Fourier Transform Calculator
Fourier transforms, Fourier sine transforms and Fourier cosine transforms in closed form: polynomial pieces on finite intervals (pulses, triangles, 1 − x²) exactly, the standard exponential and Gaussian pairs, and inverse transforms by the Fourier integral theorem, each checked by numerical integration.
About the Fourier Transform Calculator
The Fourier transform of f is F(s) = ∫ f(x) e^(isx) dx over the whole line; the cosine and sine transforms are F_c(s) = ∫₀^∞ f(x) cos(sx) dx and F_s(s) = ∫₀^∞ f(x) sin(sx) dx. Books differ in the sign of the exponent and the factor in front (1, 1/√(2π) or √(2/π)); this page uses no factor and states it.
For f made of polynomial pieces, repeated integration by parts gives F(s) as a finite sum of sin(βs)/sʲ and cos(βs)/sʲ terms. The Fourier integral theorem recovers f from F.
The formula
F(s) = ∫ f(x) e^(isx) dx; F_c(s) = ∫₀^∞ f cos sx dx; F_s(s) = ∫₀^∞ f sin sx dx
Worked example
f(x) = 1 for |x| < 1, 0 elsewhere
- F(s) = ∫₋₁¹ e^(isx) dx = [sin(sx)/s]₋₁¹
- = 2 sin(s)/s
Things to keep in mind
- Pieces must be polynomials on finite intervals with fraction endpoints; other functions come from the table of standard pairs.
Questions people ask
Which sign and factor should I use?
Whatever your syllabus uses. Changing e^(isx) to e^(−isx) replaces i with −i; the 1/√(2π) convention divides by √(2π).
What is F(0)?
The integral of f over the line (or over x > 0 for the cosine transform).