Lowest cost processor for 192k to 3.072m 16 x upsampling audio

Buy one of these cheap Cyclone IV boards:

http://www.ebay.co.uk/itm/fpga-core-board-ep4ce6e22-with-sdram-powerful-cyclone-iv-/290738985130?pt=lh_defaultdomain_0&hash=item43b164d8aa

or a Terasic DE0-Nano which also uses the Cyclone IV.

The latter costs more ($79), but includes the USB Blaster interface needed for configuring the FPGA.

I've got both, and they are equally capable of enabling you to complete your project.