Electronic – Calculate cutoff frequency of a digital IIR filter

digital filter

I have an ADC with 1 kS/s and would like to append a digital anti-aliasing filter to it as I have to downsample the data to 10 Hz.

As it's on an FPGA and I don't have any multipliers, I thought the easiest way would be a recursive averaging IIR filter:

$$ y = \alpha x[n] + (1-\alpha)y[n-1]$$

But how can I calculate the cutoff frequency of such a filter depending on \$\alpha\$ and \$f_s\$?

Best Answer

With you needing an anti-alias filter for 1/100 of the bandwidth, I'd say that both your single-pole IIR (exponentially weighted moving average) and actual moving averages are out of the question; you'd need suppression in 99% of your band sufficiently high enough to mitigate aliases.

Simultaneously, you need a filter with a steep transition between pass- and stopband, relative to the original Nyquist rate.

Neither filter approaches will achieve such a steepness.

Higher-order CICs can achieve sufficient suppression through offering a very steep (potentiated sinc) impulse response, and their use in decimators (like your 100-fold decimation) is very popular due to the elegant nature of the math: you can drag the decimation stage between the integrators and the differentiators, like

-1 kHz-->(+)------·--->[↓100]---·------(+)---10 Hz->
          ^       |             |       ^
          --[z⁻¹]--             --[z⁻ⁿ]--

As you can see: extremely low component effort!

In all honesty, though: You're building a massively decimating filter, at a sample rate that is ridiculously low for FPGAs, aiming for a sampling rate that's even lower.

Do yourself a favor. Calculate a proper 1/100-band FIR filter. At these rates, you really need but a single multiplier that you use for all coefficients. And even then it doesn't need to be a fast one. That way, you gain significantly more freedom in designing your filter response.

Found it: You should really read this article of Richard Lyons, whom you can also work with over on our Signal Processing StackExchange sister site.

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