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DaveWooff10-Marble
10-Marble
November 9, 2016
Solved

Symbolic solution to filter magnitude frequency response

  • November 9, 2016
  • 1 reply
  • 4193 views

Hi all,

please see attached.

I am attempting to derive what I thought would be a simple enough equation to describe the maxima and/or minima of a "filter" function magnitude response.  The example is a simplified version of my original problem and although in this case the answer is very obvious (w = 1), my original problem did not have so obvious an answer,  And also, although I can easily plot the response, I would like to establish the design equations to allow me to work backwards from a requirement to a solution in equation form.

In the worksheet, the symbolic solution seems to fail.  I specify the function H(s) and then substitute s = 1i.w to obtain continuous frequency response.  I suspect the problem is complicated by the use of the |magnitude| function which might make symbolic solution difficult - that's my guess at why this doesn't seem to work.  If there is a better way then please let me know!  I have been staring at this on and off for a few days so might be missing the obvious.

Thanks in advance,

D

Best answer by -MFra-

Now, it seems to me that everything is OK.

fdt1.jpg

1 reply

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-MFra-21-Topaz II
21-Topaz II
November 9, 2016

fdt.jpg

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DaveWooff10-MarbleAuthor
10-Marble
November 9, 2016

thank you for your reply F.M.

The resulting values are complex.  Since I am evaluating H(s) on the imaginary axis of the s-plane, I expect real values of w.  If I substitute the example values of wn (1) and Q (4) shown in the worksheet, I get [0.125i, 0.992+0.125i, -0.992_0.125i].  The real positive value of w which produces a peak in H(jw) is actually ~0.984.

L
LouP12-Amethyst
12-Amethyst
November 10, 2016

Another comment triggered by some of your earlier posts on differentiating the magnitude function. The magnitude function directly (as a function of w) has cusps where the magnitude has simple zeroes, and the derivative does not exist at these points, hence symbolic operations may be problematic. Conversely, the square of the magnitude (see example in preceding post) is an analytic function, and the derivative is well defined everywhere (except at poles of course).

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