Editing Fourier analysis (section)

===Discrete-time Fourier transform (DTFT)===
{{main|Discrete-time Fourier transform}}

The DTFT is the mathematical dual of the time-domain Fourier series. Thus, a convergent [[periodic summation]] in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function''':'''

:<math>S_\tfrac{1}{T}(f)\ \triangleq\ \underbrace{\sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right) \equiv \overbrace{\sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i2\pi f n T}}^{\text{Fourier series (DTFT)}}}_{\text{Poisson summation formula}} = \mathcal{F} \left \{ \sum_{n=-\infty}^{\infty} s[n]\ \delta(t-nT)\right \},\,</math>

which is known as the DTFT. Thus the '''DTFT''' of the <math>s[n]</math> sequence is also the '''Fourier transform''' of the modulated [[Dirac comb]] function.{{efn-ua|
We may also note that''':'''
:<math>\begin{align} \sum_{n=-\infty}^{+\infty} T\cdot s(nT) \delta(t-nT) &= \sum_{n=-\infty}^{+\infty} T\cdot s(t) \delta(t-nT) \\ &= s(t)\cdot T \sum_{n=-\infty}^{+\infty} \delta(t-nT). \end{align}</math>
Consequently, a common practice is to model "sampling" as a multiplication by the [[Dirac comb]] function, which of course is only "possible" in a purely mathematical sense.
}}

The Fourier series coefficients (and inverse transform), are defined by''':'''

:<math>s[n]\ \triangleq\ T \int_\frac{1}{T} S_\tfrac{1}{T}(f)\cdot e^{i2\pi f nT} \,df = T \underbrace{\int_{-\infty}^{\infty} S(f)\cdot e^{i2\pi f nT} \,df}_{\triangleq\, s(nT)}.</math>

Parameter <math>T</math> corresponds to the sampling interval, and this Fourier series can now be recognized as a form of the [[Poisson summation formula]].&nbsp; Thus we have the important result that when a discrete data sequence, <math>s[n],</math> is proportional to samples of an underlying continuous function, <math>s(t),</math> one can observe a periodic summation of the continuous Fourier transform, <math>S(f).</math> Note that any <math>s(t)</math> with the same discrete sample values produces the same DTFT.&nbsp; But under certain idealized conditions one can theoretically recover <math>S(f)</math> and <math>s(t)</math> exactly. A sufficient condition for perfect recovery is that the non-zero portion of <math>S(f)</math> be confined to a known frequency interval of width <math>\tfrac{1}{T}.</math>&nbsp; When that interval is <math>\left[-\tfrac{1}{2T}, \tfrac{1}{2T}\right],</math> the applicable reconstruction formula is the [[Whittaker–Shannon interpolation formula]].  This is a cornerstone in the foundation of [[digital signal processing]].

Another reason to be interested in <math>S_\tfrac{1}{T}(f)</math> is that it often provides insight into the amount of [[aliasing]] caused by the sampling process.

Applications of the DTFT are not limited to sampled functions. See [[Discrete-time Fourier transform]] for more information on this and other topics, including''':'''
* normalized frequency units
* windowing (finite-length sequences)
* transform properties
* tabulated transforms of specific functions