maximum detectable range for a target with a nonfluctuating RCS of $0.5{m}^{2}$ if the radar has a peak transmit power of 1 MW. Assume the transmitter gain is 40 dB and the radar transmits a pulse that is 0.5μs in duration.

```tau = 0.5e-6; G = 40; RCS = 0.5; Pt = 1e6; lambda = 3e8/1e9; SNR = 13; maxrng = radareqrng(lambda,SNR,Pt,tau,'rcs',RCS,'gain',G)```
```maxrng = 3.4516e+05 ```

The maximum detectable range is approximately 345 km.

Estimate the output SNR for a target with an RCS of $1{m}^{2}$. The radar is bistatic. The target is located 50 km from the transmitter and 75 km from the receiver. The radar operating frequency is 10 GHz. The transmitter has a peak transmit power of 1 MW with a gain of 40 dB. The pulse width is 1 μs. The receiver gain is 20 dB.

```fc = 10e9; lambda = physconst('LightSpeed')/10e9; tau = 1e-6; Pt = 1e6; TxRvRng =[50e3 75e3]; Gain = [40 20]; snr = radareqsnr(lambda,TxRvRng,Pt,tau,'Gain',Gain)```
```snr = 9.0547 ```

The estimated SNR is approximately 9 dB.