# Talk:Bilinear transform

WikiProject Systems (Rated Start-class, Mid-importance)
Start  This article has been rated as Start-Class on the project's quality scale.
Mid  This article has been rated as Mid-importance on the project's importance scale.

## 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

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)

 --> I think so. the 'T' should be inserted at appropriate locations