# Partial Fraction decomposition

1. Nov 18, 2013

### DmytriE

Hi,

I am currently in a signals and systems course where we're looking at taking the Fourier Transform with $Y(\omega) = H(\omega)X(\omega)$.

If $Y(\omega) = \frac{\omega + 2}{(\omega + 1 )^{2} (\omega + 3)} = \frac {A_{11}}{(\omega +1)} + \frac {A_{12}}{(\omega + 1)^{2}} + \frac {A_{21}}{(\omega + 3)}$

Oppenheim, author of Signals and Systems, states that you can use the following equation to solve for the value of A.

$A_{ik} = \frac {1}{(\sigma_{i} - k)!} \left[\frac {d^{\sigma_{i} - k}} {dv^{\sigma_{i} - k}} \left[(v - \rho_{i})^{\sigma_{i}}G(v)\right]\right]$

i and k are the subscripts that are associated with the values of the numerator values. This example is from Oppenheim's signal and systems book. This is NOT a homework problem.

Could someone help me understand the second equation which should be used to determine the numerator values? Once I understand how to use this equation I can them extrapolate it to my homework problems.

In the book they have the following to solve for $A_{11}$

$A_{11} = \frac {1}{(2-1)!} \left[\frac {d} {dv} \left[(v + 1)^{2}G(v)\right]\right]$

Thanks!