Digital IIR Filters from S-Domain Transfer Functions
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From: RyanW on 3 Jul 2008 15:56 I need to design a number of digital IIR filters based upon transfer functions given in the s-domain. The filters are weighting filters (as discribed in British Standard 6841)which will be used to filter some vibration data in Excel in order to assess against a human comfort criteria. I have applied the bilinear transform to give the transfer functions in the Z-domain, however I need to expand and simplify the equations to place them in the form below: <img src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-math/6961d1214399591-digital-iir-filters-form.jpg"> Once the equations are in the above form I can get the IIR filter coefficients and implement them into my filters, however I'm not quite sure how to go about it, as my maths is not very good. Is anybody willing to help me out? The three equations are as follows; 1. <img src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-math/6960d1214399514-digital-iir-filters-1.jpg"> 2. <img src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-math/6962d1214399598-digital-iir-filters-2.jpg"> 3. <img src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-math/6963d1214399604-digital-iir-filters-3.jpg"> An example is given below; <img src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-math/6965d1214399718-digital-iir-filters-eg.jpg"> Any help would be very much appreciated!
From: SteveSmith on 3 Jul 2008 20:45 Hi Ryan, I put some examples of solving these equations in my book-- mainly because I hate them so much that I didn't want to derive them ever again. Hope it helps. Steve http://www.dspguide.com/CH33.PDF
From: robert bristow-johnson on 3 Jul 2008 21:05 On Jul 3, 3:56 pm, "RyanW" <ryan_wakel...(a)hotmail.com> wrote: > I need to design a number of digital IIR filters based upon transfer > functions given in the s-domain. The filters are weighting filters (as > discribed in British Standard 6841)which will be used to filter some > vibration data in Excel in order to assess against a human comfort > criteria. > > I have applied the bilinear transform to give the transfer functions in > the Z-domain, however I need to expand and simplify the equations to place > them in the form below: > > <img > src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-m..."> > > Once the equations are in the above form I can get the IIR filter > coefficients and implement them into my filters, however I'm not quite sure > how to go about it, as my maths is not very good. Is anybody willing to > help me out? > > The three equations are as follows; > > 1. <img > src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-m..."> > > 2. <img > src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-m..."> > > 3. <img > src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-m..."> > > An example is given below; > > <img > src="http://www.mathhelpforum.com/math-help/attachments/advanced-applied-m..."> > > Any help would be very much appreciated! i'm not sure what you need help with. it looks like you did the bilinear substitution and solved the thing in nearly every 2nd-order case. (LPF, LPF with a zero, BPF, etc.) you are substituting into s where the unit angular frequency (what all those omegas are scaled against) is Nyquist/pi. are you trying to implement a generic Butterworth/Chebyshev filter in closed form? just curious. now i dunno if you need to worry about it or not, but your bilinear transform substitution is not prewarped to compensate the frequency warping of the bilinea transform. r b-j
From: RyanW on 4 Jul 2008 05:00 Thanks Steve, Your book does help a little, and I'll take another crack at it. The main problem is that when I get to my answer, I'm not confident enough in my math ability to be sure that its right....and it needs to be 100% correct or the filter is useless. r b-j I don't really know enough on the topic to say what type of filters they are from looking at the equations. The first one is a HPF, and the other two are BPFs from my understanding. I am an acoustic consultant, and DSP is not really something I have a good knowledge of. My problem is that I have some vibration data (from train pass bys) which has been recorded in order to assess against a human comfort criteria in order to say whether it is likely to cause a disturbance to people in their homes. The data I have is raw acceleration, however the British Standard criteria I am assessing against requires frequency weighted acceleration to be used for the assessment. The filters are designed to compensate to the human response to vibration (ie. we are more sensitive to vibration at some frequencies than others). The s-domain transfer functions I have given have been taken from BS6841, and my problem is trying to impement these filters into an Excel spreadsheet to filter my data. As my maths is really bad (and I mean REALLY bad) I need help getting the equations into the right form.... does that make sense? P.S I am aware that the bilinear transform substitution is not prewarped; the angular frequency I input into the equations will be warped.
From: SteveSmith on 4 Jul 2008 12:00
Hi Ryan, It seems strange that a standard would express a vibration limit as an s-domain representation, instead of giving a frequency response curve. If you would like to send me the standard you are working to, I'll be glad to look at it and see if there is a simplier way to make the filters. IIR filters are often tricky to get working. If the problem could be turned into an FIR filter, you would have much less grief. Regards, Steve Steve.Smith "at" SpectrumSDI.com |