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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

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

$$Af(\theta ,\varphi )\frac{{e}^{-ikr}}{r}$$

`step`

method are represented simply as `pattern`

method, choosing `'Type'`

parameter value as
`'efield'`

and setting the `'Normalize'`

parameter to
`false`

pattern(elem,'Normalize',false,'Type','efield');

`'Normalize'`

parameter
value to `true`

. For example, if
pattern(elem,'Polarization','H','Normalize',true,'Type','efield');

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

Physically, the radiant intensity for the electromagnetic field produced by an antenna element is given by

$${U}_{rad}(\theta ,\varphi )=\frac{{r}^{2}}{2{Z}_{0}}\left(|{E}_{H}{|}^{2}+|{E}_{V}{|}^{2}\right)$$

$${U}_{rad}(\theta ,\varphi )=\frac{{r}^{2}}{2Z}|p{|}^{2}$$

$${U}_{rad}(\theta ,\varphi )=|Af(\theta ,\varphi ){|}^{2}$$

The radiant intensity pattern is the quantity returned by the
elements `pattern`

method when the `'Normalize'`

parameter
is set to `false`

and the `'Type'`

parameter
is set to `'power'`

(or `'powerdb'`

for
decibels).

pattern(elem,'Normalize',false,'Type','power');

$${U}_{norm}(\theta ,\varphi )=\frac{{U}_{rad}(\theta ,\varphi )}{{U}_{rad,max}}=|f(\theta ,\varphi ){|}^{2}$$

The `pattern`

method returns a normalized
power pattern when the `'Normalize'`

parameter is
set to `true`

and the `'Type'`

parameter
is set to `'power'`

(or `'powerdb'`

for
decibels).

pattern(elem,'Normalize',true,'Type','power');

*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*,

$${U}_{rad}^{iso}(\theta ,\varphi )=\frac{{P}_{total}}{4\pi}$$

$$D(\theta ,\varphi )=\frac{{U}_{rad}(\theta ,\varphi )}{{U}_{rad}^{iso}}=4\pi \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{total}}$$

$$D(\theta ,\varphi )=4\pi \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{total}}=\frac{4\pi}{\Delta \Omega}$$

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

$$D(\theta ,\varphi )=\frac{3}{2}{\mathrm{cos}}^{2}\theta $$

$$G(\theta ,\varphi )=4\pi \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{total}}=4\pi \eta \frac{{U}_{rad}(\theta ,\varphi )}{{P}_{rad}}=\eta D(\theta ,\varphi )$$

Using the element’s `pattern`

method,
you can plot the directivity of an element by setting the `'Type'`

parameter
to `'directivity'`

,

pattern(elem,'Type','directivity');

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

$$y(\theta ,\varphi ,r)=Af(\theta ,\varphi )\frac{{e}^{-ikr}}{r}$$

$$z(\theta ,\varphi ,r)=A{\displaystyle \sum _{m=0}^{M-1}{w}_{m}^{*}}f(\theta ,\varphi )\frac{{e}^{-ik{r}_{m}}}{{r}_{m}}$$

$$z(\theta ,\varphi ,r)=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){\displaystyle \sum _{m=0}^{M-1}{w}_{m}^{*}}{e}^{-iku\xb7{x}_{m}}$$

$$z(\theta ,\varphi ,r)=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){w}^{H}s$$

The term ** w^{H}s** is
called the

$$s(\theta ,\varphi )=\{\dots ,{e}^{iku\xb7{x}_{m}},\dots \}$$

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,

$$z(\theta ,\varphi ,r)=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){w}^{H}s=A\frac{{e}^{-ikr}}{r}f(\theta ,\varphi ){F}_{array}(\theta ,\varphi )=A\frac{{e}^{-ikr}}{r}B(\theta ,\varphi )$$

$$Af(\theta ,\varphi ){w}^{H}s=Af(\theta ,\varphi ){F}_{array}(\theta ,\varphi )=AB(\theta ,\varphi )$$

The `pattern`

method, when the `'Normalize'`

parameter
is set to `false`

and the `'Type'`

parameter
is set to `'efield'`

, returns the magnitude of the
array field pattern at the reference distance.

pattern(array,'Normalize',false,'Type','efield');

`'Normalize'`

parameter
is set to `true`

, the `pattern`

method
returns a pattern normalized to unity.pattern(array,'Normalize',true,'Type','efield');

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 `pattern`

method, when the `'Normalize'`

parameter
is set to `false`

and the `'Type'`

parameter
is set to `'power'`

or `'powerdb'`

,
returns the array power pattern at the reference distance.

pattern(array,'Normalize',false,'Type','power');

`'Normalize'`

parameter
is set to `true`

, the `pattern`

method
returns the power pattern normalized to unity.pattern(array,'Normalize',true,'Type','power');

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

$${F}_{array}(\theta ,\varphi )={\displaystyle \sum _{m=0}^{M-1}|}{w}_{m}|{e}^{-ik(u-{u}_{0})\xb7{x}_{m}}$$

*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

$$D(\theta ,\varphi )=4\pi \frac{|Af(\theta ,\varphi ){w}^{H}s{|}^{2}}{{P}_{total}}$$

$${P}_{total}=\frac{2{\pi}^{2}}{MN}{\displaystyle \sum _{m=0}^{M-1}{\displaystyle \sum _{n=0}^{N-1}|}A}f({\theta}_{m},{\varphi}_{n}){w}^{H}s({\theta}_{m},{\varphi}_{n}){|}^{2}\mathrm{cos}{\theta}_{m}$$

Because the radiant intensity is proportional to the beampattern, *B(θ,φ)*,
the directivity can also be written in terms of the beampattern

$$D(\theta ,\varphi )=4\pi \frac{|B(\theta ,\varphi ){|}^{2}}{{\displaystyle \int |}B(\theta ,\varphi ){|}^{2}\mathrm{cos}\theta d\theta d\varphi}$$

`'Type'`

parameter of the `pattern`

methods
to `'directivity'`

,pattern(array,'Type','directivity');

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

$$G=\frac{{\text{SNR}}_{array}}{{\text{SNR}}_{element}}=\frac{|{w}^{H}s{|}^{2}}{{w}^{H}w}$$

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