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Antennas and acoustic transducers create radiated fields which propagate outwards into space or into the air and water for acoustics. Conversely, antennas and transducers react to impinging fields to produce output voltages. The electromagnetic fields created by an antenna, or the acoustic field created by a transducer (called a speaker in speech acoustics or hydrophone in ocean acoustics), depend on the distance from the sources and the direction specified by angular coordinates. The terms response pattern and radiation pattern are often used interchangeably but the term radiation pattern is mostly used to describe the field radiated by an element and the term response pattern is mostly used to describe the output of the antenna with respect to impinging wave field as a function of wave direction. By the principle of reciprocity, these two patterns are identical. When discussing the generation of the patterns, it is conceptually easier to think in terms of radiation patterns.
In radar and sonar applications, the interactions between fields and targets take place in the far-field region, often called the Fraunhofer region. The far-field region is defined as the region for which
r≫λ^{2}/L
where L represents the largest dimension of the source. In the far-field region, the fields take a special form: they can be written as the product of a function of direction (such as azimuth and elevation angles) and a geometric fall-off function, 1/r. It is the angular function that is called the radiation pattern, response pattern, or simply pattern.
Radiation patterns can be viewed as field patterns or as power patterns. We shall often add the term "field" or "power" to be more specific: contrast element field pattern versus element power pattern. The radiation power pattern describes the field's radiant intensity as a function of direction. Power units are watts/steradian.
The element field response or element field pattern represents the angular distribution of the electromagnetic field create by an antenna, E(θ,φ), or the scalar acoustic field, p(θ,φ), generated by an acoustic transducer such as a speaker or hydrophone. Because the far field electromagnetic field consists of horizontal and vertical components orthogonal, (E_{H}(θ,φ), E_{V}(θ,φ)) there may be different patterns for each component. Acoustic fields are scalar fields so there is only one pattern. The general form of any field or field component is
$$Af(\theta ,\varphi )\frac{{e}^{-ikr}}{r}$$
where A is a nominal field amplitude and f(θ,φ) is the normalized field pattern (normalized to unity). Because the field patterns are evaluated at some reference distance from the source, the fields returned by the element's step method are represented simply as A f(θ,φ). You can display the nominal element field pattern by invoking the element's plotResponse method, choosing 'Units' parameter value as 'mag' and setting the 'NormalizeResponse' parameter to false
plotResponse(elem,'NormalizeResponse', false,'Unit','mag');
You can view the normalized field pattern by setting the 'NormalizeResponse' parameter value to true. For example, if E_{H}(θ,φ) is the horizontal component of the complex electromagnetic field, its normalized field pattern is given by |E_{H}(θ,φ)|/E_{H,max}|.
plotResponse(elem,'Polarization','H','NormalizeResponse', true,'Unit','mag');
The element power response (or element power radiation pattern) is defined as the angular distribution of the radiant intensity in the far field, U_{rad}(θ,φ). When the elements are used for reception, the patterns are interpreted as the sensitivity of the element to radiation arriving from direction (θ,φ) and the power pattern represents the output voltage power of the element as a function of wave arrival direction.
Physically, the radiant intensity for the electromagnetic field produced by an antenna element is given by
$${U}_{rad}(\theta ,\varphi )=\frac{{r}^{2}}{2{Z}_{0}}\left(|{E}_{H}{|}^{2}+|{E}_{V}{|}^{2}\right)$$
where Z_{0} is the characteristic impedance of free space. The radiant intensity of an acoustic field is given by
$${U}_{rad}(\theta ,\varphi )=\frac{{r}^{2}}{2Z}|p{|}^{2}$$
where Z is the characteristic impedance of the acoustic medium. For the fields produced by the Phased Array System Toolbox™ element System objects, the radial dependence, the impedances and field magnitudes are all collected in the nominal field amplitudes defined above. Then the radiant intensity can generally be written
$${U}_{rad}(\theta ,\varphi )=|Af(\theta ,\varphi ){|}^{2}$$
The radiant intensity pattern is the quantity returned by the elements plotResponse method when the 'NormalizeResponse' parameter is set to false and the 'Unit' parameter is set to 'pow' (or 'db' for decibels).
plotResponse(elem,'NormalizeResponse',false,'Unit','pow');
The normalized power pattern is defined as the radiant intensity divided by its maximum value
$${U}_{norm}(\theta ,\varphi )=\frac{{U}_{rad}(\theta ,\varphi )}{{U}_{rad,max}}=|f(\theta ,\varphi ){|}^{2}$$
The plotResponse method returns a normalized power pattern when the 'NormalizeResponse' parameter is set to true and the 'Unit' parameter is set to 'pow' (or 'db' for decibels).
plotResponse(elem,'NormalizeResponse',true,'Unit','pow');
Element directivity measures the capability of an antenna or acoustic transducer to radiate or receive power preferentially in a particular direction. Sometimes it is referred to as directive gain. Directivity is measured by comparing the transmitted radiant intensity in a given direction to the radiant intensity transmitted by an isotropic radiator with the same total transmitted power. An isotropic radiator radiates equal power in all directions. The radiant intensity of an isotropic radiator is just the total transmitted power divided by the solid angle of a sphere, 4π,
$${U}_{rad}^{iso}(\theta ,\varphi )=\frac{{P}_{total}}{4\pi}$$
The element directivity is defined to be
$$D(\theta ,\varphi )=\frac{{U}_{rad}(\theta ,\varphi )}{{U}_{rad}^{iso}}=4\pi \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{total}}$$
By this definition, the integral of the directivity over a sphere surrounding the element is exactly 4π. Directivity is related to the effective beamwidth of an element. Start with an ideal antenna that has a uniform radiation field over a small solid angle (its beamwidth), ΔΩ, in a particular direction, and zero outside that angle. The directivity is
$$D(\theta ,\varphi )=4\pi \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{total}}=\frac{4\pi}{\Delta \Omega}$$
The greater the directivity, the smaller the beamwidth.
The radiant intensity can be expressed in terms of the directivity and the total power
$${U}_{rad}(\theta ,\varphi )=\frac{1}{4\pi}D(\theta ,\varphi ){P}_{total}$$
As an example, the directivity of the electric field of a z-oriented short-dipole antenna element is given by
$$D(\theta ,\varphi )=\frac{3}{2}{\mathrm{cos}}^{2}\theta $$
Often, the largest value of D(θ,φ) is specified as an antenna operating parameter. The direction in which D(θ,φ) is largest is the direction of maximum power radiation. This direction is often called the boresight direction. In some of the literature, the maximum value itself is called the directivity, reserving the phrase directive gain for what is called here directivity. For the short-dipole antenna, the maximum value of directivity occurs at θ = 0, independent of φ, and attains a value of 3/2. The concept of directivity applies to receiving antennas as well. It describes the output power as a function of the arrival direction of a plane wave impinging upon the antenna. By reciprocity, the directivity of a receiving antenna is the same as that for a transmitting antenna. A quantity closely related to directivity is element gain. The definition of directivity assumes that all the power fed to the element is radiated to space. In reality, system losses reduce the radiant intensity by some factor, the element efficiency, η. The term P_{total} becomes the power supplied to the antenna and P_{rad} becomes the power actually radiated into space. Then, P_{rad} = ηP_{total}. The element gain is defined by
$$G(\theta ,\varphi )=4\pi \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{total}}=4\pi \eta \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{rad}}=\eta D(\theta ,\varphi )$$
and represents the power radiated away from the element compared to the total power supplied to the element.
Using the element's plotResponse method, you can plot the directivity of an element by setting the 'Unit' parameter to 'dbi',
plotResponse(elem,'Unit','dbi');
When individual antenna elements are aggregated into arrays of elements, new response/radiation patterns are created which depend upon both the element patterns and the geometry of the array. These patterns are called beampatterns to reflect the fact that the pattern may be constructed to have a very narrow angular distribution, i.e. a beam. This term is used for an array in transmitting or receiving modes. Most often, but not always, the array consists of identical antennas. The identical antenna case is interesting because it lets us partition the radiation pattern into two components: one component describes the element radiation pattern and the second describes the array radiation pattern.
Just as an array of transmitting elements has a radiation pattern, an array of receiving elements has a response pattern which describes how the output voltage of the array changes with the direction of arrival of an plane incident wave. By reciprocity, the response pattern is identical to the radiation pattern.
For transmitting arrays, the voltage driving the elements may be phase-adjusted to allow the maximum radiant intensity to be transmitted in a particular direction. For receiving arrays, the arriving signals may be phase adjusted to maximize the sensitivity in a particular direction.
Start with a simple model of the radiation field produced by a single antenna which is given by
$$y(\theta ,\varphi ,r)=Af(\theta ,\varphi )\frac{{e}^{-ikr}}{r}$$
where A is the field amplitude and f((θ,φ) is the normalized element field pattern. This field may represent any of the components of the electric field, a scalar field, or an acoustic field. For an array of identical elements, the output of the array is the weighted sum of the individual elements, using the complex weights, w_{m}
$$z(\theta ,\varphi ,r)=A{\displaystyle \sum _{m=0}^{M-1}{w}_{m}^{*}}f(\theta ,\varphi )\frac{{e}^{-ik{r}_{m}}}{{r}_{m}}$$
where r_{m} is the distance from the m^{th} element source point to the field point. In the far-field region, this equation takes the form
$$z(\theta ,\varphi ,r)=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){\displaystyle \sum _{m=0}^{M-1}{w}_{m}^{*}}{e}^{-iku\xb7{x}_{m}}$$
where x_{m} are the vector positions of the array elements with respect to the array origin. u is the unit vector from the array origin to the field point. This equation can be written compactly is the form
$$z(\theta ,\varphi ,r)=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){w}^{H}s$$
The term w^{H}s is called the array factor, F_{array}(θ,φ). The vector s is the steering vector (or array manifold vector) for directions of propagation for transmit arrays or directions of arrival for receiving arrays
$$s(\theta ,\varphi )=\{\dots ,{e}^{iku\xb7{x}_{m}},\dots \}$$
The total array pattern consists of an amplitude term, an element pattern, f(θ,φ), and an array factor, F_{array}(θ,φ). The total angular behavior of the array pattern, B(θ,φ), is called the beampattern of the array
$$z(\theta ,\varphi ,r)=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){w}^{H}s=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){F}_{array}(\theta ,\varphi )=A\frac{{e}^{-ikr}}{r}B(\theta ,\varphi )$$
When evaluated at the reference distance, the array field pattern has the form
$$Af(\theta ,\varphi ){w}^{H}s=Af(\theta ,\varphi ){F}_{array}(\theta ,\varphi )=AB(\theta ,\varphi )$$
The plotResponse method, when the 'NormalizeResponse' parameter is set to false and the 'Unit' parameter is set to 'mag', returns the magnitude of the array field pattern at the reference distance.
plotResponse(array,'NormalizeResponse',false,'Unit','mag');
When the 'NormalizeResponse' parameter is set to true, the plotResponse method returns a pattern normalized to unity.
plotResponse(array,'NormalizeResponse',true,'Unit','mag');
The array power pattern is given by
$$|Af(\theta ,\varphi ){w}^{H}s{|}^{2}=|Af(\theta ,\varphi ){F}_{array}(\theta ,\varphi ){|}^{2}=|AB(\theta ,\varphi ){|}^{2}$$
The plotResponse method, when the 'NormalizeResponse' parameter is set to false and the 'Unit' parameter is set to 'pow' or 'db', returns the array power pattern at the reference distance.
plotResponse(array,'NormalizeResponse',false,'Unit','pow');
When the 'NormalizeResponse' parameter is set to true, the plotResponse method returns the power pattern normalized to unity.
plotResponse(array,'NormalizeResponse',true,'Unit','pow');
For the conventional beamformer, the weights are chosen to maximize the power transmitted towards a particular direction, or in the case of receiving arrays, to maximize the response of the array for a particular arrival direction. If u_{0} is the desired pointing direction, then the weights which maximize the power and response in this direction have the general form
$$w=\left|{w}_{m}\right|{e}^{-ik{u}_{0}\xb7{x}_{m}}$$
For these weights, the array factor becomes
$${F}_{array}(\theta ,\varphi )={\displaystyle \sum _{m=0}^{M-1}|}{w}_{m}|{e}^{-ik(u-{u}_{0})\xb7{x}_{m}}$$
which has a maximum at u = u_{0}.
Array directivity is defined the same way as element directivity: the radiant intensity in a specific direction divided by the isotropic radiant intensity. The isotropic radiant intensity is the array's total radiated power divided by 4π. In terms of the arrays weights and steering vectors, the directivity can be written as
$$D(\theta ,\varphi )=4\pi \frac{|Af(\theta ,\varphi ){w}^{H}s{|}^{2}}{{P}_{total}}$$
where P_{total} is the total radiated power from the array. In a discrete implementation, the total radiated power can be computed by summing intensity values over a uniform grid of angles that covers the full sphere surrounding the array
$${P}_{total}=\frac{2{\pi}^{2}}{MN}{\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}|}A}f({\theta}_{m},{\varphi}_{n}){w}^{H}s({\theta}_{m},{\varphi}_{n}){|}^{2}\mathrm{cos}{\theta}_{m}$$
where M is the number of elevation grid points and N is the number of azimuth grid points.
Because the radiant intensity is proportional to the beampattern, B(θ,φ), the directivity can also be written in terms of the beampattern
$$D(\theta ,\varphi )=4\pi \frac{|B(\theta ,\varphi ){|}^{2}}{{\displaystyle \int |}B(\theta ,\varphi ){|}^{2}\mathrm{cos}\theta d\theta d\varphi}$$
You can plot the directivity of an array by setting the 'Unit' parameter of the plotResponse methods to 'dbi',
plotResponse(array,'Unit','dbi');
In the Phased Array System Toolbox, array gain is defined to be the array SNR gain. Array gain measures the improvement in SNR of a receiving array over the SNR for a single element. Because an array is a spatial filter, the array SNR depends upon the spatial properties of the noise field. When the noise is spatially isotropic, the array gain takes a simple form
$$G=\frac{{\text{SNR}}_{array}}{{\text{SNR}}_{element}}=\frac{|{w}^{H}s{|}^{2}}{{w}^{H}w}$$
In addition, for an array with uniform weights, the array gain for an N-element array has a maximum value at boresight of N, (or 10logN in db).