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DaveWooff10-Marble
10-Marble
November 9, 2016
Solved

Symbolic solution to filter magnitude frequency response

  • November 9, 2016
  • 1 reply
  • 4161 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

D
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 9, 2016

One thing that may help with these types of problems is to define and use the squared magnitude function instead of the magnitude (as a function of w). The squared magnitude is a rational function of w, and has no radicals. This may ease the task on the symbolic engine in some cases where the magnitude function doesn't solve symbolically. This is strictly a generic comment, as I haven't tried it on on this example. I have almost always used the squared magnitude when setting up system function calculations and the magnitude is of interest. I try to avoid radicals whenever possible in symbolics, as they have multiple solutions in general, and the algebra can get quite complex(no pun intended).

Lou

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