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I am trying to get a general sense of how many multiplication operations are needed for a 4th-order Butterworth filter (IIR filter) with a rational sampling rate change. I am excluding any multiplication by 0's.

For example, for an upsampler of order 5 and downsampler of order 7 with a 99-tap FIR filter in between, the number of multiplication operations needed would be of order

$ \frac{99N}{7} $

Where $N$ is the sampling rate (per second). This is since an upsampler would effective have 4 0's for every 5 upsampled from the sampling rate of $N$.

I would like to know what the logic would be for the case of a 4th-order lowpass Butterworth filter. There are cascading designs, but there are also the canonical transpose and direct forms of implementing a Butterworth filter. What would be the difference in multiply operations (excluding multiply by 0) in each of these designs? Also, how can I calculate this? I have read somewhere essentially any IIR filter uses 6 multiplies per 2 orders of the IIR filter (4th order in this case, with $\frac{4}{2} = 2$, thus $6 * \frac{4}{2} = 12$ multiplies per second of input sample), but this is not the correct logic in my opinion.

What is the best way to think about this? I am pretty much self-teaching myself digital signal processing, so I am lost as to how to approach this.

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