Z Transform Calculator

Calculator

Mode

Pick a model, enter values, then submit.

Complex z (Real)

Complex z (Imag)

or Magnitude |z|

and Angle ∠z (deg)

Real/imag takes priority if provided.

Closed-form Use standard sequences with known transforms and ROC.

Sequence type

a

Reset

Example Data Table

Sequence x[n]	Z transform X(z)	ROC (right-sided)
a^n u[n]	z / (z - a)	\|z\| > \|a\|
r^n cos(ωn) u[n]	z (z - r cos(ω)) / (z^2 - 2 r cos(ω) z + r^2)	\|z\| > \|r\|
δ[n-2]	z^{-2}	z ≠ 0

Use the closed-form mode to reproduce these results and optionally evaluate at a complex z.

Formula Used

Bilateral Z transform: X(z) = Σ x[n] z^{-n}, summed over all n.

Unilateral Z transform: X(z) = Σ x[n] z^{-n}, n ≥ 0.

For common right-sided sequences, geometric-series identities produce rational forms and a region of convergence (ROC) such as |z| > |a|.

For finite samples, the calculator evaluates the finite sum directly and can also compute a numeric value at any complex z.

For transfer functions, it evaluates H(z) = (Σ b[k] z^{-k}) / (Σ a[k] z^{-k}), and optionally samples H(e^{jω}) on the unit circle.

How to Use This Calculator

Select a mode: closed-form, finite samples, or transfer function.
Enter parameters or your data values in the fields provided.
Optional: enter a complex z using real/imag or magnitude/angle.
Press Submit to display results above the form.
Use Download CSV or Download PDF for reporting.

Article: Practical Z Transform Analysis

1) Discrete-time physics and sampled signals

Many physics measurements arrive as sequences: sensor voltages, sampled vibrations, photon counts, or discretized simulation states. The Z transform maps sequences into the z-domain, where dynamics appear as poles and zeros. This calculator supports right-sided models and data-driven sums for real lab signals.

2) Definition and index conventions

The core definition is X(z)=Σ x[n] z^{-n}. The summation range matters: unilateral starts at n≥0, while bilateral includes negative indices. Finite-sample mode lets you set n0 and compute an exact partial sum, useful for truncated impulse responses and windowed captures. It also aligns indices with your timing origin.

3) Region of convergence as a stability gate

Every rational Z transform pairs with a region of convergence (ROC). For a right-sided exponential x[n]=a^n u[n], the ROC is |z|>|a|. If your evaluation point lies outside the ROC, the defining series diverges, even when the algebraic expression is finite.

4) Poles, zeros, and what they imply

Poles indicate modes that dominate response. For z/(z−a), the pole at z=a moves outward as |a| increases, producing slower decay. Zeros create notches and cancellations. In transfer-function mode, a[k] and b[k] encode these structures for quick comparison.

5) Closed-form models for fast insight

Engineering workflows start with exponentials, ramps, and damped sinusoids. The tool includes r^n cos(ωn) u[n] and r^n sin(ωn) u[n]; r sets damping and ω sets oscillation rate. Example: r=0.9 gives ROC |z|>0.9 for causality checks.

6) Finite sequences and direct evaluation

With measured samples, compute X(z)=Σ x[n] z^{-n} directly. When z is provided, the calculator shows z^{-n} and each term, so you can see which indices dominate. This helps with short FIR filters, experimental impulse responses, and ringdown snippets used to estimate damping.

7) Transfer functions and unit-circle frequency response

For LTI discrete systems, H(z)=(Σ b[k] z^{-k})/(Σ a[k] z^{-k}). On z=e^{jω}, H(e^{jω}) is the frequency response used in filtering and resonance studies. The frequency table samples ω from 0 to π (3–61 points), reporting magnitude, phase, and dB gain. This matches digital filter checks in instrumentation and control.

8) Reporting, repeatability, and verification

Good analysis is highly repeatable. Use the summary tiles to capture parameters (a, r, ω, n0) and computed values, then export CSV for spreadsheets or PDF for lab notes. Re-run settings to verify changes, compare designs, and document decisions with consistent outputs. Add experiment IDs to exports for stronger traceability.

FAQs

1) What is the difference between unilateral and bilateral Z transforms?

Unilateral sums start at n≥0 and match causal sequences and initial-condition problems. Bilateral sums include negative indices and are useful for two-sided signals. The ROC depends on the chosen form and sequence support.

2) Why does the region of convergence matter if I already have a closed form?

The ROC tells you where the original series converges to that closed form. Outside the ROC, the same rational expression may not represent the sequence’s Z transform. Always compare your z point to the ROC condition.

3) How do I choose a complex evaluation point z?

For frequency response, use z=e^{jω} (magnitude 1) with ω in rad/sample. For stability margin checks, pick |z| slightly larger than dominant pole magnitudes. The calculator accepts real/imag or magnitude/angle inputs.

4) What does a pole near the unit circle imply?

A pole magnitude close to 1 indicates slow decay and long memory, often seen as narrow resonances in frequency response. If poles lie outside the unit circle for a causal system, the response grows and the system is unstable.

5) Can I use finite-sample mode for measured data?

Yes. Enter your samples and the starting index n0 to match how the data was recorded. If you also enter z, the tool computes per-term contributions and the total X(z), helping validate models against measurements.

6) What coefficient ordering should I use for H(z)?

Enter b[0], b[1], … for the numerator of Σ b[k] z^{-k} and a[0], a[1], … for the denominator Σ a[k] z^{-k}. Typically a[0]=1 after normalization of a linear difference equation.

7) How many frequency table points should I pick?

Use 9–21 points for quick inspection and 41–61 points for smoother plots or tighter comparison between designs. The table covers ω from 0 to π because discrete-time spectra are symmetric beyond π.