Given of order with all its poles being distinct, it can be expressed in terms of partial fraction expansion in the form of and the resulting can be found to be where is the sampling period.
In the case when contains a pole of order , then can be written as and the resulting can be found to be .
In the case when contains a pole of order , then can be written as and the resulting can be found to be .
The following table was generated in order to obtain the general formula. This table below shows only the part of due to the multiple order pole.
It is easy to see that the denominator of has the general form where is the pole order, the hard part is to ﬁnd the general formula for the numerator. The following table is a rewrite of the above table, where only the numerator is show, and was written as to make it easier to see the general pattern
I am trying to determine the general formula to generate the above. This seems to involve some combination of binomial coeﬃcient. But so far, I did not ﬁnd the general formula.