# Talk:Transfer function

Active discussions

## Examples

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 ${\displaystyle {\sqrt {-1}}\ }$  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)

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 ${\displaystyle x(t)}$  represents the time-dependent input signal then the corresponding output signal, ${\displaystyle y(t)}$  will be related (in the most general manner) to the input signal via a function h as, ${\displaystyle y(t)=h(x(t),t)}$ . However the relationship between output and input is usually considered to be linear with respect to ${\displaystyle x(t)}$  so that, ${\displaystyle y(t)=h(t)x(t)}$ . 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, ${\displaystyle Y(s)=H(s)X(s)}$ , where ${\displaystyle X(s)={\mathcal {L}}(x(t))}$  and where ${\displaystyle {\mathcal {L}}(.)}$  is the Laplace transform. (A similar definition applies to ${\displaystyle Y(s)}$ .) The variable s is defined on the complex plane. If it is restriced to the imaginary axis then the transform is the Fourier transform.

${\displaystyle y(t)=h(t)x(t)}$  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)

{{ping:InternetMeme}] You are not improving the article by mangling the article text in order to avoid pipes. You should not change the visible article text unless you can actually improve it. It is not acceptable to make the text less intelligible. Poor piped links are better than poor prose. SpinningSpark 14:15, 18 January 2017 (UTC)

I spent 15 minutes wading through the questionable grammar of the lead section trying to fix the links, in order to satisfy your concerns. Given the respect I showed you, how about you have some thought for my effort before reverting my edits wholesale?

For example, how does the change from "inputs and outputs of black box models" to "inputs and outputs of black box models" constitute "mangling the article"? Why did you revert that?

InternetMeme (talk) 14:19, 18 January 2017 (UTC)

That one is fine. My objection is to the numerous places you have changed the article text. Of course, an improvement to the text would be good, but in most cases you have made it either more obscure or downright wrong. MOS:NOPIPE is not a hard rule that says don't use pipes. Sometimes that is the best solution. Sometimes creating a redirect solves the problem. Changing the article text to "fix" the pipe is usually the worst thing to do. On technical articles you should take especial care when doing that. SpinningSpark 14:31, 18 January 2017 (UTC)
That's a valid concern. But then what's wrong with: "... a linear time-invariant (LTI) system with zero initial conditions ...": How does that make the lead either more obscure or downright wrong? InternetMeme (talk) 14:52, 18 January 2017 (UTC)
Um, isn't that what's already in the article? SpinningSpark 18:34, 18 January 2017 (UTC)
That's odd. It should have been, but it wasn't (it's fixed now). Progress! Now, here's the last one I'm interested in:
This sentence: "With optical imaging devices, for example, ..." doesn't seem very good. The main problem is that an "optical transfer function" is in no way analogous to an "optical imaging device". My previous sentence wasn't satisfactory to you, but I'd content that it was still a marked improvement to its current state. What would you think is the best phrasing to use?
InternetMeme (talk) 16:25, 3 February 2017 (UTC)
Frankly, the whole article is a mess. The lead is far from a summary of the actual contents of the article. It seems more concerned with making a point about optics than addressing anything actually in the body of the article. And the body of the article is going to be pretty hard going for anyone who does not already know this stuff.
Anyway, back to the specific issue, my initial take was that link the was provided simply because it provided a list of optical imaging devices. Agreed, it is a bit of an Esater Egg, but what was really needed was to find a better link (I looked, but didn't find one) rather than make any changes to the visible text. However, on looking more closely, there is a stronger connection than that and this might work: "For optical imaging devices, for example, the optical transfer function is the Fourier transform of the..." SpinningSpark 14:25, 6 February 2017 (UTC)
Yep, I like it. I'll splice it in. I agree with you about the article, and the lead too. I read it three times trying to get a handle on the topic, and gave up unenlightened, and unable to improve the content. The one thing I did try and remedy, though, was the tangled snarl of hidden links (we all know how that went).
Incedentally, I read somewhere that 90% of the time, people read only the lead of the article they access. As a result, I generally concern myself with getting lead sections in order, and making sure they touch on every key topic. If you want to give me a bullet list of what really matters, I could spend some time ordering it in a sequence conducive to learning, and fix the lead. It'd be nice to do something of more substance than removing easter eggs.
InternetMeme (talk) 16:08, 6 February 2017 (UTC)
I wouldn't advise that unless you have some understanding of the material. SpinningSpark 20:07, 6 February 2017 (UTC)
Ah, sorry. I didn't realize that you had no real understanding of this topic either. Do you know if there are any users who are experts in this topic who can put together a bullet list like I mentioned? InternetMeme (talk) 08:21, 7 February 2017 (UTC)
I didn't say I had no understanding. I was advising you not to edit a highly technical article you don't understand. I'm not particularly keen on working on this article, I have other priorities on Wikipedia, and holding someone elses hand is likely to be harder work than doing the job myself. SpinningSpark 14:37, 7 February 2017 (UTC)
You're coming across as very arrogant and dismissive. You suggest that people like you have more important things to do than work on petty stuff like this; presumably leaving you with only enough time to undo the edits of other users, thus wasting their time instead. Did it occur to you that I have better things to do than argue the finer points of MOS:NOPIPE? At the very least, you could apologise for undoing the entirety of my edit when it turns out, after much discussion, that only one of the five changes I made were erroneous. In short, you made a mistake that wasted my time, due to either a lack of understanding of MOS:NOPIPE, or a lack of interest in applying it. Please show some respect for the time and skill of people other than yourself.
Anyway, with that out of the way, what I'm suggesting is that we work on getting just the lead section right. And, no, I won't need you to hold my hand. What I'm suggesting you do is take ten minutes to make a bullet list, here on the talk page, that includes each key point of the topic. I can then work out the best order in which to explain them, assemble them grammatically, etc, and explain how they relate to each other. I certainly value your knowledge on the topic, and I'd appreciate your consideration of the fact that there is some skill required to write things in a cogent manner conducive to learning.
InternetMeme (talk) 12:20, 8 February 2017 (UTC)
Reading through the page, I am pretty happy with it. I do agree that the {{citation needed}} and {{what}} links are probably right. (I often enough don't agree.) One that I think could use some work is the optics section. Optics most often deals with intensity (square of the absolute value of the amplitude), while the rest of the article is in terms of amplitude. I believe that this difference should be pointed out. Gah4 (talk) 15:01, 10 February 2017 (UTC)
As to MOS:NOPIPE, it seems to me that there should be a distinction between links to a different form of the same term, vs. a different, often more general, form of the term. One might, for example, want an adjective when the link is to the noun. MOS:NOPIPE seems mainly concerning the case where there is a redirect, and one should use that. If there isn't, and it isn't worth making one, then the pipe should be fine. Gah4 (talk) 15:16, 10 February 2017 (UTC)

## Gain, transient behavior and stability

Hi,

I edited this section to change the symbol which denotes the angular frequency of the arbitrary input from \omega_i to \omega_0.

In more detail, first I expanded this explanation very slightly to actually give a definition of what was then (and now is after my edit was reverted) named \omega_i, so that the line should read:

A general sinusoidal input to a system of frequency {\displaystyle 2\pi \omega _{0}} may be written {\displaystyle \exp(j\omega _{0}t)}.

Then instead of writing the frequency of the general sinusoidal input as 2\pi \omega_i, I substituted 0 for i, so it would read:

The Laplace transform of a general sinusoid of unit amplitude will be {\displaystyle {\frac {1}{s-j\omega _{i}}}}. The Laplace transform of the output will be {\displaystyle {\frac {H(s)}{(s-j\omega _{0})}}} and the temporal output will be the inverse Laplace transform of that function:
{\displaystyle g(t)={\frac {e^{j\,\omega _{0}\,t}-e^{(\sigma _{P}+j\,\omega _{P})t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}}

Thus the total edit was a few words explaining what \omega actually is, and to change \omega_i to \omega_0 both there and in the subsequent paragraph.

Here is the justification for the change. The subscript i just earlier actually refers to the i'th pole, and the letter i is the indexing subscript. Here we are starting out with an arbitrary sinusoidal input of amplitude 1 and fixed phase, with frequency f=2\pi\omega, and the function is e^{j \omega t}.

This \omega did need to be decorated by the original author to indicate that it is a particular constant being newly defined in this line. (In other words, a different letter like \tau could be used). The original author decided to do this by applying the subscript i, writing e^{j\omega_i t}. This occurrence of the subscript i is unfortunate since the same letter is in fact used as a subscript elsewhere in the unrelated context. Namely, i is the indexing subscript for the poles of the transfer function.

It s not absolutely incorrect to re-use the letter i (it was a bound variable in the earlier formula). But it is analagous to writing "Consider \sum_{i=1}^5 i^2. Let i denote the value of that sum. An equation like "let i = \sum_{i=1}^5 i^2" is not *wrong* because i is bound on the right side. But it is not nice grammar.

It is better to write the arbitrary input function by saying if the frequency is 2 \pi \omega_0 the input function is e^{j \omega_0 t}.

In reverting my edit the editor seemed to comment that i should be the indexing subscript for the poles here. There is some vague sense that that could be meaningful in the sense that "now we've chosen to analyze the special case when there is only one pole, let's pretend it is the i'th pole of what we started looking at, and now let's decorate every new variable we introduce with the subscript i."

But that is like saying, when we're considering the i'th pole, that the function of multiplying by 3 is defined by x_i \mapsto 3x_i.

Again, not wrong, but it would be impossible for someone to understand it without knowing the author's intentions.

This was immediately reverted with the comment that the i is correctly used as an indexing subscript for the poles in this instance. I feel somewhat surprised that my edit was immediately reverted without anyone contacting me to ask about it.Createangelos (talk) 10:30, 21 February 2018 (UTC)

Well, either way it ended up where it should in ....in talk. This involves some pretty abstract details....variable naming.....which I think is going to limit participation. But, without totally following it, the above argument sounds good. North8000 (talk) 12:57, 21 February 2018 (UTC)