Properties of the Fourier Transform#
No. |
Name |
\(f(t)\) |
\(F(j\omega)\) |
Remarks |
|---|---|---|---|---|
1. |
Linearity |
\(a_1f_1(t)+a_2f_2(t)+\cdots+a_nf_n(t)\) |
\(a_1F_1(j\omega)+a_2F_2(j\omega)+\cdots+a_nF_n(j\omega)\) |
Fourier transform is a linear operator. |
2. |
Symmetry |
\(2\pi f(-j\omega)\) |
\(F(t)\) |
|
3. |
Time and frequency scaling |
\(f(\alpha t)\) |
\(\displaystyle{\frac{1}{\lvert\alpha\rvert}F\left(j\frac{\omega}{\alpha}\right)}\) |
time compression is frequency expansion and vice versa |
4. |
Time shifting |
\(\displaystyle{f(t-t_0)}\) |
\(\displaystyle{e^{-j\omega t_0}F(j\omega)}\) |
A time shift corresponds to a phase shift in frequency domain |
5. |
Frequency shifting |
\(\displaystyle{e^{j\omega_0 t}f(t)}\) |
\(\displaystyle{F(j\omega-j\omega_0)}\) |
Multiplying a signal by a complex exponential results in a frequency shift. |
6. |
Time differentiation |
\(\displaystyle{\frac{d^n}{dt^n}\,f(t)}\) |
\(\displaystyle{(j\omega)^nF(j\omega)}\) |
|
7. |
Frequency differentiation |
\(\displaystyle{(-jt)^n f(t)}\) |
\(\displaystyle{\frac{d^n}{d\omega^n}F(j\omega)}\) |
|
8. |
Time integration |
\(\displaystyle{\int_{-\infty}^{t}f(\tau)d\tau}\) |
\(\displaystyle{\frac{F(j\omega)}{j\omega}+\pi F(0)\delta(\omega)}\) |
|
9. |
Conjugation |
\(\displaystyle{f^*(t)}\) |
\(\displaystyle{F^*(-j\omega)}\) |
|
10. |
Time convolution |
\(\displaystyle{f_1(t)*f_2(t)}\) |
\(\displaystyle{F_1(j\omega) F_2(j\omega)}\) |
Compare with Laplace Transform |
11. |
Frequency convolution |
\(\displaystyle{f_1(t)f_2(t)}\) |
\(\displaystyle{\frac{1}{2\pi}F_1(j\omega)*F_2(j\omega)}\) |
This has application to amplitude modulation as shown in Boulet pp 182—183. |
12. |
Area under \(f(t)\) |
\(\displaystyle{\int_{-\infty}^{\infty} f(t)\,dt = F(0)}\) |
Way to calculate DC (or average) value of a signal |
|
13. |
Area under \(F(j\omega)\) |
\(\displaystyle{f(0) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(j\omega)\,d\omega}\) |
||
14. |
Energy-Density Spectrum |
\(\displaystyle{E_{[\omega_1,\omega_2]}:=\displaystyle{\frac{1}{2\pi}\int_{\omega_1}^{\omega_2}\lvert F(j\omega)\rvert ^2\,d\omega.}}\) |
||
15. |
Parseval’s theorem |
\(\displaystyle{\int_{-\infty}^{\infty}\lvert f(t)\rvert^2\,dt=\displaystyle{\frac{1}{2\pi}\int_{-\infty}^{\infty}\lvert F(j\omega)\rvert ^2\,d\omega.}}\) |
Definition RMS follows from this |
See also: Wikibooks: Engineering Tables/Fourier Transform Properties and Fourier Transform—WolframMathworld for more complete references.