If this is a feature of the device and not just the software it came with (it's probably the former), then it has to be implemented in the kernel driver.1 It appears to me that normally, Realtek PCI sound cards actually use the Intel HDA driver with a patch to support the ALC codecs (for anyone who's interested:
Searching through the kernel source documentation, "noise suppression" or "echo cancellation" is not mentioned in relation to these devices. Searching the driver source itself, it's not explicitly mentioned either. I think if it were implemented, this would almost certainly be documented with an interface in
/sys which userspace software could use, or with options for the driver module.
The issue, of course, is that Realtek, like most hardware manufacturers, considers their driver source code to be secret, making it difficult or impossible for third parties to create drivers for, e.g., systems Realtek doesn't fully support, such as linux. Realtek does have a page listing a linux driver, however, the links appear to be dead. If you have better luck, someone has instructions for installing it here (that's a Mint oriented site, but the process will be the same for any distro).2
However, even if you can find a copy somewhere, I doubt very much that driver implements the features you want, since there is no mention of this that I can find. Most likely, it is more or less the same as what you already have; it may be slightly better (that Mint page mentions the kernel driver swaps the side and rear speakers), but it may also end up being worse, or completely non-functional, as out-of-tree drivers are not vetted by the kernel devs and may contain issues in relation to the kernel the authors are unaware of (note that the authors of the in-tree hda patch have
@realtek.com.tw email addresses).
So unless you hear any different, you are probably out of luck. You could look around for software solutions to this, such features are apparently implemented by Skype on linux, so they evidently can work in real time using software alone. Somebody's also written a paper on it, lol. I did not have any further luck searching around, unfortunately.
There is some discussion here of using the features of
alsamixer to accomplish some noise reduction. I think you'll have to disable pulseaudio to do that.
1. Pulseaudio is actually a userspace tool that is built on top of ALSA, the kernel's sound subsystem (meaning: pulseaudio doesn't implement the hardware access, it just works with streams coming from/going to the kernel driver).
2. If you do find it, you want the 3.0 version, not the 2.6 version. Those numbers refer to the version of the linux kernel they are compatible with and you are using a 3.x kernel. However, again, if your sound is otherwise okay, I strongly recommend against bothering with this as it is unlikely to do anything for you.