## Element and Array Radiation and Response Patterns

### Element Response and Radiation Patterns

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. Transducers are called microphones or speakers
in speech acoustics or projectors or hydrophones in ocean acoustics. The electromagnetic
fields created by an antenna, or the acoustic field created by a transducer, depend upon the
distance and direction from the radiator. 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≫L^{2}/λ

*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. The terms
“field” or “power” are often added to be more specific: contrast
element field pattern versus element power pattern. The radiation power pattern describes
the radiant intensity of a field, *U*, as a function of direction. Radiant
intensity units are watts/steradian. Sometimes, radiant intensity is confused with power
density. Power density, *I*, is the energy passing through a unit area in a
unit time. Units for power density are Watts/square meter. Unfortunately, in some
disciplines, power density is sometimes called intensity. This document always uses radiant
intensity instead of intensity to avoid confusion. For a point source, the radiant intensity
is the power density multiplied by the square of the distance from the source,
*U* =
*r*^{2}*I*.

#### Element Field Patterns

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 can 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

$$A\text{\hspace{0.17em}}f(\theta ,\varphi )\frac{{e}^{-ikr}}{r}$$

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 `step`

method are represented simply as
*A f(θ,φ)*. You can display the nominal element field pattern by
invoking the element `pattern`

method and then choosing the
`'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
*E*is the horizontal component of the complex electromagnetic field, the normalized field pattern has the form

_{H}(θ,φ)*|E*.

_{H}(θ,φ)/E_{H,max}|pattern(elem,'Polarization','H','Normalize',true,'Type','efield');

#### Element Power Patterns

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

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

where *Z _{0}* is the
characteristic impedance of free space. The radiant intensity of an acoustic field is

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

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 the 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 element
`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');

*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 `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

*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
transmitted radiant intensity of an isotropic radiator having 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

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

with a peak value of 1.5. 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 the directivity when
used as 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*becomes the power radiated into space. Then,

_{rad}*P*. The element gain is

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

and represents the power radiated away from the element compared to the total power supplied to the element.

Using the element `pattern`

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

parameter to
`'directivity'`

,

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

### Array Response and Radiation Patterns

#### Array Magnitude and Power Patterns

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 can be constructed to have a narrow angular
distribution, that is, 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 a plane incident wave. By reciprocity, the response pattern is identical to the radiation pattern.

For transmitting arrays, the voltage driving the elements can be phase-adjusted to allow the maximum radiant intensity to be transmitted in a particular direction. For receiving arrays, the arriving signals can 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 can
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*. The vector

_{array}(θ,φ)**is the**

*s**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 `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

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

which has a maximum at **u** = **u**_{0}.

#### Array Directivity

*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 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 radiant 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
`'Type'`

parameter of the `pattern`

methods to
`'directivity'`

,

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

#### Array Gain

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).

## References

[1] Balanis, Constantine A.
*Antenna Theory: Analysis and Design*. Fourth edition, Wiley,
2016.

[2] Johnson, Don H., and Dan E.
Dudgeon. *Array Signal Processing: Concepts and Techniques*. P T R
Prentice Hall, 1993.