Fourier Transform Table
Find common Fourier transform pairs, inverse transforms, and properties in one searchable reference. Choose ordinary frequency (f) or angular frequency (ω), explore standard signals, and copy the formulas you need.
Definition · Angular frequency
Here i² = −1, ω = 2πf, and sinc(v) = sin(πv)/(πv), with sinc(0) = 1. If t is measured in seconds, f is in Hz and ω is in radians per second. This is a continuous-time reference, not a Fourier series table, discrete time Fourier transform table, DFT or arbitrary symbolic solver.
Table of Common Fourier Transform Pairs
27 of 27 transform pairs
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| Function / category | x(t) | X(ω) | Conditions / type |
|---|---|---|---|
| Dirac deltaImpulse and Step Functions | δ(t) | 1 | Distribution; t on ℝ. Distribution |
The delta samples the inverse kernel at zero. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Sifting identity ∫δ(t)φ(t)dt = φ(0). Hz form: 1 | |||
| Constant functionBasic Functions | 1 | 2π δ(ω) | Tempered distributions; not an absolutely integrable function. Distribution |
A constant has only a zero-frequency spectral line. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Apply Fourier inversion to the delta pair; δ(2πf) = δ(f)/(2π). Hz form: δ(f) | |||
| GaussianGaussian Functions | exp(−a t²) | √(π/a) exp(−ω²/(4a)) | a > 0, real. Ordinary function |
For a = 1, X(0) = √π. The transform is real and positive, with zero phase. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Complete the square in the Gaussian integral; analytic continuation of the Gaussian integral. Hz form: √(π/a) exp(−π² f²/a) | |||
| Causal decaying exponentialExponential Functions | exp(−a t) u(t) | a > 0, real. Ordinary function | |
A one-sided exponential has both real and imaginary spectral components. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Integrate exp(−(a + iω)t) from zero to infinity. Hz form: | |||
| Left-sided decaying exponentialExponential Functions | exp(a t) u(−t) | a > 0, real. Ordinary function | |
Time reversal conjugates this real signal’s spectrum. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Substitute τ = −t in the causal exponential integral. Hz form: | |||
| Two-sided exponentialExponential Functions | exp(−a |t|) | a > 0, real. Ordinary function | |
A real even signal gives a real even spectrum. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Sum the left- and right-sided exponential integrals. Hz form: | |||
| Odd decaying exponentialExponential Functions | sgn(t) exp(−a |t|) | a > 0, real. Ordinary function | |
Odd symmetry produces a purely imaginary odd spectrum. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Subtract the left exponential integral from the right exponential integral. Hz form: | |||
| Rectangular pulseRectangular and Triangular Pulses | rect(t/T) | T sinc(ωT/(2π)) | T > 0; rect(v) = 1 for |v| < 1/2, 0 for |v| > 1/2; endpoints do not affect the integral. Ordinary function |
A pulse of width T has first spectral zeros at ω = ±2π/T or f = ±1/T. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Integrate exp(−iωt) between −T/2 and T/2; take the removable limit at zero. Hz form: T sinc(fT) | |||
| Triangular pulseRectangular and Triangular Pulses | tri(t/T) | T sinc²(ωT/(2π)) | T > 0; tri(v) = max(1 − |v|, 0). Ordinary function |
The triangle has support [−T, T] and area T. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Convolve two width-T rectangles, divide by T, and apply convolution. Hz form: T sinc²(fT) | |||
| Normalized sincSinc Functions | sinc(Bt) | B > 0; endpoint values of the rectangular spectrum are immaterial in L². L² / improper | |
sinc(v) = sin(πv)/(πv), with sinc(0) = 1. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Duality of the rectangular pulse pair, with correct frequency scaling. Hz form: | |||
| Squared sincSinc Functions | sinc²(Bt) | B > 0, real. Ordinary function | |
Squaring sinc convolves two rectangular spectra. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Product theorem applied to the normalized sinc pair. Hz form: | |||
| CosineTrigonometric Signals | cos(ω₀t) | π [δ(ω − ω₀) + δ(ω + ω₀)] | ω₀ > 0, real; spectral lines are distributions. Distribution |
ω₀ is in radians per unit time in both conventions. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Euler identity plus the modulated constant pair; scale each delta under ω = 2πf. Hz form: | |||
| SineTrigonometric Signals | sin(ω₀t) | (π/i) [δ(ω − ω₀) − δ(ω + ω₀)] | ω₀ > 0, real. Distribution |
The imaginary spectrum is odd; the minus sign follows the negative forward exponent. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Euler sine identity and frequency-shifted constant transform. Hz form: | |||
| Complex exponentialTrigonometric Signals | exp(iω₀t) | 2π δ(ω − ω₀) | ω₀ real. Distribution |
One spectral line, not a finite-height plotted peak. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Frequency shift of the constant pair and the delta scaling identity. Hz form: δ(f − ω₀/(2π)) | |||
| Unit stepImpulse and Step Functions | u(t) | Tempered distributions; u(t) = 1 for t > 0 and 0 for t < 0. Distribution | |
PV denotes Cauchy principal value; the value of u at zero does not change this transform. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Distributional limit of 1/(ε + iω) as ε → 0⁺. Hz form: δ(f)/2 + PV[1/(i2πf)] | |||
| Sign functionImpulse and Step Functions | sgn(t) | Tempered distributions. Distribution | |
sgn(t) = 2u(t) − 1 almost everywhere. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Linearity applied to 2u(t) − 1 cancels the delta term. Hz form: | |||
| Derivative of Dirac deltaImpulse and Step Functions | δ′(t) | iω | Distributional derivative. Distribution |
Differentiation is interpreted by its action on test functions. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: ⟨δ′, exp(−iωt)⟩ = iω. Hz form: i2πf | |||
| Time-weighted GaussianGaussian Functions | t exp(−a t²) | a > 0, real. Ordinary function | |
An odd Gaussian-shaped pulse has an imaginary odd spectrum. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Use F{tx} = i dX/dω on the Gaussian pair. Hz form: | |||
| Quadratic-weighted GaussianGaussian Functions | t² exp(−a t²) | √(π/a) [1/(2a) − ω²/(4a²)] exp(−ω²/(4a)) | a > 0, real. Ordinary function |
The spectral sign can change; a signed spectrum is not its magnitude. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Use F{t²x} = −d²X/dω² on the Gaussian pair. Hz form: √(π/a) [1/(2a) − π² f²/a²] exp(−π² f²/a) | |||
| LorentzianBasic Functions | (π/a) exp(−a |ω|) | a > 0, real. Ordinary function | |
A rational time signal has an exponentially decaying spectrum. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Duality and scaling of exp(−a|t|), or residues in the upper/lower half-plane. Hz form: (π/a) exp(−2πa |f|) | |||
| Squared LorentzianBasic Functions | a > 0, real. Ordinary function | ||
Its zero-frequency value equals π/(2a³). Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Differentiate the Lorentzian pair with respect to a and multiply by −1/(2a). Hz form: | |||
| Causal gamma pulseExponential Functions | tⁿ exp(−a t) u(t) | a > 0; n ≥ 1 integer. The n = 0 case is the causal exponential above. Ordinary function | |
A family of causal smooth-onset pulses. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Differentiate the causal exponential integral n times with respect to a. Hz form: | |||
| Causal damped cosineTrigonometric Signals | exp(−a t) cos(bt) u(t) | a > 0, b > 0 real. Ordinary function | |
A causal damped oscillation has a complex spectrum. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Split cosine into exponentials and integrate the two causal terms. Hz form: | |||
| Causal damped sineTrigonometric Signals | exp(−a t) sin(bt) u(t) | a > 0, b > 0 real. Ordinary function | |
The spectrum uses angular oscillation parameter b even in the Hz view. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Split sine into exponentials and integrate the two causal terms. Hz form: | |||
| Even damped cosineTrigonometric Signals | exp(−a |t|) cos(bt) | a/[a² + (ω − b)²] + a/[a² + (ω + b)²] | a > 0, b > 0 real. Ordinary function |
Two shifted real Lorentzian lobes. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Cosine modulation of the two-sided exponential pair. Hz form: a/[a² + (2πf − b)²] + a/[a² + (2πf + b)²] | |||
| Finite exponential pulseRectangular and Triangular Pulses | exp(−a t) [u(t) − u(t − T)] | a > 0, T > 0 real. Ordinary function | |
Finite support removes the need to infer an infinite causal tail. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Integrate exp(−(a + iω)t) from 0 to T. Hz form: | |||
| Delayed impulseImpulse and Step Functions | δ(t − τ) | exp(−iωτ) | τ real; distribution. Distribution |
The magnitude is 1 and phase is −ωτ or −2πfτ, modulo 2π. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Sifting of the transform kernel at t = τ. Hz form: exp(−i2πfτ) | |||
Inverse Fourier Transform Table
Select Inverse above to put the transform first and its corresponding time function second. Search works across both sides in either direction; all pair conditions still apply.
Fourier Transform Properties
x ↔ X and y ↔ Y use the selected convention. Integrals without limits below run over the whole real line. These operations are properties, not extra pairs.
| Property | Time-domain operation | Transform-domain expression | Conditions |
|---|---|---|---|
| Linearity | A x(t) + B y(t) | A X(ω) + B Y(ω) | Both transforms exist in a common function/distribution space. |
Linearity of integration or distributional pairing. Reference | |||
| Time shifting | x(t − τ) | exp(−iωτ) X(ω) | τ real. |
Substitute v = t − τ in the defining integral. Reference | |||
| Frequency shifting | exp(iω₀t) x(t) | X(ω − ω₀) | ω₀ real, in radians per unit time. |
Combine the exponential factors in the defining integral. Reference | |||
| Time scaling | x(at) | a ≠ 0 real; the absolute value is essential for a < 0. | |
Change variables v = at, reversing limits if a < 0. Reference | |||
| Time differentiation | (iω)ⁿ X(ω) | n ≥ 1 integer; classical boundary terms vanish, or interpret derivatives distributionally. | |
Integration by parts n times, or distributional differentiation. Reference | |||
| Multiplication by time | tⁿ x(t) | n ≥ 1 integer; enough moments for classical differentiation, otherwise distributionally. | |
Differentiate the kernel under the integral with the requisite domination. Reference | |||
| Convolution | (x ∗ y)(t) = ∫ x(τ)y(t − τ) dτ | X(ω) Y(ω) | For example x,y ∈ L¹; convolution of arbitrary distributions is not always defined. |
Fubini and v = t − τ factor the double integral. Reference | |||
| Time-domain product | x(t) y(t) | Sufficient integrability for the product and spectral convolution, e.g. Schwartz functions. | |
Insert the inverse representation of one factor. Reference | |||
| Cosine modulation | x(t) cos(ω₀t) | ω₀ real; base transform exists. | |
Euler cosine identity and frequency shifting. Reference | |||
| Duality | X(t) | 2π x(−ω) | Apply within one convention, e.g. Schwartz functions or by distributions; X(t) reuses the frequency formula as a time function. |
Apply Fourier inversion and reverse the argument. Reference | |||
| Parseval identity | ∫ |x(t)|² dt | x ∈ L²; these are equal energies, not an additional transform pair. | |
Plancherel theorem in the stated normalization. Reference | |||
How to Use a Fourier Transform Table
- Find the function by name, formula or synonym.
- Choose angular frequency ω or ordinary frequency f.
- Check parameters, sinc normalization and distributional conditions.
- Use a matching pair or a property such as shifting or convolution.
- Check the inverse normalization, signs and convergence assumptions.
For example, exp(−t²) has angular transform √π exp(−ω²/4). Delaying the Gaussian by 2 seconds multiplies its transform by exp(−i2ω). In Hz, the multiplier is exp(−i4πf). A delay changes phase, not magnitude.
Fourier Transform Conventions: Hz vs. Angular Frequency
The two selected formulas describe the same spectrum with a different horizontal coordinate: XHz(f) = Xangular(2πf). There is no additional amplitude factor for ordinary spectra in these two forward conventions. Distributional expressions must also transform their delta arguments: 2πδ(2πf) = δ(f). That is why the constant’s transform changes its apparent coefficient.
For a complex spectrum X = R + iI, magnitude is √(R² + I²) and phase is atan2(I,R), undefined where X = 0. For exp(−at)u(t), R = a/(a² + ω²), I = −ω/(a² + ω²), magnitude = 1/√(a² + ω²) and phase = −atan2(ω,a), with a > 0. Substitute ω = 2πf for the Hz form.
Classical absolute convergence is guaranteed for integrable signals. Square-integrable signals can be interpreted in L², and constants, pure sinusoids, steps and impulses require distributions. Inverse pointwise reconstruction additionally needs suitable regularity; at a jump, the usual symmetric inversion value is the average of the one-sided limits.
Common Functions: Gaussian, Rectangular, Sinc, and Delta
exp(−at²) ↔ √(π/a) exp(−ω²/(4a)), a > 0
A Gaussian remains Gaussian. A width-T rectangular pulse produces a sinc spectrum with zero-frequency value T. A triangular pulse produces sinc squared. Conversely, normalized sinc gives a rectangular spectrum in the L² sense. rect(v) has width 1; tri(v) = max(1 − |v|, 0).
δ(t) is defined through ∫δ(t)φ(t)dt = φ(0), not a finite spike height. PV means Cauchy principal value, used for the step and sign transforms. No delta is drawn as an ordinary finite curve.
Gaussian and pulse spectra: lightweight plots
Fixed examples, not a general transform calculator. Horizontal axes show t in seconds or f in Hz; the angular coordinates are ω = 2πf. Only ordinary functions are plotted. Spectral curves below are real parts, not magnitude plots; the pulse spectrum can be negative.
Frequently Asked Questions
What is a Fourier transform table?
It lists time-domain functions and their corresponding frequency-domain transforms, together with definitions and conditions that make each pair meaningful.
How do I use a Fourier transform table?
Find the signal by name or formula, select angular frequency or Hz, check its parameters and conditions, then apply the pair or a listed property. Keep the same normalization throughout.
Does the inverse Fourier transform use the same table?
Yes. Match the frequency expression and read the time-domain function. Use the inverse convention shown above the table and respect ordinary-function, L² or distributional conditions.
What is the difference between a Fourier transform table and a Fourier series table?
A continuous-time Fourier transform uses a continuous frequency variable. A Fourier series represents a periodic function using discrete harmonics. They are related but are not the same table.
Why do Fourier transform tables sometimes show different formulas?
Tables may use angular frequency ω or ordinary frequency f, different normalization constants, or the opposite sign in the forward exponent. Here ω = 2πf, and the forward exponent has a negative sign.
What is the Fourier transform of a Gaussian function?
For a > 0, exp(−at²) transforms to √(π/a) exp(−ω²/(4a)) in angular frequency, or √(π/a) exp(−π²f²/a) in Hz. Both convention forms are shown here; the table displays your selected form.
What is the difference between the Fourier transform and the discrete Fourier transform?
The continuous transform acts on a continuous-time function. A DFT acts on a finite sampled sequence and produces a finite set of frequency bins. Sampling, record length and scaling matter when relating them.
Why do some Fourier transforms contain a Dirac delta?
Constants and pure sinusoids are not integrable over all time. Their transforms are distributions: deltas encode spectral lines with specified weights, not ordinary finite-height peaks.
Sources and verification
- NIST DLMF §1.14 — Integral Transforms (normalization converted to this page’s definition)
- MIT — Transform Properties and Pairs
- Swansea University — Fourier transforms of common signals
Each row provides its mathematical derivation and reference. Smooth integrable examples are checked independently by quadrature; distributional formulas are checked through sifting, duality and scaling, not numerical spike approximations.