Constructing Pascal's Triangle

• In order to see the amplitudes of the harmonics produced by a term in the polynomial, we can look at Pascal's triangle.

• To construct Pascal's triangle, first create a $N\times N$ table, and input ones along the diagonal, starting from the top left-hand corner (i.e., create an $N$ by $N$ identity matrix).

  
  

  DIV

  $h_0$

  $h_1$

  $h_2$

  $h_3$

  $h_4$

  $h_5$

  $h_6$

  $h_7$

  $h_8$

  $h_9$

  $h_{10}$

  $h_{11}$

  $x^0$

  1

  $x^1$

  1

  $x^2$

  1

  $x^3$

  1

  $x^4$

  1

  $x^5$

  1

  $x^6$

  1

  $x^7$

  1

  $x^8$

  1

  $x^9$

  1

  $x^{10}$

  1

  $x^{11}$

  1

  
  

• The symbols along the side, $x^j$, represent the term in the polynomial.

• The symbols along the top, $h_j$, represent amplitude of the $j^{\mbox{th}}$ harmonic.