Calculator Input
Enter numerator and denominator coefficients in descending powers of z^-1.
For example, 1,-0.8 means 1 - 0.8z^-1.
Formula Used
The page computes a right-sided sequence using zero initial samples. This is the common causal expansion of a rational transform. ROC selection affects interpretation, especially for non-causal signals.
How to Use This Calculator
- Write coefficients in powers of
z^-1. - Use commas, spaces, or semicolons between numbers.
- Enter denominator coefficient
a0first. - Select the number of sequence terms to display.
- Add a gain when the whole transform needs scaling.
- Press the calculate button to show the sequence above the form.
- Review the plot, recurrence steps, pole notes, and table.
- Export the result using the CSV or PDF buttons.
Example Data Table
| Case | Numerator | Denominator | Expected sequence idea |
|---|---|---|---|
| Unit impulse | 1 | 1 | x[n] = δ[n] |
| Unit step | 1 | 1, -1 | x[n] = 1 for causal n |
| Exponential | 1 | 1, -0.8 | x[n] = 0.8nu[n] |
| Alternating exponential | 1 | 1, 0.5 | x[n] = (-0.5)nu[n] |
| FIR sequence | 2, -1, 0.25 | 1 | x[n] = 2, -1, 0.25 |
Inverse Z Transform Guide
What the Tool Does
The inverse Z transform changes a Z-domain expression into a discrete sequence. This sequence is usually written as x[n]. Engineers use it in digital filters, control models, sampled systems, and signal analysis. A rational transform can be expanded into time samples through a recurrence equation. This page follows that practical method. It accepts numerator and denominator coefficients in powers of z inverse.
Why Coefficients Matter
A coefficient list gives a compact model. The numerator describes direct feed terms. The denominator describes feedback terms. Feedback terms create infinite responses when poles exist. For example, 1 divided by 1 minus 0.8z inverse gives a decaying sequence. The first term is one. Later terms are multiplied by 0.8. This makes the sequence easy to inspect.
Understanding ROC
The same algebraic expression can describe different sequences. The region of convergence decides the time direction. A causal ROC outside the largest pole creates a right-sided sequence. An anti-causal ROC creates a left-sided sequence. Two-sided cases can also occur. This calculator generates the common causal expansion. It also shows ROC notes so the interpretation stays clear.
Practical Uses
The sequence table helps verify filters before coding them. The plot shows growth, decay, oscillation, and sign changes. The energy value helps compare response strength. The pole check gives a fast stability warning. If poles lie inside the unit circle, a causal response is usually stable. If poles are outside, the response may grow.
Best Input Practice
Use normalized denominator values when possible. Keep a0 non-zero. Increase terms when slow decay is expected. Use more precision for small poles or repeated calculations. Export the table when reports need exact samples. Export the PDF when you need a readable summary.
FAQs
1. What coefficient order should I use?
Enter coefficients by powers of z inverse. The first value is the constant term. The next value multiplies z^-1, then z^-2, and so on.
2. Does this calculate causal sequences?
Yes. The recurrence uses zero initial samples and expands the expression as a right-sided causal sequence unless you only use FIR coefficients.
3. What does ROC mean?
ROC means region of convergence. It tells which sequence matches the same transform. It separates causal, anti-causal, and two-sided interpretations.
4. Why is a0 important?
The recurrence divides by a0. If a0 is zero, the equation cannot be solved in this direct causal form.
5. Can I model an exponential sequence?
Yes. Use numerator 1 and denominator 1,-a. For example, denominator 1,-0.8 gives the causal sequence 0.8^n.
6. What does zero threshold do?
It removes very small numerical values. This helps when rounding errors create tiny values that should be treated as zero.
7. Why are poles shown?
Poles help describe growth, decay, oscillation, and stability. For a causal sequence, poles inside the unit circle usually indicate stability.
8. What can I export?
You can export sequence rows as CSV. You can also create a PDF summary with inputs, settings, and computed values.