On 23 June 2014 19:43, Andrew Simper <a...@cytomic.com> wrote: > On 23 June 2014 17:11, Ivan Cohen <ivan.co...@orosys.fr> wrote: >> Hello everybody ! >> >> I may be able to clarify a little the confusion here... > > Thanks Ivan for your great email contribution. I will only reply to > the one and only correction / clarification to what I have posted > previously. > > >> The bilinear transform is a tool allowing people to get from a Laplace >> transfer function H(s) an equivalent Z transfer function H(z). It is based >> on the fact that z is precisely equivalent to exp(sT) with T the sampling >> period. So, s equals 1/T * ln(z), which may be approximated with its first >> Taylor series term, 2/T * (z - 1) / (z + 1). > > Thanks for pointing this out clearly! In derivations I've seen of the > bi-linear transform they use trapezoidal integration to get there, > taking the first taylor series expansion of 1/T ln(z) makes more sense > to distinguish it from trapezoidal integration, even though it results > in the same thing. I would love to see the original derivation by > Tustin, in particular where the idea came from, I've searched but > can't find it. In addition Wikipedia / Oppenheim doesn't help here > either: > "...where T is the numerical integration step size of the trapezoidal > rule used in the bilinear transform derivation.[1]" > [1] Oppenheim, Alan (2010). Discrete Time Signal Processing Third > Edition. Upper Saddle River, NJ: Pearson Higher Education, Inc. p. > 504. ISBN 978-0-13-198842-2. > > Again thanks for your post, I just hope people not only read it, but > take the time to understand it.
Ok, I'm still stumped here. Can someone please show me a reference to how the bi-linear transform is created without using trapezoidal integration? I found this: http://www.josiahland.com/archives/1178 , but trapezoidal integration is used in the middle section (a - b) * (f(a) + f(b))/2 Likewise this: "http://www.d-filter.ece.uvic.ca/SupMaterials/Slides/DSP-Ch11-S6,7.pdf"; , also uses trapezoidal integration on frame 6 slide 7. And again here: http://donalprice.com/dsp/bilinear-transform/ trapezoidal used in the derivation and so on. Can anyone help out here? -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
