K-theory as elliptic complexes

Question

I would like to learn the theorem which states that K-theory defined in terms of vector bundles is isomorphic to the so called compactly supported complexes (those are complexes of vector bundles over locally compact space $X$ of the form $0 \to E_1 \to E_2 \to ... \to E_n \to 0$ where this sequence is exact except for some compact set $A \subset X$). I would be happy with the proof from which it is clear how the maps establishing isomorphism look like. I would be grateful for pointing me some references.