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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.

For two three-dimensional real maps, let $F_1,F_2$ be their Fourier transforms and let $S_q$ be a reciprocal-space shell around spatial frequency $q$. The displayed normalized coefficient inner product is the Fourier shell correlation. It is defined only when both shell-energy sums in its denominator are positive.

For a shell containing both $\mathbf{k}$ and $-\mathbf{k}$, Hermitian symmetry makes the numerator real. The Cauchy-Schwarz inequality bounds its magnitude by one. Identical nonzero shell coefficients give one; reversing their sign gives minus one. A random-noise estimate can also fluctuate below zero, so the mathematical range is not restricted to zero through one.

Independent half-map reconstruction tests which frequency components reproduce across separate data subsets. This is how the EMDB validation pipeline interprets its FSC curve. ^[ref]

Shared refinement information, applying an image mask or correlated noise can inflate agreement. The selected resolution crossing depends on an explicitly stated threshold and processing convention; a high coefficient alone does not establish that every local feature in either map is correct.

If $f$ is a real-valued function and an integrable function, its Fourier transform has Hermitian symmetry: $F(-u)=\overline{F(u)}$. The value at negative frequency is the complex conjugate of the value at positive frequency.

For real $f$, complex conjugation commutes with its absolutely convergent Fourier integral:ThusThis Hermitian symmetry pairs the positive- and negative-frequency components of a real signal; its real part is even and its imaginary part is odd.

Apply a Fourier transform or discrete Fourier transform, retain the desired low-frequency coefficients and attenuate those in the high-frequency noise band with a low-pass filter, then invert. For a real signal, apply matching gains at positive and negative frequencies to preserve Hermitian symmetry.

Parseval identity for Fourier series shows what an ideal coefficient cutoff does: the output mean-square signal power is the sum of the retained squared coefficients, and cannot exceed that of the unfiltered input. Much of the high-frequency noise can be removed, but a finite cutoff also removes genuine square wave harmonics. Its edges become less sharp, and a sharp spectral cutoff can exhibit the Gibbs phenomenon. There is no claim of exact recovery if the noise overlaps the retained band. Noise already folded into low frequencies by aliasing cannot in general be removed merely by discarding the high-frequency bins afterward.