# 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.

### List of Fourier-related transforms

**Fourier-related transformFourier-related transforms**

### Cyclostationary process

**cyclostationarityCyclostationarycyclostationary noise**

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