Maximum theoretical range estimate

## Description

example

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.

example

maxrng = radareqrng(lambda,SNR,Pt,tau,Name,Value) estimates the theoretical maximum detectable range with additional options specified by one or more Name,Value pair arguments.

## Examples

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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;
maxrng = 4.1057e+04

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 m². 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;
maxrng2 = radareqrng(lambda,SNR,Pt,tau,'Gain',Gain, ...
'RCS',RCS,'Loss',Loss)
maxrng2 = 1.9426e+05

## 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

Input signal-to-noise ratio (SNR) at the receiver, specified as a scalar or length-J real-valued vector. J is the number of targets. Units are in dB.

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 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: SNR,10

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

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

Units of the estimated maximum theoretical range, specified as one of:

• 'm' meters

• 'km' kilometers

• 'mi' miles

• 'nmi' nautical miles (U.S.)

## Output Arguments

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The estimated theoretical maximum detectable range, returned as a positive scalar. The units of maxrng are specified by unitstr. For bistatic radars, maxrng is the geometric mean of the range from the transmitter to the target and the receiver to the target.

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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.