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Fourier Transform

Math

Fourier Transform

Area

MathematicsSignal ProcessingPhysicsStatistics

History

Developed by Joseph Fourier (late 18th century – early 19th century)

Basic Features

LinearityTime/Frequency shiftingConvolution propertyParseval's Theorem

Example Signals

Square waveGaussian functionSawtooth waveHarmonic signals

The Fourier transform is a mathematical tool that decomposes a function (typically a signal defined in time or space) into its frequency components. Named after the French mathematician Jean-Baptiste Joseph Fourier, this transform is one of the fundamental analytical methods in fields ranging from engineering to statistics. Mathematically:

Where:

f(t): the signal in the time domain
F(ω): the representation in the frequency domain
ω: angular frequency (rad/s)

Transition from Time to Frequency

Signals in the time domain often appear complex. However, the Fourier transform decomposes this signal into simple waves (sine and cosine) at different frequencies.

Mathematical Properties and Theorems

Parseval’s Theorem

The energy of the signal in the time domain equals its energy in the frequency domain:

∫−∞∞∣f(t)∣2dt=2π1∫−∞∞∣F(ω)∣2dω

Convolution Property

The convolution of two functions in the time domain becomes multiplication in the frequency domain:

f(t)∗g(t)F<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMinYMin slice"><path d="M400000 241H110l3-3c68.7-52.7 113.7-120 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8 -5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247 c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202 l-3-3h399890zM100 241v40h399900v-40z"/></svg><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128 -16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85 -40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5 -12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"/></svg>F(ω)⋅G(ω)

This property is used in the analysis of system responses.

Fourier Transform with Signal Examples

Square Wave

A periodic square wave in the time domain contains harmonic components in the frequency domain (approximated by a Fourier series).

This spectrum shows that only odd harmonics are present in the signal.

Gaussian Function

The Fourier transform is again a Gaussian function:

F(ω)=π<svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"/></svg>⋅e−4ω2

This property makes the Gaussian function “ideal” in both time and frequency domains.

Relation to Characteristic Functions

In probability theory, the statistical counterpart of the Fourier transform is the characteristic function:

ϕX(t)=E[eitX]

Example: Normal Distribution

If X∼N(𝜇,𝜎2), then:

ϕX(t)=exp(iμt−21σ2t2)

This transform carries information about the moments of the distribution. It is particularly used in proving results such as the Lévy Continuity Theorem and the Central Limit Theorem.

Discrete Fourier Transform (DFT) and FFT

To apply the Fourier transform on a computer, the signal is sampled into a discrete version. In this case:

Xk=∑n=0N−1xn⋅e−2πi⋅Nkn,k=0,1,...,N−1

FFT (Fast Fourier Transform)

The FFT is an algorithm that accelerates the computation of the DFT, reducing the time complexity from O(N2) to O(NlogN).

Example applications:

FFT analysis of audio signals reveals which frequencies are dominant.
In image processing, spectral analysis identifies patterns in images.

Application Areas

Author Information

AuthorMelihcan BaşkırDecember 8, 2025 at 1:35 PM

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Transition from Time to Frequency
Mathematical Properties and Theorems
- Parseval’s Theorem
- Convolution Property
Fourier Transform with Signal Examples
- Square Wave
- Gaussian Function
Relation to Characteristic Functions
- Example: Normal Distribution
Discrete Fourier Transform (DFT) and FFT
- FFT (Fast Fourier Transform)
Application Areas

Fourier Transform

Transition from Time to Frequency

Mathematical Properties and Theorems

Parseval’s Theorem

Convolution Property

Fourier Transform with Signal Examples

Square Wave

Gaussian Function

Relation to Characteristic Functions

Example: Normal Distribution

Discrete Fourier Transform (DFT) and FFT

FFT (Fast Fourier Transform)

Application Areas

Author Information

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Discussions

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