# Oversampling, decimation and Nyquist

I know this is a bit of a fundamental DSP question but I want to finally understand this and I think I need some help to finally put it all together.

Let's say I have an RF complex sampling ADC with a sample rate of 4GSPS. The device has built in decimation with a minimum decimation of 4X so my maximum capture bandwidth is 1GHz (I know slightly less to allow for aliasing guard bands etc.).

So I would put a 1GHz analog anti alias bandpass filter in front of the ADC to avoid any aliasing, and capture the full 1GHz bandwidth with the ADC. At this point I effectively have 4X oversampling and my data is output to the FPGA at 1000MSPS IQ samples.

The question I have is if I now want to break up this bandwidth in the digital domain into smaller pieces for simultaneous processing of different channels, lets say 100MHz chunks for simplicity.

Is that possible with decimation without breaking Nyquist rules? My analog anti alias filter is still 1GHz, it would seem like if I decimate by 40x to get the 100MHz bandwidth channels from the original samples that I would now have an effective sample rate of 100MSPS.

Does this work because the original data was taken with a higher sample rate and I am decimating and digital filtering non-aliased data?

I think that is the root of what i was getting at, would those filters need to be analog and limit the BW to half the final sample rate after decimation? The real question ends up being: is it possible to sample a wide-bandwidth (1GHz) and break it up digitally by digital filtering and decimating?

• $100\mathrm{MHz} \cdot 40 \ne 1\mathrm{GHz}$. ?? – TimWescott Aug 26 '20 at 17:44
• Thats what i think i am having trouble with, it seems like if i want to decimate down i need to still make sure that my analog anti-alias filter limits my bandwidth to half the sample rate. In the case of 40X decimation my analog anti-alias filter would need to be 100MHz or smaller. – Josh W Aug 26 '20 at 17:49
• You would need filters to separate your channels before decimating to 100MHz, if that's what you're asking. If so, the question isn't clear. – TimWescott Aug 26 '20 at 17:52
• Stackexchange wants a nice tidy question/answer pair. So could you please edit your question to include that bit about whether the filters need to be analog, and the bit about whether you can break your signal up digitally. Thanks. – TimWescott Aug 26 '20 at 18:05

Let's say I have an RF sampling ADC with a sample rate of 4GSPS. The device has built in decimation with a minimum decimation of 4X (and I/Q output, per comment below) so my maximum capture bandwidth is 1GHz (I know slightly less to allow for aliasing guard bands etc.).

Ultimately the ADC is sampling at 1Gsps with I/Q output, so it's producing $$2\cdot10^9$$ independent samples per second. The Nyquist rate is twice the bandwidth, so between your analog filter and the ADC's, you'd need a final bandwidth of less than 1GHz.

You're asking if you can separate the signal digitally into a number of sub-bands and then decimate. The answer is -- yes. There's no reason that your filtering has to be in the analog domain -- you can just as easily filter in the digital domain.

So -- make a filter bank in digital-land, and decimate the filter outputs. Done properly, this'll work just fine.

As an example, say you separate that 1GHz signal into 8 signals, spaced 125MHz apart and filtered to only include the middle 100MHz of each signal. Then you decimate down to 125MHz. Now you have 16 samples (8 channels, I & Q) on a 125MHz interval. That is the same number of independent samples per second that you started with, and you've reduced the signal content. So -- you have less information, and just as many samples. You're fine.

• There are refinements, like using FFT's to speed up the filter bank computations and using inphase/quadrature decimation -- but those are beyond the scope of this question. – TimWescott Aug 26 '20 at 18:16
• This ADC is converting and outputting complex data so that's where i was getting the 1GHz bandwidth from. – Josh W Aug 26 '20 at 18:29
• So to close the loop on my understanding would this scenario work without breaking any Nyquist rules. I sample at 4GSPS with complex data coming out of the converter at 1000MSPS after the 4x decimation. The only analog filtering i need is the 1GHz bandpass anti-alias filter before the ADC. Then i can decimate down to any bandwidth required in the digital world and all i need are digital anti-alias filters? – Josh W Aug 26 '20 at 19:00