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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.

An interferometric baseline is the displacement $\mathbf{b}={\mathbf{r}}_2-{\mathbf{r}}_1$ between two antennas. For a unit line of sight ${\mathbf{s}}_0$ toward the phase centre, its projection onto the plane of the sky isHere $c$ is the speed of light. The signed delay depends on which receiver is delayed; the geometric path difference's magnitude does not. If ${\mathbf{e}}_{\theta }$ is a unit tangent in the direction being imaged, the dimensionless coordinate $u=\mathbf{b}\cdot {\mathbf{e}}_{\theta }/\lambda$ is a spatial frequency, measured in cycles per radian of small sky offset. A single pair gives fringes of angular period $1/|u|$ along that tangent. This period specifies one measured oscillation; it does not specify a whole image's angular resolution or eliminate ambiguous source arrangements.

The left panel separates the component producing the reference delay from the component measuring angular structure. The right panel shows a possible Earth-rotation aperture synthesis track, with its opposite complex conjugate track. Those opposite samples provide the symmetry required by a real brightness distribution; they are not independent extra observations.

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.