Fourier Transform

2 min read#fourier

The generalization of the Fourier series to non-periodic signals — an integral that resolves any signal into a continuous band of frequencies.

Contents

Fourier Transform

A Fourier Series only works for periodic signals, where the allowed frequencies form a discrete ladder of Harmonics. But most real signals — a single clap, a transient, a pulse that happens once and never again — are not periodic. The Fourier transform handles them by letting the period stretch to infinity, at which point the discrete ladder of harmonics fills in to a continuum. The sum becomes an integral:

X(f) = \int_{-\infty}^{\infty} x(t)\,e^{-2\pi i f t}\,dt.

Here x(t) is the signal in time and X(f) is its representation in the Frequency Domain: a complex number for every frequency f, whose magnitude says how much of that frequency is present and whose phase says where it sits. The factor e^{-2\pi i f t} = \cos(2\pi f t) - i\sin(2\pi f t) is a pure rotation — the transform correlates the signal against a sinusoid of every possible frequency at once.

A perfectly reversible change of view

Nothing is lost. The inverse transform rebuilds the signal exactly, reassembling it from its frequency content:

x(t) = \int_{-\infty}^{\infty} X(f)\,e^{+2\pi i f t}\,df.

So x(t) and X(f) are two encodings of one object. This is the time ↔ frequency duality made precise.

The properties that make it powerful

Three structural facts do most of the heavy lifting in applications:

  • Linearity — the transform of a sum is the sum of the transforms, the Superposition Principle in another guise.
  • The convolution theorem — convolving two signals in time is the same as multiplying their transforms in frequency. Filtering becomes a single multiplication.
  • Duality and scaling — squeezing a signal in time stretches its Spectrum in frequency, the seed of the Uncertainty Principle.

See also