Editing Periodogram (section)

==Computation==

[[File:Periodogram.svg|thumb|400px|A power spectrum (magnitude-squared) of two sinusoidal basis functions, calculated by the periodogram method.]]
[[File:Periodogram windows.svg|thumb|400px|Two power spectra (magnitude-squared) (rectangular and Hamming [[Window function|window functions]] plus background noise), calculated by the periodogram method.]]

For sufficiently small values of parameter {{mvar|T,}} an arbitrarily-accurate approximation for {{math|''X''(''f'')}} can be observed in the region &nbsp;<math>-\tfrac{1}{2T} < f < \tfrac{1}{2T}</math>&nbsp; of the function:

<math display="block">X_{1/T}(f)\ \triangleq \sum_{k=-\infty}^{\infty} X\left(f - k/T\right),</math>

which is precisely determined by the samples {{math|''x''(''nT'')}} that span the non-zero duration of {{math|''x''(''t'')}} &nbsp;(see [[Discrete-time Fourier transform]]).

And for sufficiently large values of parameter {{mvar|N}}, &nbsp;<math>X_{1/T}(f)</math> can be evaluated at an arbitrarily close frequency by a summation of the form:

<math display="block">X_{1/T}\left(\tfrac{k}{NT}\right) = \sum_{n=-\infty}^\infty \underbrace{T\cdot x(nT)}_{x[n]}\cdot e^{-i 2\pi \frac{kn}{N}},</math>

where {{mvar|k}} is an integer.  The periodicity of &nbsp;<math>e^{-i 2\pi \frac{kn}{N}}</math>&nbsp; allows this to be written very simply in terms of a [[Discrete Fourier transform]]:

<math display="block">X_{1/T}\left(\tfrac{k}{NT}\right) = \underbrace{\sum_{n} x_{_N}[n]\cdot e^{-i 2\pi \frac{kn}{N}},}_\text{DFT} \quad \scriptstyle{\text{(sum over any }n\text{-sequence of length }N)},</math>

where <math>x_{_N}</math> is a periodic summation: &nbsp;<math>x_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} x[n - mN].</math>

When evaluated for all integers, {{mvar|k}}, between 0 and {{mvar|N}}-1, the array:
<math display="block">S\left(\tfrac{k}{NT}\right) = \left| \sum_{n} x_{_N}[n]\cdot e^{-i 2\pi \frac{kn}{N}}  \right|^2</math>
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