Lecture 2: Introduction to the Z-Transform

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Schedule :: Intro :: Inverse :: Properties :: Solution :: Transfer Functions :: System Realization :: Bilateral :: Frequency Response :: Rubber Sheet Geometry :: FIR Filters

As you recall, we talked first about differential equations, then difference equations. The methods of solving difference equations was in very many respects parallel to the methods used to solve differential equations. We then learned about the Laplace transform, which is a useful tool for solving differential equations and for doing system analysis on continuous-time systems. Our development now continues to the Z-transform. This is a transform technique used for discrete time signals and systems. As you might expect, many of the tools and techniques that we developed using Laplace transforms will transfer over to the Z-transform techniques.

The Z-transform is simply a power series representation of a discrete-time sequence. For example, if we have the sequence , the Z-transform simply multiplies each coefficient in the sequence by a power of corresponding to its index. In this example

$\begin{displaymath}X(z) = x[0] + x[1] z^{-1} + x[2] z^{-2} + x[3]z^{-3}.\end{displaymath}$

Note that negative powers of

are used for positive time indexes. This is by convention. (Comment.)

For a general causal sequence , the Z-transform is written as

$\begin{displaymath}F[z] = \sum_{k=0}^{\infty} f[k] z^{-k}. \end{displaymath}$

For a general (not necessarily noncausal) sequence

$\begin{displaymath}F[z] = \sum_{k=-\infty}^{\infty} f[k] z^{-k}. \end{displaymath}$

As for continuous time systems, we will be most interested in causal signals and systems.

The inverse Z-transform has the rather strange and frightening form

$\begin{displaymath}f[k] = \frac{1}{2\pi j} \oint F[z]z^{k-1}dz \end{displaymath}$

where $\oint$ is the integral around a closed path in the complex plane, in the region of integration. (Fortunately, this is rarely done by hand, but there is some very neat theory associated with integrals around closed contours in the complex plane. If you want the complete scoop on this, you should take complex analysis.)

Notationally we will write

$\begin{displaymath}F[z] = \Zc\{f[k]\} \quad\quad\quad f[k] = \Zc^{-1}\{F[z]\} \end{displaymath}$

$\begin{displaymath}f[k] \leftrightarrow F[z] \end{displaymath}$

$\begin{example}Find the Z-transform of $f[k] = \gamma^k u[k]$. \begin{displaymat... ...vert\gamma\vert$. (Compare with ROC for causal Laplace transform.) \end{example}$

$\begin{example}Find the following Z-transforms. \begin{enumerate} \item $f[k] = ... ...s\beta)}{z^2 - 2z \cos \beta + 1} \end{displaymath}\end{enumerate}\end{example}$
Important and useful functions have naturally been transformed and put into tabular form.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, May 17). Lecture 2: Introduction to the Z-Transform. Retrieved November 22, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/node1.html. This work is licensed under a Creative Commons License.

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	Info Lecture 2: Introduction to the Z-Transform Document Actions Schedule :: Intro :: Inverse :: Properties :: Solution :: Transfer Functions :: System Realization :: Bilateral :: Frequency Response :: Rubber Sheet Geometry :: FIR Filters As you recall, we talked first about differential equations, then difference equations. The methods of solving difference equations was in very many respects parallel to the methods used to solve differential equations. We then learned about the Laplace transform, which is a useful tool for solving differential equations and for doing system analysis on continuous-time systems. Our development now continues to the Z-transform. This is a transform technique used for discrete time signals and systems. As you might expect, many of the tools and techniques that we developed using Laplace transforms will transfer over to the Z-transform techniques. The Z-transform is simply a power series representation of a discrete-time sequence. For example, if we have the sequence , the Z-transform simply multiplies each coefficient in the sequence by a power of corresponding to its index. In this example $\begin{displaymath}X(z) = x[0] + x[1] z^{-1} + x[2] z^{-2} + x[3]z^{-3}.\end{displaymath}$ Note that negative powers of are used for positive time indexes. This is by convention. (Comment.) For a general causal sequence , the Z-transform is written as $\begin{displaymath}F[z] = \sum_{k=0}^{\infty} f[k] z^{-k}. \end{displaymath}$ For a general (not necessarily noncausal) sequence , $\begin{displaymath}F[z] = \sum_{k=-\infty}^{\infty} f[k] z^{-k}. \end{displaymath}$ As for continuous time systems, we will be most interested in causal signals and systems. The inverse Z-transform has the rather strange and frightening form $\begin{displaymath}f[k] = \frac{1}{2\pi j} \oint F[z]z^{k-1}dz \end{displaymath}$ where $\oint$ is the integral around a closed path in the complex plane, in the region of integration. (Fortunately, this is rarely done by hand, but there is some very neat theory associated with integrals around closed contours in the complex plane. If you want the complete scoop on this, you should take complex analysis.) Notationally we will write $\begin{displaymath}F[z] = \Zc\{f[k]\} \quad\quad\quad f[k] = \Zc^{-1}\{F[z]\} \end{displaymath}$ or $\begin{displaymath}f[k] \leftrightarrow F[z] \end{displaymath}$ $\begin{example}Find the Z-transform of $f[k] = \gamma^k u[k]$. \begin{displaymat... ...vert\gamma\vert$. (Compare with ROC for causal Laplace transform.) \end{example}$ $\begin{example}Find the following Z-transforms. \begin{enumerate} \item $f[k] = ... ...s\beta)}{z^2 - 2z \cos \beta + 1} \end{displaymath}\end{enumerate}\end{example}$ Important and useful functions have naturally been transformed and put into tabular form. Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, May 17). Lecture 2: Introduction to the Z-Transform. Retrieved November 22, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/node1.html. This work is licensed under a Creative Commons License.	Reuse Course Download this course

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Lecture 2: Introduction to the Z-Transform

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