Fourier transform and frequency domain analysisbasics

Discrete Fourier transform (DFT) and Fast Fourier transform (FFT)

The Discrete Fourier transform (DFT) is obtained by decomposing a sequence of values into components of different frequencies. It converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence.

A Fast Fourier transform (FFT) is an algorithm that computes the DFT of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.

Toolbox: DFT/IDFT calculator
Signal representationTime domainFrequency domain
DFT
IDFT
DFT/IDFT calculator (modify {xn} or {Xk}, the other one is automatically updated)
Time domain: {xn}
Frequency domain: {Xk}

Frequency domain analysis

The Fourier transform is commonly used to convert a time domain signal to the frequency domain.

The time domain signal can be seen as a sum of sine waves. A FFT will separate the constructing sine waves and return information about the frequency of these sine waves.

Toolbox: Frequency domain analysis

Be careful: if f1 and f2 differs a lot (e.g. by more than 20 times), it will take very long time to process.

TimeFrequency
  • f1 [Hz] =

  • f2 [Hz] =
  • A =
Frequency domain analysis of user provided signal
sampling frequency [Hz] =