Definition and Region of Convergence
The bilateral Z-transform of a sequence $x[n]$ is the power series:
\[X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n]\, z^{-n}, \qquad z \in \mathbb{C}\]The region of convergence (ROC) is the set of $z$ for which this series converges absolutely:
\[\text{ROC} = \left\{z : \sum_{n=-\infty}^{\infty} |x[n]|\, |z|^{-n} < \infty\right\}\]| The ROC is always an annular region $r_1 < | z | < r_2$ in the complex plane (possibly including $ | z | = 0$ or $ | z | = \infty$). The Z-transform alone does not uniquely specify a sequence — the ROC is equally important. |
| Signal type | ROC structure | ||
|---|---|---|---|
| Right-sided (causal) | $ | z | > r_1$ (exterior of a circle) |
| Left-sided (anti-causal) | $ | z | < r_2$ (interior of a circle) |
| Two-sided | $r_1 < | z | < r_2$ (annulus) |
| Finite-length (FIR) | All $z$ (except possibly $0$ or $\infty$) |
Relationship to the DTFT
The discrete-time Fourier transform (DTFT) is the Z-transform evaluated on the unit circle $z = e^{j\omega}$, $\omega \in [-\pi, \pi]$:
\[X(e^{j\omega}) = X(z)\big|_{z = e^{j\omega}} = \sum_{n=-\infty}^{\infty} x[n]\, e^{-j\omega n}\]| The DTFT exists when the ROC includes the unit circle, which for a stable causal system requires all poles to lie strictly inside the unit circle. If the ROC does not include $ | z | = 1$, the DTFT does not exist in the classical sense (though it may exist in a distributional sense for marginally stable systems). |
| This geometric picture is powerful: the magnitude response $ | H(e^{j\omega}) | $ can be computed by tracing the unit circle and measuring how close poles and zeros are to each point on the circle. A pole near the unit circle causes a peak in the frequency response; a zero on the unit circle creates a null. |
Common Z-Transform Pairs
| Sequence $x[n]$ | Z-transform $X(z)$ | ROC | ||||
|---|---|---|---|---|---|---|
| $\delta[n]$ | $1$ | All $z$ | ||||
| $u[n]$ (unit step) | $\dfrac{z}{z-1} = \dfrac{1}{1-z^{-1}}$ | $ | z | >1$ | ||
| $a^n u[n]$ | $\dfrac{z}{z-a} = \dfrac{1}{1-az^{-1}}$ | $ | z | > | a | $ |
| $-a^n u[-n-1]$ | $\dfrac{z}{z-a}$ | $ | z | < | a | $ |
| $n\,a^n u[n]$ | $\dfrac{az^{-1}}{(1-az^{-1})^2}$ | $ | z | > | a | $ |
| $\cos(\omega_0 n)\,u[n]$ | $\dfrac{1 - z^{-1}\cos\omega_0}{1 - 2z^{-1}\cos\omega_0 + z^{-2}}$ | $ | z | >1$ |
The table highlights that right-sided (causal) sequences have ROCs exterior to a circle, while the same rational function with a left-sided ROC represents a different (anti-causal) time-domain sequence.
Key Properties
Linearity: $\mathcal{Z}{ax[n] + by[n]} = aX(z) + bY(z)$, ROC contains $\text{ROC}_x \cap \text{ROC}_y$.
Time shifting: $\mathcal{Z}{x[n-k]} = z^{-k} X(z)$, ROC unchanged (except possibly at $z=0$ or $z=\infty$).
Convolution theorem: If $y[n] = x[n] * h[n]$, then: \(Y(z) = X(z) \cdot H(z), \qquad \text{ROC} \supseteq \text{ROC}_x \cap \text{ROC}_h\)
This is the central result: convolution in time becomes multiplication in the $z$-domain, converting integral equations into algebraic ones.
Differentiation in $z$: $\mathcal{Z}{n\,x[n]} = -z \dfrac{dX(z)}{dz}$
Initial value theorem (causal sequences): $x[0] = \lim_{z \to \infty} X(z)$
Final value theorem (stable, causal sequences): $\lim_{n\to\infty} x[n] = \lim_{z\to 1}(z-1)X(z)$
Inverse Z-Transform via Partial Fractions
For a rational $X(z) = N(z)/D(z)$ with $M$ poles $p_k$, the partial fraction expansion (in terms of $z^{-1}$) gives:
\[X(z) = \sum_{k=1}^{M} \frac{A_k}{1 - p_k z^{-1}}\]| Each term corresponds to a geometric sequence: for a causal system ($ | z | > | p_k | $ in the ROC), the inverse transform is $A_k\, p_k^n\, u[n]$. The residue is: |
For repeated poles of order $r$, the expansion includes terms $(n+1)a^n$, $(n+1)(n+2)a^n/2!$, etc.
| Example: Given $X(z) = \dfrac{1}{(1-0.5z^{-1})(1-0.25z^{-1})}$ with ROC $ | z | >0.5$: |
Transfer Functions of LTI Systems
A causal LTI system described by the difference equation:
\[\sum_{k=0}^{N} a_k\, y[n-k] = \sum_{k=0}^{M} b_k\, x[n-k]\]has transfer function (taking Z-transforms, using the shift property):
\[H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}} = \frac{B(z)}{A(z)}\]This can always be factored in terms of poles $p_k$ and zeros $z_k$:
\[H(z) = \frac{b_0}{a_0} \cdot \frac{\prod_{k=1}^{M}(1 - z_k z^{-1})}{\prod_{k=1}^{N}(1 - p_k z^{-1})}\]The pole-zero plot in the complex $z$-plane gives a complete picture of the system. Zeros of $H(z)$ are frequencies at which the system completely blocks a sinusoidal input; poles are resonant frequencies where the response grows large.
Stability Analysis: Poles and the Unit Circle
| For a causal LTI system, BIBO (bounded-input, bounded-output) stability requires the ROC to include the unit circle $ | z | =1$, which means all poles must lie strictly inside the unit circle: |
| Pole location | Impulse response behavior | ||
|---|---|---|---|
| $ | p | <1$ inside unit circle | Decaying exponential (stable) |
| $ | p | =1$ simple pole | Persistent oscillation (marginally stable) |
| $ | p | =1$ repeated pole | Growing polynomial (unstable) |
| $ | p | >1$ outside unit circle | Exponentially growing (unstable) |
This is the discrete-time analog of the Laplace-domain rule (all poles in the left half-plane for continuous-time stability). The bilinear transform $s = 2(z-1)/[(z+1)T_s]$ maps the left half $s$-plane to the interior of the unit $z$-circle, enabling continuous-to-discrete filter conversion while preserving stability.
Jury stability criterion provides an algebraic test for stability without explicitly computing pole locations: given characteristic polynomial $A(z) = \sum_{k=0}^N a_k z^k$, construct the Jury array and check sign conditions on corner elements. This is the discrete-time analogue of the Routh-Hurwitz criterion.