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. In this equation, the signal model is assumed to be deterministic. The equation for the power at the input to the receiver is:

where the terms in the equation are:

*P*— Received power in watts._{r}*P*— Peak transmit power in watts._{t}*G*— Transmitter gain._{t}*G*— Receiver gain._{r}*λ*— Radar operating frequency 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.*R*— Range from the transmitter to the target._{t}*R*— Range from the receiver to the target. If the radar is monostatic, the transmitter and receiver ranges are identical._{r}

The equation for the power at the input to the receiver represents the signal term in the signal-to-noise (SNR) ratio. To model the noise term, assume the thermal noise in the receiver has a white noise power spectral density (PSD) given by:

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:

where *F _{n} * is
the receiver

The product of the effective noise temperature and the receiver
noise factor is referred to as the *system temperature* and
is denoted by *T _{s}*, so that

Using the equation for the received signal power and the output noise power, the receiver output SNR is:

Solving for the required peak transmit power:

The preceding equations are implemented in the Phased Array System Toolbox™ by
the functions: `radareqpow`

, `radareqrng`

, and `radareqsnr`

.
These functions and the equations on which they are based are valuable
tools in radar system design and analysis.

This example shows how to compute the required peak transmit
power using the radar equation. You implement a noncoherent detector
with a monostatic radar operating at 5 GHz. Based on the noncoherent
integration of ten one-microsecond pulses, you want to achieve a detection
probability of 0.9 with a maximum false-alarm probability of 10^{–6} for
a target with a nonfluctuating radar cross section (RCS) of 1 m^{2} at
30 km. The transmitter gain is 30 dB. Determine the required SNR at
the receiver and use the radar equation to calculate the required
peak transmit power.

Use Albersheim's equation to determine the required SNR for the specified detection and false-alarm probabilities.

Pd = 0.9; Pfa = 1e-6; NumPulses = 10; SNR = albersheim(Pd,Pfa,10)

The required SNR is approximately 5 dB. Use the function `radareqpow`

to determine the required
peak transmit power in watts.

tgtrng = 30e3; % target range in meters lambda = 3e8/5e9; % wavelength of the operating frequency RCS = 1; % target RCS pulsedur = 1e-6; %pulse duration G = 30; % transmitter and receiver gain (monostatic radar) Pt = radareqpow(lambda,tgtrng,SNR,pulsedur,'rcs',RCS,'gain',G)

The required peak power is approximately 5.6 kW.

Assume that the minimum detectable SNR at the receiver of a
monostatic radar operating at 1 GHz is 13 dB. Use the radar equation
to determine the maximum detectable range for a target with a nonfluctuating
RCS of 0.5 m^{2} if the radar has a peak transmit
power of 1 MW. Assume the transmitter gain is 40 dB and the radar
transmits a pulse that is 0.5 μs in duration.

tau = 0.5e-6; % pulse duration G = 40; % transmitter and receiver gain (monostatic radar) RCS = 0.5; % target RCS Pt = 1e6; %peak transmit power in watts lambda = 3e8/1e9; SNR = 13; % required SNR in dB maxrng = radareqrng(lambda,SNR,Pt,tau,'rcs',RCS,'gain',G)

The maximum detectable range is approximately 345 km.

Estimate the output SNR for a target with an RCS of 1 m^{2}.
The radar is bistatic. The target is located 50 km from the transmitter
and 75 km from the receiver. The radar operating frequency is 10 GHz.
The transmitter has a peak transmit power of 1 MW with a gain of 40
dB. The pulse width is 1 μs. The receiver gain is 20 dB.

lambda = physconst('LightSpeed')/10e9; tau = 1e-6; Pt = 1e6; TxRvRng =[50e3 75e3]; Gain = [40 20]; snr = radareqsnr(lambda,TxRvRng,Pt,tau,'Gain',Gain);

The estimated SNR is approximately 9 dB.

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