1.1 Discrete-time signals  (Page 6/10)

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 $\{0\le n\le (N-1\left)\right\}$ 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 $2\pi$ .

• $DTFT\left\{\delta \right(n\left)\right\}=1$ for all frequencies.
• $DTFT\left\{1\right\}=2\pi \delta \left(\omega \right)$
• $DTFT\left\{e^{j{\omega }_{0}n}\right\}=2\pi \delta (\omega -{\omega }_0)$
• $DTFT\{cos\left({\omega }_{0}n\right)\}=\pi [\delta (\omega -{\omega }_0)+\delta (\omega +{\omega }_0)]$
• $DTFT\left\{{\sqcap }_M\left(n\right)\right\}=\frac{sin(\omega Mk/2)}{sin(\omega k/2)}$

The z-transform

The z-transform is an extension of the DTFT in a way that is analogous to the Laplace transform for continuous-time 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).

Definition of the z-transform

The z-transform (ZT) is defined as a polynomial in the complex variable $z$ with the discrete-time signal values as its coefficients [link] , [link] , [link] . It is given by

and the inverse transform (IZT) is

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 $z^{n-1}F\left(z\right)$ . 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 z-transform 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 z-transform 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 z-transform.

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 discrete-time systems. The main properties are given here using the notation that the ZT of acomplex sequence $x\left(n\right)$ is $\mathcal{Z}\left\{x\right(n\left)\right\}=X\left(z\right)$ .

1. Linear Operator: $\mathcal{Z}\{x+y\}=\mathcal{Z}\left\{x\right\}+\mathcal{Z}\left\{y\right\}$
2. Relationship of ZT to DTFT: ${\mathcal{Z}\left\{x\right\}|}_{z=e^{j\omega }}=\mathcal{DTFT}\left\{x\right\}$
3. Periodic Spectrum: $X\left(e^{j\omega }\right)=X\left(e^{j\omega +2\pi }\right)$
4. Properties of Even and Odd Parts: $x\left(n\right)=u\left(n\right)+jv\left(n\right)$ and $X\left(e^{j\omega }\right)=A\left(e^{j\omega }\right)+jB\left(e^{j\omega }\right)$
   $$\begin{array}{cccc}u& v& A& B\\ & & & \\ even& 0& even& 0\\ odd& 0& 0& odd\\ 0& even& 0& even\\ 0& odd& odd& 0\end{array}$$
5. Convolution: If discrete non-cyclic convolution is defined by
   $y\left(n\right)=h\left(n\right)*x\left(n\right)={\sum }_{m=-\infty }^{\infty }h(n-m)x\left(m\right)={\sum }_{k=-\infty }^{\infty }h\left(k\right)x(n-k)$
   then $\mathcal{Z}\left\{h\right(n)*x(n\left)\right\}=\mathcal{Z}\left\{h\right(n\left)\right\}\mathcal{Z}\left\{x\right(n\left)\right\}$
6. Shift: $\mathcal{Z}\left\{x(n+M)\right\}=z^{M}X\left(z\right)$
7. Shift (unilateral): $\mathcal{Z}\left\{x(n+m)\right\}=z^{m}X\left(z\right)-z^{m}x\left(0\right)-z^{m-1}x\left(1\right)-\cdots -zx(m-1)$
8. Shift (unilateral): $\mathcal{Z}\left\{x(n-m)\right\}=z^{-m}X\left(z\right)-z^{-m+1}x(-1)-\cdots -x(-m)$
9. Modulate: $\mathcal{Z}\left\{x\left(n\right)a^n\right\}=X(z/a)$
10. Time mult.: $\mathcal{Z}\left\{n^{m}x\left(n\right)\right\}={(-z)}^m\frac{d^{m}X\left(z\right)}{d{z}^m}$
11. Evaluation: The ZT can be evaluated on the unit circle in the z-plane 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.