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Peak power estimate from radar equation

## Syntax

```Pt = radareqpow(lambda,tgtrng,SNR,Tau) Pt = radareqpow(...,Name,Value) ```

## Description

`Pt = radareqpow(lambda,tgtrng,SNR,Tau)` estimates the peak transmit power required for a radar operating at a wavelength of `lambda` meters to achieve the specified signal-to-noise ratio `SNR` in decibels for a target at a range of `tgtrng` meters. The target has a nonfluctuating radar cross section (RCS) of 1 square meter.

`Pt = radareqpow(...,Name,Value)` estimates the required peak transmit power with additional options specified by one or more `Name,Value` pair arguments.

## Input Arguments

 `lambda` Wavelength of radar operating frequency (in meters). The wavelength is the ratio of the wave propagation speed to frequency. For electromagnetic waves, the speed of propagation is the speed of light. Denoting the speed of light by c and the frequency (in hertz) of the wave by f, the equation for wavelength is:`$\lambda =\frac{c}{f}$` `tgtrng` Target range in meters. When the transmitter and receiver are colocated (monostatic radar), `tgtrng` is a real-valued positive scalar. When the transmitter and receiver are not colocated (bistatic radar), `tgtrng` is a 1-by-2 row vector with real-valued positive elements. The first element is the target range from the transmitter, and the second element is the target range from the receiver. `SNR` The minimum output signal-to-noise ratio at the receiver in decibels. `Tau` Single pulse duration in seconds.

### Name-Value Pair Arguments

 `'Gain'` Transmitter and receiver gain in decibels (dB). When the transmitter and receiver are colocated (monostatic radar), `Gain` is a real-valued scalar. The transmit and receive gains are equal. When the transmitter and receiver are not colocated (bistatic radar), `Gain` is a 1-by-2 row vector with real-valued elements. The first element is the transmitter gain and the second element is the receiver gain. Default: 20 `'Loss'` System loss in decibels (dB). `Loss` represents a general loss factor that comprises losses incurred in the system components and in the propagation to and from the target. Default: 0 `'RCS'` Radar cross section in square meters. The target RCS is nonfluctuating. Default: 1 `'Ts'` System noise temperature in kelvin. The system noise temperature is the product of the system temperature and the noise figure. Default: 290 kelvin

## Output Arguments

 `Pt` Transmitter peak power in watts.

## Examples

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Estimate the required peak transmit power required to achieve a minimum SNR of 6 dB for a target at a range of 50 km. The target has a nonfluctuating RCS of 1 m². The radar operating frequency is 1 GHz. The pulse duration is 1 μs.

```fc = 1.0e9; lambda = physconst('LightSpeed')/fc; tgtrng = 50e3; tau = 1e-6; SNR = 6; Pt = radareqpow(lambda,tgtrng,SNR,tau)```
```Pt = 2.1996e+05 ```

Estimate the required peak transmit power required to achieve a minimum SNR of 10 dB for a target with an RCS of 0.5 m² at a range of 50 km. The radar operating frequency is 10 GHz. The pulse duration is 1 μs. Assume a transmit and receive gain of 30 dB and an overall loss factor of 3 dB. The system temperature is 300 K.

```fc = 10.0e9; lambda = physconst('LightSpeed')/fc; Pt = radareqpow(lambda,50e3,10,1e-6,'RCS',0.5,... 'Gain',30,'Ts',300,'Loss',3)```
```Pt = 2.2809e+06 ```

Estimate the required peak transmit power for a bistatic radar to achieve a minimum SNR of 6 dB for a target with an RCS of 1 m². The target is 50 km from the transmitter and 75 km from the receiver. The radar operating frequency is 10 GHz and the pulse duration is 10 μs. The transmitter and receiver gains are 40 dB and 20 dB, respectively.

```fc = 10.0e9; lambda = physconst('LightSpeed')/fc; SNR = 6; tau = 10e-6; TxRng = 50e3; RvRng = 75e3; TxRvRng =[TxRng RvRng]; TxGain = 40; RvGain = 20; Gain = [TxGain RvGain]; Pt = radareqpow(lambda,TxRvRng,SNR,tau,'Gain',Gain)```
```Pt = 4.9492e+04 ```

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### Point Target Radar Range Equation

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. The model is deterministic and assumes isotropic radiators. 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:

• Pt — Peak transmit power in watts

• Gt — Transmitter gain in decibels

• Gr — Receiver gain in decibels. If the radar is monostatic, the transmitter and receiver gains are identical.

• λ — Radar operating frequency wavelength in meters

• σ — Target's nonfluctuating radar cross section in square meters

• L — General loss factor in decibels that accounts 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.

Terms expressed in decibels such as the loss and gain factors enter the equation in the form 10x/10 where x denotes the variable. For example, the default loss factor of 0 dB results in a loss term of 100/10=1.

### Receiver Output Noise Power

The equation for the power at the input to the receiver represents the signal term in the signal-to-noise 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 factor.

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 .

### Receiver Output SNR

Using the equation for the received signal power in Point Target Radar Range Equation and the output noise power in Receiver 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 peak transmit power

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

## References

[1] Richards, M. A. Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005.

[2] Skolnik, M. Introduction to Radar Systems. New York: McGraw-Hill, 1980.

[3] Willis, N. J. Bistatic Radar. Raleigh, NC: SciTech Publishing, 2005.