I apologize if this question is not well-phrased, as I am not very familiar with the subject.
Let say we have a filtration $K^0\subseteq K^1\subseteq K^2\subseteq K^3$. The $p$-persistent $k$ homology group is defined as: $$H_k^{i,p}=Z_k^i/(B_k^{i+p}\cap Z_k^i)$$ (see the authorative paper by Carlsson titled: Computing Persistent Homology http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.2471&rep=rep1&type=pdf)
How is this different from simply calculating the $k$th homology of each filtration, e.g. $H_k(K^1)$, $H_k(K^2)$, etc? This will be able to tell us which cycles are "birth" and "death" which is the key idea of persistent homology, is it not?
Thanks for any help.