Step 1: Translate the term. Doppelganger literally means double-goer, and in psychiatry it is the delusion of a subjective double.
Step 2: The patient holds the false belief that an exact copy of himself exists elsewhere. That belief is captured by the $feeling\ of\ a\ double\ of\ oneself$.
Step 3: Separate it from look-alikes: misidentifying an unknown person as someone familiar is a different delusional misidentification syndrome, and a trailing shadow is not the definition.
\[\boxed{Feeling\ of\ a\ double\ of\ oneself}\]