What is the Fourier Transform?

The Fourier Transform decomposes a time-domain function f(t) into its frequency components, producing a complex spectrum F(ω). The standard formula is F(ω) = ∫_{-∞}^{∞} f(t) e^{-iωt} dt. It is fundamental to signal processing, physics, and engineering.

How do I compute the Fourier Transform of e^{-at}u(t)?

For f(t) = e^{-at}u(t) with Re(a) > 0, the Fourier Transform is F(ω) = 1/(a + iω). The unit step u(t) restricts integration to [0, ∞). The integral ∫_0^∞ e^{-(a+iω)t}dt evaluates to 1/(a+iω) since e^{-(a+iω)t} → 0 as t → ∞ when Re(a) > 0.

What is the Fourier Transform of a Gaussian?

The Fourier Transform of e^{-at²} is √(π/a) · e^{-ω²/(4a)}. Gaussians are self-dual: the transform of a Gaussian is another Gaussian. For a = 1, ℱ{e^{-t²}} = √π · e^{-ω²/4}. This self-dual property underpins the Heisenberg uncertainty principle.

How do I enter the Heaviside step function?

Enter the unit step function as heaviside(t) or u(t). For example, a one-sided exponential is entered as e^(-a*t)*heaviside(t). The calculator automatically recognizes u(t) as the Heaviside step function.

Fourier Transform Calculator - Step-by-Step ℱ{f(t)}

Q: What is the difference between Fourier Transform and Fourier Series?

The Fourier Series applies to periodic functions, decomposing them into discrete harmonics. The Fourier Transform applies to aperiodic functions and produces a continuous spectrum. The FT can be seen as the limiting case of the Fourier Series as the period approaches infinity.

Q: Which Fourier Transform convention should I use?

Use the angular ω convention (F(ω) = ∫f(t)e^{-iωt}dt) for electrical engineering and signal processing. Use the ordinary frequency ν convention (F(ν) = ∫f(t)e^{-2πiνt}dt) for physics and optics. The symmetric 1/√(2π) convention is preferred in pure mathematics.

Q: What is the Convolution Theorem?

The Convolution Theorem states that convolution in the time domain equals pointwise multiplication in the frequency domain: ℱ{f*g} = F(ω)·G(ω). This is one of the most powerful results in signal processing, turning complex convolution integrals into simple multiplications.

The Complete Guide to Fourier Transforms

The Fourier Transform is one of the most powerful tools in mathematics, science, and engineering. It decomposes any time-domain signal $f(t)$ into its constituent frequency components, revealing how much of each frequency is present. This Fourier Transform calculator computes $\mathcal{F}\{f(t)\}$, $\mathcal{F}^{-1}\{F(\omega)\}$, and Fourier Series coefficients with complete step-by-step derivations — covering exponentials, Gaussians, rectangle and sinc functions, delta functions, and more.

The Fourier Transform Integral

The forward Fourier Transform is defined as:

$$\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-i\omega t}\, dt$$

The Inverse Fourier Transform recovers $f(t)$ from its frequency-domain representation:

$$\mathcal{F}^{-1}\{F(\omega)\} = f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega)\, e^{i\omega t}\, d\omega$$

Three Conventions — Which One to Use?

Angular Frequency ω (Engineering)

$F(\omega)=\int f(t)e^{-i\omega t}dt$ with $\frac{1}{2\pi}$ in the inverse. Most common in EE and signal processing. This is the default convention in this calculator.

Ordinary Frequency ν (Physics)

$F(\nu)=\int f(t)e^{-2\pi i\nu t}dt$ with no prefactor in the inverse. Used in optics and quantum mechanics. Relation: $\omega = 2\pi\nu$.

Symmetric $1/\sqrt{2\pi}$

$F(\omega)=\frac{1}{\sqrt{2\pi}}\int f(t)e^{-i\omega t}dt$ — both forward and inverse have the same $\frac{1}{\sqrt{2\pi}}$ prefactor. Preferred in pure mathematics.

Fundamental Transform Pairs

$$\mathcal{F}\{e^{-at}u(t)\} = \frac{1}{a+i\omega}, \quad \text{Re}(a)>0 \quad \text{(One-sided exponential)}$$

$$\mathcal{F}\{e^{-a|t|}\} = \frac{2a}{a^2+\omega^2} \quad \text{(Two-sided exponential)}$$

$$\mathcal{F}\{e^{-at^2}\} = \sqrt{\frac{\pi}{a}}\,e^{-\omega^2/(4a)}, \quad a>0 \quad \text{(Gaussian — self-dual)}$$

f(t)	F(ω) — Angular Convention	Property / Notes
e⁻ᵃᵗ u(t)	$\dfrac{1}{a+i\omega}$, Re(a)>0	One-sided causal exponential
e⁻ᵃ\|t\|	$\dfrac{2a}{a^2+\omega^2}$	Two-sided exponential, even
e^{−at²}	$\sqrt{\pi/a}\,e^{-\omega^2/(4a)}$	Gaussian → Gaussian (self-dual)
rect(t)	$\text{sinc}(\omega/2\pi) = \frac{\sin(\omega/2)}{\omega/2}$	Rectangle → sinc (time-bandwidth)
sinc(t)	$\pi\,\text{rect}(\omega/2)$	sinc → rectangle (duality)
δ(t)	$1$	Impulse → flat spectrum
1	$2\pi\,\delta(\omega)$	Constant → DC impulse
sgn(t)	$\dfrac{2}{i\omega}$	Signum — odd, distributional
cos(ω₀t)	$\pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$	Two spectral lines at ±ω₀
sin(ω₀t)	$-j\pi[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]$	Imaginary antisymmetric spectrum
1/(1+t²)	$\pi e^{-\|\omega\|}$	Lorentzian ↔ decaying exponential
t·e⁻ᵃᵗ·u(t)	$\dfrac{1}{(a+i\omega)^2}$	Frequency differentiation property

Key Properties of the Fourier Transform

Linearity

$\mathcal{F}\{af+bg\} = a\mathcal{F}\{f\} + b\mathcal{F}\{g\}$. Transforms sum by summing component transforms.

Time Shifting

$\mathcal{F}\{f(t-t_0)\} = e^{-i\omega t_0}F(\omega)$. Time delay = phase rotation in frequency.

Frequency Shifting

$\mathcal{F}\{e^{i\omega_0 t}f(t)\} = F(\omega-\omega_0)$. Modulation shifts the spectrum.

Scaling

$\mathcal{F}\{f(at)\} = \dfrac{1}{|a|}F\!\left(\dfrac{\omega}{a}\right)$. Time compression = frequency expansion.

Differentiation

$\mathcal{F}\{f'(t)\} = i\omega\, F(\omega)$. Differentiation in time = multiply by $i\omega$ in frequency.

Convolution Theorem

$\mathcal{F}\{f*g\} = F(\omega)\cdot G(\omega)$. Convolution in time = multiplication in frequency.

Parseval's Theorem

$\int|f(t)|^2 dt = \dfrac{1}{2\pi}\int|F(\omega)|^2 d\omega$. Energy is conserved across domains.

Duality

$\mathcal{F}\{F(t)\} = 2\pi\,f(-\omega)$. The rect-sinc duality is a classic example.

Worked Example: One-Sided Exponential

For $f(t) = e^{-at}u(t)$ with $a > 0$ (a causal signal, zero for $t < 0$):

The unit step $u(t)$ restricts integration: $F(\omega) = \int_0^{\infty} e^{-at} e^{-i\omega t}\,dt$
Combine exponents: $\int_0^{\infty} e^{-(a+i\omega)t}\,dt = \left[\frac{e^{-(a+i\omega)t}}{-(a+i\omega)}\right]_0^{\infty}$
At $t=0$: term is $\frac{1}{a+i\omega}$. As $t\to\infty$: $|e^{-(a+i\omega)t}| = e^{-at}\to 0$ since $a > 0$.
Result: $F(\omega) = \dfrac{1}{a+i\omega}$

Worked Example: Gaussian Self-Duality

For $f(t) = e^{-t^2}$ (the unit Gaussian, even and square-integrable):

Write: $F(\omega) = \int_{-\infty}^{\infty} e^{-t^2} e^{-i\omega t}\,dt$
Complete the square: $-t^2 - i\omega t = -\left(t + \frac{i\omega}{2}\right)^2 - \frac{\omega^2}{4}$
Factor out: $F(\omega) = e^{-\omega^2/4}\int_{-\infty}^{\infty} e^{-(t+i\omega/2)^2}\,dt$
Shift contour back to real line (Cauchy's theorem): $\int e^{-u^2}du = \sqrt{\pi}$
Result: $F(\omega) = \sqrt{\pi}\,e^{-\omega^2/4}$ — another Gaussian!

Fourier Series

For a periodic function with period $T$, the Fourier Series expands it as:

$$f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\!\left(\frac{2\pi n t}{T}\right) + b_n\sin\!\left(\frac{2\pi n t}{T}\right)\right]$$

where the coefficients are computed by:

$$a_n = \frac{2}{T}\int_0^T f(t)\cos\!\left(\frac{2\pi n t}{T}\right)dt, \quad b_n = \frac{2}{T}\int_0^T f(t)\sin\!\left(\frac{2\pi n t}{T}\right)dt$$

Applications of the Fourier Transform

The Fourier Transform is foundational across many disciplines. In signal processing, it underlies filtering, modulation, and spectral analysis. In electrical engineering, it is used for circuit analysis, frequency response, and LTI system theory. In physics, it appears in quantum mechanics (wave-particle duality), optics (diffraction), and the Heisenberg uncertainty principle. In mathematics, it is fundamental to the study of PDEs, harmonic analysis, and distribution theory. In medical imaging, the MRI machine reconstructs images using the 2D Fourier Transform.

Common Input Syntax Guide

Step Function

Enter as heaviside(t) or u(t). Example: causal exponential is e^(-a*t)*heaviside(t).

Delta Function

Enter as delta(t). The calculator uses the sifting property: $\mathcal{F}\{\delta(t)\} = 1$.

Absolute Value

Use abs(t) or |t|. Example: two-sided exp is e^(-abs(t)).

Multiplication

Always use * for multiplication: write 2*t, not 2t. Use e^(-a*t) for $e^{-at}$.

LSI Keywords: Related Concepts

This Fourier Transform calculator covers: continuous Fourier transform, discrete Fourier transform, inverse Fourier transform calculator, Fourier series coefficients, Fourier transform pairs table, angular frequency Fourier transform, ordinary frequency Fourier transform, Fourier transform of Gaussian, Fourier transform of exponential, Fourier transform with steps, signal processing calculator, frequency domain analysis, spectrum calculator, Parseval's theorem calculator, and convolution theorem calculator. Whether you're working on EE circuits, quantum mechanics, optics, or mathematical physics, this tool provides verified symbolic results.

Frequently Asked Questions

What is the difference between Fourier Transform and Fourier Series?+

The Fourier Series applies to periodic functions and decomposes them into a discrete set of harmonics with integer-multiple frequencies. The Fourier Transform applies to aperiodic functions and produces a continuous frequency spectrum. The Fourier Transform can be seen as the limiting case of the Fourier Series as the period $T \to \infty$, turning the discrete harmonics into a continuous integral.

Which Fourier Transform convention should I use?+

Use Angular frequency ω (engineering) — $F(\omega)=\int f(t)e^{-i\omega t}dt$ — for electrical engineering, signal processing, and control systems. Use Ordinary frequency ν (physics) — $F(\nu)=\int f(t)e^{-2\pi i\nu t}dt$ — for optics, quantum mechanics, and physical sciences. The symmetric $1/\sqrt{2\pi}$ convention is favored in pure mathematics because both the forward and inverse transforms have identical prefactors.

Why is the Gaussian function special for Fourier Transforms?+

The Gaussian $e^{-at^2}$ maps to $\sqrt{\pi/a}\,e^{-\omega^2/(4a)}$ — another Gaussian. This self-dual property means Gaussians are eigenfunctions of the Fourier Transform. The Heisenberg uncertainty principle is rooted in the Gaussian achieving the minimum time-bandwidth product: $\Delta t \cdot \Delta\omega \geq \frac{1}{2}$.

What is the unit step function u(t) and how do I enter it?+

The unit step function $u(t) = 1$ for $t \geq 0$ and $0$ for $t < 0$. Enter it as heaviside(t) or u(t). For example, a one-sided exponential is e^(-a*t)*heaviside(t). The calculator automatically interprets u(t) as the Heaviside step function and restricts the integration range accordingly.

How is the Dirac delta function handled?+

Enter the Dirac delta as delta(t). The calculator applies the sifting property: $\mathcal{F}\{\delta(t)\} = 1$ and $\mathcal{F}\{1\} = 2\pi\delta(\omega)$. These are fundamental distributional Fourier Transform pairs used extensively in systems analysis, sampling theory, and quantum mechanics.

What is the Convolution Theorem?+

The Convolution Theorem states that convolution in the time domain corresponds to pointwise multiplication in the frequency domain: $\mathcal{F}\{f*g\} = F(\omega) \cdot G(\omega)$. This is one of the most powerful results in signal processing — it converts a complex convolution integral into a simple multiplication, enabling efficient filtering, LTI system analysis, and fast algorithms like the FFT.

What is the difference between Fast Mode and Accurate Mode (SymPy)?+

Fast Mode uses a built-in lookup table and symbolic engine for instant results — covering all standard transform pairs including exponentials, Gaussians, sinc, rect, delta, signum, cosine, sine, and combinations. Accurate Mode loads the full SymPy symbolic mathematics library, enabling computation of arbitrary expressions not in the lookup table. The first load of Accurate Mode takes ~5–15 seconds; all subsequent calculations are instant.

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External Resources

For deeper reading on Fourier analysis, refer to OpenStax Calculus and the Wolfram MathWorld entry on Fourier Transforms. Both verify the transform pairs implemented in this calculator. Results are also cross-referenced against Oppenheim & Schafer's Discrete-Time Signal Processing and Bracewell's The Fourier Transform and Its Applications as part of our 4-layer verification methodology.

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Fourier Transform Calculator