We know that the US dollar exchange rate increased during the global recession while usually when an economy performs weakly its currency should depreciate. How was there still a huge demand for USD in foreign markets even when subprime mortgage crisis was at its peak.
Because dollar is a major reserve currency. When a financial crisis occurs in other countries, like India, Brasil, Russia, etc, everyone sells stocks and liquiates assets to raise cash. But the cash they want to raise is usually dollars or euros. When a crisis hit the US, everyone started selling assets FOR dollars. I.e. they were buying dollas for stocks, trillions of dollars worth. This demand for cash appreciated dollar. There was a multitude of other factors involved, as discussed in this paper: Dollar Appreciation in 2008: Safe Haven, Carry Trades, Dollar Shortage and Overhedging http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1519814
After Fed's relatively successful monetary policy decisions, the US economy recovered faster than the economies of the EU and Japan, other reserve currency emitters. Then it was natural for dollar to continue strengthening.