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Originally i asked this question on DSP SE yesterday but untill now couldn't get even a single comment so I am reproducing my question here as bigger audience is available here and also my question is also related to other subjects of EE such as circuit analysis and electronics

While studying the topic of filter design, I came across the following terms:

  1. Factored form

  2. SOS (second order sections) form

  3. Rational form

  4. Transfer function form

What is the difference between all these terms?

According to my understanding, the first two terms are mutually synonymous and are implemented in the same way in MATLAB, and the last two terms are also mutually synonymous and are implemented in same way in MATLAB.

Is my understanding valid?

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    \$\begingroup\$ Whose terminology? Quote the sources properly and then we may all see what is what. \$\endgroup\$ – Solar Mike Apr 18 at 17:19
  • \$\begingroup\$ I suspect the reason why you didn't get any comments on dsp.ee is because the question, as you asked it, strongly suggests you are lacking some of the basics, and I mean very basic notions, which implies you need to learn these, first, instead of pursuing a very large topic such as "filters". While asking on a (D)SP-dedicated site. It also sounds like a homework question (it may not be), and these are usually either ignored (at best) or downvoted, if there is no specific tag (homework) attached to the question, or if there is no effort shown in attempting to solve the problem. \$\endgroup\$ – a concerned citizen Apr 19 at 7:21
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A rational number is a number that can be written as a fraction, a / b, where a and b are integers. This is not a synonym of a transfer function yet all numbers used in such are rational because of truncation, rounding, fractions or integers.

Irrational numbers have decimal expansions that neither terminate nor become periodic but if a transfer function only has a limited number of significant figures due to tolerances, irrational numbers can be used but truncated to the number of "sig figs".

Using an infinite series value in a transfer function would be not only irrational in the logical sense but also insignificant in the mathematical sense.

So they are not synonyms but commonly used together.


2nd Order filters are common in RLC filters and active filters and may be represented in a number of ways including Factored Form.

enter image description here REF

Higher order filters may be cascaded to create different responses by cascading LCLCLC or 2nd order Active filters.

e.g. 4th order Chebychev. 10kHz Active and 50 Ohm passive
The transfer functions of filters are often made from 2nd order sections LC or their equivalent RC*(-RC) with negative feedback. But they can also be any odd order as well or any feasible length of polynomial order or slope of rolloff.

So they are not synonyms but it is common to convert a polynomial into factored numbers to visualize the pole locations of a filter or pole/zero transfer functions.

enter image description here

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