Rafik Mohamad Yousef, 41 -- who had done jail time for a 2004 Berlin assassination plot against former Iraqi prime minister Iyad Allawi -- had a history of aggressive and unstable behaviour, recently threatening a judge and other public officials.
"We are presuming he acted alone," rather than as part of a wider attack plot, said Berlin prosecution service spokesman Martin Steltner.
"We don't have new information on what may have been the man's motive," he added. "We are checking his background and whether he was in contact with other Islamists."
Before they arrived, he went out to threaten passers-by in his western Berlin district, until four police patrol cars arrived.
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In the ensuing confrontation, he stabbed a 44-year-old policewoman in the neck before one of her colleagues shot him dead, also accidentally hitting the policewoman. Her condition was described as stable Friday.
Hours after the attack, authorities said he was "a suspected Islamist" who had remained under police watch following his release from an eight-year term, including time served while on trial.
Since his release, he had again drawn attention with aggressive behaviour against police and by making "jihadist threats" against a judge and a staffer at the immigration department, said Steltner.
Yousef had told the immigration officer that "we will behead you", reported national news agency DPA.
In his trial, presiding judge Christine Rebsam-Bender had described Yousef as "hot-tempered and aggressive" and cited his frequent outbursts, including an attack on a prison guard that broke the officer's rib.