Linearity

$a_1{f}_1(t)+a_2{f}_2(t)+\cdots +a_n{f}_n(t)$

$a_1{F}_1(j\omega )+a_2{F}_2(j\omega )+\cdots +a_n{F}_n(j\omega )$

Fourier transform is a linear operator.

Time and frequency scaling

$f(\alpha t)$

${\displaystyle \frac{1}{|\alpha |}F\left(j\frac{\omega }{\alpha }\right)}$

time compression is frequency expansion and vice versa

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.

Frequency differentiation

${\displaystyle (-jt{)}^{n}f(t)}$

${\displaystyle \frac{d^n}{d{\omega }^n}F(j\omega )}$

Conjugation

${\displaystyle f^{\ast }(t)}$

${\displaystyle F^{\ast }(-j\omega )}$

Frequency convolution

${\displaystyle f_1(t)f_2(t)}$

${\displaystyle \frac{1}{2\pi }{F}_1(j\omega )\ast F_2(j\omega )}$

This has application to amplitude modulation as shown in Boulet pp 182—183.

Area under $F(j\omega )$

${\displaystyle f(0)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(j\omega )d\omega }$

Parseval’s theorem

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|f(t){|}^{2}dt={\displaystyle \frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|F(j\omega ){|}^{2}d\omega .}}$

Definition RMS follows from this