This article tries to explain all the different Linux DSP systems for OMAP chips, how they are similar and different.
DSP Gateway was developed by Nokia for the Maemo Internet Tablets. It's the oldest and more open of the implementations. The open code consists not only of the linux kernel side, but also the DSP operating system. Unfortunately the development is essentially halted.
It works on OMAP1 and OMAP2, it's production ready, used on the Nokia N800 and N810, it follows linux standards and it's close to upstream acceptance. There's code for OMAP3 but it hasn't been thoroughly tested.
It is maintained by Hiroshi DOYU.
There are a few user-space applications that use it. Essentially GStreamer Nokia DSP plugins and a few others developed by the Maemo community.
tidspbridge originally developed by TI, but after being released as open source it has received many contributions, predominantly by Nokia.
It's probably not production ready (at least I don't know of any released product using it), and it still doesn't meet linux standards although there has been a lot of progress. Only the ARM side is available as open source, unlike the dsp-gateway, the DSP side is completely closed.
Unfortunately there's a disagreement between TI and the community as how power management must be handled, so there are two forks: linux-omap, and omapzoom.
It's under heavy development, originally maintained by Hiroshi DOYU, but now Ameya Palande took over.
There is plans to share the mailbox and iommu that the dsp-gateway uses, but it's independent from it, as well as to move parts of it to user-space.
There are slightly more user-space applications using it, including gst-openmax through TI's OpenMAX IL implementation which is also open source. and gst-goo.
A slimmer version of the dsp-bridge targeted for the DaVinci platform also developed by TI. It supports a wider variety of chips, not only OMAP.
It is the one farther from meeting linux standards, the code hasn't even been submitted for reviewing nor is planned to to be merged at any time.
It's tightly tied to codecengine and dmai and there are GStreamer plug-ins provided by TI to use the algorithms.