"How does FED qua[n]tify inflation expectation?"
NY Fed survey, UoM survey, Philly Fed SPF, proprietary adjustments to breaks for various risk premia, etc. But overall, this question should probably be in economics stack exchange.
NY Fed survey
Philly Fed SPF
Edit: adding some references (main body is untouched)
Kenneth Rogoff and Richard Meese received an incredulous reaction to their now-famous paper showing that random-walk (RW) forecasts outperform economic models of exchange rates. Reactions were along the line of “You just cannot possibly have done it right” or "the results are obviously garbage". ...