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.

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;
maxrng2 = radareqrng(lambda,SNR,Pt,tau,'Gain',Gain,...'RCS',RCS,'Loss',Loss);

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

G_{r} —
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

R_{t} —
Range from the transmitter to the target

R_{r} —
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 10^{x/10} where x denotes
the variable. For example, the default loss factor of 0 dB results
in a loss term of 10^{0/10}=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:

$$P(f)=kT$$

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 F_{n} 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 T_{s},
so that T_{s}=TF_{n}.

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 $${R}^{4}={R}_{t}^{2}{R}_{r}^{2}$$.