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Aperture synthesis reconstructs the sky's brightness from complex visibilities measured at different interferometric baselines. Each measurement constrains a Fourier transform component of the brightness, rather than supplying a little photograph. Combining many such components can distinguish sources that each individual antenna leaves unresolved. The remarkable economy is that changing separations and the rotation of Earth can provide many measurements with only a few physical antennas.

Martin Ryle's Cambridge group made aperture synthesis a practical method of astronomical imaging at the Mullard Radio Astronomy Observatory. Cambridge's institutional history connects that work with the One-Mile Telescope and its larger successors.^[ref]

In his 1974 lecture Radio Telescopes of Large Resolving Power, Martin Ryle recalled the problem of distinguishing weak extragalactic sources and resolving their structure. He explained sequential wavefront sampling using one simultaneously observed element as a wave phase reference, and the saving obtained by letting Earth rotate the apparent baseline.^[ref]

The 1974 Nobel Prize in Physics distinguished Martin Ryle's observations and inventions, especially aperture synthesis, from Antony Hewish's role in discovering pulsars. The latter is a separate astronomical discovery, with its own observational history.^[ref]

A complex visibility is a calibrated complex correlation of the signals received at a particular interferometric baseline. In a narrow field, with scalar radiation, the far-field approximation for an outgoing source, a narrow frequency channel, negligible noncoplanar phase curvature and a steady sky of mutually incoherent emitters,Here $l,m$ are small dimensionless sky-direction offsets, $I$ is brightness in the chosen channel, and $A$ is the primary beam. This is the Fourier transform of beam-weighted brightness. The two-dimensional formula requires those approximations: a wide field generally needs the third baseline coordinate and curved sky geometry.

For an explicit one-dimensional derivation, set the primary beam to one over the small field, and place independent emitters at angular offsets ${\theta }_j$. Choose a wave phase convention in which, after removing the phase centre's delay, their received complex amplitudes areHere ${\delta }_{jk}$ is the Kronecker delta and angle brackets denote an expected value estimated by time averaging a steady signal. The last equality states mutual incoherence: different emitters have zero averaged cross term, although each emitter's two received copies remain correlated. Multiplying and averaging givesFor continuous brightness this becomes $V(u)=\int I(\theta )e^{-2\pi iu\theta }d\theta$. Real brightness implies the Hermitian symmetry $V(-u)=V(u{)}^{\ast }$; it does not require brightness to be even. In particular, an asymmetric real source usually has a nonzero imaginary complex visibility. At zero baseline, $V(0)={\sum }_j{I}_j$ is the total beam-weighted source strength. Ordinary separated-element correlations need not measure that zero-baseline value.

The dirty beam is the point spread function obtained by directly inverting the weighted sampled complex visibilities. For sampling positions $u_k$ and nonnegative weights $w_k$ with $W={\sum }_k{w}_k>0$, chooseInclude each $u_k$ with $-u_k$ and the same weight so that $B$ is real. Its unit peak convention means that an isolated unit point source contributes a shifted unit-height response. It is not a unit-integral convention. Sparse sampling can leave substantial subsidiary peaks, even when the longest interferometric baseline is large.

The dirty image is the direct weighted inverse Fourier transform of the measured complex visibilities:The final equality follows by substituting the point-source complex visibility and exchanging the finite sums. For continuous brightness it is a convolution with the dirty beam, with the primary beam included in the brightness when necessary. Negative values and subsidiary peaks can arise from that response even when every true source has positive strength.

The phase centre is the reference sky direction for removing geometric delay and its associated wave phase. An unresolved source exactly there ideally has constant complex visibility after calibration; a displaced source has a baseline-dependent wave phase.

The primary beam is the individual antenna pair's directional sensitivity envelope in the complex visibility equation. For identical, identically pointed elements it is the individual element's power pattern. Its angular width sets the useful field of view; the much longer interferometric baselines set finer measured scales. The primary beam and the dirty beam describe different responses.

Radio interferometer calibration estimates instrumental and propagation effects that alter a complex visibility. A simple narrow-field model iswhere $g_1,g_2$ are complex receiver gains and $n_{12}$ represents additive noise. A point-like astronomical source of known position and stable strength constrains these gains: its expected complex visibility has a known wave phase and amplitude. A resolved or variable reference source requires a better source model. Errors in the reference model can be transferred into the inferred sky.

In a proposed extension, Martin Ryle described simultaneous observation of a nearby reference point source to correct atmospheric wave phase errors, using an additional dish at each element. He presented this as a possible extension, rather than the standard One-Mile Telescope arrangement.^[ref]

Radio interferometry measures the spatial coherence of received radio waves. At two antennas, a receiver preserves the relative wave phase, compensates the geometric delay for a chosen phase centre, and averages a product of one signal with the complex conjugate of the other. In a complex-signal description this correlation is $⟨E_1{E}_2^{\ast }⟩$. Its real and imaginary parts carry two complementary components of the complex visibility; measuring only the received powers discards the relative wave phase needed for general imaging.

A finite frequency band makes delay compensation essential. If the residual delay is $\Delta \tau$, the residual wave phase varies by $2\pi \Delta \nu \Delta \tau$ across a band of width $\Delta \nu$. Averaging these differently oriented complex contributions reduces the correlation. For an ideal rectangular band the multiplicative factor is $\sin (\pi \Delta \nu \Delta \tau )/(\pi \Delta \nu \Delta \tau )$. A phase correction at just the central frequency cannot remove this variation across the band. Long time averages can likewise smear a changing signal.

A radio telescope is an astronomical instrument that receives radio waves using an antenna and associated receivers. A single dish measures radiation over its directional sensitivity pattern; separated elements can supply complex visibilities for aperture synthesis.

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 isAt $u=0$, $V=1.65$; 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.

The sparse case samples $u=\pm 12.5k$ for $k\in \{2,5,9,14,20,28,36,44,52,60,64\}$; the denser case samples every $k=1,\dots ,64$, 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.