Analyze unilateral signals using clear structured inputs. Check bilateral poles convergence and logic for ROC. Download practical summaries for notes reports assignments and review.
| Signal Type | Parameters | Transform | ROC |
|---|---|---|---|
| Right-sided exponential | A = 5, a = -2 | 5 / (s + 2) | Re(s) > -2 |
| Left-sided exponential | A = 3, a = 4 | -3 / (s - 4) | Re(s) < 4 |
| Polynomial exponential | A = 2, a = 1, n = 2 | 4 / (s - 1)3 | Re(s) > 1 |
| Two-sided exponential | Ar = 3, ar = -1, Al = 2, al = 4 | 3 / (s + 1) - 2 / (s - 4) | -1 < Re(s) < 4 |
| Finite rectangular pulse | A = 4, t1 = 0, t2 = 3 | 4(1 - e-3s) / s | All s |
The bilateral Laplace transform is X(s) = ∫ x(t)e-st dt over all time.
For right-sided exponentials, X(s) = A / (s - a), with ROC Re(s) > a.
For left-sided exponentials, X(s) = -A / (s - a), with ROC Re(s) < a.
For delayed right-sided exponentials, X(s) = Ae-sT / (s - a).
For right-sided polynomial exponentials, X(s) = A n! / (s - a)n+1.
For two-sided exponentials, add the right and left transforms, then intersect both convergence regions.
For a finite pulse from t1 to t2, X(s) = A(e-st1 - e-st2) / s.
Pole locations come from transform denominators. The ROC never includes poles.
The region of convergence is central in Laplace analysis. It tells you where the bilateral integral truly converges. A transform formula alone is not enough. The same algebraic expression can belong to different time-domain signals. ROC removes that ambiguity. It shows whether a signal is right-sided, left-sided, or two-sided.
This Laplace ROC calculator is built for quick study and clean checking. It handles common signal families used in mathematics and signals courses. You can test a simple exponential, a delayed term, a polynomial exponential, a two-sided exponential, or a finite pulse. Each model returns the transform, pole locations, and the valid convergence region.
Pole locations alone do not tell the whole story. The ROC tells you where the complex variable s can live. For a right-sided signal, the ROC lies to the right of the rightmost pole. For a left-sided signal, it lies to the left. For a two-sided signal, the ROC becomes a strip between poles. For finite-duration signals, the transform converges for all s.
Students often confuse unilateral formulas with bilateral convergence rules. This calculator reduces that confusion. It also checks a chosen test point s = σ + jω. That makes it easier to confirm whether a point lies inside the ROC. The magnitude output adds another layer of verification when you compare manual steps.
Use this page when revising transforms, checking homework, or preparing lecture notes. The example table shows common patterns. The formula section summarizes the main identities. The FAQ answers quick doubts. Export options help when you want a saved record. This makes the page useful for both classroom learning and independent practice.
ROC means region of convergence. It is the set of complex values of s for which the Laplace integral converges absolutely. It is essential for identifying the correct time-domain signal.
Yes. A right-sided and a left-sided signal can produce the same rational expression but different ROCs. That is why the transform formula alone does not fully define the signal.
No. Poles are excluded from the region of convergence. The transform becomes unbounded at a pole, so absolute convergence fails there.
A finite pulse has compact time support. The integral only runs over a bounded interval. Because of that, the bilateral Laplace transform converges for every complex value of s.
A delay multiplies the transform by an exponential factor such as e-sT. It changes the transform expression, but it does not move the ROC boundary for standard delayed right-sided signals.
Testing a point helps verify whether a chosen location in the complex plane lies inside the ROC. It is a fast check for homework, plotting, and conceptual understanding.
The jω-axis matters because the Fourier transform exists when that axis lies inside the ROC. This is a common test when discussing frequency response and stability ideas.
Yes. It gives structured outputs, example cases, formulas, and export options. That makes it helpful for revision sheets, quick checks, and short worked examples.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.