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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questions
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1answer
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impulse response VS zero-input response
I am new in the field of systems and signals, and I have a rather basic for the majority of the group, question:
Can we find the impulse response function of homogeneous ODE, instead of its zero-...
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votes
4answers
723 views
Can every type of linear filter be modelled by a convolution?
I have an input time series going through a filter that creates another time series as output.
If I assume in first approximation that my filter is linear, does it necessarily mean that I can model ...
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2answers
81 views
Convolution of Signal with a Gaussian Filter / Kernel
Following up on Analytical Solution for the Convolution of Signal with a Box Filter, I am now trying to convolve a Gaussian filter with the sine signal by hand.
My method is to use the definition of ...
2
votes
1answer
36 views
Initial Rest Condition for LCCDE causal LTI systems
I am self studying Alan Opennheim's course Signals and Systems. I am a math major and have no background in EE.
I understand that for a LCCDE system to be linear its auxiliary conditions must be 0.
...
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2answers
63 views
Why can adaptive IIR filters result in unstable solutions?
For adaptive filtering, both finite and infinite impulse response (FIR/IIR) filters can be utilized. As an advantage of FIR filters in this context, guaranteed stability is often mentioned, while IIR ...
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1answer
82 views
transient and steady-state response of first order system
Considering this general 1st order transfer function
$$
H(z) = \frac{b_0 + b_1z^{-1}}{1-az^{-1}}
$$
How to find (analytically) the transient and steady-state responses?
With steady-state response I ...
2
votes
2answers
41 views
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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0answers
23 views
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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1answer
27 views
Invariance of $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$
I have to test whether the following system is invariant or not: $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$, so I want to verify that, if $y[n]=T[x[n]]$, then $y[n-N]=T[x[n-N]]$.
$$T[x[n-N]]=\sum_{k=n-1}^{n+2}x[...
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1answer
68 views
Analytical Solution for the Convolution of Signal with a Box Filter
I have an exercise in which I am trying to filter an input signal $y(x) = \sin(x)$. Ideally, I would like to apply a box filter to this signal.
Previously, I successfully convolved the input signal $...
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2answers
78 views
Why does knowing the impulse response allow you to determine the output for any LTI system?
Going over some notes on LTI systems and I’m quite confused about how knowing the impulse response of a system allows us to recover the systems’ response to any signal. In the textbook screenshot ...
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1answer
43 views
Representing a continuous LTI system as a discrete one
I am aware that there are different ways to represent a continuous time system in discrete domain (e.g. bilinear transform, impulse invariance transform).
But my problem is as follows: Given an ...
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vote
2answers
49 views
Cross-correlation of filtered random processes
I have a wide-sense-stationary (WSS) process $\{x(t)\}$ and two linear filters with impulse functions $h_1$ and $h_2$.
Let $\delta(\omega)$ be the power spectrum of $\{x(t)\}$ and $$H_1:\omega\...
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votes
1answer
26 views
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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votes
1answer
36 views
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 ...
0
votes
1answer
31 views
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 ...
0
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1answer
13 views
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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1answer
76 views
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 "...
0
votes
4answers
52 views
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 $x(-2t)$...
1
vote
2answers
27 views
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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4answers
93 views
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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0answers
32 views
Does phase margin exists only for linear systems?
If possible, how to determine the phase margin for nonlinear systems?
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2answers
72 views
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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0answers
27 views
Using Kalman filter vs Extended Kalman filter for differential drive robot with IMU
I have an IMU that provides me with a heading that is pretty accurate and accurate encoders on the wheels of my differential drive robot which provides me with pretty accurate velocity but has ...
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0answers
23 views
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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0answers
47 views
System identification with limited bandwidth
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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votes
2answers
104 views
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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1answer
50 views
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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votes
3answers
373 views
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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1answer
32 views
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 ...
0
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1answer
56 views
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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4answers
790 views
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:
$$y(t)...
2
votes
1answer
50 views
How to detect algebraic loop in a system
I came across a system (screen below) that looks like it should result in a algebraic loop, but when writing the equations using the Z-transform, it obviously does not (calculation below too). But I ...
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1answer
23 views
Why does the Y component in YCbCr represent luma/intensity?
As far as I know, the RGB to YCbCr conversion is just a linear transformation, yet the Y channel successfully captures the intensity value of each pixel. How?
In other words, where do the following ...
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vote
1answer
36 views
confused in difference between calculations of convolution and multiplication?
How are two signals multiplied? And how is it different from convolving two signals??
I have read the above question on dsp stack exchange and i also tried it in my matlab , code is as follow:
...
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votes
1answer
26 views
Is it possible to use k-means algorithm with just one vector
Suppose I have a vector $X = (x_1, x_2 , . . . ,x_n)$, $x_i$ is the maximum of $X$ and $x_k$ is the minimum. Is it possible to use k-means algorithm to cluster the values in vector $X$ into two ...
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1answer
119 views
Output of an LTI system given its transfer function and input
Given the transfer function $$T(s) = \frac{100}{1 + \frac{s}{10^{6}}}$$ and the input $$v_i(t) = 0.1 \sin(100t)$$ find the output, $v_o(t)$.
My approach was to use $v_o(t) = \mathcal{L^{-1}}\left\{T(...
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2answers
56 views
time invariance concept?
First of all ,please let me know?Is cosine time invariant or time variant?
If latter is the case then if a time varying input like cos is given to a time invariant system,how will it behave?
I have a ...
1
vote
1answer
80 views
Cost function for LTI system identification
I am currently reading and trying to understand a paper (Kulkarni and Colburn, 2004) that utilizes system identification methods to approximate head-related transfer functions.
The general approach ...
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0answers
28 views
Region of convergence of transfer function
I posted this question Mathematics SE and got no answer so I have posted it here.
I learned in my signal processing class that an LTI system can be defined using a linear constant coefficient ...
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0answers
27 views
Comparison Between Mahoney Filter and Kalman Filter in Euler Angles (Yaw, Pitch, Roll) Estimation
What are the advantages of Mahoney Filter over Kalman Filter for Euler Angle (Yaw, Pitch, Roll) estimation?
Could anyone compare them for Angular State Estimation?
Resources
Mahoney Filter ...
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1answer
39 views
Why is the ROC of Laplace transform independent of imaginary part of s?
An integral is defined as converging if it yields a finite value upon application of limits of integration. It is divergent otherwise.
Now sticking to the mathematical notation of Laplace transform, ...
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1answer
23 views
Equivalence of difference equation with system of difference equations
I have this system and tried to write down its difference equation as follows
$$ y[n] = b_0v[n] + b_1v[n-1]\\
v[n] = x[n] - a_1v[n-1] $$
I was asked to prove that the system can be ...
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1answer
43 views
Consider the integrator and check for time invariance
I'm trying to check the following system for for time-invariance.
$$ y(t) = \int_0^t x(\lambda) d\lambda$$
Please explain me why it is time-invariant.
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1answer
31 views
confusion in time invariance?
I am trying to learn shift invariance(time invariance) and i came across a function/system
$$y(t)=3x(t)+2\cos(\pi t/3)$$
I am confused.If it was alone 3x,it should be definitely time invariant(i ...
2
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2answers
62 views
Contradiction while using the convolution sum for a non-LTI system
In a recent quiz, we were given the following problem:
The cascaded LTI systems $\mathcal{T}_1$ and $\mathcal{T}_2$ respectively have impulse responses $h_1[n]=\delta[n+3]$ and $h_2[n]=\delta[5−n]$....
0
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2answers
104 views
Confusion in linearity?
I am studying linear systems from the book "Signals and Systems Laboratory with MATLAB". I have performed an example on MATLAB myself in this regard.
Code is as follow:
...
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0answers
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Range of updated/new time axis after upsampling? [closed]
I am reading the book "Signals and systems laboratory with MATLAB" by Alex Palamides and I am reading system properties on pg 154 of the book but I am confused about an example shown in attached photo....
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2answers
105 views
Why can convolution only be applied to compute the output of a linear filter?
We apparently cannot compute the output of a bilateral filter (BF) using convolution (with the image) because the BF is a non-linear filter. In general, why can convolution only be applied to compute ...
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2answers
30 views
System Response Terminology
If I have a system specified by
$$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 ...
