Dirac comb

A periodic distribution formed by equally spaced Dirac delta impulses. It represents an ideal impulse train and is central to sampling, Fourier analysis, and periodic summation.

Dirac delta: A distribution concentrated at one point, with unit integral and zero value everywhere else in the distributional sense. Each tooth of the comb is a scaled or translated copy of this impulse.
Sampling theorem: A result specifying when a band-limited signal can be reconstructed exactly from regularly spaced samples. Multiplication by a comb models uniform sampling and exposes spectral replicas.
Distribution (mathematics): A generalized function defined by its action on smooth test functions rather than by pointwise values. The comb belongs to this broader framework because its impulses are not ordinary function values.
Impulse train: A sequence of impulses at specified times or positions, not necessarily equally spaced or periodic. A Dirac comb is the special periodic, equally spaced case of an impulse train.
Fourier transform: An operation that expresses a function or distribution in terms of its frequency components. The comb transforms into another comb, with spacing set by the reciprocal period.
Nyquist frequency: Half the sampling rate, marking the highest frequency that can be represented without aliasing under ideal sampling. Comb-based sampling makes this frequency the boundary for unambiguous spectral representation.
Kronecker delta: A discrete function equal to one at index zero and zero at every other integer index. It is the discrete counterpart of the continuous impulses arranged in the comb.
Finite impulse response: A discrete-time filter whose output depends on a finite number of input samples. Finite sequences can approximate a comb but lack its infinite periodic support.
Poisson summation formula: An identity equating a sum of function values on a lattice with a sum of Fourier-transform values on its reciprocal lattice. It formalizes the dual-lattice relation behind the comb's Fourier transform.
Aliasing: The misidentification of distinct frequencies or signals when sampling or representing them at insufficient resolution. Periodic spectral copies produced by sampling overlap when the sampling rate is too low.

Linked from 7 pages