# Given a wide band signal with equal SNR over the whole spectrum, what should the stop band attenuation be for a decimation filter?

Imagine an incoming signal with $$60$$ dB SNR over the whole spectrum.

I want to decimate this signal by a factor of, say, $$40$$, while maintaining a $$60$$ dB SNR in the remaining frequency range.

What should the be stop band attenuation be for a decimation filter like this?

Since 39 frequency ranges will alias onto the remaining frequency range, I assume that a stop band attenuation of $$60$$ dB is wrong, and that I need to use an attenuation of $$72$$ dB instead ($$-60$$ dB -> $$0.001 /40 = 0.000025$$ -> $$-72$$ dB).

Imagine an incoming signal with 60dB SNR over the whole spectrum.

The 60 dB isn't what matters here – what matters is that you've got the same power spectral density all over your spectrum, including the 39/40 that you'll alias onto your remaining band.

So, if you don't want the SNR of that remaining band to be affected, you'd need infinite attenuation; can't have that.

You'd need to define the amount of SNR degradation you'd want to accept. If you didn't filter at all (anti-aliasing filter with a 0 dB stopband attenuation), you'd get -16 dB (that's roughly 1/40) of SNR. The fact that you had -60 dB of noise immediately stops mattering, because you're aliasing 39 times as much power onto your band as your band has.

If you want to, say, don't reduce your SNR to below 57 dB, well, you need your 16 dB stronger aliasing energy to be suppressed to -60 dB (then you have as much noise in your band as aliases, and that halves your SNR from 60 to 57 dB), that's 76 dB of stopband attenuation.

That is a lot. It's actually hard to implement that reliably and performantly. So, you'd often design a good halfband filter with high suppression and a decimation of 2, and concatenate a couple of these before finally doing the rest with one or again cascaded FIR filters.

• Ok! So that confirms that the decimation ratio impacts the amount of stop band attenuation required. I've been trying hard to find papers or tutorials on this, but they all start out with the amount of stop band attenuation they want without telling why. Once follow up question: you say that 1/40 ~ -16dB. Why is that not -32dB? Oct 8, 2020 at 22:14
• Bandwidth factors are an integral thing, so 10 log Oct 9, 2020 at 9:42