Filtering a Square Wave

• We had seen before, that the square wave can be written as a linear combination of sinusoids
  
  $$s(t) = \frac{4}{\pi} \sin(2 \pi t) + \frac{4}{3 \pi} \sin(2 \... ...c{4}{k \pi} \sin(2 \pi k t) + \ldots \mbox{, $k$\space odd. }$$
  
  

• Let the square wave be input to the lowpass filter.
• Then, because of the linearity of the RC lowpass the resulting output is
  
  $$v(t) = \frac{4}{\pi} A(1) \sin(2 \pi t + \phi(1)) + \frac{4}{... ...) \sin(2 \pi k t + \phi(k)) + \ldots \mbox{, $k$\space odd. }$$
  
  

• Notice, that the lowpass filter supresses the higher frequency components.
• The low frequency components are retained.
• This results in a ``rounded'' square wave.
  
    

  Figure 4.7: Input and Output Signal (RC=0.1).