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 representation | Time domain | Frequency 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 analysisBe careful: if f1 and f2 differs a lot (e.g. by more than 20 times), it will take very long time to process.
| Time | Frequency | |
|---|---|---|
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| Frequency domain analysis of user provided signal | ||
| sampling frequency [Hz] = | ||