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# Documentation

Maximum theoretical range estimate

## Description

maxrng = radareqrng(lambda,SNR,Pt,Tau) estimates the theoretical maximum detectable range maxrng for a radar operating with a wavelength of lambda meters with a pulse duration of Tau seconds. The signal-to-noise ratio is SNR decibels, and the peak transmit power is Pt watts.

maxrng = radareqrng(...,Name,Value) estimates the theoretical maximum detectable range 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:$\lambda =\frac{c}{f}$ Pt Transmitter peak power in watts. 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 kelvins. The system noise temperature is the product of the system temperature and the noise figure. Default: 290 kelvin 'unitstr' The units of the estimated maximum theoretical range. unitstr is one of the following strings: 'km' kilometers'm' meters'nmi' nautical miles (U.S.) Default: 'm'

## Output Arguments

 maxrng The estimated theoretical maximum detectable range. The units of maxrng depends on the value of unitstr. By default maxrng is in meters. For bistatic radars, maxrng is the geometric mean of the range from the transmitter to the target and the receiver to the target.

## Examples

Estimate the theoretical maximum detectable range for a monostatic radar operating at 10 GHz using a pulse duration of 10 µs. Assume the output SNR of the receiver is 6 dB.

```lambda = physconst('LightSpeed')/10e9;
SNR = 6;
tau = 10e-6;
Pt = 1e6;

Estimate the theoretical maximum detectable range for a monostatic radar operating at 10 GHz using a pulse duration of 10 µs. The target RCS is 0.1 square meters. Assume the output SNR of the receiver is 6 dB. The transmitter-receiver gain is 40 dB. Assume a loss factor of 3 dB.

```lambda = physconst('LightSpeed')/10e9;
SNR = 6;
tau = 10e-6;
Pt = 1e6;
RCS = 0.1;
Gain = 40;
Loss = 3;
'RCS',RCS,'Loss',Loss);```

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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.

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 .

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

You can derive this expression using the following equations:

### Theoretical Maximum Detectable Range

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.