Table of CT Fourier Series Coefficients and Properties

Fourier series Coefficients

Function	Fourier Series	Coefficients

}

Property Name	Property	Proof
Linearity	$\mathfrak{F}(c_1g(t) + c_2h(t) = c_1G(f) + c_2H(f)$	$\mathfrak{F}(c_1g(t) + c_2h(t) = \int_{-\infty}^\infty c_1g(t) dt + \int_{-\infty}^\infty c_2h(t) dt$ $=c_1\int_{-\infty}^\infty g(t)e^{i2\pi ft} dt + c_2 \int_{-\infty}^\infty g(t)e^{i2\pi ft} dt$ $$ =c_1G(f) + c_2H(f) $$
Time Shifting	$\mathfrak{F}(g(t - a)) = e^{-i2\pi fa}*G(f)$	$\mathfrak{F}(g(t - a)) = \int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt$ $=\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du$ $=e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu}du$ $=e^{-i2\pi fa} G(f)$
Time Scaling	$\mathfrak{F}(g(ct)) = \frac{G(\frac{f}{c})}{\|c\|}$	$\mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt$ subtitute : u = ct, du = cdt $\mathfrak{F}(g(ct)) = \int_{-c\infty}^{c\infty} \frac{g(u)}{c}e^{-i2\pi f\frac{u}{c}}du$ if c is greater than 0: then no signs change. if c is less than 0: the integration must be flipped as well as the negative from the c so you still get the same equation. therefore the absolute value of c is obtained.
Frequency Shifting	$\mathfrak{F}^{-1}[X(j\omega + omega_0)] = x(t)e^{-j\omega_0t}$	$\mathfrak{F}^{-1}[X(j\omega + omega_0)] = \frac{1}{2\pi} \int_{-\infty}^{\infty}X(j(\omega +\omega_0))e^{j\omega t} d\omega$ $=\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega ')e^{j\omega (\omega ' + \omega_0)} d\omega$ $=e^{j\omega_0 t}\frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega ')e^{j\omega '} d\omega'$ $=x(t)e^{j\omega_0 t}$