Properties of Z-Transform

Introduction

The properties of the Z-Transform form the toolkit that makes discrete-time system analysis practical. Without these properties, you would need to evaluate the defining summation from scratch every time you encountered a new signal. Instead, these properties allow you to build complex transform pairs from simple ones, convert between time-domain operations and z-domain operations, and most importantly, transform difference equations into algebraic equations that can be solved with standard methods.

The single most important property is time shifting: a delay of one sample corresponds to multiplication by $z^{-1}$ in the z-domain. This is why $z^{-1}$ is universally called the "unit delay operator" in digital signal processing. Every digital filter block diagram uses $z^{-1}$ blocks to represent memory elements.

Linearity

$$\mathcal{Z}\{ax_1[n] + bx_2[n]\} = aX_1(z) + bX_2(z)$$

The ROC of the combination contains at least the intersection of the individual ROCs. Linearity is straightforward but enormously useful — it allows you to decompose complex sequences into simpler components, transform each one using known pairs, and add the results.

Example: $\mathcal{Z}\{3(0.5)^n u[n] - 2(0.8)^n u[n]\} = \frac{3z}{z-0.5} - \frac{2z}{z-0.8}$, ROC: $|z| > 0.8$

Time Shifting (Delay and Advance)

Right Shift (Delay)

$$\mathcal{Z}\{x[n-k]\} = z^{-k}X(z)$$

Each sample of delay multiplies by $z^{-1}$. This is THE fundamental property of the Z-Transform — it converts time delays into simple algebraic multiplication, which is why difference equations become polynomial equations in $z$.

Example: If $x[n] = (0.5)^n u[n]$ with $X(z) = \frac{z}{z-0.5}$, then: $$\mathcal{Z}\{x[n-2]\} = z^{-2} \cdot \frac{z}{z-0.5} = \frac{z^{-1}}{z-0.5}$$

Left Shift (Advance)

$$\mathcal{Z}\{x[n+k]\} = z^k\left[X(z) - \sum_{n=0}^{k-1}x[n]z^{-n}\right]$$

The advance property requires subtracting out the initial terms, since advancing the sequence brings in values from before the origin.

z-Domain Scaling (Multiplication by Exponential)

$$\mathcal{Z}\{a^n x[n]\} = X(z/a)$$

The ROC is scaled by $|a|$: if the original ROC was $|z| > r$, the new ROC is $|z| > |a|r$.

This property is particularly useful for modifying existing transform pairs. For instance, if you know $\mathcal{Z}\{u[n]\} = \frac{z}{z-1}$, then immediately: $$\mathcal{Z}\{a^n u[n]\} = \frac{z/a}{z/a - 1} = \frac{z}{z-a}$$

Physical interpretation: Multiplying by $a^n$ in the time domain represents exponential growth ($|a| > 1$) or decay ($|a| < 1$) applied to the original signal.

Time Reversal

$$\mathcal{Z}\{x[-n]\} = X(z^{-1})$$

The ROC inverts: if the original ROC was $|z| > r$, the new ROC becomes $|z| < 1/r$. Time reversal in the time domain corresponds to replacing $z$ with $z^{-1}$ in the transform domain.

Important consequence: A causal signal becomes anti-causal under time reversal, and the ROC flips from an exterior region to an interior region.

Convolution Property

$$\mathcal{Z}\{x_1[n] * x_2[n]\} = X_1(z) \cdot X_2(z)$$

Convolution in the time domain becomes multiplication in the z-domain. This is perhaps the most powerful computational property, since it means the output of an LTI system can be found simply by multiplying the input's Z-Transform with the system's transfer function:

$$Y(z) = H(z) \cdot X(z)$$

The ROC of the product contains at least the intersection of the individual ROCs.

Example: To find the output of a system with $h[n] = (0.5)^n u[n]$ when excited by $x[n] = u[n]$: $$Y(z) = \frac{z}{z-0.5} \cdot \frac{z}{z-1} = \frac{z^2}{(z-0.5)(z-1)}$$

Then partial fractions give the output sequence directly.

Differentiation in the z-Domain

$$\mathcal{Z}\{n \cdot x[n]\} = -z\frac{dX(z)}{dz}$$

Multiplying by $n$ in the time domain corresponds to differentiating and scaling in the z-domain. This property is useful for deriving transforms of sequences like $n \cdot a^n u[n]$.

Example: Starting with $\mathcal{Z}\{a^n u[n]\} = \frac{z}{z-a}$: $$\mathcal{Z}\{n \cdot a^n u[n]\} = -z \cdot \frac{d}{dz}\left(\frac{z}{z-a}\right) = -z \cdot \frac{-a}{(z-a)^2} = \frac{az}{(z-a)^2}$$

Accumulation (Running Sum)

$$\mathcal{Z}\left\{\sum_{k=-\infty}^{n} x[k]\right\} = \frac{z}{z-1} \cdot X(z)$$

The accumulation property is the discrete-time equivalent of integration. It states that the running sum of a sequence has a Z-Transform equal to the original transform multiplied by $\frac{z}{z-1}$. This introduces a pole at $z = 1$, which makes sense because a running sum of a constant input would grow without bound (like integration of a constant).

Initial Value Theorem

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

This theorem provides the value of the sequence at $n = 0$ directly from the transform without performing a complete inversion. It works because as $z \to \infty$, all terms with $z^{-n}$ (for $n \geq 1$) vanish, leaving only the $n = 0$ term.

Final Value Theorem

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

This theorem gives the steady-state value of a sequence directly from the transform. It is valid only when the limit actually exists — specifically, all poles of $(z-1)X(z)$ must lie inside the unit circle. If the system is unstable or oscillatory, the final value theorem does not apply.

Example: For $X(z) = \frac{z}{(z-1)(z-0.5)}$ (step response of a stable system): $$\lim_{n\to\infty} x[n] = \lim_{z\to 1}(z-1) \cdot \frac{z}{(z-1)(z-0.5)} = \frac{1}{0.5} = 2$$

Summary Table of Properties

Key Takeaways

• Time delay by one sample corresponds to multiplication by $z^{-1}$ — the fundamental building block of digital systems
• Convolution becomes multiplication, enabling system analysis via $Y(z) = H(z)X(z)$
• The z-domain scaling property generates new transform pairs from known ones
• Initial value theorem: $x[0] = \lim_{z\to\infty}X(z)$ — quick check without full inversion
• Final value theorem: $x[\infty] = \lim_{z\to 1}(z-1)X(z)$ — steady-state analysis
• Accumulation introduces a pole at $z = 1$, analogous to integration adding a pole at $s = 0$
• These properties collectively transform difference equations into algebraic equations in $z$