Hi Peter,
Sorry for the delay in replying to your query.
From your explaination, I understand that you need to lower the sampling rate of the ADC from 48KHz down to 12KHz due to high processing power requirement .
ADSP-214xx processors has an ASRC (Asynchronous Sample Rate Converter Hardware module) where the most frequently used sample rate conversions are off-loaded into hardware modules that are dedicated for filter processing and reduce the instruction processing load on the core, freeing it up for other tasks. Hence is capable of downsampling the incoming digital signals with the input/output continuous sample ratios 7.5:1.
Please go through the ADSP-214xx Hardware Reference Manual from Pg.No 649 for complete details of the ASRC Module .The link for the Manual is given below.
http://www.analog.com/static/imported-files/processor_manuals/ADSP-214xx_hwr_rev1.1.pdf
Regarding your questions on the harmonics, Maybe somebody who has expertise in this field can comment better.
Regards,
Mahesh