# 2nd order filters parameters

#### emmanuelcrova

Joined Dec 6, 2020
2
Hi everyone, my name is Emmanuel, I am from Italy, and this is my first message in this forum.

I am performing some research about digital filtering for my BS Degree thesis in Electronic and TLC Engineering.
I found myself quite stucked with the project since I can't easily figure out some important concept about the parameters of the 2nd order filters;
to give an idea of what I am talking about, the whole project is related to a digital signal generated by a ROM beeing filtered through a second order filter and its implementation on a FPGA with the use of some external components to set some parameters (I am going to use some potentiometers to set, for instance, the values of frequency and amplitude for the input wave; at this moment I want to set up the regulation of Q factor and the cutoff frequency for the filter. This last point is driving me mad, since I cannot find a correlation between the choice of some poles/zeros and those parameters)

I am going to attach a couple images reporting what I was basically searching for, this is a paper from a NASA scientist I found online. I am just wondering where these calculation come from, I tried to google them but with no success. My question is, do you have some paper or link that explain how to get these results, starting from why do you factor out 1/2 from the general equation arriving to the constraints given for certain values. I want to take those results as certain and correct, of course, but I sholud further document them in my paperwork, as you can imagine.

How can you derive beta, theta, tau and gamma, as in these 2 screen?

How can you derive tau in this third slide?

and finally, how do you get these constraints about gamma and beta?

#### Papabravo

Joined Feb 24, 2006
21,308

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#### Ian0

Joined Aug 7, 2020
10,276
I'd recommend this one:
https://www.abebooks.co.uk/book-sea...tal-signal-processing/author/lyons-richard-g/
I took a course in DSP as part of my degree back in 1986, when the software could only be run on the university mainframe. I understood enough to pass my exams, but as it wasn't really much practical use, didn't do much with it.
Now that 32-bit processors are available for £1 or less, I revisited the topic, and found this book the best of the ones I'd kept since university or acquired since.
There is a good explanation of the z-transform, how it compares to the Laplace transform, and how the IIR coefficients are derived from the prototype analogue filter by impulse invariance method or Bilinear transform.
And you can get a used copy for a tenner.

#### emmanuelcrova

Joined Dec 6, 2020
2
First of all thanks for you effort to help me @Papabravo , I will cite you in my thesis since you solved my biggest problem at the moment!

I'm sure the same derivation is present in similar textbooks, and articles on the internet. I think you may have OD'ed on videos.
If you like I can recreate the steps.
I tried to google almost anything in search of these steps but nothing seems to be driving in that direction as far as I managed to discover.

I will update this thread if I need further explanation.

Cheers
Emmanuel

#### Papabravo

Joined Feb 24, 2006
21,308
I'd recommend this one:
https://www.abebooks.co.uk/book-sea...tal-signal-processing/author/lyons-richard-g/
I took a course in DSP as part of my degree back in 1986, when the software could only be run on the university mainframe. I understood enough to pass my exams, but as it wasn't really much practical use, didn't do much with it.
Now that 32-bit processors are available for £1 or less, I revisited the topic, and found this book the best of the ones I'd kept since university or acquired since.
There is a good explanation of the z-transform, how it compares to the Laplace transform, and how the IIR coefficients are derived from the prototype analogue filter by impulse invariance method or Bilinear transform.
And you can get a used copy for a tenner.
Understanding what is happening in the digital (sampled) domain is a good deal easier when you can refer back to the analog fundamentals.

#### Ian0

Joined Aug 7, 2020
10,276
Understanding what is happening in the digital (sampled) domain is a good deal easier when you can refer back to the analog fundamentals.
Absolutely - that's why I like the way Lyons starts with the Laplace transform.

Do you think Van Valkenburg is particularly good? I'm all for buying a good book! There's still a few DSP topics in which I could do with an alternative explanation.

I'd also recommend Basic Linear Design by Analog Devices, which is a free download
https://www.analog.com/media/en/tra...andbooks/Basic-Linear-Design/Introduction.pdf
Chapter 8 on analogue filters is particularly good, and includes all the tables for Butterworth, Bessel, Chebyshev, Gaussian etc.

#### Papabravo

Joined Feb 24, 2006
21,308
Absolutely - that's why I like the way Lyons starts with the Laplace transform.

Do you think Van Valkenburg is particularly good? I'm all for buying a good book! There's still a few DSP topics in which I could do with an alternative explanation.

I'd also recommend Basic Linear Design by Analog Devices, which is a free download
https://www.analog.com/media/en/tra...andbooks/Basic-Linear-Design/Introduction.pdf
Chapter 8 on analogue filters is particularly good, and includes all the tables for Butterworth, Bessel, Chebyshev, Gaussian etc.
I do like his expository writing style which gets to the fundamental principles in a very direct way. He did not live long enough to update the 1982 version, but he is listed as a coauthor with Rolf Schaumann on a 2001 revision re-titled "Design of Analog Filters". The newer text eliminates some of the methods more appropriate to the pre-personal computer era, which is a shame. Geffe's algorithm was a notable casualty.