×
Oct/01/2022: Apple Inc (California, US) is looking for talents with 3GPP RAN2 experience and a good physical layer understanding. Contact Me if you are interested.

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] =