$$\sum _{n=0}^{N1}{\leftx\left(n\right)\right}^{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{L}\sum _{k=0}^{L1}{\leftX(2\pi k/L)\right}^{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{\pi}{\int}_{0}^{\pi}{\leftX\left(\omega \right)\right}^{2}\phantom{\rule{0.166667em}{0ex}}d\omega .$$
The second term in
[link] says the Riemann sum is equal to its limit in this
case.
Examples of dtft
As was true for the DFT, insight and intuition is developed by
understanding the properties and a few examples of the DTFT. Severalexamples are given below and more can be found in the literature
[link] ,
[link] ,
[link] . Remember that while in the case of the DFT signals
were defined on the region
$\{0\le n\le (N1\left)\right\}$ and values outside
that region were periodic extensions, here the signals are defined overall integers and are not periodic unless explicitly stated. The spectrum
is periodic with period
$2\pi $ .

$DTFT\left\{\delta \right(n\left)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}1$ for all frequencies.

$DTFT\left\{1\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}2\pi \delta \left(\omega \right)$

$DTFT\left\{{e}^{j{\omega}_{0}n}\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}2\pi \delta (\omega {\omega}_{0})$

$DTFT\{cos\left({\omega}_{0}n\right)\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\pi [\delta (\omega {\omega}_{0})+\delta (\omega +{\omega}_{0})]$

$DTFT\left\{{\sqcap}_{M}\left(n\right)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\frac{sin(\omega Mk/2)}{sin(\omega k/2)}$
The ztransform is an extension of the DTFT in a way that is analogous to
the Laplace transform for continuoustime signals being an extension of theFourier transform. It allows the use of complex variable theory and is
particularly useful in analyzing and describing systems. The question ofconvergence becomes still more complicated and depends on values of
$z$ used in the inverse transform which must be in the “region of
convergence" (ROC).
The ztransform (ZT) is defined as a polynomial in the complex variable
$z$ with the discretetime signal values as its coefficients
[link] ,
[link] ,
[link] . It is given by
$$F\left(z\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\sum _{n=\infty}^{\infty}f\left(n\right)\phantom{\rule{0.166667em}{0ex}}{z}^{n}$$
and the inverse transform (IZT) is
$$f\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\frac{1}{2\pi j}{\oint}_{ROC}F\left(z\right)\phantom{\rule{0.166667em}{0ex}}{z}^{n1}dz.$$
The inverse transform can be derived by using the residue theorem
[link] ,
[link] from complex variable theory to find
$f\left(0\right)$ from
${z}^{1}F\left(z\right)$ ,
$f\left(1\right)$ from
$F\left(z\right)$ ,
$f\left(2\right)$ from
$zF\left(z\right)$ , and in general,
$f\left(n\right)$ from
${z}^{n1}F\left(z\right)$ . Verification by substitution is more difficult than
for the DFT or DTFT. Here convergence and the interchange of order of thesum and integral is a serious question that involves values of the complex
variable
$z$ . The complex contour integral in
[link] must be taken in the
ROC of the z plane.
A unilateral ztransform is sometimes needed where the definition
[link] uses a
lower limit on the transform summation of zero. This allow the transformationto converge for some functions where the regular bilateral transform does
not, it provides a straightforward way to solve initial conditiondifference equation problems, and it simplifies the question of finding
the ROC. The bilateral ztransform is used more for signal analysis andthe unilateral transform is used more for system description and analysis.
Unless stated otherwise, we will be using the bilateral ztransform.
Properties
The properties of the ZT are similar to those for the DTFT and DFT and are
important in the analysis and interpretation of long signals and in theanalysis and description of discretetime systems. The main
properties are given here using the notation that the ZT of acomplex sequence
$x\left(n\right)$ is
$\mathcal{Z}\left\{x\right(n\left)\right\}=X\left(z\right)$ .
 Linear Operator:
$\mathcal{Z}\{x+y\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathcal{Z}\left\{x\right\}+\mathcal{Z}\left\{y\right\}$
 Relationship of ZT to DTFT:
${\mathcal{Z}\left\{x\right\}}_{z={e}^{j\omega}}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathcal{DTFT}\left\{x\right\}$
 Periodic Spectrum:
$X\left({e}^{j\omega}\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}X\left({e}^{j\omega +2\pi}\right)$
 Properties of Even and Odd Parts:
$x\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}u\left(n\right)+jv\left(n\right)$ and
$X\left({e}^{j\omega}\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}A\left({e}^{j\omega}\right)+jB\left({e}^{j\omega}\right)$
$$\begin{array}{cccc}u& v& A& B\\ & & & \\ even& 0& even& 0\\ odd& 0& 0& odd\\ 0& even& 0& even\\ 0& odd& odd& 0\end{array}$$
 Convolution: If discrete noncyclic convolution is defined by
$y\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}h\left(n\right)*x\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\sum}_{m=\infty}^{\infty}h(nm)x\left(m\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{\sum}_{k=\infty}^{\infty}h\left(k\right)x(nk)$
then
$\mathcal{Z}\left\{h\right(n)*x(n\left)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\mathcal{Z}\left\{h\right(n\left)\right\}\mathcal{Z}\left\{x\right(n\left)\right\}$
 Shift:
$\mathcal{Z}\left\{x(n+M)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{z}^{M}X\left(z\right)$
 Shift (unilateral):
$\mathcal{Z}\left\{x(n+m)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{z}^{m}X\left(z\right){z}^{m}x\left(0\right){z}^{m1}x\left(1\right)\cdots zx(m1)$
 Shift (unilateral):
$\mathcal{Z}\left\{x(nm)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{z}^{m}X\left(z\right){z}^{m+1}x(1)\cdots x(m)$
 Modulate:
$\mathcal{Z}\left\{x\left(n\right){a}^{n}\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}X(z/a)$
 Time mult.:
$\mathcal{Z}\left\{{n}^{m}x\left(n\right)\right\}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}{(z)}^{m}\frac{{d}^{m}X\left(z\right)}{d{z}^{m}}$
 Evaluation: The ZT can be evaluated on the unit circle in the
zplane by taking the DTFT of
$x\left(n\right)$ and if the signal is finite in
length, this can be evaluated at sample points by the DFT.