Maximum detectable range using search radar equation

## Syntax

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

## Description

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

example

````range = radareqsearchrng(___,Name,Value)` computes the maximum detectable range 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 maximum detectable range at which a surveillance radar can detect a target.

The radar operates at a frequency of `2.5` GHz and transmits an average power of `2.1` kW. The gain of the receiving antenna is `34` decibels. Calculate the power-aperture product using these values.

```lambda = freq2wavelen(2.5e9); pav = 2100; g = 34; a = gain2aperture(g,lambda); pap = pav*a;```

The radar traverses a search volume with azimuths in the range [–180,180] degrees and elevations in the range [0,40] degrees. Find the solid angular search volume in steradians by using the `solidangle` function.

```az = [-180;180]; el = [0;40]; omega = solidangle(az,el); ```

The antenna rotates at a rate of `12.5` RPM. Assume the system noise temperature is `487` Kelvin, the total system loss is `20` decibels, and the minimum SNR required to make a detection is `13` decibels.

```tsearch = 60 / 12.5; ts = 487; loss = 20; snr = 13;```

Compute the maximum detectable range. By default, the target RCS is 1 square meter.

```R = radareqsearchrng(snr,pap,omega,tsearch,... 'Ts',ts,'Loss',loss,'unitstr','km') ```
```R = 80.7673 ```

## Input Arguments

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Required signal-to-noise ratio (SNR), specified as a scalar or a length-J vector of real values. Units are in decibels.

Example: `13`

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

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 `radareqsearchrng` 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`

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. Units are in decibels.

Example: `[10,20]`

Data Types: `double`

Range units, specified as one of the following values:

• `'m'` — Return range using meters

• `'km'` — Return range using kilometers

• `'mi'` — Return range using statute miles

• `'nmi'` — Return range using nautical miles (US)

If you do not specify range units, then the `radareqsearchrng` function returns ranges using meters.

Data Types: `string` | `char`

## Output Arguments

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Maximum detectable range, returned as a scalar or a length-J column vector of positive values. Units are in meters.

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### Maximum Detectable Range Form of Search Radar Equation

The maximum detectable range form of the search radar equation, R, is:

`$R={\left[\frac{{P}_{av}A{t}_{s}\sigma {F}^{2}{F}_{c}}{4\pi k{T}_{s}\left(SNR\right){L}_{a}^{2}L\Omega }\right]}^{1/4}$`

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

• SNR — Required signal-to-noise ratio

• La — One-way atmospheric absorption loss

• L — Combined system losses

• Ω — Search volume in steradians

You can derive this equation by rearranging the SNR form of the search radar equation. See the `radareqsearchsnr` function for more information.

## 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.

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