Third Order High Pass Filter (Report)

[NascentOxygen]

Calculating and plotting the response of an 8 pole Butterworth filter to the same scale as the left hand image of figure 8.13 gives this result

Electrician, are you able to get your software to replicate Fig 8.14 of that document to show a comparison of the step response of your Butterworth LPF with that of a comparable Bessel?

 

[The Electrician]

Electrician, are you able to get your software to replicate Fig 8.14 of that document to show a comparison of the step response of your Butterworth LPF with that of a comparable Bessel?

I can do this, but in the process of doing so I noticed something a little off about the Bessel frequency response in figure 8.13 (left image). The response shown apparently isn't for an 8 pole Bessel filter. Here's an image showing the response of a 1 pole (red), 2 pole (magenta), 3 pole (blue) and 8 pole (Green) Bessel filter. What is shown in figure 8.13 is either a 1 pole or a 2 pole filter; it seems clear that it isn't an 8 pole filter.

Here's the step response of an 8 pole Butterworth filter:

This is the theoretically perfect step response of an 8 pole Butterworth filter. It doesn't exactly match the step response shown in figure 8.14, but that is apparently a real hardware filter and it doesn't have an accurate 8 pole Butterworth frequency response, so we shouldn't expect the step response to be exactly accurate either.

Here is the 8 pole Butterworth step response and the 2 pole Bessel step response together

. I show the 2 pole Bessel because that more or less matches the frequency response of the Bessel in figure 8.13:

 

[NascentOxygen]

Electrician, if you scaled your 8 pole Bessel's frequency so its -3dB point co-incided with f=1.0 then you'd have a curve matching that in Fig 8.13, I think. It's difficult to estimate filter order in the passband without reference to the high-frequency gain rolloff or output phase range. Remarkably, it looks like a scale factor of exactly ⅓ may be what's needed, that will certainly be close enough for my purposes, anyway. The Analog Devices filters all co-incide at -3dB and all appear to rolloff at similar rates where f >> fo, so I'd say they seem fine.

 

[The Electrician]

OK. I tracked down the problem with the Bessel response. I was using Bessel poles normalized to have unit group delay.

Here is the response of the 8 pole Butterworth and 8 pole Bessel using standard Bessel poles:

And here is the step response for the two:

These are the theoretically ideal curves. The frequency response of the Butterworth filter shows no ripple as stated on page 8.21, and the step response of the Bessel shows only a tiny bit of overshoot as the ideal should have. The Gaussian filter is the one with no step response overshoot.

Figure 8.15 on page 8.31 shows the Butterworth response and it appears to be calculated rather than measured. The Butterworth responses have no ripple.

 

[LvW]

Regarding overshoot of a second-order function, here is the formula (in %):

γ=100*exp[-π/SQRT(4Qp²-1)]

As you can see, the overshoot depends on the pole-Q and is zero for Qp=o.5 only.

 

[NascentOxygen]

OK. I tracked down the problem with the Bessel response. I was using Bessel poles normalized to have unit group delay.

Here is the response of the 8 pole Butterworth and 8 pole Bessel using standard Bessel poles:

And here is the step response for the two:

These are the theoretically ideal curves. The frequency response of the Butterworth filter shows no ripple as stated on page 8.21, and the step response of the Bessel shows only a tiny bit of overshoot as the ideal should have. The Gaussian filter is the one with no step response overshoot.

Thank you for showing the plots are all in agreement with the Analog Devices plots, apart from their misleading Butterworth. It is remarkable to see the alacrity with which the more heavily damped 8 pole Bessell responds to a step input, in comparison with the ponderous though less damped 8 pole Butterworth. This is in stark contrast with the performance of second-order filters where it's the Butterworth that responds smartly to a step input but the Bessell is seen to become progressively slower.

 

[LvW]

Thank you for showing the plots are all in agreement with the Analog Devices plots, apart from their misleading Butterworth. .

Some days ago, I mentioned my opinion that the "misleading" Butterworth response could be a measured one. Meanwhile, I do not think this is true. We should not overlook the fact that we speak about an 8th-order (simulated) response.
There are two possible sources of error:
* Perhaps the simulation was based on real opam models (limited GBW)
* For an 8th-order filter just a small deviation of the filter coefficients from their ideal values may lead to the observed deviation.
(a1...a8: 5.126...13.138... 21.848... 25.691... 21.848... 13.138... 5.126... 1).

By the way, at the beginning of chapter 8.4 (Analog Devices) they confirm that Butterworth is identical to max. flat.

 

[NascentOxygen]

Analog Devices are publishers of great tutorial-style material, as evidenced by the helpful detail in the chapter we have been referred to by rude man. I said their graph is misleading simply because they include a plot which shows a departure from theory but they fail to acknowledge this discrepancy and don't offer an explanation. Possibly it is the dominant pole of an amplifier, but they need to explain this.

 

[LvW]

Possibly it is the dominant pole of an amplifier, but they need to explain this.

Yes - i am of the same opinion.