This table is adapted and reworked from Dr Chris Jobling's resources, see this page. Other links: Properties of the Fourier Transform (Wikpedia) or Wikibooks: Engineering Tables/Fourier Transform Properties and Fourier Transfom—WolframMathworld for more complete references..
| Name | $x(n)$ | $X(f)$ | Remarks | |
| 1 | Linearity | $a_1x_1(n)+a_2f_2(t)+\cdots+a_kf_k(n)$ | $a_1X_1(f)+a_2X_2( f)+\cdots+a_kX_k(f)$ | Fourier transform is a linear operator. |
| 2 | Duality | $$x(-f)$$ | $$X(n)$$ | |
| 3. | Time and frequency scaling | $$x(\alpha n)$$ | $$\frac{1}{|\alpha|}S\left(\frac{ f}{\alpha}\right)$$ | time compression is frequency expansion and vice versa |
| 4. | Time shifting | $$x(n-n_0)$$ | $$e^{-j2\pi f n_0}X(f)$$ | A time shift corresponds to a phase shift in frequency domain |
| 5. | Frequency shifting | $$e^{j2\pi f_0 n}x(n)$$ | $$X(f-f_0)$$ | Multiplying a signal by a complex exponential results in a frequency shift. |
| 7. | Frequency differentiation | $$(-jn)^k x(n)$$ | $$\frac{d^k}{d f^k}X(f)$$ | |
| 8. | Time integration | $$\sum_{m=-\infty}^{n}f(m)$$ | $$\frac{X(f)}{j2\pi f}+\pi X(0)\delta(f)$$ | |
| 9. | Conjugation | $$s^*(n)$$ | $$S^*(- f)$$ | |
| 10. | Time convolution | $$x_1(n)*x_2(n)$$ | $$X_1(f) X_2(f)$$ | |
| 11. | Frequency convolution | $$x_1(n)x_2(n)$$ | $$\frac{1}{2\pi} X_1(f)*X_2(f)$$ | Application to amplitude modulation. |
| 12. | Sum of $x(n)$ | $$\sum_{n=-\infty}^{\infty} x(n) $$ | $X(0)$ | Average value of a signal |
| 13. | Area under $X(f)$ | $$f(0) $$ | $$\frac{1}{2\pi}\int_{-\infty}^{\infty} X(f)\,df$$ | |
| 15. | Parseval's theorem | $$\sum_{-\infty}^{\infty}|x(n)|^2$$ | $$\frac{1}{2\pi}\int_{-\infty}^{\infty}|X(f)|^2\,df.$$ |