User:Zhennan/FTcomparison
The table shows members of Fourier transform, with the characteristics of input signal and output signal [1].
| Technique | Time domain | Frequency domain |
|---|---|---|
| Continuous Fourier Transform | Continuous, non-periodic | Continuous, non-periodic |
| Fourier Series | Continuous, periodic | Discrete, non-periodic |
| Discrete-time Fourier transform | Discrete, non-periodic | Continuous, periodic |
| Discrete Fourier transform | Discrete, periodic | Discrete, periodic |
It shows that,
- Discrete signal in one domain means periodic signal in the other domain.
- Continuous signal in one domain means non-periodic signal in the other domain.
Briefly speaking, Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials. Fourier transform can be viewed as the limit of the Fourier series of a function with the period approaches to infinity, so the limits of integration change from one period to (−∞,∞). Discrete-time Fourier transform is sampling of Continuous Fourier Transform in time domain. And Discrete Fourier transform is even sampling of Discrete-time Fourier transform in frequency domain.
Reference