How to use the Fourier Transform Table and Properties
- 1. Pick a functionChoose from Fourier, Fourier cosine and Fourier sine transform pairs.
- 2. Set a and ωEnter the constant a and a value of ω to evaluate the transform.
- 3. CompareWhere possible the tool integrates numerically and shows the difference from the formula.
Fourier transform, cosine and sine transforms
The Fourier transform turns a function of time into a function of frequency: F(ω) = ∫₋∞^∞ f(t) e^(−iωt) dt, with inverse f(t) = (1/2π) ∫ F(ω) e^(iωt) dω. For functions defined on t > 0, the cosine transform Fc(ω) = ∫₀^∞ f(t) cos ωt dt and the sine transform Fs(ω) = ∫₀^∞ f(t) sin ωt dt are used.
Books differ in convention. Some (for example Grewal) use e^(isx) in the forward transform, which replaces ω by −ω; others put 1/√(2π) in front of both integrals. For even functions the first difference disappears. Always check which convention your syllabus uses before copying a pair.
Worked example
Find the Fourier transform of the rectangular pulse f(t) = 1 for |t| < a and 0 otherwise. F(ω) = ∫₋ₐ^ₐ e^(−iωt) dt = [e^(−iωt)/(−iω)] from −a to a = (e^(iaω) − e^(−iaω))/(iω) = 2 sin(aω)/ω, with value 2a at ω = 0.
With a = 1.5 and ω = 0.7 the formula gives 2 sin(1.05)/0.7 = 2.47835, and the tool's numerical integral gives the same value. Applying the inverse transform at t = 0 gives the well-known result ∫₀^∞ sin x / x dx = π/2.
Using the properties
The properties table saves work: shifting gives transforms of delayed signals, modulation handles f(t) cos ω₀t, and convolution turns into multiplication. For periodic functions use the Fourier series calculator, and for one-sided transforms the Laplace transform table.
Frequently asked questions
Which definition of the Fourier transform is used?
F(ω) = ∫ f(t) e^(−iωt) dt, common in engineering and signals. For the e^(isx) convention, replace ω by −ω.
Why can't some values be computed?
Transforms like 2πδ(ω) for f(t) = 1 are delta functions, not ordinary numbers. The tool says so instead of showing a value.
How does the numerical check work?
For functions that decay fast enough the tool integrates f(t) e^(−iωt) (or the cosine or sine version) with Simpson's rule and compares the result with the formula.
What is the difference between the Fourier and Laplace transforms?
Laplace uses e^(−st) over t > 0 and suits initial value problems. Fourier uses e^(−iωt) over all t and gives the frequency content.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.