# FFT Beamforming

Started by March 5, 2011
```Hi,

Im an undergraduate student and I have researched how to estimate the
direction of arrival of a signal using a FFT. I understand how to use it
and how to implement the algorithm, but im unsure as to why when performing
an FFT on spatially separated data this yields an angle?

Thanks,

Jan

```
```On 3/5/2011 4:14 AM, JanMichica wrote:
> Hi,
>
> Im an undergraduate student and I have researched how to estimate the
> direction of arrival of a signal using a FFT. I understand how to use it
> and how to implement the algorithm, but im unsure as to why when performing
> an FFT on spatially separated data this yields an angle?
>
> Thanks,
>
> Jan
>
>

Jan,

Consider the physics:
Assume a single frequency.
Consider single samples taken from each sensor at exactly the same time.
Consider those samples as a "sequence".
The sequence is in space, not time.
A single FFT of that sequence goes from spatial samples to wave number
samples (and, by itself, not an "angle").
[Wave number = the number of wavelengths per unit distance]

If a sinusoid hits the array broadside then all of the spatial samples
will be the same and the wave number sample with the greatest energy
would be at zero.

As the direction of arrival moves away from broadside then you can
visualize that the arriving sinusoid experiences a delay from one sensor
to the next.  At a single point in time, each sensor captures a
different sinusoidal value.  So, looking down the array you would see
sinusoidal variation from sensor to sensor.  This sinusoidal variation
is in *space* and not in time.   With appropriate element spacing
related to signal frequency the sinusoid thus generated will go from
wave number=0 as above to the highest possible wave number when the
signal arrives endfire.

So, when you compute an FFT of a single-time sample sequence, you are
converting the spatial sequence into a wave number sequence which you
know a priori what angles are represented at the chosen frequency, etc.
Then, to get *the* angle, you have to pick the highest-valued wave
number sample.  Then, you can interpolate over samples or ... whatever.

Consider that the typical signal FFT / IFFT relates time<>frequency and
frequency has a reciprocal relationship to time - short times high
frequencies, etc.
This spatial FFT relates wavelength<>wave number with the same sort of
reciprocal relationship.

From this I expect you can figure out that a good system design for
this purpose might meet Shannon's sampling criterion so that:
Given the signal wavelength, the spatial samples would have to grab more
than 2 samples per wavelength which is most demanding at endfire.  So,
the array elements need to be less than 1/2 wavelength apart.  There are
no doubt tricks and exceptions to this....

Fred
```
```On Mar 5, 1:14=A0pm, "JanMichica" <janmichica@n_o_s_p_a_m.hotmail.com>
wrote:
> Hi,
>
> Im an undergraduate student and I have researched how to estimate the
> direction of arrival of a signal using a FFT. I understand how to use it
> and how to implement the algorithm, but im unsure as to why when performi=
ng
> an FFT on spatially separated data this yields an angle?

Work through the arithmetics of DoA of point sources located in
the far field of the array.

Rune
```
```On Saturday, 5 March 2011 17:44:04 UTC+5:30, JanMichica  wrote:
> Hi,
>
> Im an undergraduate student and I have researched how to estimate the
> direction of arrival of a signal using a FFT. I understand how to use it
> and how to implement the algorithm, but im unsure as to why when performing
> an FFT on spatially separated data this yields an angle?
>
> Thanks,
>
> Jan

Hi,

Will it be possible for you to share resources and references from your research. Thanks.
```
```On Thursday, January 31, 2019 at 10:01:18 PM UTC-8, polari...@gmail.com wrote:
> On Saturday, 5 March 2011 17:44:04 UTC+5:30, JanMichica  wrote:
> > Hi,
> >
> > Im an undergraduate student and I have researched how to estimate the
> > direction of arrival of a signal using a FFT. I understand how to use it
> > and how to implement the algorithm, but im unsure as to why when performing
> > an FFT on spatially separated data this yields an angle?
> >
> > Thanks,
> >
> > Jan
>
> Hi,
>
> Will it be possible for you to share resources and references from your research. Thanks.

Hi
Will it be possible to Google "fft beamformer, doa"  and not select comp.dsp?
```
```On Thursday, January 31, 2019 at 11:11:36 PM UTC-8, dbd wrote:
> On Thursday, January 31, 2019 at 10:01:18 PM UTC-8, polari...@gmail.com wrote:
> > On Saturday, 5 March 2011 17:44:04 UTC+5:30, JanMichica  wrote:
> > > Hi,
> > >
> > > Im an undergraduate student and I have researched how to estimate the
> > > direction of arrival of a signal using a FFT. I understand how to use it
> > > and how to implement the algorithm, but im unsure as to why when performing
> > > an FFT on spatially separated data this yields an angle?
> > >
> > > Thanks,
> > >
> > > Jan
.> >
.> > Hi,
.> >
.> > Will it be possible for you to share resources and references from your research. Thanks.
.>
.> Hi
.> Will it be possible to Google "fft beamformer, doa"  and not select comp.dsp?

Max may be confused on two points. One is that deleting a message removes quoted occurrences of it. The other has to do with what I meant by "not select comp.dsp". I explained that to him in my response to his quaint e-mail, both quoted below from my e-mail software:

----------------------------------------
----------------------------------------
Max

If you had the competence or integrity to try the suggested Google search, you would have found that your post to comp.dsp would be one of the top responses and it would have taught you nothing to "select comp.dsp" from the Google results. That was my "other advice".

--------------------------------------------
On Fri, 2/1/19, max <polarizadmax@gmail.com> wrote:

Subject: dsp.com attracts pricks like you
To: d.dalrymple@sbcglobal.net
Date: Friday, February 1, 2019, 8:04 AM

Hi,
Thanks
and learn some "etiquettes

-----------------------------------------
-----------------------------------------
```
```On Fri, 1 Feb 2019 09:35:08 -0800 (PST), dbd
<d.dalrymple@sbcglobal.net> wrote:

>On Thursday, January 31, 2019 at 11:11:36 PM UTC-8, dbd wrote:
>> On Thursday, January 31, 2019 at 10:01:18 PM UTC-8, polari...@gmail.com w=
>rote:
>> > On Saturday, 5 March 2011 17:44:04 UTC+5:30, JanMichica  wrote:
>> > > Hi,
>> > >=20
>> > > Im an undergraduate student and I have researched how to estimate the
>> > > direction of arrival of a signal using a FFT. I understand how to use=
> it
>> > > and how to implement the algorithm, but im unsure as to why when perf=
>orming
>> > > an FFT on spatially separated data this yields an angle?
>> > >=20
>> > > Thanks,
>> > >=20
>> > > Jan
>.> >=20
>.> > Hi,
>.> >=20
>.> > Will it be possible for you to share resources and references from you=
>r research. Thanks.
>.>=20
>.> Hi
>.> Will it be possible to Google "fft beamformer, doa"  and not select comp=
>.dsp?
>
>Max may be confused on two points. One is that deleting a message removes q=
>uoted occurrences of it. The other has to do with what I meant by "not sele=
>ct comp.dsp". I explained that to him in my response to his quaint e-mail, =
>both quoted below from my e-mail software:
>
>----------------------------------------
>----------------------------------------
>Max
>
>If you had the competence or integrity to try the suggested Google search, =
>you would have found that your post to comp.dsp would be one of the top res=
>ponses and it would have taught you nothing to "select comp.dsp" from the G=
>oogle results. That was my "other advice".
>
>over excite yourself.
>
>--------------------------------------------
>On Fri, 2/1/19, max <polarizadmax@gmail.com> wrote:
>
> Subject: dsp.com attracts pricks like you
> To: d.dalrymple@sbcglobal.net
> Date: Friday, February 1, 2019, 8:04 AM
>=20
> Hi,=20
>  Thanks
> and learn some "etiquettes
>
>-----------------------------------------
>-----------------------------------------

I'd have thought that "etiquettes" might have included taking what you
can get after resurrecting a seven-year-old thread in a newsgroup with
very little current traffic.

I thought your initial response was reasonable and informative.

```