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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Linear time invariance RLC
Why is a combination of R, L, C linear time-invariant system? But not diode, or transistor?
I looked over V-I equation for R, L, C, but still don't know how to prove it.
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Why does knowing the impulse response allow you to determine the output for any LTI system?
Going over some notes on LTI systems and I’m quite confused about how knowing the impulse response of a system allows us to recover the systems’ response to any signal. In the textbook screenshot ...
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Representing a continuous LTI system as a discrete one
I am aware that there are different ways to represent a continuous time system in discrete domain (e.g. bilinear transform, impulse invariance transform).
But my problem is as follows: Given an ...
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Cross-correlation of filtered random processes
I have a wide-sense-stationary (WSS) process $\{x(t)\}$ and two linear filters with impulse functions $h_1$ and $h_2$.
Let $\delta(\omega)$ be the power spectrum of $\{x(t)\}$ and $$H_1:\omega\...
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Difference between the two forms of input-output relationships of an LTI system
$$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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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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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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The Least Norm Solution of Under Determined Linear System
Suppose I have a matrix
$$A=
\begin{pmatrix}
1 & 0 & 1 & 0\\
0 & 1 & 1& 0\\
\end{pmatrix}
$$
where the variables are channel information like assume $X_1$, $X_2$, $X_3$ ...
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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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Linear Difference Equation and Method of Least Squares
I'm reading the book "Fault-Diagnosis Systems" by Isermann in the par. 9.2.1a.
The author explains how to estimate the parameter of a linear difference equation using Least Squares.
We start with a ...
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Estimate a Transfer Function from ARX Models vs. ARIMAX?
There is diffrent models which can be used to create a dynamical model by using least squares. Those models are following:
ARX
ARMAX
ARIMAX
OE
BJ
But if my goal with creating a dynamical model is to ...
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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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Time-invariant and Time-varying Systems
Determine whether the following system is time-invariant or not:
$y(t)=x(t)\sin 10\pi t$
Solution:
Given: $y(t)=x(t)\sin 10\pi t$
$y(t)=T[x(t)]=x(t)\sin 10\pi t$
The output due to input delayed ...
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Is the first derivative operation on a signal a causal system?
Please help me sort this issue out.
Consider a system whose output $y(t)$ is the first derivative of the input signal $x(t)$.
We can write the first derivative of an input signal as follows:
$$y(t)...
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How to detect algebraic loop in a system
I came across a system (screen below) that looks like it should result in a algebraic loop, but when writing the equations using the Z-transform, it obviously does not (calculation below too). But I ...
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expected value of two LTI output signals multiplied (cross correlation)
I have an input signal x (assumed to be iid Gaussian with $\mu=0$, $\sigma^2$) which is fed into two linear systems:
$y_1 = h_1 * x$
$y_2 = h_2 * x$
Now I would like to calculate $\mathbb{E}[y_1 y_2]...
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Proof of linearity
I have this system:
$$y[n] − 4y[n − 1] + 4y[n − 2] = 20x[n] + 10x[n − 1]$$
I have no idea how to prove if the system is linear because it depends on future outputs.
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Why does the Y component in YCbCr represent luma/intensity?
As far as I know, the RGB to YCbCr conversion is just a linear transformation, yet the Y channel successfully captures the intensity value of each pixel. How?
In other words, where do the following ...
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confused in difference between calculations of convolution and multiplication?
How are two signals multiplied? And how is it different from convolving two signals??
I have read the above question on dsp stack exchange and i also tried it in my matlab , code is as follow:
...
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Is it possible to use k-means algorithm with just one vector
Suppose I have a vector $X = (x_1, x_2 , . . . ,x_n)$, $x_i$ is the maximum of $X$ and $x_k$ is the minimum. Is it possible to use k-means algorithm to cluster the values in vector $X$ into two ...
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Output of an LTI system given its transfer function and input
Given the transfer function $$T(s) = \frac{100}{1 + \frac{s}{10^{6}}}$$ and the input $$v_i(t) = 0.1 \sin(100t)$$ find the output, $v_o(t)$.
My approach was to use $v_o(t) = \mathcal{L^{-1}}\left\{T(...
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Linearity of the given system
I am given the following system and I am checking the additive property:
$$y(t)=x(e^t)$$
where $y(t)$ is the output and $x(t)$ is the input given to the system.
Now this is what I did so far:
\...
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time invariance concept?
First of all ,please let me know?Is cosine time invariant or time variant?
If latter is the case then if a time varying input like cos is given to a time invariant system,how will it behave?
I have a ...
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Cost function for LTI system identification
I am currently reading and trying to understand a paper (Kulkarni and Colburn, 2004) that utilizes system identification methods to approximate head-related transfer functions.
The general approach ...
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Why is the ROC of Laplace transform independent of imaginary part of s?
An integral is defined as converging if it yields a finite value upon application of limits of integration. It is divergent otherwise.
Now sticking to the mathematical notation of Laplace transform, ...
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Equivalence of difference equation with system of difference equations
I have this system and tried to write down its difference equation as follows
$$ y[n] = b_0v[n] + b_1v[n-1]\\
v[n] = x[n] - a_1v[n-1] $$
I was asked to prove that the system can be ...
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Consider the integrator and check for time invariance
I'm trying to check the following system for for time-invariance.
$$ y(t) = \int_0^t x(\lambda) d\lambda$$
Please explain me why it is time-invariant.
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confusion in time invariance?
I am trying to learn shift invariance(time invariance) and i came across a function/system
$$y(t)=3x(t)+2\cos(\pi t/3)$$
I am confused.If it was alone 3x,it should be definitely time invariant(i ...
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Contradiction while using the convolution sum for a non-LTI system
In a recent quiz, we were given the following problem:
The cascaded LTI systems $\mathcal{T}_1$ and $\mathcal{T}_2$ respectively have impulse responses $h_1[n]=\delta[n+3]$ and $h_2[n]=\delta[5−n]$....
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Why can convolution only be applied to compute the output of a linear filter?
We apparently cannot compute the output of a bilateral filter (BF) using convolution (with the image) because the BF is a non-linear filter. In general, why can convolution only be applied to compute ...
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System Response Terminology
If I have a system specified by
$$P(D)y(t)=Q(D)x(t)$$
and I specify initial conditions $y(0^-)=a, \ y'(0^-)=b,\ x(0^-)=c$ does the term $x(0^-)=c$ correspond to the zero state response or zero ...
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confused in practical example of shift invariance?
I am reading book of w smith ,"The scientist and engineer guide to digital signal processing" 2 edition
I am trying to understand topic of shift invariance from theory given on pg 92
I have attached ...
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What Are Some Examples of Applications of LTI Systems
I'm giving a lecture on LTI systems. I encountered some questions:
for a discrete LTI system H with impulse response h, is the system applied on signal x(t) equals x*h - normal discrete convolution ...
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Causal unstable system turn into stable anticausal?
I would appreciate it very much if someone would be able to provide some clarity, help or comment on this problem.
I have been reading several papers on time series identification such as
https://www....
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Higher-order moment of output of LTI system
Assume a very simple LTI system. Assume $x$ is white Gaussian i.i.d. with variance $\sigma^2$.
The output variance is straightforward to obtain. For example, for a continuous-time system:
$$\mbox{...
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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 ...
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Eigen Function of Linear Time Invariant (LTI) System
Whenever I try to find the derivation of eigen signal of a LTI system, they always start with assumption that the input signal is complex exponential. Is there any other way to prove that this is the ...
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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) + \...
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Causal LTI system having exponential input
I know that for an LTI system having complex exponential input,
i.e, $x(t)=\exp(j w_o t)$ & $h(t) \to $ LTI System ;
then, its output { $y(t) \} =M \exp(j w_o t + \phi)$ , where $M= |H(j w)|_{|w= ...
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Moving average and linearization of two piecewise linear systems
I have 2 oversampling ADC's running parallelly, each to process data in a specific range of the input as shown below:
Each ADC can process only half cycle range of a sine wave. Each ADC adds its own ...
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Can $\delta(t+\infty)$ be a legitimate signal?
Mathematically speaking, when I try to use some signal to disprove a system is invertible, can I use the signal like $\delta(t+\infty)$ ($\delta$ representing the Dirac distribution)? For example, the ...
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How to obtain the steady state response of the sampled periodic signal data?
Below the green plot Vin is 5Hz sine voltage signal from a function generator output sampled at 12kHz sampling rate:
The blue plot Vout is the processed/filtered data by using the following code in ...
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