2nd order filter transfer function normalization

In summary, the conversation discusses a guide by Texas Instruments on active filter design. It mentions equations for a second order lowpass filter, including the coefficient form and the normalized form. The conversation also addresses a mistake in the equations, specifically in the second term of the last equation, which should be a square of FSF instead of just FSF. The mistake is corrected and it is clarified that \omega_c = \sqrt{a_0}.
  • #1
bitrex
193
0
I'm looking at a guide by Texas Instruments on active filter design. In it are the following equations for a second order lowpass filter, verbatim:

The coefficient form of the denominator: [tex]s^2 + a_1s + a_0[/tex]

Normalized: [tex]P(s) = (\frac{s}{\sqrt{a_0}*\omega_c})^2 + \frac{a_1s}{a_0*\omega_c} + 1[/tex]

Substituting [tex] s = j2\pi f, \omega_c = 2\pi f_c, a_1 = \frac{1}{Q}, \sqrt{a_0} = FSF [/tex][tex]P(f) = -(\frac{f}{FSF*f_c})^2 + \frac{1}{Q}\frac{jf}{FSF*fc} + 1 [/tex]

Maybe I'm missing something obvious here, but why is it that it is not FSF^2 in the second term of the last equation, if [tex]\sqrt{a_0} = FSF[/tex]?
 
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  • #2
I made an error in the LaTeX of the first equation, I've corrected it. :redface:
 
  • #3
I think I see the problem, it must be a misprint on their part. The equation should be:

[tex]P(s) = (\frac{s}{\sqrt{a_0}*\omega_c})^2 + \frac{a_1s}{\sqrt{a_0}*\omega_c} + 1[/tex]

Also remembering that [tex] \omega_c = \sqrt{a_0} [/tex] helps.
 

1. What is a 2nd order filter transfer function?

A 2nd order filter transfer function is a mathematical representation of a filter's output in response to an input signal. It describes the relationship between the input signal and the output signal in terms of frequency.

2. Why is normalization important in 2nd order filter transfer functions?

Normalization is important in 2nd order filter transfer functions because it allows for comparison and analysis of different filter designs. By normalizing the transfer function, the effects of the filter's parameters, such as gain and frequency, can be isolated and evaluated.

3. How is normalization achieved in 2nd order filter transfer functions?

Normalization is achieved by dividing the transfer function by its maximum value. This results in a normalized transfer function with a maximum value of 1.0. This allows for easier comparison between different filter designs.

4. What is the purpose of normalizing the transfer function in decibels?

Normalizing the transfer function in decibels allows for a more intuitive understanding of the filter's frequency response. It also allows for easier comparison between different filter designs, as decibels are a commonly used unit of measurement in signal processing.

5. Can normalization affect the performance of a 2nd order filter?

No, normalization does not affect the performance of a 2nd order filter. It only changes the scale of the transfer function and does not alter the filter's behavior or characteristics. Normalization is simply a mathematical tool used for analysis and comparison purposes.

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