sarwiki

Transfer Function

Y(s)=H(s)\cdot X(s)

$H(s)=\frac{Y(s)}{X(s)}=\frac{\mathcal L\{y(t)\}}{\mathcal L\{x(t)\}}$

Y(z)=H(z)\cdot X(z)

$H(z)=\frac{Y(z)}{X(z)}=\frac{\mathcal Z\{y[n]\}}{\mathcal Z\{x[n]\}}$

G(\omega)=\frac{|Y|}{|X|}=|H(j\omega)|

$H(j\omega)$ , Frequency response

\phi(\omega)=arg(Y)-arg(X)=arg\left(H\left(j\omega\right)\right)

$\phi(\omega)$ , Phase shift

\tau_\phi(\omega)=-\frac{\phi(\omega)}{\omega}

$\tau_\phi$ , Phase delay
Frequency-dependent amount of delay introduced to the sinusoid by $H$

\tau_g(\omega)=-\frac{d\phi(\omega)}{d\omega}

$\tau_g$ , Group delay
Frequency-dependent amount of delay introduced to the envelope of the sinusoid by $H$

Partial Fractions Partial Fractions for Laplace Inverse Z Transform

Discrete Time Systems:Impulse responses and convolution; An introduction to the Z-transform