Digital signal processing can be used many functional building blocks of the advanced communication systems. The soft ware defined radio technology is now a common practice for implementing advanced wireless transceivers standards in mobile communication networks, wi phi and wimax. The digital processing can be implemented either using field programmable arrays FPGAs of digital signal processors DSPs devices or a plat forms composed of the two chips in addition to general purpose processors. 

As for the functions that can be implemented are the source coding, channel codeine, encryption, multiplexing. filtering, modulation, mixing for frequency up conversion and the inverse processes. Normally the modulation is carried out at the base band model.

It can also realized in the pass band but one has to choose the carrier frequency as low as possible to save in the required computations as as the sampling frequency increases the computations required increases.

Logically the inverse processes in the receiver are also implemented using the same platforms.

So, the mixing can be realized using DSPs but you have to compromise the carrier frequency used  to save computations and power of the DSP.

It is advisable to carry out the mixing processes in the analog front end of the transceiver and use zero IF or low if mixing to lower the sampling frequency and save processing effort and power consumption. 

The system design helps choosing the most suitable system architecture of the wireless transceivers.

Best wishes 

Sampling itself replaces the function of a mixer (for down-conversion), but there must be a good anti-aliasing filter before the sampler.  If you have a good and narrow anti-aliasing filter you can use a low sampling rate - but you still need very fast sampling gates.

Well Srikar, mixer gives a range of frequency (linear /non-linear). It all depends on how you modulate say using a DSP for add or multiply the frequencies then you can use the DSP to demodulated them.

Along with this , you can use lpf /anti-aliaising / decimate etc etc

If your hardware has the capability to process the native input (mostly bandwidth, conversion rate and processing power), there is no need for an external mixer.