Joshua Long ECE301Fall2008mboutin - Rhea

Partial Fractions

This page is meant as a comprehensive review of partial fraction expansion. Partial fraction expansion allows us to fit functions to the known ones given by the known Fourier Transform pairs table.

First, the denominator must be of a higher degree than the numerator. If this is not the case, then perform long division to make it such. Note: for the remainder of this guide it is assumed that the denominator is of a higher degree than the numerator.

The Four Cases to take into Account

Case 1: Denominator is a product of distinct linear factors.

$ \frac{(Polynomial)}{(a_1x + b_1)(a_2x + b_2)...(a_kx + b_k)} = \frac{A_1}{(a_1x + b_1)}+\frac{A_2}{(a_2x + b_2)}+...+\frac{A_k}{(a_kx + b_k)} $

Case 2: Denominator is a product of linear factors, some of which are repeated.

$ \frac{(Polynomial)}{(a_1x + b_1)^r} = \frac{A_1}{a_1x + b_1}+\frac{A_2}{(a_1x + b_1)^2}+...+\frac{A_r}{(a_1x + b_1)^r} $

Case 3: Denominator contains irreducible quadratic factors, none of which is repeated.

$ \frac{(Polynomial)}{ax^x + bx + c} = \frac{Ax + B}{ax^+bx+c} $

Case 4: Denominator contains a repeated irreducible quadratic factor.

$ \frac{(Polynomial)}{(ax^x + bx + c)^r} = \frac{A_1x + B_2}{ax^+bx+c}+\frac{A_2x + B_2}{(ax^+bx+c)^2}+...+\frac{A_rx + B_r}{(ax^+bx+c)^r} $

F.T. Logic

                                      Is the Signal Periodic?
                     Yes                                                  No
              Is it in the table?                               Can you Integrate?
   Yes                               No               Yes                               No
   Cool                  Try to break it up and       Cool                        Is it in the table?
                          check the table again                      Yes                              No
                                                                    Cool            Try to break it up and reevaluate

Example of a F.T.

$ x(t) = \cos( 4t + \frac{4\pi}{3}) $

$ X(\omega) = F(x(t)) $

$ X(\omega) = F(\cos( 4t + \frac{4\pi}{3})) $

$ X(\omega) = F( $