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I am working on a project that needs a lowpass filter with a -7.5 dB/Octave rolloff. This is somewhere between the rolloff of a one-pole and two-pole filter, so I wanted to ask what some standard methods are for generating this type of response.

Is it appropriate to have a two pole filter (or two one-pole filters in series) and do a wet/dry mix? Or would is there a more robust method I should try instead?

I am hoping I am able to accomplish this using a simple IIR filter like a biquad.

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  • $\begingroup$ To my knowledge, anything not a multiple of 20dB/dec is not done via a single, standard stage, but with approximations from multiple stages (e.g. 3dB/oct, or pink noise filters). I have not done something "exotic" like this in the digital domain, but I suspect it would be similar to the analog counterpart. If I got it wrong, please correct me, anyone. $\endgroup$ – a concerned citizen Jan 26 '18 at 8:10
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As far as I know there is no standard procedure for designing such filters. A simple approach would be to design a standard second-order low pass filter with the given cut-off frequency (such as a Butterworth filter), and then re-design the numerator coefficients such that the total filter approximates the desired magnitude roll-off. This is basically a (linear phase) FIR filter design problem, which is very straightforward to solve.

Let $M(\omega)$ be the desired magnitude response. On a log-log scale, $M(\omega)$ is piecewise linear ($0$ dB in the passband, linear decay $r$ dB/decade in the stopband). Furthermore, let $A(\omega)$ be the frequency response of the denominator coefficients of the original second-order filter. Then the desired response for the new numerator coefficients is given by

$$D(\omega)=M(\omega)\cdot |A(\omega)|\cdot e^{-j\omega}\tag{1}$$

where the exponential term is a linear phase term corresponding to the delay of a second-order linear phase FIR filter ($1$ sample).

Any method can be used to design that $3$-tap FIR filter (the numerator coefficients). I used a least squares design and the result is shown in the figure below (with the original Butterworth filter in blue, the desired magnitude response in red, and the modified Butterworth filter with re-designed numerator coefficients in green):

enter image description here

If you need a better approximation you can increase the order of the numerator (without changing the second-order denominator coefficients). So you'd get a biquad concatenated with a (short) FIR filter. The figure below shows the approximation with a $10^{th}$-order numerator polynomial and the second-order Butterworth denominator coefficients (in green):

enter image description here

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