WHITE SULPHUR SPRINGS, W.Va. — Even as the punts sailed the sky and then down into his grasp Tuesday, Saints safety Jairus Byrd concentrated on how this would make him better at victimizing NFL quarterbacks.

“Get a punt going that high and that deep, you can track that ball,” said the three-time Pro Bowler, who practiced on the field with the Saints for the first time Tuesday since undergoing back surgery in late May. “That emulates what a pass is going to be like: a deep ball. That’s something a lot of safeties can work on.”

There’s no denying Byrd started his first Saints training camp in less-than-ideal football shape. The surgery to repair a problematic spinal disk caused him to miss weeks of voluntary organized team activities as well as minicamp this offseason. He wasn’t cleared to hit the practice field until five days into training camp at The Greenbrier resort in West Virginia — and even then, he was limited to lining up during a walkthrough, completing some skeleton drills and fielding punts.

But Byrd’s attitude toward punts illustrates why the Saints expect his integration into the team he joined in March via a six-year, $54 million free-agent deal will be a smooth one.

“I would love to stand here and say ... I’m the one who gave him (his) ... skills, (his) instincts,” Saints defensive backs coach Wesley McGriff said. “But he was born with those things — and he improves them by studying, being a good student of the game.”

As most are aware, Byrd’s NFL coursework was steaming along quite finely before his arrival in New Orleans. He’s amassed 22 interceptions (the most among league safeties over the previous five years), 11 forced fumbles and five loose-ball recoveries since Buffalo chose him in the second round of the 2009 draft. The numbers explain why a team that held opponents to the fourth-fewest yards in the NFL but generated the fourth-fewest turnovers signed him this spring to a big-money deal.

Yet the splashy acquisition wasn’t without its complications. He had back surgery less than three months later.

Byrd declined to answer questions about the exact kind of surgery he had Tuesday. He also didn’t say whether it had anything to do with a back injury he suffered at the Pro Bowl in January that knocked him out of that game early.

“It’s an issue that needed to be taken care of,” said Byrd, who has run two of his NFL interceptions back for scores. “It’s all I’ll say.”

However, even if the surgery sidelined him for a significant portion of the offseason, it doesn’t mean Byrd’s load lightened at the office.

That’s not solely because there was rehab and conditioning to go through. There were meetings to attend, teammates’ brains to pick and checks to learn on snaps he could only observe from afar.

“He’s in the meeting room. He’s studying,” said Rafael Bush, Byrd’s fellow safety. “He’s asking me and (safety) Kenny (Vaccaro) questions.”

It showed Tuesday, said defensive coordinator Rob Ryan, who many imagine salivating at the thought of aligning Byrd with 2013 Saints standouts Bush and Vaccaro. Byrd called out for adjustments in the drills he partook in. He coolly hauled in a series of booming, arcing punts from Thomas Morstead.

The point wasn’t that he was fielding punts, which he last consistently did in college at Oregon, running back 33 there for 383 yards and a touchdown. The point was that he brandished the ball skills that bagged him his lucrative contract from the Saints.

“Mentally, he’s right there,” Ryan said, before adding, “If that ball hits his hands, he’s going to catch it.”

Ryan said Byrd still had to log many more reps with his teammates to reach his true capacity. Byrd, who was noncommittal about playing in Saturday’s intrasquad scrimmage or the preseason opener at St. Louis, knows that.

And he won’t let the Saints’ sizable investment in him or preceding reputation as a ball-hawk rush his return to prime form.

“It’s not about me — it’s about our team, and I look forward to making an impact on this defense,” Byrd said. “Whatever category that falls, whatever category that is, that’s what I look forward to doing.”