# Talk:Linear filter

To-do list for Linear filter:

## Circuit diagrams

Wouldn't it be nice to have some circuit diagrams for each type of filter?

The high-pass filter entry describes the 'simplest' circuit, but perhaps someone could illustrate it?

I can make circuit diagrams for a few cases if we want. Which cases to show, though? First-order LP HP BP BS? I made an article for Butterworth filter. Chebyshev filter already has a short article. I'm not sure if they should each have their own article or if they should just be combined into this one. How much detail should we go into? - Omegatron

## This could use expansion

Each type of filter should have their own article with equations, schematics, pole-zero plots, etc. But I am too lazy/busy to do it.  :-) I started Butterworth filter. We need articles for:

- Omegatron 14:55, Jul 27, 2004 (UTC)

## Digital signal processing

I started some cleanup. The section looked rather out-of-date and did describe only one implementation method of one algorithm. The specialised article Digital filter should give the details, but this will some work. -- Pjacobi 16:44, 5 Aug 2004 (UTC)

## No treatment of complex analysis for filter design

This is a very important field that has been completely missed out from this article. Also, the z-transform for discrete-time filters.

## Ideal filters

Ok, we can't filter something ideally (rectangular frequency response) because the corresponding convolution in the time domain would require prediction of the future, right? But what if we already know the entire pre-recorded signal? (Like a piece of audio surrounded by zeros to infinity) Then can we apply an ideal filter? This occurred to me because of FFT filtering, but that is not perfectly ideal. It is limited to the effective bandwidth of the FFT bins, right? - Omegatron 18:24, Jan 26, 2005 (UTC)

Ideal filtering would require infinite (and no less) length to filter upon (both data and the time-domain sinc function). You can better approximate it by lengthening the filter but at some point your coefficients will be less than the quantization resulting in filtering of zeros. Cburnett 20:06, 26 Jan 2005 (UTC)
I was just talking mathematically/theoretically. So it can be done realistically, too? Neat. - Omegatron 20:14, Jan 26, 2005 (UTC)

## The title is unfortunate

Filters are not confined to electronics. For instance, one might have a software filter that does the same thing. This article should have a title that reflects the broad field of signal processing. --Smack (talk) 05:11, 4 May 2005 (UTC)

Well, that is why we have the more generic Category:Filters to accommodate things like Mail filter. I suppose if we get more software-filter related articles we could create a separate category for them too.--Hooperbloob 05:17, 4 May 2005 (UTC)

Sorry if I was unclear, but that's not at all what I meant to say. The discipline of signal processing originated in electronics, but like a creature from one of M. C. Escher's tesselations, it has broken free of the matrix where it was spawned and become a separate entity. At any rate, that's what they tell us at Berkeley, though I understand that it's a bit of an innovation and that not everyone sees things this way.

We define a "system" as an abstract object that operates on signals. The set of filters is a loosely defined subset of the set of systems. It's completely abstract, so much so that the topic almost becomes a branch of mathematics. As a result, it's implementation-independent. For instance, consider a simple lowpass filter. It can be implemented electrically (for instance, as a capacitor to ground), in software (in any number of sophisticated ways), or even mechanically (for instance, as a damped oscillator). From this perspective, it's parochial to speak of electronic filters as anything other than a special case. --Smack (talk) 04:25, 6 May 2005 (UTC)

OK, I see where you're coming from. Yes, filters can be implemented in a variety of forms/mediums from dampners to DSPs and I'm open to suggestions if you can come up with an alternate naming scheme. Articles about the electronic signal ones seem to be in the majority at the moment though.--Hooperbloob 05:16, 6 May 2005 (UTC)

I don't get the issue here. This article is specifically about electronic filters, not all filters. So you're intentionally making this article to be larger than what it is. The intro makes it very clear to me that this article applies to electronic signals and, last I knew, sequences of values in a DSP constitutes a signal.......otherwise I need to hand back my MS in DSP.

So what *specifically* is your "complaint"? That an article on electronic filters doesn't cover mechanical filters???? Cburnett 07:36, May 6, 2005 (UTC)

That is not my complaint, though I will admit that the difference between my meaning and your interpretation is subtle. This is not actually an article on electronic filters. It's an article on filters, with no qualifying adjective, as abstract and implementation-independent objects, that claims to be an article on electronic filters specifically.
Now, you ask, shouldn't there be an article on electronic filters? I say that yes, indeed there should, but its content should be limited to details of implementation. Much of what is written here is applicable to filters in general, and much of the remainder also has a place here as a cursory survey of the technology. --Smack (talk) 04:57, 7 May 2005 (UTC)
Excellent point. Specifically the Frequency response and Mathematics of filter design. Perhaps filter needs to become an article covering filtering in general (chemistry, optics, computer/email/whatever, electronic, etc. Basically that filtering is a process of removing unwanted information. Cburnett 06:21, May 7, 2005 (UTC)
Perhaps there should be some general description at Filter, but I believe that that article should still be primarily disambiguatory in nature. IMO, most of the distinctions it draws are valid. Filters that act on real-number signals, or on digital signals representing real numbers, share some common ground with signals that act on "data streams" such as email, but I'd say that even these two categories are distinct. The unification that I propose is limited to signal processing, where a signal is a function mapping one numerical set (time, position, etc.) to another. --Smack (talk) 05:23, 8 May 2005 (UTC)
I guess I'm confused as to exactly what you are proposing here.... Care to make a bulleted list or something? Cburnett 06:48, May 8, 2005 (UTC)
I propose to generalize signal-processing articles to eliminate bias toward electronics. However, I believe that filters from outside the realm of signal processing are fundamentally unrelated and can be left alone. --Smack (talk) 01:42, 9 May 2005 (UTC)
Still not very clear to me. First, you're saying that signal filters in software...are not electronic? Secondly, what would you call this if not "electronic filter"? Cburnett 04:55, May 9, 2005 (UTC)
Carriage return.

First, I wasn't referring to all signal filters in software. An image-processing filter, for example, is a perfectly good filter from the realm of signal processing. In particular, many interesting signal filters are LTI, or can be (blurrers and color shifters, to name a few). I wanted to distinguish these kinds of software filters, which operate on streams of numbers, from software filters that operate on streams of data and perform semantic analysis. Yes, the latter can be said to lie within the field of signal processing because of the trivial fact that they process signals, but (as far as I can tell) they yield to none of the powerful analytical techniques of signal processing. An email spam filter doesn't have a frequency response. We can model it as a state machine, but I don't see how that would shed any light on the sophisticated algorithms governing its operation.

Furthermore, I want to remove the intellectual barrier that separates the former variety of software filters from electronic filters. Consider a simple causal blur filter, such that its impluse response is a decaying exponential. We can implement this as a computer program, or we can implement it as a lossy integrator. From an abstract standpoint, they're exactly the same.

My complaint is not just that the title of this article is poorly chosen. My complaint is that it fails to make the leap of abstraction that unifies all signal-processing devices - irrespective of implementation - within the analytical framework of information theory. --Smack (talk) 04:02, 10 May 2005 (UTC)

Perhaps signal filter might be a better name? Or linear filter for the very common case of such filters? -- The Anome 08:46, May 10, 2005 (UTC)
OK, the article is now called linear filter -- however, the category it belongs to is still called Category:Electronic filters. We should probably have two new categories: Category:Linear filters and Category:Non-linear filters, with both being sub-categories of Category:Filter theory, but I haven't the time to make the changes right now. -- The Anome 08:58, May 10, 2005 (UTC)

## Article refactored into two

I've now refactored the article completely into two articles: linear filter (this one) and electronic filter (the technology bits removed from this one, with an added intro). We should now address the glaring lack of detailed discussion of the details of pole/zero filter design theory. -- The Anome 10:46, May 10, 2005 (UTC)

I think Filter (signal processing) or Filter (information theory) would be better choices, though each is a bit of a mouthful. --Smack (talk) 04:58, 11 May 2005 (UTC)
Umm, there are non-linear filters so I can't agree with the current split.... I also think electronic filter should go to filter (signal processing). Cburnett 06:07, May 11, 2005 (UTC)
Ok, so I just found non-linear filter. I think we need just a filter article since the basic concept of filtering is not unique to linear or non-linear. And electronic filter doesn't cut it as Smach pointed out above. Cburnett 06:15, May 11, 2005 (UTC)

The Analogue filter article has been a stub for a long time and does not contain any significant information that is not here. Analogue filter/linear filter seem to be the same beast to me. A non-linear analogue filter might be theoretically possible but I don't know of any actual examples. The non-linear filter article seems to be describing signal processing of a different sort altogether from frequency filtering. I don't see the analogue filter article ever going anywhere so we may as well merge. SpinningSpark 23:39, 8 May 2008 (UTC)

Linear filter describes any filter which is linear, which includes non-analogue filters (i.e. digital filters). And as you've already pointed out, there's nothing that says that an analogue filter has to be linear (although in the vast majority of cases, it will be). Therefore, I would be tempted to avoid this merge. Oli Filth(talk) 00:04, 9 May 2008 (UTC)
It's about time that somebody did something with it then. The trouble as I see it, is that any expansion of that article would mostly be duplicating information here and the information here already has a large amount of overlap with Electronic filter. This is because most of the material is on linear analogue filters (for which there is no article - it re). Any suggestions on the way forward? Could just leave it alone, but I don't really like the idea of an article that is going to be permanently a stub. SpinningSpark 09:25, 9 May 2008 (UTC)
Good question. IMO, this article needs some serious work. There are several mistakes, but more importantly, the majority of the article is actually discussing LTI filters, and not linear filters in general (for instance, the concepts of transfer function, impulse response and frequency response really only make sense if the filter is time-invariant). I'll see about rectifying this over the weekend.
The overriding problem here is that the set of filter articles on Wiki seem to be very confused. We have Linear filter, Electronic filter, Digital filter, Analogue filter, FIR filter, IIR filter (and probably others) that all confuse and overlap various aspects of linear vs. non-linear, time-variant vs time-invariant, analogue vs. digital, continuous-time vs. discrete-time. I'm not sure how best to tackle this problem. Oli Filth(talk) 19:01, 9 May 2008 (UTC)
I agree, it is very confused. My immediate concern is that I am in the process of writing a series of articles on image designed filters and was unsure how to dove-tail these in to the existing material. However, it has soon become apparent that there are bigger problems than that. Starting from the top, the first problem as I see it is that there is no top-level article to help navigate around the rest. Filter is a disambiguation page covering a lot more than our meaning of filter so the first question is what should such a page be called. Do you think that we should start a filter project page (or maybe signal processing project page) to co-ordinate and discuss these activities? SpinningSpark 19:23, 9 May 2008 (UTC)

## Compact?

In general, a filter with a compact frequency response will have an infinite impulse response and a filter with a compact impulse response will have an infinite frequency response.

What does compact mean? --Abdull (talk) 15:20, 3 May 2009 (UTC)

The statement is provable if compact means having a finite support region; that is, if the response in one domain is zero outside some finite region, then in the other domain it's not. It's possible that there's some more formal definition of compact intended here; let's see if any mathematicians clarify... Dicklyon (talk) 18:26, 3 May 2009 (UTC)
The statement was a misunderstanding of the Fourier uncertainty principle. Consider an impulse response that is a Dirac comb. Clearly it's IIR; however, it's frequency support is also unbounded. Hence, unbounded in one does not imply unbounded in the other; however, high spread in one allows for low spread in the other (and if you restrict one to a single point, like an impulse, you're guaranteeing the other one will have infinite spread). "Spread" is more important than support, but compact support does imply finite spread. Either way, I've removed most of that language and tried to clean up some of the other mess. Why does this article exist? Isn't there an LTI system article that covers much of this? Aren't there lots of other articles that cover this? —TedPavlic (talk) 19:49, 4 May 2009 (UTC)
Ted, I reverted your removal; it sounds like you interpreted the statement as its inverse, or converse, or something, which would have been untrue. All it says is that compact in one domain implies infinite support in the other; it doesn't say infinite in one implies compact in the other. You can re-do some of the minor changes if you think they're important. Dicklyon (talk) 06:39, 5 May 2009 (UTC)
OK; I agree I misread. I'm not sure I'm comfortable with the term "infinite frequency response." In fact, although "frequency response" appears to be an engineering colloquialism for Fourier transform, it's confusing. An "impulse response" is a time-domain response to a single "impulse," and so a "frequency response" should be a time-domain response to a single sinusoid. So maybe I'll make a minor change. —TedPavlic (talk) 18:58, 5 May 2009 (UTC)
Actually, here's what I hated about the previous version: "In general, a filter with a compact frequency response will have an infinite impulse response and a filter with a compact impulse response will have an infinite frequency response." It's true that a compact impulse response (FIR) will have an unbounded Fourier support. However, the presence of this statement seems to imply that an IIR filter should have a compact time support. In fact, that's almost never going to be the case! Unless the IIR is a pure sinusoid (or a sum of them), it's almost definitely going to have unbounded frequency support. —TedPavlic (talk) 19:16, 5 May 2009 (UTC)
Well, it's simply not appropriate to be bothered by a not-stated converse that wouldn't be true. Dicklyon (talk) 23:34, 5 May 2009 (UTC)
It's not relevant to discuss compact/unbounded support topics with IIR filters. That's all I'm trying to say. —TedPavlic (talk) 01:04, 6 May 2009 (UTC)
Compact (for reals, bounded) support in one domain does not imply infinite support (whatever that means) in the other. As discussed in Fourier uncertainty principle, the product of the spreads (think variance or second moment) must be greater than some finite lower bound. Hence, you can have an infinite spread in both domains, and you can have a compact spread in both domains.
As far as I know, it's not possible that the support in both domains is finite, and that's all the statement was saying; you're saying it's not true. Do you have an example or a source? Actually, I'm pretty sure it's provable that the product of the standard deviations is bounded, and that the bound is achieved by Gaussian functions, but those aren't compact. Dicklyon (talk) 23:38, 5 May 2009 (UTC)
Did I ever say you could have a compact support in both domains; as far as I can tell, I said "spread"? Obviously, you cannot have a compact support in both domains. It wouldn't make any sense for a "sum" of a finite number of sinusoids to decay to zero in finite time. However, knowing that one of the supports is unbounded tells you nothing about the other support. That's why I'm saying that this whole discussion only adds confusion when it's brought up in the context of IIR filters. An IIR filter may or may not have a compact frequency support. —TedPavlic (talk) 01:04, 6 May 2009 (UTC)
On the other hand, I do agree that the LTI article covers this better, and the section we're quibbling about is lame, as this division between IIR and FIR is only normally applied to discrete-time filters, isn't it? Dicklyon (talk) 06:41, 5 May 2009 (UTC)
I agree that it's lame. And I also agree that this whole discussion needs to focus on the discrete-time case. In the changes you reverted, I fixed several details about this. I'll add them again and then fix things as per the discussion above (about the conjugate waveforms). I would not argue with someone else removing this entire section entirely... —TedPavlic (talk) 18:58, 5 May 2009 (UTC)

Move math from lead to Mathematics of filter design. Equations are not required to understand the concept. Lead should summarize body. --Kvng (talk) 18:15, 6 March 2011 (UTC)

Hi, while suggestions are appreciated, I sort of disagree here. In the first place, it appears that your involvement has more to do with the WikiProject Professional sound production (am I wrong?), which I have to say isn't the main point of the article. Indeed there should perhaps be a page about audio filtering, but the concept of linear filters is much much broader. (Actually it's broader than this article which concentrates on "classical" analog IIR filters, but that's another issue). The concept of linear filters is TOTALLY mathematical, so you cannot remove the math from the lede. If you mean just to remove the equations, that doesn't help because then you have to EXPLAIN "convolution" (would surely take more space!) or assume that the reader already is familiar with it (making the article yet more specialized). So I wrote that just to lay out what we're talking about, what makes a filter linear (and time invariant and causal) apart from the range of all possible filters. It was to NARROW the scope of the article. I don't know what you could say in words that would make the article more useful: if the article were to be "dumbed down" it wouldn't say anything. The detailed math is not in the lede, just the basics of what we're talking about.

Could you either remove the tag, or express your disagreement (and suggestions)? Thanks, Interferometrist (talk) 22:35, 6 March 2011 (UTC)

As a signal processing guy, I agree that it's not great to try to put the filtering math into the lead paragraph. The integral and summation formulas will be gibberish to most readers. To explain things like that x(t) is the input and y(t) is the output and h(t) is the impulse response, etc., should take a bit more space than is available in the lead. As this is by no means the uniquely correct way to describe or summarize linear filters. Dicklyon (talk) 22:52, 6 March 2011 (UTC)
Well again, if someone can't handle convolution (with a quick refresh given by the equation that defines it) then I doubt they're going to handle rational functions. So I guess my assumption was that this is the type of an article that isn't oriented toward the "general public" but someone looking for more specialized information (I think WP guidelines differentiate between different levels of article in terms of the background you'd assume for the reader). I can only think of 2 ways of defining "linear filter" (which the lede is supposed to do, after all!). 1) Convolution; or 2) Explain what linear systems are in general, or specifically for electronics which components are linear. (2) seems much more difficult to do in 2 sentences, no?! If not one or the other, then you haven't defined them or even described them in a way that a person could differentiate between linear systems and non-linear systems. Nor is the scope of the article then clear by reading the lede as it should be.
The version as of Luckas-bot 17:51, 11 November 2010 only said that "A linear filter applies a linear operator ...." but that just hides the math in a buzz word that someone would have to look up anyway. It was the following version (long before I got into it) of Javalenok 20:41, 15 November 2010 that introduced an equation which was a poor attempt to depict an impulse response in z-transform terms that was even more unreadable, and it was that that I tried to improve by writing an equation that actually describes convolution (plus naming it!). So I feel it's a bit unfair to pick on the current version if no one objected to the version that was posted for 3 months before I cleaned it up.
There's no problem with further explaining convolution inside the body, but that seems somewhat superfluous since most of the discussion relates to the frequency domain. But I'm interested in more opinions on what would make this a useful article to the people who are going to be looking it up. And by the way, I see that there IS an article on Audio filters which is and SHOULD BE less technical, though that article really needs a lot of work itself. Anyone looking up filters including a term such as "linear" I assume already has an inkling of its significance. Comments, suggestions? Interferometrist (talk) 23:50, 6 March 2011 (UTC)
There is no evidence for your statements about the make-up of the article readership, but you are probably right that they are unlikely to be completely non-technical. By the same token, professional designers are probably not looking at this article either. My guess is that most hits are from electronic hobbyists. If that is true, the maths is certainly going to be off-putting in the majority of cases. SpinningSpark 00:05, 7 March 2011 (UTC)
(ec) Yes, I go along with Dicklyon, it is not necessary to understand convolution in order to grasp the basic idea of linear filtering. Ledes should be accessible to the majority of the readership. At the very least, it is not necessary to open with the convolution formula in the first paragraph. Also, the lede should be a summary of the article, but neither of these formulae are actually used in the body of the article - see WP:LEAD SpinningSpark 23:56, 6 March 2011 (UTC)

What's your suggestion then: to move the part about convolution into the first part of the body? Why don't you suggest what should remain in the lede then? Also, you're wrong about:
" neither of these formulae are actually used in the body of the article"
Well they are very much used! The entire section about the impulse response doesn't make any sense without understanding what the impulse response does after all. The reason you don't see any further math about convolution is because it is totally defined by the equation in the lede and anything more would be a repetition (or trying to explain it in words, or pictures, but I'd rather just refer someone to the article on convolution which it does link to). I understand the concern about putting an equation in the first sentence that many general readers won't appreciate, but how else do you approach the topic (and say something informative)? Interferometrist (talk) 00:15, 7 March 2011 (UTC)

Not to belabour the point, but it's hard to define something like "linear filter" without reference to mathematics beyond the understanding of the "general reader." I suggest you look at the ledes of the articles on Linear systems and Linear operators and tell me if they are more accessible and still informative. Interferometrist (talk) 00:22, 7 March 2011 (UTC)

" My guess is that most hits are from electronic hobbyists. If that is true, the maths is certainly going to be off-putting in the majority of cases"

I suppose we could reduce this to electronics and say "A linear filter is a filter consisting only of capacitors, inductors, resistors and (linear) amplifiers, but no nonlinear components such as diodes." Many people would understand that (whether they totally appreciate why these are "linear" is another question, but we would have pointed them in the right direction). The problem is that this doesn't help anyone outside of electronics. So is that an improvement or not? Interferometrist (talk) 00:31, 7 March 2011 (UTC)

The guidelines for writing a lead WP:MOSINTRO tell us that the lead should summarize the rest of the article. By my reading, then, there should be no information in the lead that does not also appear in the body. The convolution equations appear in the lead but nowhere else in this article. So we need to put them in the body and they can also be in the lead if they're deemed to be that central to the topic. I still believe strongly that they are not. Convolution is but one way to understand a linear filter. --Kvng (talk) 05:58, 7 March 2011 (UTC)
I took a stab at it. Please improve as needed. Dicklyon (talk) 06:08, 7 March 2011 (UTC)
I had been working (offline) on a version to address some of the concerns (though I'm still not convinced that having an integral in the lede is necessarily wrong....). I will look at your changes carefully and might merge in parts where I really think something is missing. Right now there is one thing definitely missing: the entire lede never specifies why some (albeit most) filers are "linear." I had attempted to do that by introducing convolution, and I really disagree with Kvng, convolution is the most succinct definition for a linear and time/shift-invariant system. Any textbook starts with that. But I understand the desire for the lede to be widely accessible. I don't think just using the word "convolution" does the trick. But on the other hand I had never really proposed that there be a section about convolution (as has resulted from moving that text down), only that it needs to be mentioned in order to talk about the impulse response. I will probably try to address at least those two problems, but feel free to revert it (but if so DO explain exactly what issues you have with my edit, so we can converge).
Also, this is more of a question: the focus of the article (as we all agree) is only on a certain class of linear filters (one dimensional, time invariant). Exactly where in the lede should the scope of the article be delineated? Do people looking for other sorts of linear filters (image processing etc.) need to be pointed elsewhere? (I havent seen a comparable article on WP on image processing filters. And the general article on "Filter (signal processing)" covers only the same territory as this one.). Interferometrist (talk) 13:49, 7 March 2011 (UTC)
Perhaps a merge or bigger reorg is indicated. As to the "why" it's hard to say, since in reality most filters are not linear. Linear is an approximation we use for a lot of good reasons; convolution is a linear operator, not a reflection of reality. Dicklyon (talk) 15:56, 7 March 2011 (UTC)
Well I have done a little reorganization and changed some of what you wrote, so please look at the version I'm about to upload. But no, you are really wrong about the application of "linearity." For all practical purposes (in other words, as well as one could actually measurs) many natural systems are absolutely linear and circuits consisting only of capacitors, resistors, air-core inductors, and even amplifiers (with some limitations) are absolutely linear until driven beyond some reasonable limit (such as when a capacitor breaks down or an amplifier clips). The applicability of "linear" is more exact than you imply. Perhaps what you are thinking of is "ideal" inasmuch as practical components are not always linear in the exact way described (capacitors have some resistance and series inductance) but linear nonetheless. In electronics that is important in a particular sense: linear circuits do NOT produce new frequencies, whereas nonlinearity in a circuit leads to "distortion" and new frequencies (see intermodulation distortion for instance.
Please leave feedback concerning my new version (unless you consider it so awful it needs to be reverted!) and we can further converge. Interferometrist (talk) 16:46, 7 March 2011 (UTC)
Thanks everyone for your contributions. The article is much improved. --Kvng (talk) 00:48, 8 March 2011 (UTC)
Yes, much better. Next task is to cite some sources.... As for linearity, I'd say that "absolutely linear until driven beyond some reasonable limit" is on the optimistic side, but close enough; my point is just that "absolutely" is not true, and that there are limits. Air-core inductors, for example, are not linear, because they warm up due to losses, and then the resistance goes up; same with resistors; capacitors aren't linear because the electrostatic forces distort the dielectric, which isn't really linear anyway. Close enough for practical purposes, but not "absolutely" linear; we shouldn't claim more. Dicklyon (talk) 06:24, 8 March 2011 (UTC)
(More on the article, below) We both agree this isn't a crucial issue, but on linearity, I can see what you are saying but I would not have been inclined to call a change in an inductors resistance with temperature (yes, a function of time) "nonlinearity" since it occurs over a much longer time scale than anything that would (for instance) produce harmonic distortion. I would have called it a "change in the system over time" or if you must a non-time-invariant linear system. Since the heat is a function of the signal itself, yes I guess that's technically "non-linear" if you were going to analyze the "distortion" of a sine wave by doing an FFT of it over many millions of cycles, but not anything you'd measure directly as harmonic distortion. Anyway, we're probably arguing over definitions and what time scale "linearity" applies over so never mind.
Regarding capacitors I'm not familiar with what you're saying: that a capacitor will vibrate according to the charge on it? I wouldn't have expected that at RF frequencies but I guess it's possible since crystals do. And you can measure that? It would cause harmonic distortion. I guess it would only be important at higher voltages, perhaps in radio transmitters? As far as I know, the atomic polarizability of dielectrics used are linear until you're just about tearing electrons away from the atoms and the dielectric breaks down. But otherwise? Oh, I could easily believe electrolytic capacitors being measurably nonlinear, but they're not usually used as linear elements in the sense we were discussing.
Thanks for the copy edit and improving the English! More below, Interferometrist (talk) 13:11, 8 March 2011 (UTC)

## Latest edits

Thanks Dicklyon for the improvements in the language! Just a few issues. Your new statement:

The frequency response, given by the filter's transfer function H(?), is an alternative characterization of the filter.

I'm not sure if there's an exact accepted terminology, but I was calling H(f) the transfer function, but frequency response could (and in common speaking usually does) refer to |H(f)|. If that's so, then the sentence could be slightly misleading. Or should frequency response == transfer function and specify "magnitude" when needed? Anyway, this isn't exactly what I meant to say and will make the wording more precise:

The magnitude of the frequency response | H(?) | is an important alternative characterization of a filter.

You also twice removed a reference to the frequency response being more important (to the application) than the impulse response, which is an overgeneralization of course but surely has to be mentioned. Otherwise the whole discussion of Butterworth etc. LPF filters isn't so important. I realize there are applications where the phase response is of great importance, but usually just to specify that it doesn't ruin the pulse shape etc. with the freq amplitude response being the point of the filter, and frankly when designing analog (and digital IIR) filters the designer wouldn't normally even calculate the impulse response itself. In an FIR filter (less emphasized in the article) of course you have it whether you were interested or not. I'd call the magnitude of the transfer function which approximates the sought-for frequency response generally MORE important that the impulse response, no? I'm changing the wording a little, tell me if you agree with the rewording.

Also this is semantics, but shift-invariant MEANS time-invariance in the context of time-domain (realtime) filters which this article is restricted to, so I think adding that term does more harm than does it clarify. Likewise the DFT is an FT and probably doesn't need to be named separately (especially since the FT is never detailed in the article, just refered to).

Citations: Granted I didn't add any citations, though in my view I was only rewriting what already had been there without citations. I don't know where they are exactly needed, if there are any particular places that something unfamiliar needs to be backed up rather than just listing books on the subject of which there is one in a footnote, and others under Further reading. References would be from textbooks not research papers, and I only have 2 such textbooks on my shelf; maybe you have more?

## Todo

Based on all this discussion and editing, I have removed one item from the todo list (at the top of this page) and added three new ones. If anything in the list looks controversial feel free to discuss. --Kvng (talk) 15:19, 8 March 2011 (UTC)

I will also try (sometime) to improve the last section, Mathematics of filter design. Interferometrist (talk) 14:32, 8 March 2011 (UTC)

Thanks Kvng. I can see by browsing the various related articles that there is overlap and, more importantly, underlap. I don't have time right now to go through it all (I'm much better just writing about something than organizing it, especially organizing it across a fair number of articles!). If someone has some opinions about where the delineations should lie, please post those. The title of this article doesn't exactly reflect the scope of it, but I don't think it should have a more complicated title either. 40 years ago this would have been much simpler ;-)
If "overlap" were considered tragic, then I'd have to judge WP a terrible failure. (But of course it's a great success! :-) Also, I'm uncertain as to where citations need to be specified and how frequently repeating the same reference, etc. Interferometrist (talk) 16:48, 8 March 2011 (UTC)

Also thanks to Dicklyon for the copyediting which ^H^H^H^H^H^H that makes it more readable :-) Interferometrist (talk) 16:53, 8 March 2011 (UTC)

And don't hesitate to change the wording further even where it isn't so important. I'm certainly not offended when my exact words are modified. But of course if I detect even a sliver of change in the meaning, that's different! Interferometrist (talk) 17:08, 8 March 2011 (UTC)
On the referencing question, Wikipedia tends to want a higher density of referencing than the typical scholarly paper, at least, it does if the aim is to get to one of the Wikipedia quality standards. In practice this means a citation for every paragraph at least, with specific facts individually referenced as well; even if this means citing the same ref in multiple places. There is a method of making the same citation reference multiple places, but the first thing is to get some references on the table and worry about formatting later.
On overlap, this is not a sin as such, and a certain amount is going to be inevitable. It is more important that each article is able to stand on its own. The advantage of reducing overlap is easier maintenance. Often, the solution is a summary of one article in another plus a link to the more detailed piece.
SpinningSpark 21:44, 8 March 2011 (UTC)

## ???

"Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity. This results from systems composed solely of components (or digital algorithms) classified as having a linear response." I have no idea what this means. Could we try writing a first sentence that defines linear filters in terms of something someone can see or hear, using familiar terminology, before we start speaking Klingon?--Atlantictire (talk) 17:59, 6 May 2011 (UTC)

Look, I don't think what you ask is really possible, since this is an article about mathematics, not things you can "see or hear." I gather you're concerned because equalization uses linear filters and you want to link to something more accessible, but this isn't it. Taking the mathematical language out of the lede would just mislead someone who started reading it if they weren't looking for such an article. What's more, the lede has been somewhat toned down (in response to criticism) from what I wrote in this version: [1]
Which was far superior to the previous version: [2]
If you need to point to something one can see or hear, then I could mention Electronic filter, or Active filter (almost all of the circuits in modern day equipment are active filters -- unless they are digital filters that is) or Audio filter which is extremely general and doesn't say anything that would enhance the eq article. Anyway thanks for calling my attention again to this page, which I promissed to do further work on. But it won't get any less mathematical because that is what it is about. Sorry to break it to you, but concepts like "love", "honesty" and "linear filters" can't be summed up in terms of things you can see or hear. - Interferometrist (talk) 19:32, 6 May 2011 (UTC)
An explanation along the lines of this one might be more accessible. And actually, I agree with Atlantictire that the current opening does not really tell the reader anything if they do not already know what linearity means; it almost comes down to a truism - "it is linear because it is made of linear parts". The essence of the matter to my mind is that superposition can be applied to a linear circuit and this is how this particular book goes about defining it (although it does not mention the term). Thinking about it, this could be clearer if presented as a block diagram rather than a set of equations. I would be happy to produce such a diagram if people think that would be helpful. SpinningSpark 23:02, 6 May 2011 (UTC)

Alright listen, I was trying to offer an explanation to Atlantictire as to why the current lede is as it is, as it had (apparently) been agreed to by the people who debated the matter and rewrote it in early March (see discussion above under #Disagree with "task" tag recently added). I am hardly concerned if someone wants to change it in any reasonable manner as long as it isn't inaccurate. So go ahead and offer your text if you wish. Yes, I went to the google books reference and see section 15.7 -- is that what you're refering to? Let me make a few points though:

1. Section 15.7 of the text is a fine definition of linear systems but doesn't say anything about filters per se. The version I had written that was rejected DID say what a linear filter was: something that implemented a convolution, but that was judged too mathematical. Thus we wound up with the current version.
2. Correct, the current version doesn't explain anything, only saying that a linear filter is a filter that's linear. That was the most we could agree on since it didn't (still doesn't) make sense to explain the concept of linearity in 2 or 3 sentences in the lede. Does it?
3. Another tact would be to describe what a linear filter would consist of in practice. I could say "resistors, inductors, capacitors, and linear amplifiers" but that would only be talking about Electronic filters not the whole field of time domain filters (mechanical, acoustic, purely mathematical for data analysis, etc.) let alone non time domain filters (which SHOULD really be in an article with this title but we decided to limit the scope of it).
4. Another tact would be to say that it is a filter with a frequency response but that is also tautological since the frequency response only exists for an LTI system and so you've just hidden what types of systems have a frequency response in the first place.

For reference, here are the previous versions of the first paragraph:

15 November version: A linear filter applies a linear operator to a time-varying input signal. Mathematically, it is a combination ----- followed by a very poor writing of the Z transform of an FIR filter, discrete time case only and non-causal!
1 March (my edit):A linear filter applies a linear operator to a time-varying input signal. In the usual case of a causal time invariant filter, the output signal y(t) can be expressed as the convolution of the input signal x(t) with the filter's impulse response h(t) ---- followed by convolution equations.
7 March by Dicklyon: A linear filter applies a linear operator to an input signal to produce an output signal. The input and output signals can be functions of time, as in sound and radio signals, or of space, as in image signals, or of other domains
7 March till now (my edit): Linear filters in the time domain process time-varying input signals to produce output signals, subject to the constraint of linearity.This results from systems composed solely of components (or digital algorithms) classified as having a linear response.

Again, I'm not picky because I don't think you can say much in 2 sentences that is particularly informative. The best you can do is to narrow the subject down so people reading it understand what it covers, perhaps using terms that they won't understand (and might not need to). If someone tthinks they can improve it, then please do so. Interferometrist (talk) 16:35, 7 May 2011 (UTC)

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