SNR estimate from radar equation

Since R2021a

## Syntax

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

## Description

example

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

example

``` `SNR = radareqsnr(lambda,tgtrng,Pt,tau,Name,Value)` estimates the output `SNR` at the receiver with additional options specified by one or more Name,Value pair arguments.```

## Examples

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Estimate the output SNR for a target with an RCS of 1 m² at a range of 50 km. The system is a monostatic radar operating at 1 GHz with a peak transmit power of 1 MW and pulse width of 0.2 μs. The transmitter and receiver gain is 20 dB. The system temperature has the default value of 290 K.

```fc = 1.0e9; lambda = physconst('LightSpeed')/fc; tgtrng = 50e3; Pt = 1e6; tau = 0.2e-6; snr = radareqsnr(lambda,tgtrng,Pt,tau)```
```snr = 5.5868 ```

Estimate the output SNR for a target with an RCS of 0.5 m² at 100 km. The system is a monostatic radar operating at 10 GHz with a peak transmit power of 1 MW and pulse width of 1 μs. The transmitter and receiver gain is 40 dB. The system temperature is 300 K and the loss factor is 3 dB.

```fc = 10.0; T = 300.0; lambda = physconst('LightSpeed')/10e9; snr = radareqsnr(lambda,100e3,1e6,1e-6,'RCS',0.5, ... 'Gain',40,'Ts',T,'Loss',3)```
```snr = 14.3778 ```

Estimate the output SNR for a target with an RCS of 1 m². The radar is bistatic. The target is located 50 km from the transmitter and 75 km from the receiver. The radar operating frequency is 10.0 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 = 10.0e9; lambda = physconst('LightSpeed')/fc; tau = 1e-6; Pt = 1e6; txrvRng =[50e3 75e3]; Gain = [40 20]; snr = radareqsnr(lambda,txrvRng,Pt,tau,'Gain',Gain)```
```snr = 9.0547 ```

## Input Arguments

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Wavelength of radar operating frequency, specified as a positive scalar. The wavelength is the ratio of the wave propagation speed to frequency. Units are in meters. 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:

`$\lambda =\frac{c}{f}$`

Data Types: `double`

Target ranges for a monostatic or bistatic radar.

• Monostatic radar - the transmitter and receiver are co-located. `tgtrng` is a real-valued positive scalar or length-J real-valued positive column vector. J is the number of targets.

• Bistatic radar - the transmitter and receiver are separated. `tgtrng` is a 1-by-2 row vector with real-valued positive elements or a J-by-2 matrix with real-valued positive elements. J is the number of targets. Each row of `tgtrng` has the form ```[TxRng RxRng]```, where `TxRng` is the range from the transmitter to the target and `RxRng` is the range from the receiver to the target.

Units are in meters.

Data Types: `double`

Transmitter peak power, specified as a positive scalar. Units are in watts.

Data Types: `double`

Single pulse duration, specified as a positive scalar. Units are in seconds.

Data Types: `double`

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `'RCS',5.0,'Ts',295`

Radar cross section specified as a positive scalar or length-J vector of positive values. J is the number of targets. The target RCS is nonfluctuating (Swerling case 0). Units are in square meters.

Data Types: `double`

System noise temperature, specified as a positive scalar. The system noise temperature is the product of the system temperature and the noise figure. Units are in Kelvin.

Data Types: `double`

Transmitter and receiver gains, specified as a scalar or real-valued 1-by-2 row vector. When the transmitter and receiver are co-located (monostatic radar), `Gain` is a real-valued scalar. Then, the transmit and receive gains are equal. When the transmitter and receiver are not co-located (bistatic radar), `Gain` is a 1-by-2 row vector with real-valued elements. If `Gain` is a two-element row vector it has the form `[TxGain RxGain]` representing the transmit antenna and receive antenna gains.

Example: `[15,10]`

Data Types: `double`

System losses, specified as a scalar. Units are in dB.

Example: `1`

Data Types: `double`

Atmospheric absorption losses for the transmit and receive paths.

• When the absorption is a scalar or length-J column vector, the loss specifies the atmospheric absorption loss for a one-way path.

• When the absorption is a 1-by-2 row vector or J-by-2 column vector, the first column specifies the atmospheric absorption loss for the transmit path and the second column of contains the atmospheric absorption loss for the receive path

Example: `[10,20]`

Data Types: `double`

Propagation factor for the transmit and receive paths.

• When the propagation factor is a scalar or length-J column vector, the propagation factor is specified for a one-way path.

• When the propagation factor is a 1-by-2 row vector or J-by-2 column vector, the first column specifies the propagation factor for the transmit path and the second column of contains the propagation factor for the receive path

Units are in dB.

Example: `[10,20]`

Data Types: `double`

Custom loss factors specified as a scalar or length-J column vector of real values. J is the number of targets. These factors contribute to the reduction of the received signal energy and can include range-dependent STC, eclipsing, and beam-dwell factors. Units are in dB.

Example: `[10,20]`

Data Types: `double`

## Output Arguments

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Minimum output signal-to-noise ratio at the receiver, returned as a scalar. Units are in dB.

Data Types: `double`

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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 — Transmit antenna gain

• Gr — Receive antenna gain. If the radar is monostatic, the transmit and receive antenna gains are identical.

• λ — Radar 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. This value is denoted by Ts, so that Ts=TFn .

### Receiver Output SNR

Define the output SNR. The receiver output SNR is:

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

You can derive this expression using the following equations:

### Theoretical Maximum Detectable Range

Compute the maximum detectable range of a target.

For monostatic radars, the range from the target to the transmitter and receiver is identical. Denoting this range by R, you can express this relationship as ${R}^{4}={R}_{t}^{2}{R}_{r}^{2}$.

Solving for R

`$R={\left(\frac{N{P}_{t}\tau {G}_{t}{G}_{r}{\lambda }^{2}\sigma }{{P}_{r}{\left(4\pi \right)}^{3}k{T}_{s}L}\right)}^{1/4}$`

For bistatic radars, the theoretical maximum detectable range is the geometric mean of the ranges from the target to the transmitter and receiver:

`$\sqrt{{R}_{t}{R}_{r}}={\left(\frac{N{P}_{t}\tau {G}_{t}{G}_{r}{\lambda }^{2}\sigma }{{P}_{r}{\left(4\pi \right)}^{3}k{T}_{s}L}\right)}^{1/4}$`

## 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.

## Version History

Introduced in R2021a