# Talk:Transfer function

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## Examples

How about a few examples for transfer functions please? mickpc

i is the imaginary unit in general mathematical notation, but not in electrical engineering, where i is already taken for denoting current. Hence electrical engineers tend to use j as the notation for the imaginary unit. Which is correct depends on the field being discussed. Transfer functions are generally an engineering field (contrast with Greens functions) -- The Anome

Makes sense. --AxelBoldt

Removed "digital" from (digital) signal processing - transfer functions are continuous as well as digital! User:Extro

Ditto (someone had put "(digital)" back without explanation). BTW, shouldn't this page be merged with frequency response? Jorge Stolfi 03:25, 25 Mar 2004 (UTC)

I think they should stay separate. They are certainly related, but not the same thing. - Omegatron 16:18, Feb 7, 2005 (UTC)

## LTI systems only?

So the way I've always thought of it, there is a transfer function, which is related to the frequency response of a linear, time-invariant system, and there is the transfer characteristic, which is related to things like non-linear amplifiers, clipping, distortion, companding, transistor quiescent points, tape saturation, etc. A relation tying input voltage to output voltage. But now I'm thinking "aren't they the same thing?" I know that some people refer to my characteristic as a function.[1] More accurately, you could divide it into linear transfer functions and non-linear transfer functions, but they are both variations of the same thing. is this right? should we have two separate sections? - Omegatron 16:18, Feb 7, 2005 (UTC)

I never thought of transfer functions only being for LTI systems. Google turns up quite a few hits for "nonlinear transfer function", so I don't think it's just me who includes non-LTI systems. -- Oarih 21:17, 7 Feb 2005 (UTC)
I've been using the phrase "locally linear" for signals that have otherwise LTI processes applied to them but with differing amounts over time (like audio level compression). Is this valid? - Omegatron 19:01, Feb 8, 2005 (UTC)
"transfer function" in the context of this article, means the ratio of the Laplace (or Fourier or Z) transform of the output Y to the Laplace transform of the input X. this has meaning only for LTI systems. general input/output transfer characteristic for linear or non-linear systems is another topic. try looking up Volterra series about that. i would suggest asking questions about stuff like this at the USENET newsgroup, comp.dsp . i don't think that wiki talk pages is the best place you could go for that. r b-j 20:43, 8 Feb 2005 (UTC)
I'm asking because it should be included in the article. When I find out myself I'll add it. - Omegatron 21:36, Feb 8, 2005 (UTC)
if you're saying that there should be some treatment of non-linear transfer characteristics of non-linear systems in the article, i strongly disagree. to maybe accentuate the difference between LTI transfer functions and this non-linear stuff, maybe that's okay. but let's not "crap this up" with something much bigger than it is. another cliche is "opening a can of worms". r b-j 18:46, 9 Feb 2005 (UTC)
If the phrase "transfer function" is used for both, then we cover both. If so, we can move the linear stuff to the LTI system theory page or something if we need to for some reason. - Omegatron 21:56, Feb 9, 2005 (UTC)
it is not common use of the term "transfer function" to apply it to non-linear systems. that's why i believe that adding that variation of meaning to the wiki article only contributes to the lack of clarity of definition. it does not add clarity. r b-j 17:11, 10 Feb 2005 (UTC)
Results 1 - 10 of about 154,000 for nonlinear OR non-linear "transfer function".
Results 1 - 10 of about 318,000 for linear "transfer function".
Results 1 - 10 of about 5,820 for "nonlinear transfer function" OR "non-linear transfer function".
Results 1 - 10 of about 9,960 for "linear transfer function".
And the "linear" results find articles with "non-linear" as well, so it's roughly the same amount for each. - Omegatron 17:48, Feb 10, 2005 (UTC)
common fallacy. put it in quotes so you have context.
Results 1 - 10 of about 3,820 for "nonlinear transfer function"
Results 1 - 10 of about 720,000 for "transfer function"
i'm not saying that the usage isn't there, but it is not what we mean in the electrical engineering discipline. "transfer function" is an LTI concept almost all of the time and it changes the lexicon and confuses others to confuse the two concepts. r b-j 22:11, 10 Feb 2005 (UTC)
Those were in quotes. Try them yourself. "nonlinear transfer function" is used roughly equally to "linear transfer function". Yes, I would expect that "transfer function" would come up a lot more times than either of the phrases that contain it.
Like I said, I originally thought transfer function only meant linear, and "transfer characteristic" was what you used when referring to a nonlinear kind of thing, like input current to output current of a BJT. But now that I look into it, "transfer function" is a term that applies to both. Just like we don't use j for the imaginary number on wikipedia, we also shouldn't limit our definitions to the type used in electronics courses. Non-linear electronics, optical transfer functions, and whatever else there is should be covered. - Omegatron 23:54, Feb 10, 2005 (UTC)
we do use j for $\sqrt{-1} \$ in wikipedia. physics and math articles will have i most of the time but electrical engineering ones will have j most of the time. just like there is english english and american english. above you pointed out the relationship of "transfer function" to "frequency response" and they are not exactly the same thing but so closely related that they are often used interchangably. but there is no "frequency response" for nonlinear systems. not without fudging the concept. it does not make sense since a nonlinear system creates frequencies that did not go in to the input. what is the gain at those frequencies? please, let's not crap this up. let's do articles about nonlinear stuff (Volterra series would be a good one), but let's not add issues regarding nonlinearities to what is, in the discipline, a linear time-invariant system concept. r b-j 02:55, 11 Feb 2005 (UTC)
So start a transfer characteristic article and include a disambig notice at the top of each page? - Omegatron 19:56, Feb 26, 2005 (UTC)
if you want, it's fine by me. (it ain't my encyclopedia, i can only rely on persuasion and popular concensus to resolve these things in the way i think they should be.) r b-j 01:13, 27 Feb 2005 (UTC)

Omegatron 19:42, 9 April 2007 (UTC)

Just to let you guys know most transfer functions of non-linear components can be converted using the small signal model. Unfortunately, that article skims over the fact that anything (Not just electronics) can be converted. I have books that describe this if you would like more information. Adam Y (talk) 03:02, 17 November 2007 (UTC)

## Clearing Up Red Links

It would help if common LTI transfer functions (integrator,first order lag, second order lag, lead/lag) were listed here, as they are frequently needed for control system descriptions (e.g. autopilots, etc.). Gordon Vigurs 19:27, 29 May 2006 (UTC)

## Mason's Gain Theorem

I was looking for a discussion of Mason's Gain theorem, commonly used in circuit theory and control theory to quickly calculate transfer functions of complicated networks from their signal flow graphs, and was surprised to find that Wikipedia does not have an article on it. It seems that the Transfer Function page is the most appropriate place to put it. If no one is already thinking of adding it, I will put in a paragraph on it in the next few days. Mraj 15:22, 13 July 2006 (UTC)mraj

go fer it! i never heard of Mason's Gain Theorem and would be happy for an article on it. r b-j 05:55, 14 July 2006 (UTC)
I have never heard of this Mason's gain theorem either, so it would be interesting to see what its about! 8-)--Light current 13:34, 14 July 2006 (UTC)
Check this: Mason's rule Tnae 11:11, 22 August 2006 (UTC)
So its actually Masons Rule! AHHH! We dont need a new page then--Light current 15:56, 22 August 2006 (UTC)
I added redirects for "Mason's theorem" and "Mason's gain theorem" so there shouldn't be a problem any more. PAR 17:28, 22 August 2006 (UTC)

## Meaning in game theory

The term "transfer function" is also used in Game theory, maybe a disambiguation pointer should be added. —Preceding unsigned comment added by 169.237.10.220 (talk) 23:26, 11 June 2008 (UTC)

If there's an appropriate article to link to, then that may be appropriate. Oli Filth(talk) 23:33, 11 June 2008 (UTC)
This term appears in the article mechanism design. Mct mht (talk) 05:43, 25 April 2013 (UTC)

## Transfer Functions for Dummies

There is a lot of great info in this article, but for a tyro (and I speak from experience), there's a great big brick wall right at the beginning. What the heck does "s" represent and what is its form? I have read half a dozen Wiki articles and others from around the web, and bought two text books. Still it is not clear.

It is clear (finally, at long last) is that the domain of a transfer function includes pure imaginary numbers s = j*ω, where ω is a frequency expressed in radians per unit-time (seconds). It is also clear that the range of a transfer function consists of complex numbers whose modulus (aka absolute value, aka L2 norm) represents amplitude or gain, and whose argument (aka angle) represents phase. What is not at all clear to me is whether the domain also includes numbers that are not pure imaginary, and if so what they represent. Everywhere I go I see the formula s = σ + j*ω, without explanation. It seemed to me that σ represents phase-shift in the domain. I expected that when I used a complex number σ + j*ω with a non-zero σ as input to a transfer function, the amplitude output would be unchanged and the phase output would differ from the output for j*ω by σ. Not the case. If I make σ much different from 0, I get nonsense results. Test cases are formulae for audio crossovers and closed box loudspeaker systems whose responses are well-known to me. I know that a loudspeaker's frequency response in SPL does not depend on the phase of the signal.

So my question, which I think this Wiki page should clarify, is what specifically is the domain of a transfer function? Does it include only numbers of the form s = j*ω? If it includes numbers of the form s = σ + j*ω for non-zero σ, what do the those inputs and the associated responses represent?

[I posted essentially the same question on the talk page for "frequency domain."]

Jive Dadson (talk) 08:45, 17 December 2010 (UTC)

s is the complex-frequency domain, given by the Laplace transform. The does indeed correspond to pure complex sinusoids, i.e. the Fourier transform. The σ denotes exponential decay/growth. Oli Filth(talk|contribs) 12:12, 17 December 2010 (UTC)

## A Whimsical Definition

The introductory paragraph is hard for non-specialists to understand. Looking at the article's history someone has tried to improve it by appending a plainly worded sentence. I suggest the following be considered. Of course, the rest of the article can continue with technical details but the beginning should be written for those who just want to get to the nut of the idea. So here goes:

Transfer functions are used (mostly by engineers) to relate an output signal to an input signal. If $x(t)$ represents the time-dependent input signal then the corresponding output signal, $y(t)$ will be related (in the most general manner) to the input signal via a function h as, $y(t) = h(x(t),t)$. However the relationship between output and input is usually considered to be linear with respect to $x(t)$ so that, $y(t) = h(t)x(t)$. In many fields, such as communications, one usually works in the frequency domain so that x and y are replaced by their Laplace transforms and we have instead, $Y(s) = H(s)X(s)$, where $X(s) = \mathcal{L}(x(t))$ and where $\mathcal{L}(.)$ is the Laplace transform. (A similar definition applies to $Y(s)$.) The variable s is defined on the complex plane. If it is restriced to the imaginary axis then the transform is the Fourier transform.

$y(t) = h(t)x(t)$ is not correct, which points out a glaring omission. The main body of the article does not mention convolution or the convolution theorem.--Bob K (talk) 14:26, 9 April 2013 (UTC)