Range-dependent SNR using search radar equation

## Syntax

``snr = radareqsearchsnr(range,pap,omega,tsearch)``
``snr = radareqsearchsnr(___,Name,Value)``

## Description

````snr = radareqsearchsnr(range,pap,omega,tsearch)` computes the available signal-to-noise ratio (SNR), `snr`, for a surveillance radar based on the range, `range`, power-aperture product, `pap`, solid angular search volume, `omega`, and search time, `tsearch`.```

example

````snr = radareqsearchsnr(___,Name,Value)` computes the available SNR with additional options specified by one or more name-value arguments. For example, `'Loss',6` specifies system losses as 6 decibels.```

## Examples

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Compute the available signal-to-noise ratio (SNR) for a search radar at a target range of `1000` kilometers with a power-aperture product of $3×{10}^{6}$ $\mathrm{W}\cdot {\mathrm{m}}^{2}$. Assume the search time is `10` seconds, the RCS of the target is `–10` dBsm, the system noise temperature is `487` Kelvin, and the total system loss is `6` decibels.

```range = 1000e3; pap = 3e6; tsearch = 10; rcs = db2pow(-10); ts = 487; loss = 6;```

The radar surveys a region of space with azimuths in the range [0,30] degrees and elevations in the range [0,45] degrees. Find the solid angular search volume in steradians by using the `solidangle` function.

```az = [0;30]; el = [0;45]; omega = solidangle(az,el); ```

Calculate the available SNR.

`snr = radareqsearchsnr(range,pap,omega,tsearch,'RCS',rcs,'Ts',ts,'Loss',loss)`
```snr = 13.8182 ```

Plot the available signal-to-noise ratio (SNR) as a function of the range for a search radar with a power-aperture product of $2.5×{10}^{6}$ $\mathrm{W}\cdot {\mathrm{m}}^{2}$. Incorporate path loss due to absorption into the calculation of the SNR.

Specify the ranges as 1000 linearly-spaced values in the interval [0,1000] kilometers. Assume the search volume is `1.5` steradians and the search time is `12` seconds.

```range = linspace(1,1000e3,1000); pap = 2.5e6; omega = 1.5; tsearch = 12;```

Find the path loss due to atmospheric gaseous absorption by using the `gaspl` function. Specify the radar operating frequency as `10` GHz, the temperature as `15` degrees Celsius, the dry air pressure as `1013` hPa, and the water vapour density as `7.5` $\mathrm{g}/{\mathrm{m}}^{3}$.

```freq = 10e9; temp = 15; pressure = 1013e2; density = 7.5; loss = gaspl(range,freq,temp,pressure,density);```

Compute the available SNR. By default, the target RCS is 1 square meter.

`snr = radareqsearchsnr(range,pap,omega,tsearch,'AtmosphericLoss',loss);`

Plot the SNR as a function of the range. Before plotting, convert the range from meters to kilometers.

```plot(range*0.001,snr) grid on ylim([-10 60]) xlabel('Range (km)') ylabel('SNR (dB)') title('SNR vs Range')```

## Input Arguments

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Range, specified as a scalar or a length-J vector of positive values, where J is the number of range samples. Units are in meters.

Example: `1e5`

Data Types: `double`

Power-aperture product, specified as a scalar or a length-J vector of positive values. Units are in W·m2.

Example: `3e6`

Data Types: `double`

Solid angular search volume, specified as a scalar. Units are in steradians.

Given the elevation and azimuth ranges of a region, you can find the solid angular search volume by using the `solidangle` function.

Example: `0.3702`

Data Types: `double`

Search time, specified as a scalar. Units are in seconds.

Example: `10`

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: `'Ts',487` specifies the system noise temperature as 487 Kelvin

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

Data Types: `double`

System noise temperature, specified as a positive scalar. Units are in Kelvin.

Data Types: `double`

System losses, specified as a scalar or a length-J vector of real values. Units are in decibels.

Example: `1`

Data Types: `double`

One-way atmospheric absorption loss, specified as a scalar or a length-J vector of real values. Units are in decibels.

Example: `[10,20]`

Data Types: `double`

One-way propagation factor for the transmit and receive paths, specified as a scalar or a length-J vector of real values. Units are in decibels.

Example: `[10,20]`

Data Types: `double`

Custom loss factors, specified as a scalar or a length-J vector of real values. These factors contribute to the reduction of the received signal energy and can include range-dependent sensitivity time control (STC), eclipsing, and beam-dwell factors. Units are in decibels.

Example: `[10,20]`

Data Types: `double`

## Output Arguments

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Available signal-to-noise ratio, returned as a scalar or a length-J column vector of real values, where J is the number of range samples. Units are in decibels.

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### SNR Form of Search Radar Equation

The signal-to-noise ratio form of the search radar equation, SNR, is:

`$SNR=\frac{{P}_{av}A{t}_{s}\sigma {F}^{2}{F}_{c}}{4\pi k{T}_{s}{R}^{4}{L}_{a}^{2}L\Omega }$`

where the terms of the equation are:

• Pav — Average transmit power in watts

• A — Antenna effective aperture in square meters

• ts — Search time in seconds

• σ — Nonfluctuating target radar cross section in square meters

• F — One-way propagation factor for the transmit and receive paths

• Fc — Combined range-dependent factors that contribute to the reduction of the received signal energy

• k — Boltzmann constant

• Ts — System temperature in Kelvin

• R — Target range in meters. The equation assumes the radar is monostatic.

• La — One-way atmospheric absorption loss

• L — Combined system losses

• Ω — Search volume in steradians

You can derive this equation based on assumptions about the SNR form of the standard radar equation. For more information about the SNR form of the standard radar equation, see the `radareqsnr` function. These are the assumptions:

• The radar is monostatic, so that R = Rt = Rr, where Rt is the range from the transmitter to the target and Rr is the range from the receiver to the target.

• The search time is the time the transmit beam takes to scan the entire search volume. As a result, you can express the search time, ts, in terms of the search volume, Ω, the area of the beam in steradians, Ωt, and the dwell time in seconds, Td.

`${t}_{s}={T}_{d}\frac{\Omega }{{\Omega }_{t}}$`

• The transmit antenna beam has an ideal rectangular shape. As a result, you can express the transmit antenna gain, Gt, in terms of the angular area of the antenna beam.

`${G}_{t}=\frac{4\pi }{{\Omega }_{t}}$`

• The receive antenna is ideal. This means you can express the receive antenna gain, Gr, in terms of the antenna effective aperture, A, and the radar operating frequency wavelength, λ.

`${G}_{r}=\frac{4\pi A}{{\lambda }^{2}}$`

## References

[1] Barton, David Knox. Radar Equations for Modern Radar. Artech House Radar Series. Boston, Mass: Artech House, 2013.

[2] Skolnik, Merrill I. Introduction to Radar Systems. Third edition. McGraw-Hill Electrical Engineering Series. Boston, Mass. Burr Ridge, IL Dubuque, IA: McGraw Hill, 2001.

## Version History

Introduced in R2021a