Introduction
The Fourier Transform for Discrete-time signals and systems (FTD), is a complex function of the angular frequency $\theta$. It provides a frequency domain representation of discrete-time signals and LTI-systems. Moreover, because of the convergence conditions, in many cases the FTD of a sequence may not exist, and as a result, it is not possible to make use of such frequency domain characterization in these cases.
A generalization of the FTD leads to the Z-transform, abbreviated as ZT, which is a function of the complex variable $z$. It can exist for many sequences for which the FTD does not exist. Also, the use of ZT techniques permits simple algebraic manipulations. Consequently, the ZT has become an important tool in the analysis and design of digital filters and it can be viewed as the discrete-time counterpart of the Laplace transform for continuous-time signals and systems.
Finally, it is important to note that the ‘time’-domain representation is the domain where the signals are generated and processed, and where the implementation of the filters takes place. On the other hand, the frequency domain has physical significance when analyzing frequencies, while the $z$-domain exist primarily for its convenience in mathematical analysis and synthesis.
Screencast video [⯈]
Module overview
This module covers the following topics:
- Z-transform - This section will explain what the Z-transform actually does and how it relates to other transforms.
- Properties [⯈] - As with all other transforms, there are some commonly used properties which ease the calculations. This section will discuss the properties of the Z-transform in more detail.
- Inverse Z-transform [⯈] - This section will describe the inverse Z-transform. The formal calculation rule is relatively difficult, however, there exist several other methods that can ease the calculation.
Summary
- Definition ZT: $$ \boxed{X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}} \label{eq:ZT} $$
- Region Of Convergence (ROC): All values of $z$ for which ZT converges.
- FTD $\equiv$ ZT for $|z|=1$ only when the unit circle is in the ROC
- The ZT $X(z)$ is only uniquely related to a sequence $x[n]$ if the ROC is known. Hence, the ZT must always be specified with its ROC.
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ROC's of divergent and convergent sequences:
- If rational $X(z)$ has poles with $R$ distinct magnitudes $\Rightarrow$ $R+1$ distinct sequences $x[n]$ having same rational $X(z)$.
- Linear property: $$ w[n] = \alpha x[n] + \beta y[n] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad W(z) = \alpha X(z) + \beta Y(z) \text{ ROC } R_w=R_x \cap R_y $$
- Shift property: $$ x[n-n_0] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad z^{-n_0} X(z) \text{ with ROC } R_x $$
- Time reversal property: $$ x[-n] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad X(z^{-1}) \text{ with ROC } 1/R_x $$
- Multiply by $a^n$: $$ a^n x[n] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad X(a^{-1} z) \text{ with ROC } R_y= |a|R_x $$
- Convolution property: $$ x[n] \star y[n] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad X(z) \cdot Y(z) \hspace{3mm} \text{with } R_w=R_x \cap R_y $$
- Conjugation property: $$ x^\ast[n] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad X^\ast(z^\ast) \text{ ; } R_x $$
- Derivation property: $$ n \cdot x[n] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad -z \frac{\text{d}}{\text{d} z} X(z) \text{ ; } R_x $$
- Initial value property: $$ x[0] = \lim_{z \rightarrow \infty} \left \{ X(z) \right \} $$
- One sided ZT: $$ \boxed{\tilde{X}(z)=\sum_{{\color{red}{n=0}}}^{\infty} x[n] z^{-n}} $$
- One sided ZT shift property: $$ x[n-k] \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad {\color{red}{\sum_{p=-k}^{-1} x[p] z^{-(p+k)}}} + z^{-k} \cdot \tilde{X}(z) $$
- IZT method 1: Table Lookup
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IZT method 2: Long Tail Division
Descending ordered polynomials: $$ X(z) = \frac{N(z)}{D(z)}=\sum_{k }^{} c_k z^{-k} \hspace{3mm} \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \hspace{3mm} x[n] = \sum_{k }^{} c_k \delta[n-k] $$ Ascending ordered polynomials: $$ X(z)=\frac{N(z)}{D(z)}=z^{M-N} \cdot \sum_{k=0}^{\infty} d_k z^{k} \quad\circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad x[n] =\sum_{k=0}^{\infty} d_k \delta[n+(M-N)+k] $$
- IZT method 3: Partial Fraction; Single poles: $$ X(z) =\frac{N(z)}{D(z)} = \frac{\sum_{p=0}^{N} b_p z^{-p}}{\prod_{k=1}^{M} (1 - \alpha_k z^{-1})} $$ Case $M>N$: $$ X(z) =\sum_{k=1}^{M} \frac{A_k}{1 - \alpha_k z^{-1}} \quad\circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad x[n] =\sum_{k=1}^{M} A_k (\alpha_k)^n u[n] $$ $A_k$ via coefficient matching or residuals: $$ A_k = \left [ (1 - \alpha_k z^{-1}) X(z) \right ] \mid_{z=\alpha_k} $$ Case $M \leq N$: $$ X(z) = \sum_{k=0}^{N-M}B_k z^{-k} + F(z) \quad \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \quad x[n] =\sum_{k=0}^{N-M}B_k \delta[n-k] +f[n] $$ Order denominator $F(z)$ larger than order numerator. Coefficients $B_k$ via ascending ordered long tail division.
- IZT method 3: Partial Fraction; Multiple order poles: $$ X(z) = \frac{\sum_{p=0}^{N} b_p z^{-p}}{\left ( \prod_{k=1}^{M-K} (1 - \alpha_k z^{-1})\right ) \cdot \left ( 1 - \tilde{\alpha} z^{-1} \right )^K } $$ $$ X(z) = \sum_{k=0}^{N-M}B_k z^{-k} +\sum_{k=1}^{M-K} \frac{A_k}{1 - \alpha_k z^{-1}} + \sum_{p=1}^{K} \frac{C_p}{(1 - \tilde{\alpha} z^{-1})^p} $$ General equation to compute residuals: $$ C_p = \frac{1}{(K-p)! \cdot (-\tilde{\alpha})^{(K-p)}} \cdot \frac{\text{d}^{K-p}}{\text{d} (z^{-1})^{K-p}} \bigg[ (1-\tilde{\alpha} z^{-1})^K \cdot F(z) \bigg] \mid_{z=\tilde{\alpha}}. $$
- Definition Inverse Z-Transform (IZT): $$ \boxed{ x[n] = \frac{1}{2 \pi j} \oint_{{\color{red}{C}}} X(z) z^{n-1} \text{d} z } $$
- Cauchy's residu theorem: $$ \frac{1}{2 \pi j} \oint_{C} F(z) \text{d} z = \sum \left [ \text{Residues $F(z)$ @ poles inside $C$} \right ] $$
- General equation to compute the residu: $$ \text{Res} \left [ F(z) \text{ @ } z=z_0 \right ] = \frac{1}{(K-1)!} \cdot \frac{\text{d}^{K-1}}{\text{d} z^{K-1}} \left [ (z-z_0)^K \cdot F(z) \right ] \mid_{z=z_0} $$
- Cauchy'integral theorem: $$ \frac{1}{2 \pi j} \oint_{C} z^{n-1} \text{d} z = \delta[n] $$