I didn't follow the 3440 much and don't know what if any control signals it is capable of sending, but many actual "
Tascam" branded dbx units could only go into encode/decode when receiving appropriate control signals from the machines they were designed to work with.
Some of the Tascam dbx units did have global in/out switches for the encode/decode functions, but these didn't affect the actual individual channel functions...those were user inaccessible except by control signals to the dbx unit from the tape recorder via ribbon cable or similar. If the dx4 is anything like the old Tascam dx8 dbx units, it won't work at all unless it gets those appropriate control signals...which I don't know that a 3440 can send.
You may want to call Jimmy in tech support at Montebello and have him look this stuff up. He'll be able to tell you if the dx4's encode/decode only worked with relay switching...and whether it can be integrated with a 3440.
I DO know that you can pick up any number of actual "Dbx" brands of nr and integrate those with a 3440. The actual dbx type 1 units (such as the Dbx 155 etc) did not depend on control signals for encoded/decode switching. They were designed to work with pretty much any make of multitrack. Of course you will have to manually switch the dbx units in and out of encode/decode.....but you WILL have noise reduction. You might want to keep that in mind if Jimmy tells you no-go on the dx4.
If you get one of the actual Dbx brands of noise reduction from Ebay etc, make sure it's a type 1 type. Type 2 or 3 are not the professional circuit and are not very good. Type 1 is what was found in all Tascam dbx types as well as all the other pro versions.