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image showing a J-type thermocoupe connected to an instrumentation amplifier

Given a circuit similar to the above image, where you have a DC-coupled signal that occasionally goes below the in-amp's negative rail (ground in this case because I'm interested in designing for single supply operation), what is the best way to offset the input's voltage so that it's centered around Vs/2 (~= 2.5V here) rather than ground? I'd prefer to stick with DC-coupling but would be interested in AC-coupled examples as well.

As a possibly relevant aside, the attached image is a design reference for an instrumentation amplifier whose input range extends -100mV below ground when used in single supply mode. As-shown the circuit is fine due to the minuscule negative voltage generated by a J-type thermocouple but I'm interested in how to properly design the circuit for a wider voltage swing.

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Here's a few ways to do it where V+/2 can be an RC filter from V+ and 0V.

enter image description here

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