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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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Introduction to Clustering and Classification of Linear Sequence

I'm a programmer/engineer that is getting into developing prototype wearable technology. I would like to be in a position to interpret raw accelerometer data and categorised the data into activities ...
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Impulse response of the inverse system to the backward difference

I'm currently stuck on a problem. I have a DT LTI system $\mathcal{T}$ that depicts the first backward difference: $$y[n] = \mathcal{T} \{ {x[n] \} } = x[n] - x[n-1]$$ Its impulse response is ...
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Compute output of a LTI system

Is it possible to compute the output from one linear time invariant system which unit impulse sequence is: $$h(n) = \left(\frac{1}{3}\right)^nu(n)$$ and the entry in the system is: $$x(n)=u(n)-u(n-...
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How to find the output of a system with the unit-step response and the input?

Convolution is used to find the output when the input and the impulse response is known. I am curious if there is a similar operator, like convolution, or a method that can determine the output of a ...
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1answer
4k views

How to determine if a system is LTI?

If the difference equation that defines the DT system is, $$z[n] = (1-x[n]) + (1-y[n]) $$ How can we determine in the system is LTI or not when: $x[n]$ is the input $y[n]$ is the input both are ...
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1answer
683 views

First-order low pass filter with bounded derivative

Given a function $x(t)$ and a first order low-pass filter $\frac{dy}{dt} = \frac{1}{\alpha} (x(t) - y(t))$, is there a way to choose $\alpha$ analytically such that the derivative of $y(t)$ is bounded,...
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2answers
362 views

Causality of Linear Phase Filter

If a digital LTI filter has phase response on the form $arg[H(e^{jw})] = -\alpha\omega$, what could one say about it's causality (and consequently, the causality of it's impulse response)? Also is ...
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176 views

Conceptual Question from Signal Processing - Impulse Response and AR Coefficients

In continuation to the previous question Conceptual questions from signal processing I have a doubt which is: Consider an Autoregressive model (AR(2)): $$ y(t) = ay(t-1) + by(t-2) $$ and a FIR (...
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1answer
103 views

Linear classification and overlapping problem reduction

I am study linear classification(Y=aX+b) But it is still weakness that is overlapping problem. Could you suggest to me how to reduce overlapping problem in linear classification or how to find these ...
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1answer
229 views

Issues in generating AR model with a constraint

I am new to the topic of system identification and looking for a large Autoregressive (AR) model. Can anybody point out a large stable AR model which has more than 2 coefficients AND there should be a ...
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1answer
70 views

Is the following system LTI (finite speed of signal propagation)?

I wonder if the following acoustic system is i) considered to be linear time-invariant, and ii) if it is fully described by it's frequency response? A speaker produces waves at the input location of ...
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1answer
3k views

Relationship between Discrete Deconvolution and Toeplitz Matrices

I have 2 vectors, $a$ & $c$, both of length M. I know they are related by $a*b=c$. My goal is to recover $b$. Obviously $b=$deconv$(c,a)$. I am only interested in the first M elements of the ...
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Coefficient Cancelation on Particular Solution

I am stuck on a problem about finding the complete solution of a Discrete Linear System by using the parametrization $y_{c}(n) = y_{h}(n) + y_{p}(n)$ where $y_{h}(n)$ and $y_{p}(n)$ are, respectively, ...
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2answers
148 views

Impulse Response with non-delayed feedback

I've been trying to solve a Discrete LTI System problem where the overall impulse response of the following System, whose input and output sequences are, respectively, $x(n)$ and $y(n)$, has to be ...
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1answer
4k views

How to calculate the zero state response?

In the following question and answer I kind of get it how it is calculated at surface but I don't get it fully. Can someone please explain the answer in depth?
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2answers
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Criteria to distinguish FIR and IIR filters from difference equation?

What criteria should be used to safely decide if it is an IIR filter or FIR filter from a difference equation? FIR filter are always stable, meaning all poles are inside the unit circle AND have pole-...
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6answers
2k views

Why do we always characterize a LTI system by its impulse response?

Why do we always characterize a LTI system by its impulse response and not by another response, like the step response? What does the impulse response have that is so special?
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2answers
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Can I construct a linear system so that it results in a moving average?

I've just learned about linear systems and impulse response functions. I know that in a linear system consisting of $h_1[k]$ and $h_2[k]$, the impulse response of the system is $h_1[k] \ast h_2[k]$. ...
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3answers
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What's the Difference between Convolution Kernel and Point Spread Function in the Context of Image Convolution?

When you use the deconvolution method to make the blurry image sharper, you will have to estimate the Point Spread Function. Is there a difference between this PSF and an image kernel? Second ...
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1answer
312 views

The impulse response, $h(t)$, of a system

I have this differential equation $$ y''(t) + y'(t) + y(t) = x(t) - 3x'(t) $$ I found the system response of the system: $$ H(f) = \frac{1 - 6\pi fj}{(2\pi fj)^2 +2\pi fj+1} $...
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3answers
216 views

Deconvolution with an underdetermined system of equations?

If I have a measured signal $\mathbf{y}$ which is the result of the true signal $\mathbf{x}$ convolved with another function (linear and not circular convolution), I always seem to get an ...
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271 views

Solving linear system and find impulse response

I have this linear system that is defined by the differential equation $$ y''(t) + 4y'(t) + 5y(t) = 2x(t) + 3x'(t) $$ with x(t) the input and y(t) the output. I'm asked to find two things:...
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Convolution of a signal by itself properties [closed]

I'm studying for a test and I can't really explain this question very well. If we convolve a signal by itself, will it have the same properties (time-invariant, linear, stable, casual)? I know how ...
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4answers
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Are complex exponentials the only eigenfunctions of LTI systems?

Is there an example of an eigenfunction of a linear time invariant (LTI) system that is not a complex exponential? Justin Romberg's Eigenfunctions of LTI Systems says such eigenfuctions do exist, but ...
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892 views

Is there a way to obtain the impulse response of a discrete system by just knowing it's response to the discrete unit step function?

In continuous time it was possible; $$ u(t){\longrightarrow} \boxed{\quad\textrm{system}\quad} {\longrightarrow} y(t)\implies \delta(t)=\frac{du(t)}{dt}{\longrightarrow}\boxed{\quad\textrm{system}\...
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1answer
92 views

What filter is better with -1/3 2/3 -1/3 or with -1/4 1/2 -1/4 coefficients?

Professor asked me what filter is better -1/3 2/3 -1/3 or -1/4 1/2 -1/4, the answer was second. But what those coefficients means? It was during discussion of linear filter but i'm not sure was it ...
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1answer
2k views

Causal systems - Paradox? Confusion of definition?

For a system such that $y[n]=\cos(x[n])$, I'd like to determine whether the system is causal or not. Definition 1: A system is causal if and only if the output $y[n]$ is a function of a linear ...
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1answer
276 views

Determining linearity, causality, memory, and time invariability from a picture of the signal

So, this is a picture of saccadic eye movement reflected by an EOG. How can I determine if the eye movement system is linear, causal, memoryless, and time invariant from this displayed waveform? ...
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3answers
1k views

Sum of Sine and Cosine with Random Phase as LTI System

I have the following system: Where $ {H}_{1} \left( f \right) = {H}_{2} \left( f \right) $ and $ \theta \sim U[0, 2\pi]$ independent of any other factor in the system. Given the input is identical, ...
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1answer
91 views

Choosing inverse Z-transform equation, given that $|a|<1$

Given that $|a|<1$, then which of those inverse-Z-transform equations are we to use? I am leaning towards the first because (as I understand it), $z$ is merely a complex number that is evaluated ...
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1answer
124 views

Ideas on Matrix Factorization / Transformations for $ {L}_{1} $ Minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where $\...
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What Are Linear and Circular Convolution?

I have some basic understanding of signals and convolution. As far as I know it shows the similarities of two signals. Could I get some explanation in plain English of: what are the linear and ...
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2answers
262 views

Situations where complexity is too big to exceed linearity and gaussianity

I'm studing about Kalman Filter and Particle Filter in multiple target tracking in computer vision (tracking pedestrians). Reading sientific papers I'm colliding with a lot of sentences like: The ...
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3answers
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Why unit impulse function is used to find impulse response of an LTI system?

Hello i am working in digital image restoration field, recently i have studied concept of convolution, i studied that to find the impulse response/point-spread function of an LTI system, an unit ...
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Panorama global registration

I'm trying to do global registration algorithm for panorama like described here. But I don't understand how they deal with false pairs. Also if expand the system u1*I*P1=u1*I => P1=I u2*I*P2=u2*...
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2answers
184 views

Frequency response of system with feedback and feedforward co-efficients?

For seismological application, I am playing with an elliptical filter, and once I have both sets of feedback and feedforward co-efficients, how do I make it so that I can plot the frequency response ...
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2answers
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What is the difference between Lumped and Distributed systems?

What are the salient differences between Lumped and Distributed systems? In what contexts are distributed systems the appropriate model and in what are lumped systems the appropriate model? Also, ...
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2answers
243 views

State space representation in s-domain

I was supposed to find state space representation and its matrices of this system: and I have no idea, how to do this. We were told not to transfer the system to time domain, but I can only do state ...
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1answer
444 views

Impulse Response Question

For the differential equation $$\frac{d^2y(t)}{dt^2}+\frac{3dy(t)}{dt} + 2y = x(t),$$ I was able to find the frequency response as $$H(j\omega)= \frac{1}{-\omega^2+3j\omega+2}.$$ However, I am not ...
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1answer
269 views

How can frequency dispersion in a vibrating string be modelled using an LTI filter?

According to JOS on stiff-string synthesis, stiff strings (like on a piano) introduce inharmonicity (i.e. the harmonics of the tone are not all in tune) due to different frequency components of the ...
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1answer
302 views

How to perform model fitting for system identification

I am having a really hard time in understanding how to formulate a model say linear AR model to represent a communication channel or maybe any motion. I have the experimental data representing the ...
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1answer
2k views

Problem with the definition of Linearity

From the high school maths we know that y=mx+c is a linear equation. However, in DSP the linear system must satisfy Additivity properties which y=mx+c does not hold because of +c. So, is the ...
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1answer
105 views

Calculating a delayed response of a system

Suppose I have a system: $$y[n] = x[n] + x[n-1] + x[0]$$ To see if this system is time-variant, we can compare $H\{x[n-1]\}$ to $y[n-1]$ and check that they are different. I think the delayed (by ...
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Group Delay of a system

Say I have an LTI system whose impulse response h[n] is: $$ h[n] = \delta[n] + \delta[n-4] $$ I want to find the group delay of this system. I know the group delay ($grd$) is defined as: $$ ...
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1answer
341 views

Why is the state space representation, $dx/dt = A x(t) + B u(t)$ linear?

One thing puzzles me about the state space representation and that is why is it claimed that such systems are linear and time-invariant but particularly why are they claimed to be linear? The right-...
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How to Deduce a Linear System's Impulse Response from a Set of Input and Output Signals?

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
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1answer
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Concept of finding forced response from difference equation?

I know how the process works, first from difference equation make a homogeneous equation and then find natural response , then find a particular response (in the form of input) substitute it back in ...
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1answer
162 views

ARMA (Auto Regressive Mean Average) Process Representation as AR (Auto Regressive)

Let's say we have an ARMA (Auto Regressive Moving Average Model) process where the transfer function is a minimum phase system (Namely Invertible). By Wold's Decomposition it is guaranteed to have MA ...
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1answer
418 views

Inverse Discrete Time Fourier Transform of a given Function using the definition

I would like to compute the Inverse Fourier Transform of a given function using the DTFT definition $$H(e^{jω})=\frac{1}{(2−e^{jω})(2−e^{−jω})}$$ But I have to use the definition of the DTFT $$H(e^{...
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302 views

How do I compute the impulse response of a robot?

I am trying to model a robot (my question in robotics.SE), so that I may train it using some reinforcement learning techniques. I was wondering if I can compute an impulse response of a four wheel ...