liminfoZ-Transform Reference
Free web tool: Z-Transform Reference
| x[n] | X(z) | ROC |
|---|---|---|
| δ[n] | 1 | All z |
| u[n] | z/(z-1) | |z| > 1 |
| n u[n] | z/(z-1)² | |z| > 1 |
| n² u[n] | z(z+1)/(z-1)³ | |z| > 1 |
| aⁿ u[n] | z/(z-a) | |z| > |a| |
| n aⁿ u[n] | az/(z-a)² | |z| > |a| |
| -aⁿ u[-n-1] | z/(z-a) | |z| < |a| |
| cos(ω₀n) u[n] | z(z-cosω₀)/(z²-2z cosω₀+1) | |z| > 1 |
| sin(ω₀n) u[n] | z sinω₀/(z²-2z cosω₀+1) | |z| > 1 |
| rⁿ cos(ω₀n) u[n] | z(z-r cosω₀)/(z²-2rz cosω₀+r²) | |z| > r |
| rⁿ sin(ω₀n) u[n] | rz sinω₀/(z²-2rz cosω₀+r²) | |z| > r |
| δ[n-k] | z⁻ᵏ | z ≠ 0 (k>0) |
| aⁿ⁻ᵏ u[n-k] | a⁻ᵏ z/(z-a) | |z| > |a| |
About Z-Transform Reference
The Z-Transform Reference is a free, searchable reference tool covering the most important z-transform pairs and properties used in digital signal processing (DSP) and discrete-time systems analysis. The tool is divided into two tabs: Transform Pairs, which lists 13 common signal-domain entries alongside their z-domain closed-form expressions and Regions of Convergence (ROC); and Properties, which summarises 8 fundamental theorems including Linearity, Time Shift, Time Reversal, Scaling, Convolution, Differentiation, Accumulation, and the Initial Value theorem.
DSP engineers, control systems designers, electrical engineering students, and anyone studying difference equations or digital filter design will find this reference invaluable. Instead of flipping through a textbook index, users can instantly search for a specific signal like "cos" or "aⁿ" and immediately see the correct X(z) expression and its ROC constraint — the most error-prone part of z-transform work.
The pairs are organised into three filterable categories: Basic (unit impulse, unit step, ramp, n², aⁿ, n·aⁿ, causal vs anti-causal exponential), Trigonometric (cosine, sine, and damped sinusoids), and Shifted (delayed impulse and delayed exponential). The search bar matches against both the time-domain notation x[n] and the z-domain expression X(z). All data is embedded directly in the component and filtered client-side in real time using React useMemo, so there is zero latency and no network dependency.
Key Features
- 13 z-transform pairs spanning Basic, Trigonometric, and Shifted categories
- Region of Convergence (ROC) shown for every transform pair
- 8 z-transform properties table: Linearity, Time Shift, Time Reversal, Scaling, Convolution, Differentiation, Accumulation, Initial Value
- Keyword search filtering by time-domain expression x[n] or z-domain expression X(z)
- Category filter buttons for Basic, Trigonometric, and Shifted pairs
- Bilingual interface — toggle between English and Korean labels for all entries
- 100% client-side — all data is embedded; no network request needed
- Clean tabular layout with monospace fonts for unambiguous mathematical notation
Frequently Asked Questions
What is the Z-transform?
The Z-transform converts a discrete-time signal x[n] into a complex-frequency domain representation X(z). It is the discrete-time analogue of the Laplace transform and is fundamental to analysing digital filters, control systems, and difference equations.
What is the Region of Convergence (ROC)?
The ROC is the set of complex values of z for which the Z-transform sum converges (i.e., is finite). For a causal exponential aⁿu[n], the ROC is |z| > |a|. The ROC determines whether a system is causal, stable, or anti-causal and is therefore essential for a complete Z-transform specification.
How do I search for a specific transform pair?
Type any part of the time-domain expression or z-domain formula into the search bar at the top of the Pairs tab. For example, typing "cos" will filter to show only the cosine and damped cosine entries. You can also use the category buttons to restrict the view to Basic, Trigonometric, or Shifted pairs.
What does the Properties tab contain?
The Properties tab lists 8 key Z-transform theorems in a three-column table: the property name, the time-domain form, and the corresponding Z-domain form. These include Linearity (aX₁+bX₂), Time Shift (z⁻ᵏX(z)), Convolution (X₁(z)·X₂(z)), and others essential for solving difference equations.
How is the Z-transform different from the Laplace transform?
The Laplace transform operates on continuous-time signals whereas the Z-transform is the discrete-time equivalent. The relationship is z = e^{sT} where T is the sampling period. A pole at s = jω in the s-plane maps to a pole on the unit circle in the z-plane.
What is the Z-transform of the unit impulse δ[n]?
The Z-transform of δ[n] (the Kronecker delta / unit impulse) is X(z) = 1, and the ROC is all z. This is because the sum reduces to a single non-zero term at n = 0, which equals 1 regardless of z.
How do I find the inverse Z-transform?
The inverse Z-transform can be computed by partial fraction expansion (for rational X(z)), power series expansion, or the contour integral formula. This reference tool focuses on the forward transform pairs and properties; for a step-by-step inverse calculation you would use the table entries in reverse, guided by the ROC to select the correct causal or anti-causal form.
Is this reference suitable for exam preparation?
Yes. The pairs and properties tables cover the standard entries found in most DSP and signals-and-systems textbooks and university exam cheat sheets. The search and filter functionality helps you quickly locate and review any specific entry without scrolling through a printed table.