1. What does this z transform tool calculate?
It evaluates the finite z transform of a discrete sequence at a chosen complex z value. It also applies optional scaling, exponential weighting, and index shifting before summing each contribution.
Z transform calculator
Enter a finite sequence, set indexing and transformation options, and evaluate the result at a complex z value.
Example data table
This example uses x[n] = [1, 2, 3], n₀ = 0, c = 1, a = 1, k = 0, and z = 2 + 0i.
| n | x[n] | z-n | Contribution |
|---|---|---|---|
| 0 | 1 | 1 | 1 |
| 1 | 2 | 0.5 | 1 |
| 2 | 3 | 0.25 | 0.75 |
| Total | 2.75 | ||
Formula used
Direct finite-sequence evaluation
X(z) = Σ x[n] z-n
X(z) = Σ x[n] z-n
Modified sequence used in this tool
The tool applies scaling, exponential weighting, and index shifting as:
y[n+k] = c·anx[n]
so the evaluated transform becomes:
Y(z) = Σ c·anx[n]z-(n+k) = c·z-k·X(z/a)
The tool applies scaling, exponential weighting, and index shifting as:
y[n+k] = c·anx[n]
so the evaluated transform becomes:
Y(z) = Σ c·anx[n]z-(n+k) = c·z-k·X(z/a)
Complex output measures
Magnitude: |Y(z)| = √(Re(Y)2 + Im(Y)2)
Phase: ∠Y(z) = tan-1(Im(Y) / Re(Y))
Magnitude: |Y(z)| = √(Re(Y)2 + Im(Y)2)
Phase: ∠Y(z) = tan-1(Im(Y) / Re(Y))
How to use this calculator
- Enter the finite sequence values separated by commas.
- Set the starting index for the first value.
- Add an index shift if the sequence moves left or right.
- Set the scale factor and exponential weight base if needed.
- Choose rectangular or polar entry for the complex z value.
- Select the display precision and submit the form.
- Read the transform value, magnitude, phase, ROC note, and symbolic form.
- Use the CSV or PDF buttons to save the breakdown.
Frequently asked questions
2. Can I enter fractions in the sequence?
Yes. Values such as 1/2, 3/4, 2, and -1.25 are accepted. Separate every term with a comma so the parser can assign the correct sequence index.
3. What is the starting index used for?
The starting index tells the tool which integer belongs to the first entered term. The rest of the terms follow in order, one index step at a time.
4. How does the index shift affect the result?
A positive shift moves the sequence right, changing every exponent in the z transform. The tool reflects that change in the term table and in the relation Y(z) = c·z^{-k}·X(z/a).
5. Why are magnitude and phase shown?
The evaluated transform is usually complex. Magnitude shows overall size, while phase shows angular direction in the complex plane. Together they help interpret stability, resonance, and response behavior.
6. What does the ROC note mean here?
For finite sequences, the region of convergence is easy to summarize from the index range. The note helps you see whether zero is excluded and whether all finite z values are included.
7. When should I use polar input for z?
Polar input is useful when z is defined by magnitude and angle, such as frequency-response style work. Rectangular input is better when real and imaginary parts are already known.
8. What do the export buttons save?
The CSV export saves summary metrics and the full term table. The PDF export creates a clean report containing the same result summary and detailed breakdown.