# Documentation

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## Syntax

`SNR = radareqsnr(lambda,tgtrng,Pt,tau)SNR = radareqsnr(...,Name,Value)`

## Description

`SNR = radareqsnr(lambda,tgtrng,Pt,tau)` estimates the output signal-to-noise ratio (SNR) at the receiver based on the wavelength `lambda` in meters, the range `tgtrng` in meters, the peak transmit power `Pt` in watts, and the pulse width `tau` in seconds.

`SNR = radareqsnr(...,Name,Value)` estimates the output SNR at the receiver 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. `Pt` Transmitter peak power in watts. `tau` Single pulse duration in seconds.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

 `'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'` Target 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 effective noise temperature and the noise figure. Default: 290 kelvin

## Output Arguments

 `SNR` The estimated output signal-to-noise ratio at the receiver in decibels. `SNR` is 10log10(Pr/N). The ratio Pr/N is defined in Receiver Output SNR.

## Examples

Estimate the output SNR for a target with an RCS of 1 square meter at a range of 50 kilometers. The system is a monostatic radar operating at 1 gigahertz with a peak transmit power of 1 megawatt and pulse width of 0.2 microseconds. The transmitter and receiver gain is 20 decibels and the system temperature is 290 kelvin.

```lambda = physconst('LightSpeed')/1e9; tgtrng = 50e3; Pt = 1e6; tau = 0.2e-6; snr = radareqsnr(lambda,tgtrng,Pt,tau);```

Estimate the output SNR for a target with an RCS of 0.5 square meters at 100 kilometers. The system is a monostatic radar operating at 10 gigahertz with a peak transmit power of 1 megawatt and pulse width of 1 microsecond. The transmitter and receiver gain is 40 decibels. The system temperature is 300 kelvin and the loss factor is 3 decibels.

```lambda = physconst('LightSpeed')/10e9; snr = radareqsnr(lambda,100e3,1e6,1e-6,'RCS',0.5,... 'Gain',40,'Ts',300,'Loss',3);```

Estimate the output SNR for a target with an RCS of 1 square meter. The radar is bistatic. The target is located 50 kilometers from the transmitter and 75 kilometers from the receiver. The radar operating frequency is 10 gigahertz. The transmitter has a peak transmit power of 1 megawatt with a gain of 40 decibels. The pulse width is 1 microsecond. The receiver gain is 20 decibels.

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

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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 — Nonfluctuating target radar cross section in square meters

• L — General loss factor in decibels that accounts for both system and propagation losses

• Rt — Range from the transmitter to the target in meters

• Rr — Range from the receiver to the target in meters. 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 value in decibels. For example, the default loss factor of 0 dB results in a loss term equal to one in the equation (100/10).

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 .

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

You can derive this expression using the following equations:

## 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.