Editing Z-transform (section)

==Relationship to Fourier series and Fourier transform==
{{further|Discrete-time Fourier transform#Relationship to the Z-transform}}

For values of <math>z</math> in the region <math>|z| {=} 1</math>, known as the [[unit circle]], we can express the transform as a function of a single real variable <math>\omega</math> by defining <math>z {=} e^{j \omega}.</math> And the bi-lateral transform reduces to a [[Fourier series]]:

{{Equation box 1 |title =
|indent= |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk|:|
<math>\sum_{n=-\infty}^{\infty} x[n]\ z^{-n} = \sum_{n=-\infty}^{\infty} x[n]\ e^{-j\omega n},</math>  
|{{EquationRef|Eq.1}} }} }}

which is also known as the [[discrete-time Fourier transform]] (DTFT) of the <math>x[n]</math> sequence. This <math>2\pi</math>-periodic function is the [[periodic summation]] of a [[continuous Fourier transform|Fourier transform]], which makes it a widely used analysis tool. To understand this, let <math>X(f)</math> be the Fourier transform of any function, <math>x(t)</math>, whose samples at some interval <math>T</math> equal the <math>x[n]</math> sequence. Then the DTFT of the <math>x[n]</math> sequence can be written as follows.

{{Equation box 1 |title =
|indent= |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk|:|
<math>\underbrace{\sum_{n=-\infty}^{\infty} \overbrace{x(nT)}^{x[n]}\ e^{-j 2\pi f nT}
}_{\text{DTFT}} = \frac{1}{T}\sum_{k=-\infty}^{\infty} X(f-k/T),</math>  
|{{EquationRef|Eq.2}} }} }}

where <math>T</math> has units of seconds, <math>f</math> has units of [[hertz]]. Comparison of the two series reveals that <math> \omega {=} 2\pi fT</math> is a [[Normalized frequency (digital signal processing)#Alternative normalizations|normalized frequency]] with unit of ''radian per sample''. The value <math>\omega{=}2\pi</math> corresponds to <math display="inline"> f {=} \frac{1}{T}</math>. And now, with the substitution <math display="inline"> f{=}\frac{\omega }{2\pi T},</math> {{EquationNote|Eq.1}} can be expressed in terms of <math>X( \tfrac{\omega - 2\pi k}{2\pi T} )</math> (a Fourier transform):

{{Equation box 1 |title =
|indent= |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk|:|
<math>
\sum_{n=-\infty}^{\infty} x[n]\ e^{-j\omega n} = \frac{1}{T}\sum_{k=-\infty}^{\infty} \underbrace{X\left(\tfrac{\omega}{2\pi T} - \tfrac{k}{T}\right)}_{X\left(\frac{\omega - 2\pi k}{2\pi T}\right)}.
</math>  
|{{EquationRef|Eq.3}} }} }}

As parameter ''T'' changes, the individual terms of {{EquationNote|Eq.2}} move farther apart or closer together along the ''f''-axis. In {{EquationNote|Eq.3}} however, the centers remain 2{{pi}} apart, while their widths expand or contract. When sequence <math>x(nT)</math> represents the [[impulse response]] of an [[LTI system]], these functions are also known as its [[frequency response]]. When the <math>x(nT)</math> sequence is periodic, its DTFT is divergent at one or more harmonic frequencies, and zero at all other frequencies.  This is often represented by the use of amplitude-variant [[Dirac delta]] functions at the harmonic frequencies. Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler [[discrete Fourier transform]] (DFT). (See ''{{slink|Discrete-time Fourier transform|Periodic data}}''.)