25-th order approximation

• With 25 sinusoids the aprroximating signal is given by
  
  $$s(t) = \frac{4}{\pi} \sin(2 \pi t) + \frac{4}{3 \pi} \sin(2 \... ...c{4}{k \pi} \sin(2 \pi k t) + \ldots \mbox{, $k$\space odd. }$$
  
  

• As the figure below shows, the approximation is very good already.
• The formula above can be used to compute even better approximations to the square wave.
• By including more sinusoids of higher frequencies (k) it is possible to capture more of the details of the square wave.
• Ultimately, the approximation becomes perfect.

  

Figure 4.4: Approximation of square wave by sum of 25 sinusoids.