Postecoglou insists bringing success to the club will be no overnight thing.
The Australian is the first manager to last a full season at Spurs since Mauricio Pochettino in the 2018-19 campaign.
He joked: "Yes, they had a cake for me.
"It (the negativity) doesn't seep into my environment. The fans are free to feel how they want to feel and it doesn't impact me or what we are trying to achieve.
"For me, you can get overly impacted in both ways, we just have to stay clear-eyed and focused on what we want to achieve and what kind of team we want to be."
He added: "I came in here to try to do something that hasn't been done for a while.
"You can't just think me walking in will change things, you've got to change things and it can be unsettling but it is necessary for us to achieve what we want to."