System object: phased.SumDifferenceMonopulseTracker2D
Package: phased
Perform monopulse tracking using URA
ESTANG = step(H,X,STANG)
estimates
the incoming direction ESTANG
= step(H
,X
,STANG
)ESTANG
of the input signal, X
,
based on an initial guess of the direction.
Note:
The object performs an initialization the first time the 

Tracker object of type 

Input signal, specified as a row vector whose number of columns corresponds to number of channels. 

Initial guess of the direction, specified as a 2by1 vector
in the form 

Estimate of incoming direction, returned as a 2by1 vector
in the form 
Determine the direction of a target at around 60 degrees azimuth and 20 degrees elevation of a URA.
ha = phased.URA('Size',4); hstv = phased.SteeringVector('SensorArray',ha); hmp = phased.SumDifferenceMonopulseTracker2D('SensorArray',ha); x = step(hstv,hmp.OperatingFrequency,[60.1; 19.5]).'; est_dir = step(hmp,x,[60; 20]);
The sumanddifference monopulse algorithm is used to the estimate the arrival direction of a narrowband signal impinging upon a uniform linear array (ULA). First, compute the conventional response of an array steered to an arrival direction φ_{0}. For a ULA, the arrival direction is specified by the broadside angle. To specify that the maximum response axis (MRA) point towards the φ_{0} direction, set the weights to be
$${w}_{s}=\left(1,{e}^{ikd\mathrm{sin}{\varphi}_{0}},{e}^{ik2d\mathrm{sin}{\varphi}_{0}},\dots ,{e}^{ik(N1)d\mathrm{sin}{\varphi}_{0}}\right)$$
where d is the element spacing and k = 2π/λ is the wavenumber. An incoming plane wave, coming from any arbitrary direction φ, is represented by
$$v=\left(1,{e}^{ikd\mathrm{sin}\varphi},{e}^{ik2d\mathrm{sin}\varphi},\dots ,{e}^{ik(N1)d\mathrm{sin}\varphi}\right)$$
The conventional response of the this array to any incoming plane wave is given by $${w}_{s}^{H}v\left(\phi \right)$$ and is shown in the polar plot below as the Sum Pattern. The array is designed to steer towards φ_{0} = 30°.
The second pattern, called the Difference Pattern, is obtained by using phasedreversed weights. The weights are determined by phasereversing the latter half of the conventional steering vector. For an array with an even number of elements, the phasereversed weights are
$${w}_{d}=i\left(1,{e}^{ikd\mathrm{sin}{\varphi}_{0}},{e}^{ik2d\mathrm{sin}{\varphi}_{0}},\dots ,{e}^{ikN/2d\mathrm{sin}{\varphi}_{0}},{e}^{ik(N/2+1)d\mathrm{sin}{\varphi}_{0}},\dots ,{e}^{ik(N1)d\mathrm{sin}{\varphi}_{0}}\right)$$
(For an array with an odd number of elements, the middle weight is set to zero). The multiplicative factor –i is used for convenience. The response of the difference array to the incoming vector is
$${w}_{d}^{H}v\left(\phi \right)$$
and is show in the polar plot below
The monopulse response curve is obtained by dividing the difference pattern by the sum pattern and taking the real part.
$$R(\phi )=Re\left(\frac{{w}_{d}^{H}v(\phi )}{{w}_{s}^{H}v(\phi )}\right)$$
To use the monopulse response curve to obtain the arrival angle of a narrowband signal, x, compute
$$z=Re\left(\frac{{w}_{d}^{H}x}{{w}_{s}^{H}x}\right)$$
and invert the response curve, φ = R^{1}(z), to obtain φ.
The response curve is not single valued and can be inverted only when arrival angles lie within the mainlobe. The figure below shows the center portion of the monopulse response curve in the mainlobe for a 4element ULA array.
There are two desirable properties of the monopulse response curve. The first is that it have a steep slope. A steep slope insures robustness against noise. The second property is that the mainlobe be as wide as possible. A steep slope is ensure by a larger array but leads to a smaller mainlobe. You will need to trade off one property with the other.
For further details, see [1].
[1] Seliktar, Y. SpaceTime Adaptive Monopulse Processing. Ph.D. Thesis. Georgia Institute of Technology, Atlanta, 1998.
[2] Rhodes, D. Introduction to Monopulse. Dedham, MA: Artech House, 1980.