Is there an actual definition for "direction of arrival"?

Hi,

This is a beamforming question, but I thought perhaps some of the SP
people might know the answer to this question, as it is a commonly
enough occurring term in SP applications.
I am a mathematician working on some interferometric problems, and I
find myself having trouble getting an actual definition of "direction
of arrival" (or DOA).

Some older analyses using linear antenna arrays seem to use the
inclination angle, theta, (i.e. the angle with the +z-axis or
"azimuthal angle") for the DOA, but for a source that is not coplanar
with the array, there is a factor of Cos(phi) in the phase portion of
the steering vector (where phi is the angle with the +x-axis or polar
angle).

Some analyses seem to define a generic phase angle from the inner
product of the source orientation with the array position vector. This
definition is consistent for a linear array, but it is not for a planar
array.

Is there a rigorous definition of what is meant for "direction of
arrival" (or angle of arrival) that holds for a general array and
source? You can respond directly to this post or via my e-mail address.

TIA,

Matt

Brenneman skrev:
> Hi,
>
> This is a beamforming question, but I thought perhaps some of the SP
> people might know the answer to this question, as it is a commonly
> enough occurring term in SP applications.
> I am a mathematician working on some interferometric problems, and I
> find myself having trouble getting an actual definition of "direction
> of arrival" (or DOA).
>
> Some older analyses using linear antenna arrays seem to use the
> inclination angle, theta, (i.e. the angle with the +z-axis or
> "azimuthal angle") for the DOA, but for a source that is not coplanar
> with the array, there is a factor of Cos(phi) in the phase portion of
> the steering vector (where phi is the angle with the +x-axis or polar
> angle).

In the case of the linear array and point source, the analysis is done
in the plane defined by the line of the array, and the point source.
In certain practical applications, this definition is somewhat
arbitrarily expanded to include a medium of some thickness.
In passive sonar, for instance, DoA is usually refered to the
horizontal plane, depth information is seldom included.

> Some analyses seem to define a generic phase angle from the inner
> product of the source orientation with the array position vector. This
> definition is consistent for a linear array, but it is not for a planar
> array.

Correct. For planar arrays one needs to define an (azimuth, elevation)
pair of angles.

> Is there a rigorous definition of what is meant for "direction of
> arrival" (or angle of arrival) that holds for a general array and
> source? You can respond directly to this post or via my e-mail address.

Not other than the geometrical references. When the physical
position and orientation of the array is known, one wants to
detect what (azimuth,elevation) direction a signal arrives from.

I am not aware of any generic mathematical definitions of the DoA.

Rune

Brenneman wrote:
> Hi,
> 
> This is a beamforming question, but I thought perhaps some of the SP
> people might know the answer to this question, as it is a commonly
> enough occurring term in SP applications.
> I am a mathematician working on some interferometric problems, and I
> find myself having trouble getting an actual definition of "direction
> of arrival" (or DOA).
> 
> Some older analyses using linear antenna arrays seem to use the
> inclination angle, theta, (i.e. the angle with the +z-axis or
> "azimuthal angle") for the DOA, but for a source that is not coplanar
> with the array, there is a factor of Cos(phi) in the phase portion of
> the steering vector (where phi is the angle with the +x-axis or polar
> angle).
> 
> Some analyses seem to define a generic phase angle from the inner
> product of the source orientation with the array position vector. This
> definition is consistent for a linear array, but it is not for a planar
> array.
> 
> Is there a rigorous definition of what is meant for "direction of
> arrival" (or angle of arrival) that holds for a general array and
> source? You can respond directly to this post or via my e-mail address.
> 
> TIA,

DoA makes sense when the receiving antenna subtends an infinitesimal 
angle at the source and the source subtends an infinitesimal angle at 
the receiving antenna. The mathematics becomes very complex otherwise.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

>
> DoA makes sense when the receiving antenna subtends an infinitesimal
> angle at the source and the source subtends an infinitesimal angle at
> the receiving antenna. The mathematics becomes very complex otherwise.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> =AF=AF

Hi Jerry,

Thank you for your response. Let me ask you one follow up question to
make sure I understand what you are saying.
I am interested in antenna arrays in the far-field of point sources
that emit plane wave radiation (like a GPS source for example). I am
wondering if this is the same context to which you are referring. When
you say that BOTH source and receiver "subtend an infinitesimal angle"
with respect to each other, the only scenario I can envision that fits
that criertion is perhaps when the source is on the horizon, so that
the angle to which you are referring is essentially the complement of
theta. Even then, I still don't understand what is meant by the
reciever subtending a small angle wrt the source. If the source emits
plane radiation, in the far-field approximation we think of the
relative geometrical relationship of antenna and source as being
characterized by a single line from the point source to some fixed
point wrt the array (i.e. origin). I don't understand where the "other
side" that would define an angle comes from.
Sorry if this seems like a dumb question or if I seem a bit slow: I'm
new to this field, and I guess I'm a stickler for getting down things
like definitions (occupational hazard, I guess).

Thanks again,

Matt

Brenneman wrote:
>>DoA makes sense when the receiving antenna subtends an infinitesimal
>>angle at the source and the source subtends an infinitesimal angle at
>>the receiving antenna. The mathematics becomes very complex otherwise.
>>
>>Jerry
>>--
>>Engineering is the art of making what you want from things you can get.
>>��
> 
> 
> Hi Jerry,
> 
> Thank you for your response. Let me ask you one follow up question to
> make sure I understand what you are saying.
> I am interested in antenna arrays in the far-field of point sources
> that emit plane wave radiation (like a GPS source for example). I am
> wondering if this is the same context to which you are referring. 

Yes

> When
> you say that BOTH source and receiver "subtend an infinitesimal angle"
> with respect to each other, the only scenario I can envision that fits
> that criertion is perhaps when the source is on the horizon, so that
> the angle to which you are referring is essentially the complement of
> theta. Even then, I still don't understand what is meant by the
> reciever subtending a small angle wrt the source. If the source emits
> plane radiation, in the far-field approximation we think of the
> relative geometrical relationship of antenna and source as being
> characterized by a single line from the point source to some fixed
> point wrt the array (i.e. origin). I don't understand where the "other
> side" that would define an angle comes from.
> Sorry if this seems like a dumb question or if I seem a bit slow: I'm
> new to this field, and I guess I'm a stickler for getting down things
> like definitions (occupational hazard, I guess).

Knowing what we -- and the other guy -- means is a always a good thing.

I meant that the transmitter seems like a point source to the receiver, 
and that it is sufficiently remote that the bearing to it is the same 
from every point on the receiving array. If these conditions are met, 
they would also be met if the role of receiver and transmitter were 
interchanged. "Plane wave" encompasses these conditions, but doesn't 
serve well as a definition. "Point source" and "plane wave" are in fact 
mutually contradictory, but serve well as local approximations. A true 
plane wave doesn't have inverse-square intensity.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Brenneman skrev:
> >
> > DoA makes sense when the receiving antenna subtends an infinitesimal
> > angle at the source and the source subtends an infinitesimal angle at
> > the receiving antenna. The mathematics becomes very complex otherwise.
> >
> > Jerry
> > --
> > Engineering is the art of making what you want from things you can get.
> > =AF=AF
>
> Hi Jerry,
>
> Thank you for your response. Let me ask you one follow up question to
> make sure I understand what you are saying.
> I am interested in antenna arrays in the far-field of point sources
> that emit plane wave radiation (like a GPS source for example).

These are contradictions in terms. The frame of refernce is the
reciever system. A "point source" is a source which is so small
that it resembles a point when viewed from the reciever. A simple
example is a star shining in the sky. It resembles a point.
Good examples of "distributed sources" are the sun and the
moon, that have clear angular extents.

A "point source" emits a spherical or cylindrical wave, which
energy dilutes as function of traveled distance. Also, the local
curvature of the wave front reduces as the distance to the
source increases. At some distance there is no point in dealing
with local curvature, and the plane wave approximation is used
instead.

> I am
> wondering if this is the same context to which you are referring. When
> you say that BOTH source and receiver "subtend an infinitesimal angle"
> with respect to each other, the only scenario I can envision that fits
> that criertion is perhaps when the source is on the horizon, so that
> the angle to which you are referring is essentially the complement of
> theta.

The source is never a point source, it has some physical extension.
However, for the far field to be valid, the angular extension of that
source, when viewed from the reciever, must be infitesimal.

Again, a star as example. The star Betelgeuze has a diameter
on the order of the diameter of the earth's orbit around the sun
http://en.wikipedia.org/wiki/Betelgeuze. Nevertheless, when viewed
from earth, it is seen as just a point.

Rune

>
> I meant that the transmitter seems like a point source to the receiver,
> and that it is sufficiently remote that the bearing to it is the same
> from every point on the receiving array. If these conditions are met,
> they would also be met if the role of receiver and transmitter were
> interchanged.

Thank you for the explanation: I understand exactly what you mean now
and the distinction you make is an important one. Since the case to
which you are referring is the one of interest to me let me go back to
your first post now. When you say that the "DOA makes sense" under
these conditions, what do YOU mean by DOA: the two spherical coordinate
angles (phi,theta)?

Thanks,

Matt

Brenneman skrev:
> >
> > I meant that the transmitter seems like a point source to the receiver,
> > and that it is sufficiently remote that the bearing to it is the same
> > from every point on the receiving array. If these conditions are met,
> > they would also be met if the role of receiver and transmitter were
> > interchanged.
>
> Thank you for the explanation: I understand exactly what you mean now
> and the distinction you make is an important one. Since the case to
> which you are referring is the one of interest to me let me go back to
> your first post now. When you say that the "DOA makes sense" under
> these conditions, what do YOU mean by DOA: the two spherical coordinate
> angles (phi,theta)?

Don't know about Jerry, but *I* think of the pair of angles (phi,
theta) as 
the DoA in the case of the planar array. 

Rune

Jerry Avins wrote:

> I meant that the transmitter seems like a point source to the receiver, 
> and that it is sufficiently remote that the bearing to it is the same 
> from every point on the receiving array. If these conditions are met, 
> they would also be met if the role of receiver and transmitter were 
> interchanged. "Plane wave" encompasses these conditions, but doesn't 
> serve well as a definition. "Point source" and "plane wave" are in fact 
> mutually contradictory, but serve well as local approximations. A true 
> plane wave doesn't have inverse-square intensity.
> 


Jerry,
Another way of looking at it is this:

Those two terms: "point source" and "plane wave" are not mutually 
contradictory if you take into account the 'optical' location of the 
source.

If you trace back the 'nearly' plane-wave radiated by a large dish or 
array, the wave appears to be coming from a point that is located well 
behind the antenna.  The less curvature you have on that wave-front the 
further behind the antenna the source appears to be.

In the case of a theoretical-perfect plane-wave, the source appears to 
be located at an infinite distance behind the antenna aperture.  As the 
apparent distance between the signal source and the receiving antenna is 
already infinite it is not possible to change this distance to any 
significant degree by changing the physical separation between the 
antennas.  The inverse-square law is working correctly when it shows us 
that there is no change in received signal under these circumstances.

Regards,
John

Brenneman wrote:
>>I meant that the transmitter seems like a point source to the receiver,
>>and that it is sufficiently remote that the bearing to it is the same
>>from every point on the receiving array. If these conditions are met,
>>they would also be met if the role of receiver and transmitter were
>>interchanged.
> 
> 
> Thank you for the explanation: I understand exactly what you mean now
> and the distinction you make is an important one. Since the case to
> which you are referring is the one of interest to me let me go back to
> your first post now. When you say that the "DOA makes sense" under
> these conditions, what do YOU mean by DOA: the two spherical coordinate
> angles (phi,theta)?

Matt,

I'm sorry I didn't get back to you sooner. Two linear antenna arrays are 
needed to get two angles. (Of course, a plane structure serves also.) 
The best illustration of the One-dimensional situation I can think of is 
the usual freshman physics explanation of a diffraction grating. That 
explanation also makes clear why an extended line of closely spaces 
antennas reduces ambiguity (sidelobes).

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������