Editing Fourier analysis (section)

===Summary===
For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components (Fourier series), and the transforms diverge at those frequencies. One common practice (not discussed above) is to handle that divergence via [[Dirac delta]] and [[Dirac comb]] functions. But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finite-duration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact.

It is common in practice for the duration of ''s''(•) to be limited to the period, {{mvar|P}} or {{mvar|N}}.&nbsp; But these formulas do not require that condition.
{| class="wikitable" style="text-align:left"
|+ <math>s(t)</math> transforms (continuous-time)
|-
! !! Continuous frequency !! Discrete frequencies
|-
! Transform
| <math>S(f)\, \triangleq\, \int_{-\infty}^{\infty} s(t) \cdot e^{-i2\pi f t} \,dt</math>
|| <math>\underbrace{\frac{1}{P}\cdot S\left(\frac{k}{P}\right)}_ {S[k]}\, \triangleq\, \frac{1}{P} \int_{-\infty}^{\infty} s(t) \cdot e^{-i2\pi \frac{k}{P} t}\,dt \equiv \frac{1}{P} \int_P s_{_P}(t) \cdot e^{-i2\pi \frac{k}{P} t} \,dt</math>
|-
! Inverse
| <math>s(t) = \int_{-\infty}^{\infty} S(f) \cdot e^{ i2\pi f t}\, df</math>
||<math>\underbrace{s_{_P}(t) = \sum_{k=-\infty}^{\infty} S[k] \cdot e^{i2\pi \frac{k}{P} t}}_{\text{Poisson summation formula (Fourier series)}}\,</math>
|}

{| class="wikitable" style="text-align:left"
|+ <math>s(nT)</math> transforms (discrete-time)
|-
! !! Continuous frequency !! Discrete frequencies
|-
! Transform
| <math>\underbrace{S_\tfrac{1}{T}(f)\, \triangleq\, \sum_{n=-\infty}^{\infty} s[n]\cdot e^{-i2\pi f nT}}_{\text{Poisson summation formula (DTFT)}}</math>
||
<math>
\begin{align}
\underbrace{S_\tfrac{1}{T}\left(\frac{k}{NT}\right)}_ {S[k]}\, &\triangleq\, \sum_{n=-\infty}^{\infty} s[n]\cdot e^{-i2\pi \frac{kn}{N}}\\
&\equiv \underbrace{\sum_{N} s_{_N}[n]\cdot e^{-i2\pi \frac{kn}{N}}}_{\text{DFT}}\,
\end{align}
</math>
|-
! Inverse
| <math>s[n] = \underbrace{T \int_\frac{1}{T} S_\tfrac{1}{T}(f)\cdot e^{i2\pi f nT} \,df}_{\text{Fourier series coefficient}}</math>
<math>\sum_{n=-\infty}^{\infty} s[n]\cdot \delta(t-nT) = \underbrace{\int_{-\infty}^{\infty} S_\tfrac{1}{T}(f)\cdot e^{i2\pi f t}\,df}_{\text{inverse Fourier transform}}\,</math>
||
<math>
s_{_N}[n] = \underbrace{\frac{1}{N} \sum_{N} S[k]\cdot e^{i2\pi \frac{kn}{N}}}_{\text{inverse DFT}}
</math>
|}