Editing Discrete-time Fourier transform (section)

== Relation to Fourier Transform ==

Let <math>s(t)</math> be a continuous function in the time domain.
We begin with a common definition of the continuous [[Fourier transform]],
where <math>f</math> represents frequency in hertz and <math>t</math> represents time in seconds:

:<math>S(f) \triangleq \int_{-\infty}^\infty s(t)\cdot e^{-i 2\pi ft} dt.</math>

We can reduce the integral into a summation by sampling <math>s(t)</math> at intervals of <math>T</math> seconds
(see {{slink|Fourier_transform|Numerical_integration_of_a_series_of_ordered_pairs}}).
Specifically, we can replace <math>s(t)</math> with a discrete sequence of its samples, <math>s(nT)</math>, for integer values of <math>n</math>,
and replace the differential element <math>dt</math> with the sampling period <math>T</math>.
Thus, we obtain one formulation for the discrete-time Fourier transform (DTFT):

:<math>S_{1/T}(f) \triangleq \sum_{n=-\infty}^{\infty} \underbrace{T\cdot s(nT)}_{s[n]}\ e^{-i 2\pi f T n}.</math>

This [[Fourier series#Definition|Fourier series]] (in frequency) is a continuous periodic function, whose periodicity is the sampling frequency <math>1/T</math>.
The subscript <math>1/T</math> distinguishes it from the continuous Fourier transform <math>S(f)</math>,
and from the angular frequency form of the DTFT.
The latter is obtained by defining an angular frequency variable, <math>\omega \triangleq 2 \pi f T</math> (which has [[Normalized frequency (digital signal processing)|normalized units]] of ''radians/sample''), giving us  a periodic function of angular frequency, with periodicity <math>2\pi</math>:{{efn-la
|Oppenheim and Schafer,<ref name=Oppenheim/> p 147 (4.17), where: &nbsp;<math>x[n] \triangleq s(nT) = \tfrac{1}{T}s[n],</math> therefore <math>X(e^{i\omega}) \triangleq \tfrac{1}{T}S_{2\pi}(\omega).</math>
}}

{{Equation box 1
|indent=: |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk||<math>
S_{2\pi}(\omega) = S_{1/T}\left(\tfrac{\omega}{2\pi T}\right) = \sum_{n=-\infty}^{\infty} s[n] \cdot e^{-i \omega n}.
</math> &nbsp; &nbsp;|{{EquationRef|Eq.1}}}}
}}

[[File:Fourier transform, Fourier series, DTFT, DFT.svg|thumb|500px|Fig 1. Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT).]]

The utility of the DTFT is rooted in the [[Poisson summation formula]], which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform''':'''{{efn-la
|Oppenheim and Schafer,<ref name=Oppenheim/> p 147 (4.20), p 694 (10.1), and Prandoni and Vetterli,<ref name=Prandoni/> p 255, (9.33), where: &nbsp;<math>\omega \triangleq 2\pi f T,</math>&nbsp; and &nbsp;<math>X_c(i 2\pi f) \triangleq S(f).</math>
}}{{Equation box 1|border|title=Poisson summation|indent=:|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6|equation={{NumBlk||<math>
S_{1/T}(f) =
\sum_{n=-\infty}^{\infty} s[n]\cdot e^{-i 2\pi f T n}\;
= \sum_{k=-\infty}^{\infty} S\left(f - k/T\right).
</math> &nbsp; &nbsp;|{{EquationRef|Eq.2}}}}}}
The components of the periodic summation are centered at integer values (denoted by <math>k</math>) of a [[Normalized frequency (signal processing)|normalized frequency]] (cycles per sample).  Ordinary/physical frequency (cycles per second) is the product of <math>k</math> and the sample-rate, <math>f_s=1/T.</math> &nbsp; For sufficiently large <math>f_s,</math> the <math>k=0</math> term can be observed in the region <math>[-f_s/2, f_s/2]</math> with little or no distortion ([[aliasing]]) from the other terms.&nbsp; '''Fig.1''' depicts an example where <math>1/T</math> is not large enough to prevent aliasing.

We also note that <math>e^{-i2\pi fTn}</math> is the Fourier transform of <math>\delta(t-nT).</math>  Therefore, an alternative definition of DTFT is''':'''{{efn-ua
|In fact {{EquationNote|Eq.2}} is often justified as follows''':'''<ref name=Oppenheim/>{{rp|p.143, eq 4.6}}
<math display="block">\begin{align}
\mathcal{F}\left \{\sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot \delta(t-nT)\right \} &=\mathcal{F}\left \{s(t)\cdot T \sum_{n=-\infty}^{\infty} \delta(t-nT)\right \}\\
&= S(f) * \mathcal{F}\left \{T \sum_{n=-\infty}^{\infty} \delta(t-nT)\right \} \\
&= S(f) * \sum_{k=-\infty}^{\infty} \delta \left(f - \frac{k}{T}\right) \\
&= \sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right).
\end{align}</math>
}}
{{Equation box 1
|indent=: |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk||<math>
S_{1/T}(f) = \mathcal{F}\left \{\sum_{n=-\infty}^{\infty} s[n] \cdot \delta(t-nT)\right \}.
</math> &nbsp; &nbsp;|{{EquationRef|Eq.3}}}}
}}

The modulated [[Dirac comb]] function is a mathematical abstraction sometimes referred to as ''impulse sampling''.<ref name=Rao/>