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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Analog LTI system impulse response

I have the following question in my mock exam! My Answer: I can apply the dirac delta to know the impulse response. Or use the transfer function $H(z) = \frac{Y(z)}{X(z)}$. I am not sure about the ...
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Slope of phase function

Would appreciate some help understanding if I take the phase function of some transfer function and derivative it in the linear part of it which is around the resonant frequency what does this slope ...
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Poles and zeros form of a transfer function

I know that a transfer function for a discrete-time LTI system can be written in the form $$ H(z) = \frac{Y(z)}{X(z)} = \frac { \displaystyle\sum_{m=0}^M {b_m z^{-m}}} {1 + \displaystyle\sum_{n=1}^...
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Output of a LTI system [closed]

I'm trying to resolve this exercise but I don't even know where to start. Do I have to apply convolution? **It appears that this question is classified as homework - this an exame problem which I was/...
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Linearity/Non-Linearity of $y(t) = x(t) +\cos(w_0 t)$

I have the following System with (input x(t), output(y(t)), that I have to check for linearity and time-invariance. $y(t) = x(t) + \cos(w_o t )$ I am able to show that it is time-variant. For ...
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Impulse Response Formula

How can you determine the impulse response if you know the output of the system? You should change the input signal with the dirac function with the argument equal to $t$ or $t-\tau$? I have this ...
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Determine if the system is linear, time invariant

I have a system described by the following impulse response: $h(t,τ) = g(t-2τ)$ Where $g(t) = e^{-at}u(t)$ I have to determine if the system is linear time invariant causal Then I have to determine ...
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Determining whether a system is linear or not having the impulse response

I got many exercises where I have to determine the response of a system to an input signal The only things i have are the impulse response and the input signal. I was thinking I could use the ...
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Entropy Loss in Linear Filters

I've been trying to figure out the result for the entropy loss/entropy gain in linear systems derived in "The Mathematical Theory of Communication" by Claude Shannon. Claude Shannon states ...
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Is the system $y\left(t\right)=\int _{t^3-1}^{t^3}x\left(s\right)ds\:$ invertible? [duplicate]

I have the following system: $$y\left(t\right)=\int _{t^3-1}^{t^3}x\left(s\right)ds\:$$ I was told to determine if it's invertible system, casual system, memoryless system and linear system. I was ...
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Why do transients exist when we input a sine signal to an LTI system?

Let $x[n]=a\cos(\omega_{0}n)$, if we pass it towards an LTI system, we should get as an output: $$ y[n]=a|X(e^{j\omega_{0}})|\cos(\omega_{0}n+\phi_{X}(\omega_{0}))+\operatorname{transients} $$ My ...
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L1-norm of a stable, proper but not strictly proper transfer function

in the literature I find the statement, that the L-norms exist for every strictly proper and stable transfer function. I ask myself, if the L1-norm also exists for a transfer function, that has as ...
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Fourier Transform: $x(t)=2\sin(2\omega_0t)\cos(3\omega_0t)$

I'm currently studying Fourier Transforms and do not understand the Fourier Transform of $x(t)=2\sin(2\omega_0t)\cos(3\omega_0t)$ My solution states that it is $X(\omega)=\frac{2}{2\pi}(-j\pi[\delta_0(...
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Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?

Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions? I had this question when I saw this claim in my textbook: for ...
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How to determine the impulse response matrix, when zero state response is given

My understanding is that since its zero state response, the system is at rest. But what should I do after that? is there any online material to read about this ? Thank you
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Why the convolution of the impulse response doesn't match the system's output?

If you define an LTI system sys in scipy, you may conveniently feed an input x to it to get its output ...
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How to solve Hilbert Transform with empirical discrete data in frequency domain?, from zero to infinity

I have a filter/LTI system frequency response in form of list of values in the frequency domain. I want to get the phase curve/data from magnitude data. Input data can have either linear spaced points ...
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Resolution of linear constant coefficient difference equation (homogeneus + particular)

I'm studying how to solve linear constant coefficients difference equations but I have some troubles with a step of the procedure. Consider the system \begin{equation} \begin{cases} y[n]+y[n-1]-6 y[n-...
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Linear Time-Invariant system without State-space form

I would like to know whether there is any LTI system that does not have state-space representation, but do have a convolution representation. and vice-versa. Can you name some examples for me to ...
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Sufficient conditions for invertibility of discrete LTI systems [duplicate]

Is $h[0] \neq 0$ a sufficient condition for the invertibility of a discrete, LTI, causal system? Can we get to similar results (i.e. get to some other sufficient condition(s)) for noncausal or ...
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Does causality imply linearity in a discrete system described by difference equations?

In my textbook, it is stated that for a discrete system, where the input and output are expressed by difference equations, to be causal, there needs to be initial rest. It is also stated that for the ...
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Calculate response of an LTI system with infinite DC gain to an finite length input in time domain

I have given the transfer function $P(z)$ of an LTI system in z-domain. Furthermore, I have a time-discrete finite length input signal $r$. I want to calculate the system response $y$ in time domain. ...
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Linearity of the system with Pearson correlation sliding window

I have a sampling system that consists of linear and non-linear components(analog filters, ADC, CPU, and so on). I made some HW changes and I want to verify that the linearity was preserved. My method:...
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Find moments of a discrete process using $\mathcal{Z}$-Transform method

For a linear process relating a variable $Y_i$ to random, independent variables $X_i$ using the equation: \begin{equation} Y_i = aY_{i-1} + (1-a)X_i \end{equation} which has the solution: \begin{...
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3 answers
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Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions?

Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions $y(t)$?? With output I am meaning that $y(t)$ is such ...
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How to find the impulse response from the following input/output relation

Suppose we have a system defined with input $x(t)$, and output $y(t)$, related by $$ y(t) = \sum_i \alpha_i(t)x(t-\tau_i(t)) \tag 1 $$ This is a linear system, (according to the textbook), so we can ...
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Is the filter $1/(1-s)$ anti-causal?

The filter with the response function $$ H(s) = \frac{1}{1 - s} $$ Produces a positive phase shift and a negative group delay for all frequencies Is it anti-causal? Is there a way to deduce such ...
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Time Invariance of Continuous Time System

I am trying to check whether the system given by the following input-output relation is LSI or not? $$y(t) = \cos(x(t)-x(0))$$ My work: The given relation: $$y(t) = \cos(x(t)-x(0))$$ If the input ...
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State-space model and/or TF for a $K$-th order linear system given the signal sequence

The discrete-time signal $x_n$ is a sum of $K$ discrete-time complex damped exponential: \begin{equation} x_n = \sum_{k=1}^{K} \underbrace{(a_k e^{j\phi_k})}_{c_k} {\underbrace{e^{\{(- \...
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Is this system linear or non linear? $y[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4})$ [closed]

$y[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4})$ If I let $(\frac{5\pi}{8}*n+\frac{\pi}{4})=x[n]$, I get the system as non linear. But one of my teacher says it is linear, whereas another teacher says it is ...
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Can we tell if a system is linear and time-invariant from its frequency response?

Given a system with a known frequency response in the S-domain. Is there a way to find whether the system is linear and time invariant? My current understanding is that we need to take the inverse ...
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What are the characteristics of a smartphone speaker?

What are the characteristics of a smartphone speaker? Whether it is linear or non-linear?
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1 answer
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FInd the amplitude function to a system

I'm studying a course in signal analysis and have come across an exercise where I find the analytical part a bit tricky. I am to find the amplitude function $|H( f )|$ for a system with the impulse ...
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Filtering a sum of cosines

The block diagram below represents a linear modulation system operating at the frequency of $1000 Hz$, $f_C = 1000 Hz$, transmitting the message $m(t) = 2\cos(400πt)$ At point B, i got the signal: $$...
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How to find a system without an input $x[n]$ is linear or non linear

The question is this: $$ y[n] = \cos\left(\frac{5\pi}{8}n + \frac{\pi}{4}\right)$$ This is what my teacher said when I asked him for help-: In any system, inputs are not given, then we have to assume ...
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2 votes
1 answer
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Why aren't the integrator and the differentiator inverse systems?

I have a statement that leads to a paradox, but I'm incapable of finding the part where I'm wrong. The integrator system $$x(t) \mapsto y(t)=\int_{-\infty}^{t}{x(\tau) \, {\rm d} \tau}$$ is a linear ...
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How to update point spread function of blind deconbolution by conjugate gradient?

There is an unblurred image $g$ and a blurred image $x$. Their relationship is expressed by the following formula using $psf$(point spread fucntion, size is $5×5$ kernel). $g = x \otimes psf\tag 1$ ...
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What is the importance of obtaining a linear signal?

I am a computer scientist who has started doing some work in the electrical engineering space – in particular, photonics. While reading about interferometric systems, I have noticed that there seems ...
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1 vote
1 answer
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Identify out signal from in signal and impulse response

Hi there I am studying a course in signal processing and systems. I was given an excecise which I am having a great deal of trouble solving. I am allowed to solved using matlab, but for top marks I ...
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Definition of frequency response

I studied various signal processing materials for a long time, and I have a question. Considering an LTI filter, one can define its frequency response by evaluating its transfer function $H(z)$ on the ...
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Mixing Kalman filter and least-squares

I'm not sure it is the right department. I try my chance I am wondering if there is a way to make a hybrid formulation of a least-square problem and a Kalman filter. Let me explain what I mean: The (...
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Zeros and poles from transfer function

I have a transfer function $$ H(z) = \frac{Y(z)}{X(z)} = 1 - 0.5z^{-1} \text{.}$$ I'm interested in zeros and poles. I know I need to adjust the function to $$ H(z) = \frac{\prod_i(z-n_i)}{\prod_i(z-...
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Difference equation to FIR filter coefficients

I have a difference equation $$y[n] = y[n-1] + 0.5x[n] - 0.5x[n-2] \text{.}$$ According to this answer, the filter should have finite impulse response. I just can't figure out how to get the FIR ...
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time invariance $ y (t) = x(\frac{t}{3})$

check time variance of this system $$y (t) = x(\frac{t}{3})$$ my solution $$y_1(t) = x_1(t/3)\\ y_2(t) = x_2(t)= x_1(\dfrac{t}{3}-t_o)$$ where $$ x_2(t) = x_1(t-t_o)\\ y_1(t - t_o) = x_1(\dfrac{t-...
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How to prove that this system is an invertible system or not?

How could i go throw proving that this system $y(t)=\int_{-\infty}^{t}e^{-(t-\tau)}x(\tau)d\tau$ is invertible system or not ?
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2 votes
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Linear system stability criteria

Suppose we have a closed-loop system $H(s)=\frac{A(s)}{1+A(s)f(s)}=\frac{A(s)}{1+T(s)}$. I've seen the stability of the system stated a couple ways: If $H(s)$ has any poles in the RHP, then it is ...
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When is a LTI IIR $H(z)$ system minimum phase

EDIT: I failed to mention that the system's inverse also needs to be causal and stable. I cannot wrap my mind around on how when a system and its inverse are both causal and stable and LTI IIR it is ...
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Signal separation

This is probably a blind signal separation problem of sorts, but it seems like it should be easier than I am finding it. Let’s say I have N time series, each of length [M x 1] that are a superposition ...
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2 votes
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Why is particular solution zero for an impulse excitation signal?

We were being taught the impulse response for a series RC Circuit- consisting of simply one resistance, one capacitor, and an impulse excitation all in series. I get that the homogeneous part of the ...
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2 answers
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Where to start with DSP?

I have a DSP exam coming up this summer and I got an abysmal mark on an assignment earlier this year. Obviously, the lecture slides are not enough for me to understand DSP, so I am wondering where can ...
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