Formula Used
For a right sided term:
x[n] = K C(n - d + m - 1, m - 1) a^(n - d) u[n - d]
For a left sided term:
x[n] = K (-1)^m C(d - n - 1, m - 1) a^(n - d), where n <= d - m
For finite coefficients:
X(z) = b0 + b1z^-1 + b2z^-2 + ... gives x[0] = b0, x[1] = b1, x[2] = b2
The calculator adds all enabled parts. It then prints x[n] across the selected sample range.
How to Use This Calculator
- Enter the starting n value and the number of samples.
- Add FIR coefficients if your transform has direct polynomial terms.
- Enter each partial fraction term using gain, pole, delay, and multiplicity.
- Choose right sided or left sided behavior for every pole term.
- Press the calculate button to view the sequence above the form.
- Use CSV for spreadsheets or PDF for reports.
Example Data Table
| Case | Gain K | Pole a | Delay d | Multiplicity m | Side | Meaning |
|---|---|---|---|---|---|---|
| Causal delayed pole | 3 | 0.6 | 2 | 1 | Right | Starts at n = 2 |
| Repeated pole | -1 | 0.2 | 0 | 2 | Right | Uses a linear binomial factor |
| Left sided pole | 2 | 1.5 | 0 | 1 | Left | Produces samples before n = 0 |
Understanding This Inverse Z Transform Tool
An inverse z transform converts a function of z into a discrete time sequence. That sequence describes samples used in digital filters, control systems, signal analysis, and difference equations. This calculator focuses on common rational forms. It also supports delayed terms and repeated poles. You can test causal sequences, left sided sequences, and finite impulse response coefficients in one place.
Why the Inputs Matter
Each pole term needs a gain, pole value, delay, multiplicity, and side choice. The gain scales the sequence. The pole controls exponential growth or decay. The delay moves the sequence along the n axis. Multiplicity adds polynomial weight to the exponential term. The side choice defines the region of convergence. That choice changes the inverse result even when the algebraic expression looks similar.
Advanced Use Cases
Use the partial fraction rows when your transform is already decomposed. Enter one row for each fraction. For example, a term like 3z^-2 divided by 1 - 0.6z^-1 uses gain 3, pole 0.6, delay 2, and multiplicity 1. A repeated pole such as (1 - 0.5z^-1)^2 uses multiplicity 2. The calculator then applies the binomial sequence factor automatically.
Reading the Output
The result table lists n, the finite coefficient part, the pole term sum, and the final sequence value. This makes checking easier. It helps you see which part caused each sample. The region of convergence note gives a quick interpretation of the selected sidedness. Use it before applying the result in a filter model.
Practical Notes
This tool is meant for numeric exploration and study. It does not replace symbolic algebra. Still, it is useful when checking textbook work, designing simple discrete systems, or preparing examples. Export the CSV for spreadsheets. Export the PDF for reports. Keep sample counts reasonable when poles have large magnitude, because values may grow fast.
Accuracy Tips
Start with a wide n range when you are unsure about delays. Then reduce the range after the active samples appear. Compare the first samples with hand expansion. For left sided cases, include negative n values. For causal cases, begin near zero or near the largest delay. Save both reports when sharing results with classmates or project reviewers later.
FAQs
What does this calculator find?
It converts common z-domain terms into discrete sequence samples. It supports FIR coefficients, delayed pole terms, repeated poles, and sided sequence choices.
Can it solve every symbolic transform?
No. It is a numeric calculator for common decomposed forms. For complex symbolic expressions, first rewrite the transform into partial fraction terms.
What is multiplicity?
Multiplicity is the power on the denominator factor. A repeated pole such as (1 - az^-1)^2 has multiplicity 2.
What does right sided mean?
Right sided means the sequence begins at the delay index and continues forward. It usually represents causal system behavior.
What does left sided mean?
Left sided means the sequence extends toward negative n values. It represents a different region of convergence for the same algebraic term.
How are FIR coefficients used?
FIR coefficients are direct samples. The first coefficient is x[0], the second is x[1], and so on.
Why can results grow very large?
Large pole magnitudes, repeated poles, and long sample ranges can create fast growth. Reduce samples or inspect the pole values.
Can I export the answer?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for printable notes, reports, and classroom submissions.