stem/Signal Proc/Transfer Function.md

tags
signals

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