Talk:Linear time-invariant system

message: Need to put an example for the discrete time case to make things a little clearer.
*I say this side focuses too much on continuous time (CT). The concept of LTI should be explained independent of whether or not the time is continuous. For example, it is difficult to link from a side which discusses digital signal processing algorithms to this LTI side, because it only talks about CT signal processing. Also, LSI (Linear Shift Invariance), which means basically the same thing, must be mentioned. Faust o 20:25, 23 January 2006 (UTC)
thread_title:  Example 

message: but i'm glad the article is there.  this should be written so that it can be understood by someone who doesn't already understand it.  to begin with, i think there needs to be a better introduction as to what \mathbb{H} is.  what are the fundamental properties of this LTI operator \mathbb{H}? first exactly what does it mean for \mathbb{H} \{ x_1(t) + x_2(t) \} to be linear (the additive superposition property) and then what does it mean for \mathbb{H} \{ x(t-\tau) \} to be time-invariant.  then from that derive the more general superposition property, then introduce the dirac delta impulse as an input and define the output of \mathbb{H} \{ \delta(t) \} to be the impulse response.  since the article is LTI, there is no need to introduce h(t_1,t_2).  all that does is obfuscate.
Mark, i hope you don't mind if i whack at this a bit in the near future.  i gotta figure out how to draw a png image and upload it.  r b-j 04:53, 28 Apr 2005 (UTC)
replies: text: I disagree.  I think h(t_1,t_2) should be there, as it shows how much simpler things become when translation invariance is imposed. I think you can represent any linear system with \int_{-\infty}^\infty h(t_1,t_2) x(t_2) d t_2, but I'm not 100% sure.  Certainly any nice system. - Jpkotta 06:40, 12 February 2006 (UTC)
replies: text:  you're correct when you say that "you can represent any linear system with \int_{-\infty}^\infty h(t_1,t_2) x(t_2) d t_2", but that is more general than LTI and unnecessarily complicated.  that expression includes Linear, time-variant systems also.  perhaps the article should be just "Linear systems" and deal with both time-variant and time-invariant. r b-j 07:25, 14 February 2006 (UTC)
thread_title:  i think there is a lot more that can be done to make this clearer. 

message: Is the bb font  \mathbb{H} at all common for an operator?  I've never seen that before, and I think bb font should be reserved for sets like the reals and complexes.  A clumsy but informative notation that one of my professors uses is this: \mathcal{L}_{t_o}\{x(t_i)\}_{t_i} ,where \mathcal{L} is the operator, to is the output variable, and ti is the input variable.  To say that a system is TI,  \mathcal{L}_{t}\{x(t + \tau)\}_{t} = \mathcal{L}_{t+\tau}\{x(t)\}_{t}.I'm not sure if it's a good idea to use it here though... -- Jpkotta 06:49, 12 February 2006 (UTC)
thread_title:  Notation 

message: Should discrete time be folded in with continuous time, or should there be two halves of the article?  
By folded, I mean:
* basics
** what it means to be LTI in C.T.
** what it means to be LTI in D.T.
* transforms
** laplace
** z
By two halves, I mean
* C.T.
** what it means to be LTI
** Laplace
* D.T.
** what it means to be LTI
** z transform
I vote for the two halves option, because then it would be easier to split into two articles in the future. -- Jpkotta 06:46, 12 February 2006 (UTC)
thread_title:  Discete time 

message: This page has the equation
:y(t_1) = \int_{-\infty}^{\infty} h(t_1, t_2) \, x(t_2) \, d t_2
which looks an awful lot like the application of a Green function
: u(x) = \int_0^\ell f(s) g(x,s) \, ds 
however this page doesn't even mention Green functions. Can someone explan when the two approaches can be applied? (My hunch right now is that Green functions can be used for linear systems that are not necessarily time-invariant.) —Ben FrantzDale 03:24, 17 November 2006 (UTC)
replies: text: Yes, a Green's function is essentially an impulse response.  Different fields have developed different terms for these things.  But the Green's function is also more general, as you note, than is needed for time-invariant systems; as is that h(t1,t2) integral. Dicklyon 06:14, 17 November 2006 (UTC)
thread_title:  Comparison with Green function 

message: The first example starts out describing the delay operator then describes the difference operator.  203.173.167.211 23:03, 3 February 2007 (UTC)
replies: text: I fixed it.  And changed z to inverse z for the delay operator.  I'm not sure where that came from or whether I've left some discrepancy.  Dicklyon 01:30, 4 February 2007 (UTC)
thread_title:  Discrete example confusion 

message: Generally, it is not true that for a linear operator L (such that L(ax_1 + bx_2) = aL(x_1) + bL(x_2)), that 
L\left(\sum_n a_nx_n \right) = \sum_n a_n L(x_n)
over arbitrary index sets (i.e. infinite sums). This is used heavily in LTI analysis.
The result does not follow from induction. So why should it be true for linear systems? I think linearity itself is not strong enough a condition to warrant the infinite-sum result. Are there deeper maths behind systems analysis that provide this result? (For example, restriction of linear systems to duals of certain maps is a sufficiently strong condition to imply this result.) 18.243.2.126 (talk) 01:42, 13 February 2008 (UTC)
replies: text: Are you saying it's not true?  Or that you don't know how to prove it?  Do you have a counter-example?  Dicklyon (talk) 05:07, 13 February 2008 (UTC)
replies: text: It is plainly not true, if linearity is the only condition being imposed (the constant signal x[n] = 1 is linearly independent from the unit impulses and all their shifts -- while this is not the case if you allow infinite sums); I have yet to construct a viable time-invariant counterexample; time-invariant functionals tend to be a lot more restrictive. For example, any time invariant system which only outputs constant signals is identically the zero system.  18.243.2.126 (talk) 00:21, 19 February 2008 (UTC)
replies: text: I'm not following you.  What is the concept of "linearly independent" and how does it relate to the question at hand? Dicklyon (talk) 06:10, 19 February 2008 (UTC)
replies: text:  The terminology comes from linear algebra (the wiki article explains it better than I can in a short paragraph). Note that the set of real-valued signals forms a real vector space. 18.243.2.126 (talk) 19:28, 20 February 2008 (UTC)
replies: text: I understand about linear algebra and vector spaces, but there's nothing in this article, nor in linear algebra about this concept you've brought up, so tell us why you think it's relevant. Dicklyon (talk) 20:05, 20 February 2008 (UTC)

text: I see that linear independence says "In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection."  This renders your above statement "while this is not the case if you allow infinite sums" somewhat meaningless.  So I still don't get your point. Dicklyon (talk) 20:10, 20 February 2008 (UTC)

text: In what way does it "not follow from induction"? Oli Filth(talk) 12:02, 13 February 2008 (UTC)
replies: text: Induction can prove it for all finite subsequences of the infinite index sequences, without bound, but not for the infinite sequence itself. Dicklyon (talk) 16:24, 13 February 2008 (UTC)
thread_title:  Problem with linearity 

message: I've removed the recently-added footnote, because I'm not sure what it says is relevant.  The explanation of the delay operator is purely an example of "it's easier to write", as by substitution, z = e^{sT}.  The differentiation explanation is irrelevant, because when using z, we're in discrete time, and so would never differentiate w.r.t. continuous time.  Oli Filth(talk) 21:28, 10 April 2008 (UTC)
thread_title:  New footnote 

message: In the section LTI_system_theory#Time_invariance_and_linear_transformation we say:
:y(t_1) = \int_{-\infty}^{\infty} h(t_1, t_2) \, x(t_2) \, d t_2.
If the linear operator \mathcal{H} is also time-invariant, then
: h(t_1, t_2) = h(t_1 + \tau, t_2 + \tau)  \qquad  \forall \, \tau \in \mathbb{R}.
For the choice 
: \tau = -t_2, \, 
it follows that
:h(t_1, t_2) = h(t_1 - t_2, 0).  \, 
But we can't make that choice, because t_2\, is the variable of integration, and \tau\, is a constant time offset.
The "proof" has been fudged so as to time-reverse the weighting function (as it was defined above) so that it looks like an impulse response.  But an impulse response is a weighting function whose value at \tau\, is the weight applied to the value of the input at time t-\tau\,, where t\, is the time of the desired output value.  That means it is a time-reversed version of our definition of h(t_1, \tau).\,
So one way to fix the problem would be to start with the definition:
But the -\tau\, will probably confuse people.  So an alternative is to show that if h\, is defined by:
:y(t) = \int_{-\infty}^{\infty} h(t, \tau) \, x(\tau) \, d \tau,
the impulse response is h(0,-\tau).\,  Then we could point out that defining h(0,\tau)\, as an impulse response leads to Eq.2 (and vice versa).
--Bob K (talk) 03:16, 19 June 2008 (UTC)
thread_title:  math error 

message: In section LTI_system_theory#Time_invariance_and_linear_transformation we say:
:y[n_1] = \sum_{n_2=-\infty}^{\infty} h[n_1, n_2] \, x[n_2],
If the linear operator \mathcal{H} is also time-invariant, then
: h[n_1, n_2] = h[n_1 + m, n_2 + m]  \qquad  \forall \, m \in \mathbb{Z}.
If we let 
: m = -n_2, \, 
then it follows that
:h[n_1, n_2] = h[n_1 - n_2, 0]. \, 
But we can't "let m = -n_2,\," because n_2\, is the variable of integration, and m\, is a constant offset.
--Bob K (talk) 14:06, 19 June 2008 (UTC)
thread_title:  another math error 

message: The included LTI.png image:
:
shows a transformation from "Time domain" to "Frequency domain." However, clearly the s (or "Laplace") domain is shown, which is a generalization of the frequency domain. Either each s should be changed to a j\omega or the "Frequency domain" should be changed to something like "Laplace domain" or "s domain." —TedPavlic (talk) 22:55, 28 January 2009 (UTC)
replies: text: It's not so bad.  The Laplace domain (complex frequencies) is probably the most commonly used frequency domain in this application. Dicklyon (talk) 00:07, 29 January 2009 (UTC)
thread_title:  LTI.png shouldn't say "Frequency domain" 

message: Zero state response also discusses the analysis of linear systems, but does not make a specific restriction of the problems to time-invariant system. There is, as yet, no top-level article on Linear system theory that deals with both this and the more general case of time-variant linear systems. Is there a possible route for refactoring/merging this material? -- The Anome (talk) 02:54, 17 February 2010 (UTC)
thread_title:  [[Zero state response]] 

message: This paragraph:
:As an example, the ideal low-pass filter with impulse response equal to a sinc function is not BIBO stable, because the sinc function does not have a finite L1 norm. Thus, for some bounded input, the output of the ideal low-pass filter is unbounded. In particular, if the input is zero for t < 0\, and equal to a sinusoid at the cut-off frequency for t > 0\,, then the output will be unbounded for all times other than the zero crossings.
It is really hard to imagine how a filter with a perfect LPF frequency response (as a sinc filter has) could be classified as unstable. Any sinusoidal input, for instance, is a bounded input with an obviously bounded output (either the same sinusoid or zero, depending on its frequency). However it indeed appears that the L1 criterion mentioned here would be violated by the sinc impulse response, which is stated to be an absolute test for stability/instability.
But never mind: I believe I see the problem. The sinc function extends to infinity in both negative and positive time so it cannot possibly be implemented as a causal filter, and this section is about causal filters (otherwise the concept of instability breaks down inasmuch as stable impulse responses with right hand plane zeros, if reversed in time, describe unstable systems). So I don't think the example is applicable.
And in any case, it could only be confusing to an average WP reader who is trying to LEARN about systems (since it's confusing to ME and I sort of thought I knew all about filter theory!). If the claim is true in some sense, then it stands more as a paradox or riddle than as useful information. Could someone remove this and put in a better example? And possibly (but here I'm not certain) restate the L1 criterion with a statement that it only applies to causal systems, or whatever qualifications are missing which make this result paradoxical or (I think) simply wrong? Interferometrist (talk) 12:14, 3 March 2011 (UTC)
replies: text: You are correct in saying "It is really hard to imagine how a filter with a perfect LPF frequency response (as a sinc filter has) could be classified as unstable."  It is not unstable; neither is its time reversal.  But it is also not bounded-input–bounded-output stable.  So what?  It is still a good illustration of the idea of BIBO stability.  Dicklyon (talk) 06:17, 7 March 2011 (UTC)
thread_title:  Dubious (and misleading!) example: "ideal LPF not stable" 

message: I worked for about 30 years in this subject and never once heard it referred to as LTI systems theory. Is the name just the invention of one person? What strange things can happen in Wikipedia.JFB80 (talk) 05:48, 24 January 2016 (UTC)
replies: text: Try Googling "LTI system". --Bob K (talk) 00:19, 13 July 2017 (UTC)
thread_title:  LTI is not a widely known abbreviation - why use it?  

message: I am not able to find a proper mathematical definition what a system in general is (a function that maps function (called input signals here) to functions (called output signals here)). There is this article here about LTI systems and an article about |time-invariant systems but there is no article about systems in general. I porpose that there should be a separate article discussing systems in general and defining some funcamental properties (time-invariance, linearity, ...) and showing examples of such systems. Fvultier (talk) 18:14, 12 July 2017 (UTC)
replies: text: A better place for this suggestion is Talk:System.  --Bob K (talk) 00:27, 13 July 2017 (UTC)
thread_title:  Article about systems/system theory in general  

message: I propose a more understandable introduction to this topic as follows:
:In signal analysis and other fields of study, a linear time-invariant system is a system that meets the following criteria: 1) the output signal is proportional to the input signal; 2) the scale of the proportion doesn't vary throughout the range of inputs under analysis; 3) the relationship between the input and output (including the scale) doesn't vary over time; and 4) in the case of multiple input signals, the addition operation on the inputs is preserved in the output. Linear time-invariance also finds application in image processing and field theory, where the LTI systems have spatial dimensions instead of, or in addition to, a temporal dimension. These systems may be referred to as linear translation-invariant to give the terminology the most general reach. In the case of generic discrete-time (i.e., sampled) systems, linear shift-invariant is the corresponding term. A good example of LTI systems are electrical circuits that can be made up of resistors, capacitors, and inductors.Hespanha 2009, p. 78.. Sparkie82 (t•c) 11:59, 19 September 2020 (UTC)
thread_title:  More understandable introduction to the concept 

message: I proposed that Time-invariant system be removed as a separate topic (it isn't) and its current contents be merged into this article. It may already be sufficiently dealt with, but if anyone sees content on the other page that would help here, please go ahead and edit that in. Interferometrist (talk) 20:51, 5 July 2021 (UTC)
replies: text: Oppose, on the grounds that if anything the merge should be in the reverse direction (to the broader topic). However, Linear time-invariant system is a sufficiently important topic that it warrants separate coverage. Hence, I'd recommend keeping the current structure. Klbrain (talk) 11:59, 19 December 2021 (UTC)

text: Oppose Time-variant system should be merged into this instead  — Preceding unsigned comment added by Marcusmueller ettus (talk • contribs) 15:00, 28 March 2022 (UTC)
thread_title:  Merge from [[Time-invariant system]]  

message: I proposed that Time-variant system be removed as a separate topic (it isn't. It isn't even a term used in practice) and its current contents be merged into this article. It may already be sufficiently dealt with, but if anyone sees content on the other page that would help here, please go ahead and edit that in. Interferometrist (talk) 14:49, 6 July 2021 (UTC)
replies: text: Oppose, as above. Klbrain (talk) 12:01, 19 December 2021 (UTC)

text: Support, as time-variant is "default unless specified otherwise", and any system that is not one is automatically the other. Also, the Linear time-variant article is just really really not good, the definition in there is confusing to students (I say as someone who taught stochastic systems and basics of digital comms), and it would be much better to give that definition in a section of this article.  — Preceding unsigned comment added by Marcusmueller ettus (talk • contribs) 15:03, 28 March 2022 (UTC)
thread_title:  Merge from [[Time-variant system]]