ApertureLab · Why Synthetic Aperture?
Step 1 of 7
Start with one element.
An element is a single transducer: the smallest piece of<br>a sonar that turns an electrical signal into sound, and turns a returning<br>echo back into a voltage. A microphone and a loudspeaker in one part, tuned<br>to a narrow band and coupled to water instead of air.
It has a physical size. The element used throughout this page is 33 mm<br>long. The sonar drives it at 300 kHz, and sound travels at roughly 1500 m/s<br>in seawater, so one wavelength is 5 mm.
Those two lengths, the size of the element and the wavelength, decide<br>almost everything that follows. Every array on this page, real or synthetic,<br>is built out of elements like this one.
33 mm
electrical drive
one element<br>acoustic wave, λ = 5 mm<br>300 kHz in seawater
One transducer element. Its length D and the acoustic<br>wavelength λ are the only two numbers the rest of this page needs.
Step 2 of 7
One element cannot tell two things apart.
Sound leaving an element does not travel as a pencil. It spreads into a<br>cone, and the angular width of that cone is set by the ratio of wavelength<br>to element size:
θ ≈ λ / D = 5 mm / 33 mm = 0.15 rad = 8.6°
Now put a target in the water. The echo comes back at a time that pins<br>down its range precisely, and a wideband pulse separates two<br>objects a few centimeters apart in range no matter how far away they are.<br>Range resolution does not degrade with distance.
Direction is another matter. All the echo tells you is that something is<br>somewhere inside the cone. Two objects at the same range, anywhere across<br>that cone, send back echoes that arrive at the same instant and add<br>together. They are one measurement, not two.
And the cone widens with distance, so the ambiguity grows in proportion<br>to range:
0.30 m<br>across the beam at 2 m
3.8 m<br>at 25 m
7.5 m<br>at 50 m
15 m<br>at 100 m
θ ≈ λ/D = 8.6°
3.8 m<br>7.5 m<br>15 m
25 m<br>50 m<br>100 m
two targets, same range, both inside the beam;<br>the echoes arrive together and add
along-track
range
Drawn to scale in both axes: the wedge really is 8.6 degrees<br>wide, and the footprint really does reach 15 m across at 100 m. Range<br>resolution stays fixed while along-track resolution degrades linearly with<br>distance.
So the picture a single element produces is sharp in one direction and<br>smeared in the other, and the smearing gets worse the further out you look.<br>That asymmetry is the entire reason synthetic aperture exists.
A note on conventions
Beamwidth can be quoted as the half-power width, the first-null width,<br>or the nominal λ/D. They differ by factors near one. This page uses<br>λ/D throughout so the numbers stay comparable from step to step.
Step 3 of 7
Directionality is not a property of a transducer.
Take a second element and place it a distance d from the first, then add<br>the two received signals together.
For a wave arriving straight ahead , the crests reach both<br>elements at the same moment. The two voltages are in step, and the sum is<br>twice either one.
For a wave arriving off to the side , one element is<br>further along the incoming wavefront than the other. The extra distance the<br>wave has to cover is d sin θ. When that extra distance is<br>half a wavelength, the two voltages are exactly opposite and the sum is<br>zero. The pair is blind in that direction.
Sweep the angle below and watch it happen.
Arrival angle of the incoming wave
0° straight ahead<br>30° first null<br>90° endfire
geometry
broadside
d = λ<br>amber: the extra distance<br>the far element has to cover
what each element receives
A + B<br>time →
amplitude of the sum, against arrival angle
2×<br>sum
-90°<br>-60°<br>-30°<br>0°<br>30°<br>60°<br>90°
24°<br>arrival angle
0.41 λ<br>extra path d sin θ
146°<br>phase difference
0.59 ×<br>sum, vs. one element
Drag the angle and watch B slide against A. Where the<br>extra path reaches half a wavelength the two are opposite and the sum<br>collapses to nothing; the curve underneath is that cancellation plotted<br>across every angle, which is the beam pattern of a pair of elements.<br>Spacing here is one wavelength, which puts the first null at a visible<br>30 degrees. The second peak at 90 degrees is a real ambiguity, and<br>suppressing it is one of the jobs the extra elements in step 4 do.
Nothing about either element changes as you sweep. Neither one knows<br>which way it is pointing, and neither one is more sensitive in any<br>direction than it was before. The whole response comes from the phase<br>relationship between two separated measurements.
This is the pivot the whole subject turns on. Directionality is something<br>you construct out of separated measurements. It is not something a<br>transducer has.
Step 4 of 7
More elements make one longer aperture.
Add more elements along the same line. More cancellation directions<br>appear, and the one direction where everything still adds gets narrower. For<br>N elements spaced d apart, the beam narrows to roughly λ/L, where<br>L = N·d is the total length of the array.
The count is not what matters. The length is. Two arrays<br>of the same...