Calculator inputs
Use the stacked page layout below. The input grid becomes three columns on large screens, two on medium screens, and one on phones.
Example data table
| Term | Amplitude Aᵢ | Shift aᵢ | Transform term | Contribution at s = 2 |
|---|---|---|---|---|
| 1 | 4 | 0 | 4 | 4.0000 |
| 2 | -1.5 | 1.25 | -1.5e-1.25s | -0.1231 |
| 3 | 2 | 2.5 | 2e-2.5s | 0.0135 |
| Summed transform value | 3.8903 | |||
Formula used
This calculator treats the delta symbol as a distribution. It is ideal for impulsive inputs, sampling models, delayed hits, and control-system forcing terms.
How to use this calculator
- Choose how many impulse terms your signal contains.
- Enter each amplitude Aᵢ and non-negative shift aᵢ.
- Enter the real evaluation point s where you want the transform value.
- Pick the number of decimals for displayed answers.
- Press Calculate to place the result above the form.
- Review the symbolic transform, numeric total, derivative, and delay metrics.
- Use Download CSV for spreadsheet analysis or Download PDF for reports.
- Use the example button to populate a tested dataset instantly.
FAQs
1) What does this calculator solve?
It evaluates the unilateral Laplace transform of a finite sum of shifted delta impulses. It also computes the transform value at a chosen real s, derivative, total impulse weight, and delay statistics.
2) Why must the shift be non-negative?
The page follows the usual unilateral Laplace convention, where delayed impulses appear at a ≥ 0. Negative shifts belong to different modeling assumptions and are excluded here for consistency.
3) What does the amplitude represent?
Amplitude is the weight attached to each impulse. Positive amplitudes add to the transform, while negative amplitudes subtract. At s = 0, the total transform equals the sum of all amplitudes.
4) What is the weighted delay sum?
The weighted delay sum is Σ(Aᵢaᵢ). It combines strength and timing into one number, which is useful when comparing delayed impulse sets or estimating a weighted average location.
5) When is the amplitude-weighted delay undefined?
It becomes undefined when the total amplitude ΣAᵢ is zero. In that case, the numerator may still exist, but dividing by a zero total weight has no valid average interpretation.
6) Can I use decimals and negative amplitudes?
Yes. Decimal amplitudes and decimal delays are supported. Negative amplitudes are also valid and simply reverse the sign of a term’s contribution in both the symbolic transform and the numeric total.
7) Does this page compute inverse Laplace transforms?
No. This page focuses on forward transforms of shifted delta terms. It is designed for quick direct analysis, not symbolic inversion of more general transform expressions.
8) Why include CSV and PDF export?
Export makes it easier to document calculations, share worked examples, and attach transform summaries to assignments, technical notes, or client reports without retyping the results.