radareqrng

R2026b

Maximum theoretical range estimate

Syntax

maxrng = radareqrng(lambda,SNR,Pt,tau)

maxrng = radareqrng(lambda,SNR,Pt,tau,Name,Value)

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(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

Estimate Maximum Detectable Range

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 = radareqrng(lambda,SNR,Pt,tau)

maxrng = 
4.1057e+04

Estimate Maximum Detectable Range With Target RCS

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

lambda — Radar wavelength (m)

positive scalar

Radar wavelength, specified as a positive scalar. Units are in meters.

To convert operating frequency to wavelength, use freq2wavelen. 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:

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

Data Types: double

SNR — Input signal-to-noise ratio at receiver

scalar | length-J vector

Input signal-to-noise ratio (SNR) at the receiver, specified as a scalar or length-J real-valued vector, where J is the number of targets. This input argument corresponds to the P_r/N term in the Point Target Radar Range Equation. Units are in dB.

Data Types: double

Pt — Transmitted peak power

positive scalar

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

Data Types: double

tau — Pulse width or time step (s)

positive scalar

Pulse width, 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: SNR,10

RCS — Target radar cross section (m²)

1 (default) | positive scalar | positive length-J vector

Target radar cross section specified as a positive scalar or length-J vector of positive values, where J is the number of targets. The target RCS is nonfluctuating (Swerling case 0). Units are in meters squared (m²).

Data Types: double

Ts — System noise temperature (K)

290 (default) | positive scalar

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

Data Types: double

Gain — Transmitter and receiver gains (dB)

20 (default) | scalar | real-valued 1-by-2 row vector

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. Units are in decibels (dB).

Example: [15,10]

Data Types: double

Loss — System losses (dB)

0 (default) | scalar | length-J real-valued vector

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

Example: 1

Data Types: double

CustomFactor — Custom factor (dB)

0 (default) | scalar | length-J column vector

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 Sensitive Time Control (STC), eclipsing, and beam-dwell factors. Units are in dB.

Example: [10,20]

Data Types: double

unitstr — Units of the estimated maximum theoretical range

'm' (default) | 'km''mi''nmi'

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

• 'm' meters

• 'km' kilometers

• 'mi' miles

• 'nmi' nautical miles (U.S.)

Output Arguments

maxrng — Estimated theoretical maximum detectable range

positive scalar

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.

More About

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

$$P_r=\frac{P_t{G}_t{G}_r{\lambda }^2\sigma }{{(4\pi )}^3{R}_t^2{R}_r^{2}L},$$

where the terms in the equation are:

• P_t — Peak transmit power in watts.

• G_t — Transmit antenna gain, as a dimensionless loss factor expressed in linear units.

• G_r — 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 to account for both system and propagation loss, as a dimensionless loss factor expressed in linear units.

• R_t — Range from the transmitter to the target in meters.

• R_r — Range from the receiver to the target in meters. If the radar is monostatic, the transmitter and receiver ranges are identical.

You can convert losses expressed in decibels to a loss factor in dimensionless linear units using 10^x/10, where x denotes the loss in dB. For example, the default loss of 0 dB results in a loss factor of 10^0/10=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:

$$P(f)=kT,$$

where k is the Boltzmann constant and T is the effective noise temperature. The receiver integrates noise over the duration of the received pulse. For a pulse of duration τ, the resulting noise power is inversely proportional to the pulse duration 1/τ, which is equivalent to the processing gain of a matched filter for a rectangular pulse. The total noise power at the output of the receiver is:

$$N=\frac{kT{F}_n}{\tau },$$

where F_n is the receiver noise factor or ratio of the input SNR to output SNR.

The product of the effective noise temperature and the receiver noise factor is referred to as the system temperature. This value is denoted by T_s, so that T_s=TF_n.

Receiver SNR

The receiver SNR is:

$$\frac{P_r}{N}=\frac{P_t\tau \text{}{G}_t{G}_r{\lambda }^2\sigma }{{(4\pi )}^{3}k{T}_s{R}_t^2{R}_r^{2}L}.$$

You can derive this expression using the following equations:

• Received signal power in Point Target Radar Range Equation

• Output noise power in Receiver Output Noise Power

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

$$R={(\frac{N{P}_t\tau G_t{G}_r{\lambda }^2\sigma }{P_r{(4\pi )}^{3}k{T}_{s}L})}^{1/4}$$

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

$$\sqrt{R_t{R}_r}={(\frac{N{P}_t\tau G_t{G}_r{\lambda }^2\sigma }{P_r{(4\pi )}^{3}k{T}_{s}L})}^{1/4}$$

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Does not support variable-size inputs.

Version History

Introduced in R2021a