Periodic summation

periodic sumperiodization
In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P.wikipedia
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Poisson summation formula

Poisson series
That identity is a form of the Poisson summation formula.
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform.

Discrete-time Fourier transform

convolution theoremDFTDTFT § Properties
Similarly, a Fourier series whose coefficients are samples of s(t) at constant intervals (T) is equivalent to a periodic summation of S(f), which is known as a discrete-time Fourier transform.
From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.

Fourier transform

continuous Fourier transformFourierFourier transforms
When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, at intervals of 1/P.
The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform.

Convolution

convolvedconvolvingconvolution kernel
Likewise, the periodic summation of an integrable function is its convolution with the Dirac comb.
The summation is called a periodic summation of the function f.

Periodic function

periodicperiodperiodicity
In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P.

Dirac comb

Sampling functionimpulse traininfinite impulse train
The periodic summation of a Dirac delta function is the Dirac comb.
The Dirac comb can be constructed in two ways, either by using the comb operator (performing sampling) applied to the function that is constantly 1, or, alternatively, by using the rep operator (performing periodization) applied to the Dirac delta \delta.

Circular convolution

Cyclic convolutioncircular (or cyclic) convolutionPeriodic convolution
The circular convolution, also known as cyclic convolution, of two aperiodic functions (i.e. Schwartz functions) occurs when one of them is convolved in the normal way with a periodic summation of the other function.

Signal processing

signal analysissignalsignal processor
In signal processing, any periodic function, s_P(t) with period P, can be represented by a summation of an infinite number of instances of an aperiodic function, s(t), that are offset by integer multiples of P.

Fourier series

Fourier coefficientFourier expansionFourier coefficients
When s_P(t) is alternatively represented as a complex Fourier series, the Fourier coefficients are proportional to the values (or "samples") of the continuous Fourier transform, at intervals of 1/P.

Dirac delta function

Dirac deltadelta functionimpulse
The periodic summation of a Dirac delta function is the Dirac comb.

Quotient space (linear algebra)

quotient spacequotientquotient vector space
If a periodic function is represented using the quotient space domain

Domain of a function

domaindomainsdomain of definition
If a periodic function is represented using the quotient space domain

Equivalence class

quotient setequivalence classesquotient
The arguments of \varphi_P are equivalence classes of real numbers that share the same fractional part when divided by P.

Real number

realrealsreal-valued
The arguments of \varphi_P are equivalence classes of real numbers that share the same fractional part when divided by P.

Fractional part

mod 1
The arguments of \varphi_P are equivalence classes of real numbers that share the same fractional part when divided by P.

Discrete Fourier transform

DFTcircular convolution theoremFourier transform
which is the convolution of the \mathbf{x} sequence with a \mathbf{y} sequence extended by periodic summation:

Wrapped distribution

wrapped probability distributionwrappedperiodization
which is a periodic sum of period 2\pi.

Nyquist–Shannon sampling theorem

sampling theoremNyquist-Shannon sampling theoremNyquist theorem
the Poisson summation formula indicates that the samples, x(nT), of x(t) are sufficient to create a periodic summation of X(f).

Multidimensional sampling

multi-dimensional samplingmultidimensional signal processing
The generalization of the Poisson summation formula to higher dimensions can be used to show that the samples, of the function f(\cdot) on the lattice \Lambda are sufficient to create a periodic summation of the function.

Upsampling

Digital up converterhigher sampling rateTime expansion
Then the discrete-time Fourier transform (DTFT) of the x[n] sequence is the Fourier series representation of a periodic summation of X(f):

Fourier analysis

FourierFourier synthesisanalyse the output wave into its constituent harmonics
, is expressed as a periodic summation of another function,

Downsampling (signal processing)

downsamplingdecimationdownsampled
The abscissa of the top pair of graphs represents the discrete-time Fourier transform (DTFT), which is a Fourier series representation of a periodic summation of X(f):

Undersampling

bandpass samplingundersampled
After sampling, only a periodic summation of the Fourier transform (called discrete-time Fourier transform) is still available.

Cyclostationary process

cyclostationarityCyclostationarycyclostationary noise
The last summation is a periodic summation, hence a signal periodic in t.