# phased.SumDifferenceMonopulseTracker System object

Package: phased

Sum and difference monopulse for ULA

## Description

The `SumDifferenceMonopulseTracker` object implements a sum and difference monopulse algorithm on a uniform linear array.

To estimate the direction of arrival (DOA):

1. Define and set up your sum and difference monopulse DOA estimator. See Construction.

2. Call `step` to estimate the DOA according to the properties of `phased.SumDifferenceMonopulseTracker`. The behavior of `step` is specific to each object in the toolbox.

## Construction

`H = phased.SumDifferenceMonopulseTracker` creates a tracker System object™, `H`. The object uses sum and difference monopulse algorithms on a uniform linear array (ULA).

`H = phased.SumDifferenceMonopulseTracker(Name,Value)` creates a ULA monopulse tracker object, `H`, with each specified property Name set to the specified Value. You can specify additional name-value pair arguments in any order as (`Name1`,`Value1`,...,`NameN`,`ValueN`).

## Properties

 `SensorArray` Handle to sensor array Specify the sensor array as a handle. The sensor array must be a `phased.ULA` object. Default: `phased.ULA` with default property values `PropagationSpeed` Signal propagation speed Specify the propagation speed of the signal, in meters per second, as a positive scalar. Default: Speed of light `OperatingFrequency` System operating frequency Specify the operating frequency of the system in hertz as a positive scalar. The default value corresponds to 300 MHz. Default: `3e8`

## Methods

 clone Create ULA monopulse tracker object with same property values getNumInputs Number of expected inputs to step method getNumOutputs Number of outputs from step method isLocked Locked status for input attributes and nontunable properties release Allow property value and input characteristics changes step Perform monopulse tracking using ULA

## Examples

Determine the direction of a target at around 60 degrees broadside angle of a ULA.

```ha = phased.ULA('NumElements',4); hstv = phased.SteeringVector('SensorArray',ha); hmp = phased.SumDifferenceMonopulseTracker('SensorArray',ha); x = step(hstv,hmp.OperatingFrequency,60.1).'; est_dir = step(hmp,x,60);```

## Algorithms

The sum-and-difference 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 φ0. For a ULA, the arrival direction is specified by the broadside angle. To specify that the maximum response axis (MRA) point towards the φ0 direction, set the weights to be

${w}_{s}=\left(1,{e}^{ikd\mathrm{sin}{\varphi }_{0}},{e}^{ik2d\mathrm{sin}{\varphi }_{0}},\dots ,{e}^{ik\left(N-1\right)d\mathrm{sin}{\varphi }_{0}}\right)$

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

$v=\left(1,{e}^{ikd\mathrm{sin}\varphi },{e}^{ik2d\mathrm{sin}\varphi },\dots ,{e}^{ik\left(N-1\right)d\mathrm{sin}\varphi }\right)$

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 φ0 = 30°.

The second pattern, called the Difference Pattern, is obtained by using phased-reversed weights. The weights are determined by phase-reversing the latter half of the conventional steering vector. For an array with an even number of elements, the phase-reversed weights are

${w}_{d}=-i\left(1,{e}^{ikd\mathrm{sin}{\varphi }_{0}},{e}^{ik2d\mathrm{sin}{\varphi }_{0}},\dots ,{e}^{ikN/2d\mathrm{sin}{\varphi }_{0}},-{e}^{ik\left(N/2+1\right)d\mathrm{sin}{\varphi }_{0}},\dots ,-{e}^{ik\left(N-1\right)d\mathrm{sin}{\varphi }_{0}}\right)$

(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.

$R\left(\phi \right)=Re\left(\frac{{w}_{d}^{H}v\left(\phi \right)}{{w}_{s}^{H}v\left(\phi \right)}\right)$

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 4-element 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].

## References

[1] Seliktar, Y. Space-Time Adaptive Monopulse Processing. Ph.D. Thesis. Georgia Institute of Technology, Atlanta, 1998.

[2] Rhodes, D. Introduction to Monopulse. Dedham, MA: Artech House, 1980.