# beamdwellfactor

Range-dependent loss for rapidly scanning beam

## Description

example

fbd = beamdwellfactor(r,hpbw,scanrate) calculates the range-dependent beam-dwell factor fbd for an antenna at the specified range r, half-power beamwidth hpbw, and scan rate scanrate. The beamdwellfactor function assumes that the transmitter and receiver antennas have equal beamwidth and an ideal Gaussian antenna pattern with no side lobes.

## Examples

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Calculate the beam-dwell factor for a surveillance radar at 100 linearly-spaced ranges in the interval [0,100000] meters. Specify the beamwidth as 1 degree and the scan rate as 120 degrees per second.

r = linspace(0,100000);
hpbw = 1;
scanrate = 120;
fbd = beamdwellfactor(r,hpbw,scanrate);

Plot the beam-dwell factor as a function of range. Before plotting, convert the range from meters to kilometers.

plot(r*0.001,fbd)
grid on
xlabel('Range (km)')
ylabel('Beam-dwell Factor (dB)')

## Input Arguments

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Range in meters, specified as a scalar or vector.

Data Types: double

Half-power beamwidth of the antenna in degrees, specified as a scalar or vector. If hpbw is a vector, then scanrate must be a scalar or a vector of the same size.

Data Types: double

Scan rate of the antenna in degrees per second. If scanrate is a vector, then hpbw must be a scalar or a vector of the same size.

Data Types: double

## Output Arguments

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Range-dependent beam-dwell factor in dB, returned as a j-by-k matrix such that j is the size of r and k is the size of hpbw or scanrate, whichever is larger.

The rows of fbd correspond to the ranges in r. The columns depend on the sizes of hpbw and scanrate.

• If hpbw is a vector and scanrate is a scalar, then the columns of fbd correspond to the half-power beamwidths in hpbw.

• If hpbw is a scalar and scanrate is a vector, then the columns of fbd correspond to the scan rates in scanrate.

• If hpbw and scanrate are both vectors, then the columns of fbd correspond to both the half-power beamwidths in hpbw and the scan rates in scanrate.

Data Types: double

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### Beam-Dwell Factor

The beam-dwell factor accounts for the misalignment between transmitter and receiver beam axes when a scanning system has a high scan rate and long-range targets.

The equation for the beam-dwell factor, Fbd, is

${F}_{bd}=L\underset{-\pi }{\overset{\pi }{\int }}{f}^{2}\left(\theta \right){f}^{2}\left(\theta -\delta \right)d\theta$

where the terms in the equation are:

• L — Normalizing factor that brings Fbd to unity for = 0

• = td / t0 — Fractional beamwidth scanned during the delay, where:

• td = 2R / c — Time delay for a target, where R is the range and c is the wave propagation speed

• t0 = θ3 / ωs — The time the system takes to continuously scan through one beamwidth, where θ3 is the half-power beamwidth and ωs is the scan rate

• f(θ) — Antenna pattern

The beamdwellfactor function assumes an ideal Gaussian antenna pattern with no side lobes. The equation for the ideal Gaussian antenna pattern with no side lobes, f(θ), is:

$f\left(\theta \right)=\mathrm{exp}\left[-2\left(\mathrm{ln}2\right)\frac{{\theta }^{2}}{{\theta }_{3}^{2}}\right]$

## References

[1] Barton, David Knox. "Beam-Dwell Factor Fbd." In Radar Equations for Modern Radar, 362. Artech House Radar Series. Boston, Mass: Artech House, 2013.

[2] Barton, David Knox. "Antenna Patterns." In Radar Equations for Modern Radar, 147. Artech House Radar Series. Boston, Mass: Artech House, 2013.