Process ADC data, moved by DMA, using CMSIS DSP: what's the right way?

Hi to you all,
I've a firmware running on a NXP LPCLink2 (LPC4370: 204 Mhz Cortex M4 MCU) board which basically does this:

• Fills the ADC FIFO @40msps.
• Copies the data into memory using the built-in DMA Controller and 2 linked buffers.
• Processes one buffer while the other is being filled.

My problem is that my code is too slow, and every now and then and overwrite occurs.

Using the DMA I'm saving the ADC data, which I get in Twos complement format (Offset binary is also available), in a uint32_t buffer and try to prepare them for the CMSIS DSP function by converting the buffer into float32_t: here's where the overwrite occurs. It's worth saying that I'm currently using Floating point Software, not hardware.

The CMSIS library also accepts fractional formats like q31_t, q15_t and so on, and since I don't strictly need floating point maths I could even use these formats if that could save me precious time.
It feels like I'm missing something important about this step, that's no surprise since this is my first project on a complex MCU, any help/hint/advise would be highly appreciated and would help me in my thesis.

I'll leave here the link for the (more datailed) question I asked in the NXP forums, just in case: LPC4370: ADCHS, GPDMA and CMSIS DSP | NXP Community .

Thanks in advance!