= Two-point-source aperture synthesis
{title2=$V(u)=e^{2\pi i u(0.00625)}+0.65e^{-2\pi i u(0.009375)}$}
Choose two <point-like astronomical sources> with $(\theta_1,I_1)=(-0.00625,1)$ and $(\theta_2,I_2)=(0.009375,0.65)$, where the angles are in <radians> and strengths use arbitrary common units. Their separation is $\Delta\theta=0.015625$ <radians>. These are invented model parameters, not historical measurements. Their <complex visibility> follows immediately from the preceding derivation, and its squared magnitude is
$$
|V(u)|^2=1+0.65^2+1.30\cos(2\pi u\Delta\theta).
$$
At $u=0$, $V=1.65$; at $u=400$, $V=-1+0.65i$; at $u=800$, $V=0.35$. The unequal strengths prevent the <complex visibility> magnitude from reaching zero. Its modulation contains the source separation, while a common displacement of both sources changes <wave phase> without changing this magnitude. Amplitude alone loses that location information.
\Image[/aperture-synthesis-two-source-sampling.png]
{title=An original two-source model: complex visibility, sampling gaps, unit-peak dirty beams, and dirty images}
The sparse case samples $u=\pm12.5k$ for $k\in\{2,5,9,14,20,28,36,44,52,60,64\}$; the denser case samples every $k=1,\ldots,64$, also with both signs. All weights are one. Both reach $|u|=800$, so their shortest fringe period is the same $0.00125$ <radians>. Their <dirty beams> and false subsidiary features differ because the intermediate <spatial frequencies> differ. Neither case includes $u=0$ or frequencies above that maximum. The figure uses a periodic angular interval of length $0.08$ <radians>; this artificial boundary makes a discrete <Fourier transform> calculation exact for the chosen grid, rather than modelling an all-sky observation.
The plotted <dirty image> is the sum of two shifted <dirty beams>, not a fitted or deconvolved estimate of the source. Adding samples in this example reduces prominent spurious peaks; it does not establish a universal monotonic improvement for every weighting or sky. A missing zero baseline leaves the periodic <dirty image> with zero mean, so it cannot by itself recover the total positive brightness. Missing short spacings also weaken constraints on extended smooth emission.
Reconstruction is an <inverse problem>. A perturbation whose <Fourier transform> vanishes at every sampled coordinate can be added without changing the measurements: incomplete coverage leaves a <null space>. Nonnegative brightness, a restricted source region, or a model with a small number of components can reduce ambiguity, but each is additional information. Removing <dirty beam> artefacts means estimating a sky consistent with both data and such assumptions; a narrow central peak alone does not certify a unique image. <Measurement error>, uncertain calibration and source variability add further limitations.
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