Inverse Z-Transform Calculator

Calculator Inputs

Enter the numerator and denominator coefficients in ascending powers of z^-1. Example: 1, -0.6 means 1 - 0.6z^-1.

Numerator coefficients b0, b1, b2, ... Ascending powers of z^-1. Example: 1, 0.5

Denominator coefficients a0, a1, a2, ... First denominator term must not be zero.

Number of samples Displayed sequence length for x[n].

Numerator scale factor Multiplies all numerator coefficients before solving.

Sample period T_s Used to show discrete-time positions t = nT_s.

Display decimals Controls table, metric, and pole-zero formatting.

Reset

Formula Used

Rational input model

X(z) = B(z) / A(z)

with coefficients entered as B(z) = b₀ + b₁z^-1 + b₂z^-2 + ... and A(z) = a₀ + a₁z^-1 + a₂z^-2 + ....

Inverse z-transform recurrence

If X(z) = Σ x[n]z^-n, then the displayed sequence is generated from:

a₀x[n] = b[n] - a₁x[n-1] - a₂x[n-2] - ... - a_Nx[n-N]

Engineering interpretation

This page treats the inverse z-transform as a causal discrete sequence generated from a rational transfer form. It also reports poles, zeros, a causal ROC note, displayed energy, and an estimated settling length when the dominant pole lies inside the unit circle.

How to Use This Calculator

Enter numerator coefficients in ascending powers of z^-1.
Enter denominator coefficients in ascending powers of z^-1.
Choose how many sequence samples you want displayed.
Set an optional numerator scale factor for quick sensitivity checks.
Enter the sample period if you need time labels.
Pick the number of decimals for readable engineering output.
Press the calculate button to show results above the form.
Review the sequence table, pole-zero summary, metrics, and graph.
Use CSV or PDF export to save the generated results.

Example Data Table

Example transfer function: X(z) = 1 / (1 - 0.6z^-1). Its inverse z-transform is the causal sequence x[n] = 0.6ⁿu[n].

n	x[n]	Comment
0	1.0000	Initial impulse-response sample
1	0.6000	One-step decay from the pole value
2	0.3600	Second sample from repeated multiplication
3	0.2160	Third sample shows stable decay
4	0.1296	Fourth sample continues geometric behavior

Frequently Asked Questions

1) What does this inverse z-transform calculator return?

It returns a discrete sequence x[n] from a rational z-domain expression. The page also shows poles, zeros, a causal ROC note, engineering metrics, and a Plotly sequence graph.

2) How should I enter coefficients?

Enter coefficients in ascending powers of z^-1. For example, 1, -0.8, 0.15 represents 1 - 0.8z^-1 + 0.15z^-2.

3) Does this page support causal engineering sequences?

Yes. The numeric solver is designed for causal right-sided sequences from rational forms. That matches many control, DSP, and sampled-data engineering tasks.

4) Why do poles matter in inverse z-transform work?

Pole locations determine growth, decay, oscillation, and stability behavior. Their magnitudes relative to the unit circle help explain whether a causal sequence settles or diverges.

5) What does the sample period T_s change?

T_s does not change the sequence values themselves. It changes the displayed time labels, which is useful when comparing discrete samples to real engineering timelines.

6) Can I export the computed values?

Yes. The page includes CSV and PDF buttons for saving the calculated table and summary, which helps with design notes, reports, and lab documentation.

7) What happens if the first denominator coefficient is zero?

The recurrence would be undefined because each new sample divides by a₀. The calculator blocks that case and asks for a valid denominator.

8) Is the displayed settling value exact?

No. It is an engineering estimate based on the dominant pole magnitude for stable causal systems. It is helpful for quick design judgment, not formal proof.