21.1 Z-transform and its properties

Z-transform and ROC

Definition and Calculation

The Z-transform converts a discrete-time signal into a function of a complex variable $z$, letting you work in the frequency domain instead of the time domain. Think of it as the discrete-time counterpart to the Laplace transform you may have seen for continuous-time systems.

The definition is:

$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$

Here, $x[n]$ is your discrete-time signal and $z$ is a complex variable. You're essentially weighting each sample $x[n]$ by $z^{-n}$ and summing them all up. The result, $X(z)$, is a function of $z$ that encodes the same information as the original signal but in a form that's much easier to manipulate algebraically.

Region of Convergence (ROC) is the set of values of $z$ in the complex plane for which that infinite sum actually converges to a finite value. The ROC matters just as much as the expression for $X(z)$ itself, because:

• For a causal system (output depends only on present and past inputs), the ROC is the exterior of a circle centered at the origin: $|z| > r$ for some radius $r$.
• For an anti-causal system, the ROC is the interior of a circle: $|z| < r$.
• A system is stable if and only if the ROC includes the unit circle $|z| = 1$. This connects directly to whether the system's impulse response is absolutely summable.

The ROC is always bounded by poles of $X(z)$, so identifying the poles is your first step when sketching the ROC.

Inverse Z-transform

The inverse Z-transform recovers the original time-domain signal $x[n]$ from $X(z)$. Two common methods:

1. Partial fraction expansion — Decompose $X(z)$ into a sum of simpler rational terms, each associated with a single pole. Then use known Z-transform pairs to read off the corresponding time-domain sequences.
2. Contour integration — Evaluate a complex integral along a closed contour within the ROC. This is more general but usually reserved for cases where partial fractions get messy.

One critical point: the inverse Z-transform is not unique unless you specify the ROC. The same algebraic expression $X(z)$ paired with different ROCs will give you different time-domain signals. Always state the ROC alongside $X(z)$.

Properties of Z-transform

Linearity and Time-Shifting

Linearity means the Z-transform respects addition and scalar multiplication. If $x_1[n] \leftrightarrow X_1(z)$ and $x_2[n] \leftrightarrow X_2(z)$, then:

$ax_1[n] + bx_2[n] \leftrightarrow aX_1(z) + bX_2(z)$

where $a$ and $b$ are constants. The ROC of the result contains at least the intersection of the two individual ROCs.

Time-shifting is one of the most frequently used properties. If $x[n] \leftrightarrow X(z)$, then:

$x[n - k] \leftrightarrow z^{-k}X(z)$

Delaying a signal by $k$ samples just multiplies its Z-transform by $z^{-k}$. This is exactly why the Z-transform is so useful for solving difference equations: every delay becomes a simple multiplication by $z^{-1}$.

Scaling and Convolution

Scaling (sometimes called the "z-domain scaling" or "exponential weighting" property): multiplying your signal by an exponential sequence $a^n$ rescales the $z$ variable. If $x[n] \leftrightarrow X(z)$, then:

$a^n x[n] \leftrightarrow X\left(\frac{z}{a}\right)$

where $a$ is a non-zero constant. The ROC scales accordingly: if the original ROC was $|z| > r$, the new ROC is $|z| > |a|r$.

Convolution is the property that makes the Z-transform so powerful for system analysis. Convolution in the time domain becomes multiplication in the Z-domain:

$x_1[n] * x_2[n] \leftrightarrow X_1(z) \cdot X_2(z)$

The ROC of the product contains at least the intersection of the individual ROCs. This property is why you can characterize an entire LTI system by its transfer function $H(z)$: the output Z-transform is simply $Y(z) = H(z) \cdot X(z)$.

Z-transform Theorems

Initial Value Theorem

The initial value theorem gives you $x[0]$ directly from $X(z)$ without computing the full inverse transform:

$x[0] = \lim_{z \to \infty} X(z)$

This works provided the limit exists, which is typically the case for causal signals (where the ROC includes $z = \infty$). It's a quick sanity check when you've derived a Z-transform and want to verify the starting value of your sequence.

Final Value Theorem

The final value theorem tells you the steady-state behavior of a signal as $n \to \infty$:

$\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z - 1)X(z)$

This holds only if the limit on the left actually exists and all poles of $(z - 1)X(z)$ lie strictly inside the unit circle. If those conditions aren't met, the theorem doesn't apply and will give you a wrong answer.

This is particularly useful for finding the steady-state response of a stable system to a step input, since you can skip solving for the entire time-domain response and jump straight to the final value.