Z-Transform Table
Use this Z-transform table to find common discrete-time sequence pairs, inverse Z-transforms, and regions of convergence (ROC). Search impulse, step, exponential, polynomial, and sinusoidal signals, then review transform properties and worked examples.
A bilateral reference for discrete-time signals—not a standard-normal probability table or a symbolic algebra solver. By default, u[0] = 1.
Definition and notation
X(z) = Σ x[n] z−n, summed from n = −∞ to ∞
- x[n]
- discrete-time sequence
- X(z)
- Z-transform
- z
- complex transform variable
- n
- integer time index
- u[n]
- unit-step sequence
- δ[n]
- unit-impulse sequence
- ROC
- region of convergence
- a
- real or complex sequence parameter
- ω
- angular frequency in radians per sample
Common Z-Transform Pairs Table
The following table lists frequently used discrete-time sequences and their Z-transforms. Each pair includes its region of convergence, which specifies where the defining series converges. For bilateral transforms, the same algebraic expression can represent different sequences when the ROC changes.
22 transform pairs. Filters also apply to the inverse table.
No matching transform pairs. Clear the search or choose All to see the full reference.
Selected pair: Right-sided exponential
Formula, conditions & ROC
Right-sided exponential
Right-sided · Bilateral transform
x[n] = (0.5)ⁿu[n]
X(z) = z/(z−(0.5))
ROC: |z| > 0.5
a may be nonzero real or complex; the inspector uses real a. Zero excluded, infinity included.
Sum the geometric series Σ(a/z)ⁿ for n ≥ 0. Its ratio must have magnitude less than 1.
Hatching: convergence region. Dashed circles: excluded boundaries. ×: pole locations (not multiplicity). Conceptual z-plane; the exterior continues beyond the frame.
Select a sequence to open its formula, conditions and ROC. On narrow screens, scroll the table horizontally.
| Category | Sequence x[n] | Z-transform X(z) | ROC |
|---|---|---|---|
| Impulse/step · Finite | δ[n] | 1 | All finite z |
| Impulse/step · Finite | δ[n−m] | z⁻ᵐ | z ≠ 0 |
| Impulse/step · Finite | δ[n+m] | zᵐ | All finite z |
| Impulse/step | u[n] | |z| > 1 | |
| Impulse/step · Left-sided | −u[−n−1] | |z| < 1 | |
| Exponential | aⁿu[n] | |z| > |a| | |
| Exponential · Left-sided | −aⁿu[−n−1] | |z| < |a| | |
| Polynomial | n u[n] | |z| > 1 | |
| Polynomial | n²u[n] | |z| > 1 | |
| Polynomial · Exponential | n aⁿu[n] | |z| > |a| | |
| Polynomial · Exponential · Left-sided | −n aⁿu[−n−1] | |z| < |a| | |
| Sinusoidal | cos(ωn)u[n] | |z| > 1 | |
| Sinusoidal | sin(ωn)u[n] | |z| > 1 if ω ∉ πℤ; all finite z if ω ∈ πℤ | |
| Sinusoidal · Exponential | aⁿcos(ωn)u[n] | |z| > |a| | |
| Sinusoidal · Exponential | aⁿsin(ωn)u[n] | |z| > |a| if ω ∉ πℤ; all finite z if ω ∈ πℤ | |
| Finite | 1 for 0 ≤ n < N; 0 otherwise | Σ z⁻ᵏ, k = 0,…,N−1 | z ≠ 0 for N > 1; all finite z for N = 1 |
| Polynomial · Exponential | n(n−1)aⁿu[n] | |z| > |a| | |
| Polynomial | (n+1)u[n] | |z| > 1 | |
| Polynomial · Exponential | n²aⁿu[n] | |z| > |a| | |
| Polynomial | n³u[n] | |z| > 1 | |
| Exponential · Left-sided | (0.5)ⁿu[n] − 2ⁿu[−n−1] | z/(z−0.5) + z/(z−2) | 0.5 < |z| < 2 |
| Finite | x[0]=1, x[1]=2, x[2]=3; 0 otherwise | 1 + 2z⁻¹ + 3z⁻² | z ≠ 0 |
Inverse Z-Transform Table
Open inverse reference · 22 pairs
Back to forward lookup and filters
An inverse Z-transform recovers a discrete-time sequence from its Z-domain expression. To identify the correct sequence, match both the algebraic transform and its region of convergence. A rational expression alone may not uniquely determine the inverse Z-transform.
| Z-transform X(z) | ROC | Inverse sequence x[n] |
|---|---|---|
| 1 | All finite z | δ[n] |
| z⁻ᵐ | z ≠ 0; m ≥ 1 integer | δ[n−m] |
| zᵐ | All finite z; m ≥ 1 integer | δ[n+m] |
| |z| > 1 | u[n] | |
| |z| < 1 | −u[−n−1] | |
| |z| > |a|; a ≠ 0 | aⁿu[n] | |
| |z| < |a|; a ≠ 0 | −aⁿu[−n−1] | |
| |z| > 1 | n u[n] | |
| |z| > 1 | n²u[n] | |
| |z| > |a|; a ≠ 0 | n aⁿu[n] | |
| |z| < |a|; a ≠ 0 | −n aⁿu[−n−1] | |
| |z| > 1; real ω | cos(ωn)u[n] | |
| |z| > 1 if ω ∉ πℤ; all finite z if ω ∈ πℤ | sin(ωn)u[n] | |
| |z| > |a|; a ≠ 0, real ω | aⁿcos(ωn)u[n] | |
| |z| > |a| if ω ∉ πℤ; all finite z if ω ∈ πℤ; a ≠ 0 | aⁿsin(ωn)u[n] | |
| Σ z⁻ᵏ, k = 0,…,N−1 | z ≠ 0 for N > 1; all finite z for N = 1 | 1 for 0 ≤ n < N; 0 otherwise |
| |z| > |a|; a ≠ 0 | n(n−1)aⁿu[n] | |
| |z| > 1 | (n+1)u[n] | |
| |z| > |a|; a ≠ 0 | n²aⁿu[n] | |
| |z| > 1 | n³u[n] | |
| z/(z−0.5) + z/(z−2) | 0.5 < |z| < 2 | (0.5)ⁿu[n] − 2ⁿu[−n−1] |
| 1 + 2z⁻¹ + 3z⁻² | z ≠ 0 | x[0]=1, x[1]=2, x[2]=3; 0 otherwise |
Why the Region of Convergence Matters
The same rational expression X(z) can correspond to different time-domain sequences. For example, X(z) = z/(z−0.5) represents the right-sided sequence (0.5)^n u[n] when |z| > 0.5. With |z| < 0.5, it represents the left-sided sequence −(0.5)^n u[−n−1]. The ROC makes the inverse interpretation unambiguous.
Right-sided exponential
X(z) = z/(z−0.5)
ROC: |z| > 0.5
x[n] = (0.5)ⁿu[n]
Left-sided exponential
X(z) = z/(z−0.5)
ROC: |z| < 0.5
x[n] = −(0.5)ⁿu[−n−1]
Notation & Assumptions
All entries use bilateral transforms. The generic a-families require a ≠ 0. For right-sided exponentials, a = 0 can instead be defined as δ[n] using 0⁰ = 1; left-sided a = 0 is undefined and is not obtained by substitution. Sine rows explicitly identify the zero sequence at ω ∈ πℤ. Rational formulas are interpreted after removable-factor cancellation.
Conditions for all 22 reference pairs
- Unit impulse
- All finite z. The single nonzero sample is at n = 0. Infinity is also included.
- Delayed impulse
- z ≠ 0; m ≥ 1 integer. A positive-time sample gives a pole at zero; infinity is included.
- Advanced impulse
- All finite z; m ≥ 1 integer. Zero is included; infinity is excluded.
- Unit step
- |z| > 1. Pole at 1; zero excluded, infinity included.
- Negative left-sided step
- |z| < 1. Zero is included; infinity excluded. This is not the unilateral transform.
- Right-sided exponential
- |z| > |a|; a ≠ 0. a may be nonzero real or complex; the inspector uses real a. Zero excluded, infinity included.
- Negative left-sided exponential
- |z| < |a|; a ≠ 0. Zero is included; infinity excluded. The ROC chooses a different inverse.
- Ramp
- |z| > 1. Double pole at 1; zero excluded, infinity included.
- Squared ramp
- |z| > 1. Triple pole at 1; zero excluded, infinity included.
- Weighted exponential
- |z| > |a|; a ≠ 0. Zero excluded, infinity included.
- Left-sided weighted exponential
- |z| < |a|; a ≠ 0. The leading minus sign is essential. Zero included, infinity excluded.
- Cosine
- |z| > 1; real ω. At ω = kπ, cancel the removable factor; ROC stays |z| > 1. Zero excluded, infinity included.
- Sine
- |z| > 1 if ω ∉ πℤ; all finite z if ω ∈ πℤ. For ω = kπ every sample is zero: X(z) = 0, with no poles; infinity also included. Otherwise zero excluded, infinity included.
- Damped cosine
- |z| > |a|; a ≠ 0, real ω. a may be complex in the identity. At ω = kπ one factor cancels; zero excluded, infinity included.
- Damped sine
- |z| > |a| if ω ∉ πℤ; all finite z if ω ∈ πℤ; a ≠ 0. At ω = kπ the sequence is identically zero, so X(z) = 0 everywhere, including infinity. Otherwise zero excluded, infinity included.
- Finite rectangular sequence
- z ≠ 0 for N > 1; all finite z for N = 1. N is a positive integer. At z = 1 the finite sum equals N: a quotient representation has only a removable singularity there. Infinity included.
- Falling-factorial exponential
- |z| > |a|; a ≠ 0. Zero excluded, infinity included.
- Offset ramp
- |z| > 1. Zero excluded, infinity included.
- Squared weighted exponential
- |z| > |a|; a ≠ 0. Zero excluded, infinity included.
- Cubic ramp
- |z| > 1. Zero excluded, infinity included.
- Two-sided exponential example
- 0.5 < |z| < 2. Both zero and infinity are excluded. The unit circle lies in this ROC.
- Three finite samples
- z ≠ 0. Infinity included. Finite sums do not need an infinite-series convergence condition.
Z-Transform Properties Table
Z-transform properties simplify discrete-time signal calculations by replacing operations on sequences with algebraic operations in the Z-domain. Their ROC conditions and assumptions must be checked, especially when combining or shifting sequences.
| Property | Time domain | Z-domain | Assumptions / ROC |
|---|---|---|---|
| Linearity | A x₁[n] + B x₂[n] | A X₁(z) + B X₂(z) | At least the common ROC, when nonempty. Cancellation can enlarge it. |
| Time shifting | x[n−m] | z⁻ᵐX(z) | Bilateral; integer m. Zero/infinity inclusion can change. |
| Exponential scaling | aⁿx[n] | X(z/a) | a ≠ 0; z/a must lie in the original ROC. Radii scale by |a|. |
| Time reversal | x[−n] | X(1/z) | Reciprocal ROC: r₁ < |z| < r₂ becomes 1/r₂ < |z| < 1/r₁. Check zero/infinity. |
| Multiplication by n | n x[n] | −z dX(z)/dz | Valid on the open ROC; cancellation can change exceptional endpoints. |
| Convolution | x₁[n] * x₂[n] | X₁(z)X₂(z) | At least a nonempty common absolutely convergent region; cancellations can enlarge the ROC. |
| First difference | x[n]−x[n−1] | (1−z⁻¹)X(z) | Bilateral; at least the original ROC excluding zero; cancellation can enlarge it. |
| Initial-value theorem | x[0], for x[n]=0 when n<0 | lim(z→∞) X(z) | Causal sequence with transform converging for sufficiently large |z|. Not for arbitrary bilateral sequences. |
Do not apply bilateral shifting rules to unilateral transforms without checking initial-condition terms.
Region of Convergence (ROC) of the Z-Transform
The region of convergence is the set of complex values z for which the Z-transform series converges. The ROC is an essential part of the transform pair, not an optional detail. It helps distinguish right-sided, left-sided and two-sided sequences that may have the same algebraic X(z).
For a typical right-sided exponential sequence, the ROC lies outside the outermost pole. For a typical left-sided exponential sequence, it lies inside the innermost pole. Two-sided sequences can have an annular ROC between pole magnitudes. The ROC excludes poles of the transform.
For a discrete-time linear time-invariant system, stability and causality depend on the impulse response and its ROC. A stable system requires an absolutely summable impulse response, corresponding to an ROC containing the unit circle for its transfer function.
How to Find the ROC
- Write the Z-transform expression and identify its poles.
- Determine whether the original sequence is right-sided, left-sided or two-sided.
- Choose the corresponding region relative to the poles.
- Exclude poles and check exceptional points such as z = 0.
- Verify that the defining series converges in the proposed region.
Pole locations alone do not determine the ROC: sequence support is also required. Right-sided means the sequence vanishes below some finite index; left-sided means it vanishes above some finite index. Neither condition alone means the sequence is stable. A two-sided sequence may converge only in an annulus, or have no nonempty ROC.
Unilateral vs. Bilateral Z-Transform
The bilateral (two-sided) Z-transform sums over all integer time indices, from negative infinity to positive infinity. The unilateral (one-sided) Z-transform starts at n = 0. They agree for many causal sequences, but the unilateral transform has different shift relationships when initial conditions are involved.
Bilateral: X(z) = Σ x[n]z^(−n), n from −∞ to ∞. Unilateral: X⁺(z) = Σ x[n]z^(−n), n from 0 to ∞.
Bilateral transform tables are particularly useful when comparing causal and anti-causal sequences through their regions of convergence. Unilateral transforms are commonly used with initial-value problems and discrete-time difference equations. Always identify the convention before applying a shifting rule.
For example, the unilateral transform of x[n−1] is z⁻¹X⁺(z) + x[−1]. The extra initial-condition term is absent from the bilateral shifting identity.
How to Use a Z-Transform Table
- Identify the discrete-time sequence, including any unit-step factors and time shifts.
- Choose the matching transform family in the table or search for its formula.
- Substitute the parameters, such as a, ω or a delay index.
- Check the region of convergence and any restrictions on the parameters.
- For an inverse lookup, match both X(z) and the ROC before identifying x[n].
For example, the right-sided exponential sequence x[n] = (0.5)^n u[n] matches X(z) = z/(z−0.5), with ROC |z| > 0.5. The ROC confirms that the sequence is right-sided.
Z-Transform Examples with Steps
Example 1: Z-Transform of the Unit Step
Find the bilateral Z-transform of x[n] = u[n], where u[n] = 1 for n ≥ 0 and 0 otherwise.
- X(z) = Σ z^(−n), n = 0 to ∞
- = 1 + z^(−1) + z^(−2) + ...
- = 1/(1−z^(−1))
- = z/(z−1)
X(z) = z/(z−1), ROC: |z| > 1
The infinite geometric series converges when |z^(−1)| < 1, which is equivalent to |z| > 1.
Example 2: Z-Transform of an Exponential Sequence
Find the Z-transform of x[n] = (0.5)^n u[n].
- X(z) = Σ (0.5)^n z^(−n), n = 0 to ∞
- = Σ (0.5/z)^n
- = 1/(1−0.5z^(−1))
- = z/(z−0.5)
X(z) = z/(z−0.5), ROC: |z| > 0.5
At z = 2: X(2) = 2/(2−0.5) = 4/3 ≈ 1.333333
Example 3: Inverse Z-Transform with Two Different ROCs
Consider X(z) = z/(z−0.5). The algebraic expression alone does not tell us whether the inverse sequence is right-sided or left-sided.
- Case A: ROC: |z| > 0.5; x[n] = (0.5)^n u[n]
- Case B: ROC: |z| < 0.5; x[n] = −(0.5)^n u[−n−1]
Match the expression and the ROC together.
The two sequences have the same rational Z-domain expression but different support and different regions of convergence. Both parts of the transform pair are needed for an unambiguous inverse.
Example 4: Z-Transform of a Finite Sequence
Suppose x[0] = 1, x[1] = 2 and x[2] = 3, with all other samples equal to zero.
- X(z) = 1·z^0 + 2·z^(−1) + 3·z^(−2)
X(z) = 1 + 2z^(−1) + 3z^(−2); ROC: z ≠ 0
For a finite right-sided sequence with nonzero positive-time samples, the transform is a finite polynomial in z^(−1), so it has no convergence problem except at z = 0.
Z-Transform Applications in Signal Processing
Z-transforms are used to analyze discrete-time signals, digital filters and sampled-data control systems. Delays become powers of z^(−1), while convolution becomes multiplication in the Z-domain. This makes many difference equations easier to analyze and helps describe the poles, zeros and stability of discrete-time systems.
For a causal discrete-time LTI system, the transfer function H(z) describes the Z-transform relationship between input and output under zero initial conditions. The location of the poles and the associated ROC help determine whether a stable frequency response exists on the unit circle.
Frequently Asked Questions
What is a Z-transform table?
A Z-transform table is a reference list connecting discrete-time sequences x[n] with their Z-domain expressions X(z). A complete table also states the region of convergence for each transform pair.
How do you use a Z-transform table?
Identify the sequence, find the matching standard form, substitute its parameters, and verify the region of convergence. For inverse Z-transforms, match both the transform expression and the ROC.
What is the Z-transform of a unit step?
The bilateral Z-transform of u[n] is X(z) = z/(z−1), with region of convergence |z| > 1.
What is the Z-transform of a unit impulse?
The Z-transform of δ[n] is 1. A delayed impulse δ[n−m] has transform z^(−m). For a positive integer delay m, the ROC excludes z = 0.
What is the Z-transform of a^n u[n]?
For a nonzero constant a, the right-sided exponential a^n u[n] has Z-transform z/(z−a), with ROC |z| > |a|.
What is an inverse Z-transform?
An inverse Z-transform recovers a discrete-time sequence from its Z-domain representation. The algebraic expression and region of convergence must both be considered to determine the correct sequence.
Why is the region of convergence important?
The ROC identifies where the defining Z-transform series converges. It distinguishes sequences that have the same rational expression but different time support, and it is important when studying causality and stability.
Can one Z-transform expression have two different inverse sequences?
Yes. For example, z/(z−a) describes a right-sided exponential when |z| > |a| and a corresponding left-sided sequence when |z| < |a|, assuming a is nonzero.
What is the difference between unilateral and bilateral Z-transforms?
The bilateral transform sums over all integer time indices. The unilateral transform starts at n = 0. They often agree for causal sequences, but initial conditions can affect unilateral shifting identities.
What is the Z-transform of a finite sequence?
For samples x[0] through x[N−1], the transform is X(z) = Σ x[n]z^(−n), with n ranging from 0 to N−1. This is a finite expression in powers of z^(−1).
How does time shifting work in the Z-transform?
For the bilateral transform, shifting x[n] by m samples gives Z{x[n−m]} = z^(−m)X(z), with possible changes in whether the ROC includes zero or infinity.
Is the Z-transform the same as the Z-score?
No. A Z-transform represents discrete-time signals in the complex Z-domain. A Z-score measures how many standard deviations a value is from the mean of a statistical distribution.
What is the difference between the Z-transform and the Laplace transform?
The Z-transform is used primarily for discrete-time sequences, while the Laplace transform is used for continuous-time signals. Both are important tools for analyzing linear systems and transform-domain relationships.
How is the Z-transform used in digital filters?
Digital filter difference equations can be represented algebraically in the Z-domain. Their transfer functions and regions of convergence help describe frequency response, causality and stability.
Can I find an inverse Z-transform from a table without solving an equation?
Yes, when the expression matches a known transform pair and the ROC is specified. More complicated rational expressions may require partial fractions, series expansion or another inversion technique before matching standard pairs.
Sources
- MIT — Tables of Transform Properties and Common Z-Transform Pairs (Z-transform tables 3 and 4)
- Swansea University — Table of Z-Transforms
- KU Leuven — Discrete-Time Systems and the Z-Transform
Reviewed: October 2026. Identities are checked against geometric-series derivations and direct finite partial sums. Parameter substitution stays in your browser.