Every antenna pair (i,j) forms a baseline with equatorial components Bx, By (the ground-plane separation, here treated as coplanar with the equator, Bz=0 — the standard simplifying assumption for an intro synthesis-imaging demo). As Earth rotates through hour angle H, each baseline traces an ellipse in the Fourier (u,v) plane set by the source declination δ:
u(H) = Bx·sinH + By·cosH
v(H) = -Bx·sinδ·cosH + By·sinδ·sinH
(u,v in wavelengths λ = c/f)
The dirty beam — the array's response to a point source, i.e. its point-spread function — is the direct Fourier sum over every sampled (u,v) point (plus its Hermitian conjugate −u,−v, since a real sky has real visibilities):
beam(l,m) = (1/N) ∑ cos( 2π(u·l + v·m) )
Longer hour-angle coverage and more baselines fill the (u,v) plane more densely, narrowing the central lobe toward the theoretical θ≈λ/Bmax resolution and suppressing the negative sidelobes (the "dirty" ringing) visible around it — exactly the trade-off real arrays like the VLA and ALMA make between observing time and image fidelity.
- Array pane — the antenna ground layout and its baselines.
- u-v pane — every sampled Fourier point, coloured by hour angle.
- Dirty beam pane — the reconstructed point-spread function; blue = negative sidelobe, white/yellow = positive.