Laplace vs Fourier Transform Comparison

Visualize and compare Laplace and Fourier transforms side by side. Understand the differences between these two fundamental mathematical tools used in engineering and physics. No coding required!

L

Laplace Transform

Time-Domain Function f(t): exp(-2*t)*sin(3*t)

f(t) = e^(-2t) · sin(3t)

Time Range (t): Format: start:step:end (e.g., 0:0.1:10 means from 0 to 10 with step 0.1)

Laplace Preset Functions:

Damped Sine

Decaying Exponential

Pure Sine Wave

Constant (Step)

Ramp Function

Damped Ramp

Laplace Transform: F(s) = ∫₀^∞ f(t)·e^(-st) dt

F

Fourier Transform

Time-Domain Function f(t) for Fourier: exp(-2*t)*sin(3*t)

f(t) = e^(-2t) · sin(3t)

Frequency Range (ω): Format: start:step:end (negative to positive frequencies)

Fourier Preset Functions:

Damped Sine

Pure Sine Wave

Gaussian

Constant

Double Exponential

Sinc Function

Fourier Transform: F(ω) = ∫_{-∞}^{∞} f(t)·e^(-iωt) dt

Comparison Results

Laplace Transform (s-domain)

Fourier Transform (ω-domain)

Key Differences

Domain

Laplace: Complex frequency domain (s = σ + jω)

Fourier: Real frequency domain (ω only)

Time Range

Laplace: 0 to ∞ (causal systems)

Fourier: -∞ to ∞

Convergence

Laplace: Broader (handles growing signals)

Fourier: Requires absolute integrability

Applications

Laplace: Control systems, circuit analysis

Fourier: Signal processing, communications

Property	Laplace Transform	Fourier Transform
Time Function	-	-
Transform Result	-	-
Poles (Singularities)	-	-
Region of Convergence	-	-

💡 Understanding the Results

Laplace Transform is shown in the complex s-plane. The real part (σ) represents damping/growth, while the imaginary part (ω) represents oscillation frequency.

Fourier Transform shows frequency content. The magnitude plot shows which frequencies are present, and the phase plot shows timing relationships.

For functions that are zero for t < 0 and have exponential damping, the Fourier transform is essentially the Laplace transform evaluated on the imaginary axis (s = jω).

Laplace vs Fourier Transform Comparison Tool | Designed for Blogger | No Coding Required

This tool uses Chart.js for visualization and Math.js for mathematical computations.