# Questions tagged [linear-systems]

A linear system operates on inputs with only linear operators so the response to a complex input can be analysed as the sum of the response to a set of simpler inputs. This mathematical property makes the analysis of linear systems much simpler than non-linear systems where this summation or superposition does not hold. Linear systems are generally further classified as time invariant, meaning that there characteristics do not change over time.

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### Are Chebyshev Type 1 filters very unstable at very high orders?

I am currently experimenting with a digital implementation of a Chebyshev Type 1 filter of mine, for audio purposes. I have written the code myself so I was wondering if the behavior I've noticed is ...
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### Determining if a 2D system is LSI

After a long time away from school, I'm a bit rusty and struggling with this question: Determine if the following discrete system is LSI: $y(m,n) = mn*x(m+n) + mn*x(m-n)$ So here's what I've done ...
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### Difference between the two forms of input-output relationships of an LTI system

$$y(t)=f(t)*h(t)\tag{1}$$ $$y(t)=H(s)e^{st}\tag{2}$$ $$H(s)=\int_{-\infty}^{\infty}h(t)e^{-st}dt\tag{3}$$ Let $f(t)$ in Eqn $(1)$ be $e^{st}$. In many worked out examples, I have found that the two ...
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### how do I derive the system equation for a simple delay with feedback?

I am a software engineer, and just learning digital signal processing formally, though I've hacked around before a fair amount. I'm implementing a delay audio VST and I'm trying to wrap my head ...
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### Controllability's dependence on B matrix

Assuming the LTI system: $\dot{x}=Ax+Bu\\$ , where $x\in R^n$ I understand that the $B$ matrix is usually given, and it is up to us to derive the optimal $u$ given an initial state of the system ...
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### External/Internal Description of a System

When an LTI system is NOT completely controllable/observable, does the linear differential equation governing the system give the internal or the external description of a system?
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### Unit Impulse : Bounded or Unbounded?

As the title suggests, I have a confusion here. In a Systems text I am going through, it mentions of the Unit Impulse as an unbounded signal. Yes, the unit impulse "height" is unbounded, but the "...
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### Determining LTI system response to a scale change

We know that for an LTI system, if $y(t)$ is the output for $x(t)$ then the response for $x(t-2)$ will be $y(t-2)$ and so on. But my question is what will be the system response for the input $x(-2t)$...
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### What is the difference between a controller and a compensator?

I recently revisited some texts on control theory. In some textbooks the term compensator is used, seemingly interchangeably with controller (see e.g. The Control Handbook by Levine, chapter 14 on ...
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### LTI system and initial conditions

I suspect this is rather obvious once explained; I must be misunderstanding something. From reading various answers here (such as this one), I understand that an LTI system must have zero output for ...
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### Does phase margin exists only for linear systems?

If possible, how to determine the phase margin for nonlinear systems?
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### Why the following discrete type system,variable accumulator, is time invariant?

$y[n]=T{x[n]}=\displaystyle\sum_{k=n-n_{0}}^{n+n_{0}} x[k]$ it is some sort of moving summer which computes $n^{\text{th}}$ output sample by adding all samples lying within length $n_{0}$ around some ...
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### Inversion problem for big dataset

Suppose my goal is to solve for $x$ in $Ax = b$ using CG method and $A$ is Toeplitz. The problem is that $b$ is extremely large in size and I can't just read in the whole vector $b$ and then solve for ...
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### System Identification with a Limited Bandwidth Input Signal and Region of Interest

Given a FIR filter $h[n]$. Its action can described as: $$\mathbf{y} = \mathbf{H} \mathbf{x} \\ \mathbf{y} = \mathbf{X} \mathbf{h}$$ where $\mathbf{H}$ and $\mathbf{X}$ is a Toeplitz matrix. If $h$...
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### What is the difference between y[n] and y(k) (using square brackets)?

Can I modify y[n] = αy[n-1] + x[n] the same way I would do it with y(k) --> Y(z)? I need H(z), but I don't know what is the difference between using (k), and (n). I often encountered [k] notation, ...
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### A question about Fourier transformation

Hello, this is my first time actually asking in stackexchange. I am a computer engineering student and currently i am doing a linear system course (i don't really know how this is equivalent in ...
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### How to Prove a System Is Invertible?

what i know is that for a system to be invertibel it should be one-one , but I am confused that if i am given a transfer function of a LTI system how can I prove or verify if it is invertible. ...
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### Reconstructed output mismatch for LTI system

I have a system with measured input (u) and output (y). I assume that this is an linear time-invariant (LTI) system and I want ...
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### Computing response of LTI system in terms of its step response

I have solved a problem, Kindly help me in determing if I solved it correctly. I will post the question and my own working below. Problem My working
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### Can all systems be described by differential/difference equations?

All LCCODEs (linear, constant-coefficient ordinary differential equation) with initial rest conditions describe causal LTI systems (is this a correct statement?) Is the converse true, i.e. can all ...
### Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System?
Is the system $y[n]=x[n]+2=T\{x[n]\}$ an LTI-System? Linearity: \$ \alpha \cdot T\{x_1[n] \} + \beta \cdot T\{x_2[n] \} = T\{\alpha \cdot x_1[n]\ + \beta \cdot x_2[n] \} \\\alpha \cdot (x_1[n]+2) + \...