Talk:Bilinear transform

Question

s = (2/T) * ((1 - z) / (1 + z))

I take it that z is the "transfer function in the analog domain"? What is T? Is z a complex function? s is the transformed transfer function? What are the arguments of these functions? --AxelBoldt

Do the changes answer your questions? -- Ap

Yes, it is a lot clearer (I made some cosmetic changes now that I understand what is going on). Articles about "stable filter" and "transfer function" would really be useful for context, though. --AxelBoldt

Formula

What is the correct formula for this transform? The current formula does not seem to agree with the "Background" paragraph. --AxelBoldt

I found two essentially equivalent definitions:

s = (2/T) * ((1 - z^-1) / (1 + z^-1))

and

s = (2/T) * ((z - 1) / (z + 1)).

The first one is from Feedback control of dynamic systems by Franklin et al. The second is from the help text of the bilinear operation in Octave.

What exactly do you mean by does not seem to agree? -- Ap

In the Background paragraph of the main article, it is explained that the transform maps complex numbers from the interior of the unit circle to the negative half plane. But I think the above transform maps complex numbers from the interior of the unit circle to the positive half plane. Or am I confused? --AxelBoldt

Oops, yes I am confused indeed. The above is correct, and gives values in the negative half plane. --AxelBoldt

True, both equations are the same, but I find it easier to convert the first form (above) into executable code. The second one requires a little algebra to make it useful in the real world, IMHO. Am I wrong? Madhu 23:57, 4 July 2006 (UTC)

---

Multiply the first equation by ${\frac {z}{z}}$ and you get the second equation. They are the same. Cburnett 05:31, 28 Nov 2004 (UTC)

"When the Laplace transform of a discrete-time signal with each element of the discrete-time sequence attached to an appropriately delayed unit impulse, the result is precisely the Z transform of the discrete-time sequence with the substitution......"

-There is something wrong grammatically here. Is there a verb missing?

"is" is the verb ("the result" is the subject), but it is poorly worded. for some reason i think it's my fault. i'll work on fixing it. r b-j 23:59, 8 December 2005 (UTC)

I understand how your first approximation z=((1+sT/2)/(1-sT/2) arises and leads to the bilinear transform, but the infinite series you show for ln(z) really puzzles me. I can't concieve of any Taylor series expansion to produce it. Can you shed some light on where it comes from? Thanks. [SW] 155.104.37.17 17:19, 30 January 2007 (UTC)

See Logarithm#Series_for_calculating_the_natural_logarithm. Oli Filth 12:54, 15 April 2007 (UTC)

Shouldn't there be a short definition for T somewhere, perhaps a link to http://en.wikipedia.org/wiki/Sampling_(signal_processing)#Sampling_interval ? It took me an extremely long time to understand this page because I had assumed that T stood for the sampling frequency, not the sampling period (reciprocal of sampling frequency). DavidCreswick 05:32, 20 October 2007 (UTC)

T is already defined in the article as the sample time. Oli Filth^(talk) 11:56, 20 October 2007 (UTC)

Question

In the Frequency wrapping section, the T is left out in the formula's (i.e. exp(jwT)-> exp(jw)), is that ok?? Arthurdenhaan (talk) 13:55, 1 November 2011 (UTC)

 --> I think so. the 'T' should be inserted at appropriate locations
     in sentences and equation images. So, I corrected them.

http://en.wikipedia.org/wiki/Discretization — Preceding unsigned comment added by 128.39.12.150 (talk) 12:03, 27 June 2011 (UTC)

Reversion of 952723881

I have reverted 952723881 as the maths in the revision was incorrect. I believe the edit was in good faith. Frankplow (talk) 10:55, 14 August 2022 (UTC)

I have rewritten the section as, although much of the working was incorrect, the results were partially correct, useful and not shown elsewhere on the page. Frankplow (talk) 11:11, 16 August 2022 (UTC)

Derivation of frequency warping

The warping equation in Digital signal processing, John G Proakis, $\Omega ={\frac {2}{T}}\tan \left({\frac {\omega }{2}}\right)$ is different than the derived one in the article. Any insights into this? Spartie19 (talk) 12:50, 17 June 2024 (UTC)

The units don't work in that formula; it is probably wrong. Constant314 (talk) 13:27, 17 June 2024 (UTC)

Hi Constant314. Always good to have your input. That's concerning since I have a module currently teaching it with that formula. From what differences I can spot it seems they used $z=re^{j\omega }$ instead of $z=re^{j\omega T}$ in the book. Also some courses did use the formula in the book from what I could find online. Which implies that either I'm missing something or there is a bunch of institutions teaching the wrong formula. Spartie19 (talk) 13:49, 17 June 2024 (UTC)

I find this lot in academics where units are simply ignored as an unnecessary distraction. Also, T=1 is sometimes assumed and then forgotten. You see this especially in earlier works on digital signal processing. Once it is in the computer, everything is just a number. ω is assumed to be normallized tto range between +/- 0.5. Constant314 (talk) 15:47, 17 June 2024 (UTC)

That definitely seems to be the case. Thank you for clearing up my confusion and for the heads up on normalizing $\omega$, it's something I would have probably overlooked. Spartie19 (talk) 16:37, 17 June 2024 (UTC)