Filters

Ideal Filters will filter out all signal content in certain frequency regions

Frequency Domain

Impedance

• The frequency-dependent resistance to current flow through the element
• Impedance can be a complex number with a real and imaginary part, commonly called the resistance and the reactance
• $\vec{Z}=R+jX$
• Passive Elements
  • Resistors
    • $Z_R=\frac{V}{I}=R$
  • Capacitors
    • $I=CsV$
    • $Z_C=\frac{V}{I}=\frac{1}{sc}=\frac{1}{j\omega C}=-\frac{j}{\omega C}$
  • Inductors
    • $v=LsI$
    • $Z_L=\frac{V}{I}=Ls=j\omega L$
  • Via the Laplace transform: $\frac{d}{dt}x\to sX$

Frequency Response

Transfer function $H(s),H(j\omega )$: relations of the output to the input in the frequency domain

The bandwidth is defined to be 0Hz to the cut off frequency, $f_c$

Low-Pass Filter

• $H(s)=\frac{{\omega }_c}{s+{\omega }_c}$
• Inductor
  • 
  • ${\omega }_c=\frac{R}{L}$
• Capacitor
  • 
  • $w_c=\frac{1}{RC}$