Freed U.S. couple held at Qatar airport

• A U.S. couple has been cleared of the death of their child


• They have been told they are free to go, but Qatari officials have blocked them


• Diplomatic sources say their release is coming soon

Doha, Qatar (CNN) -- A U.S. couple cleared in Qatar of the death of their child have been told they are free to return home, but roadblocks remain.

An appeals judge in Qatar on Sunday cleared them of starving their adopted daughter to death in 2013 in a case that drew global attention to the Middle Eastern nation's justice system.

Diplomatic sources told CNN that Matthew and Grace Huang are expected to return to the United States soon. But the couple has been unable to leave.

The Huangs were blocked from leaving Qatar on Sunday at Hamad International Airport by immigration officials, who confiscated their passports, according to the Doha News agency, which cited a family spokesman Eric Volz.

A new warrant had been issued for their arrest on unknown charges, the agency cited Volz as saying.

But diplomatic sources said Monday the departure was delayed due only to legal procedures once people have been cleared of a crime. Neither U.S. nor Qatari sources anticipate problems, the sources said.

But Matthew Huang told CNN's Brooke Baldwin Monday that each day that passes without their exit from Qatar is the continuation of an injustice.

"We're tired, we're exhausted, and we want this to end," Huang said.

He said he was unsure about the reports of a new warrant. As far as the couple knows, they have been acquitted and have been told they are free to leave, he said.

"However, we are still stuck in Qatar, and we have not been permitted to leave," Huang said.

For the moment, he said, the State Department is unable to secure their release.

"We just want to resume our lives in the U.S. with our family and the Qatari government is just not allowing us to do so."

State Department spokeswoman Jen Psaki said Secretary of State John Kerry called Qatar's foreign minister to encourage the Qataris to let the Huangs leave.

"As we understand it, there is some paperwork that needs to be filed. That is something that would of course be in the hands of the lawyers," Psaki said.

CNN's Mariano Castillo and Josh Levs contributed to this report.