A Fourier transform and 3 variations caused by periodic sampling (at interval T) and/or periodic summation (at interval P) of the underlying time-domain function.
In mathematics, any integrable function can be made into a periodic function with period P by summing the translations of the function by integer multiples of P. This is called periodic summation:
When is alternatively represented as a Fourier series, the Fourier coefficients are equal to the values of the continuous Fourier transform, at intervals of .[1][2] That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of at constant intervals (T) is equivalent to a periodic summation of which is known as a discrete-time Fourier transform.
The periodic summation of a Dirac delta function is the Dirac comb. Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb.
^Pinsky, Mark (2001). Introduction to Fourier Analysis and Wavelets. Brooks/Cole. ISBN 978-0534376604.
^Zygmund, Antoni (1988). Trigonometric Series (2nd ed.). Cambridge University Press. ISBN 978-0521358859.
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) {\displaystyle s(t)} by integer multiples of P. This is called periodicsummation: s P ( t ) = ∑ n = − ∞ ∞ s ( t + n P ) {\displaystyle s_{P}(t)=\sum...
mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodicsummation of a function to values...
symmetry restored. The DTFT samples, labeled DFT8 periodicsummation, are an example of using periodicsummation to sample it at the same frequencies as the...
function. Since periodicsummation of the function means discretizing its frequency spectrum and discretization means periodicsummation of the spectrum...
discrete sequences is the periodic convolution of the DTFTs of the individual sequences. And each DTFT is a periodicsummation of a continuous Fourier transform...
data Periodic sequence PeriodicsummationPeriodic travelling wave Quasiperiodic function Seasonality Secular variation Wavelength List of periodic functions...
} Any s P ( t ) {\displaystyle s_{_{P}}(t)} can be expressed as a periodicsummation of another function, s ( T ) {\displaystyle s(T)} : s P ( t ) ≜ ∑...
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g. electrostatic interactions) in periodic systems...
groups. The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodicsummation of a function to values...
{\displaystyle P} -periodic functions u P {\displaystyle u_{_{P}}} and v P , {\displaystyle v_{_{P}},} which can be expressed as periodicsummations: u P ( x...
factor of N {\displaystyle N} ) to the inverse DFT of one cycle of the periodicsummation, S N . {\displaystyle S_{N}.} : p.542 (eq 8.4) : p.77 (eq 4.24) ...
{\displaystyle x[n]} sequence is the Fourier series representation of a periodicsummation of X ( f ) : {\displaystyle X(f):} When T {\displaystyle T} has units...
In thermal quantum field theory, the Matsubara frequency summation (named after Takeo Matsubara) is a technique used to simplify calculations involving...
functions are symmetrical around the 0 Hz axis. After sampling, only a periodicsummation of the Fourier transform (called discrete-time Fourier transform)...
}}n{\text{-sequence of length }}N)},} where x N {\displaystyle x_{_{N}}} is a periodicsummation: x N [ n ] ≜ ∑ m = − ∞ ∞ x [ n − m N ] . {\displaystyle x_{_{N}}[n]\...
Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small...
_{n=0}^{\infty }a(n)x^{n}} where x is a real variable (see Z-transform). Replacing summation over n with integration over t, a continuous version of the power series...
kT} , convolution with the Dirac comb corresponds to replication or periodicsummation: ( Ш 1 T ∗ X ) ( f ) = ∑ k = − ∞ ∞ X ( f − k T ) {\displaystyle...
-kT_{0})p^{*}(t-kT_{0}).\end{aligned}}} The last summation is a periodicsummation, hence a signal periodic in t. This way, x ( t ) {\displaystyle x(t)} is...
pulse train or rectangular wave is a non-sinusoidal waveform that is the periodic version of the rectangular function. It is held high a percent each cycle...
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