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');

`'NormalizeResponse'`

parameter value to `true`

.
For example, if 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

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');

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

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

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

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

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

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

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

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,

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

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');

`'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');

`'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).

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