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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1answer
221 views

Changing limit of integral make time-variant to time-Invariant system

$S_1:x(t)\longrightarrow y(t) = \int_{-\infty}^{3t}{x(\tau)\,d\tau}$ $x(t-t_0)\longrightarrow y_1(t) = \int_{-\infty}^{3t}{x(\tau-t_0)\,d\tau}$ $= \int_{-\infty}^{3t-t_0}{x(z)dz} \neq y(t-t_0) $ ...
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1answer
695 views

Linearity, Causality and Stability of a System

Consider a system: $$ y[n] = y[n-1] + u[n], $$ where $y[n]$ is the output and $u[n]$ is the unit step function. Is this system causal, linear, time-invariant and stable ? My attempt at the ...
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442 views

A system that perfoms Fourier Transform operation - is it an LTI system?

If a system takes input as the time domain signal and outputs the frequency domain signal, is such a system an LTI system? For if the input time domain signal can be represented as a linear ...
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1answer
441 views

Convolution of $\sin(\omega t)$ and $\cos(\omega t)$?

If $x(t)=\sin(\frac{\pi t}{4})$ and $y(t)=\cos(\frac{\pi t}{4})$ then i need to find the Convolution $$z(t)=x(t) \circledast y(t)$$ So convolution will be $$\begin{align} z(t) &=\int_{-\infty}^{...
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55 views

follow up to previous question “ is this an impulse response or step response”

This question is a follow up to the gorgeous explanation of how the leaky integrator model becomes a differential equation. The link to that question and answer is below. should this be viewed as an ...
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2answers
159 views

should this be viewed as an impulse response or step response

I'm trying to teach myself the relation between simple discrete ODE's and the impulse response-step response concept. Getting back to the question: I don't expect anyone to read the whole thing but I'...
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1answer
473 views

Exponential decaying step response in LTI System

I'm attempting to better understand the relationship between step responses, impulse responses, and convolutions. Say that I have a system where if I apply a constant input, my output decays from a ...
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1answer
124 views

Response of Linear System to Stochastic Process

Somehow I am getting the variance{u(n)} equal to '0' !! This is the case when I take the coefficient 'a' as real. As it is not mentioned in the question I need to find the solution to this question ...
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2answers
3k views

Given the input and the output, how to determine the impulse response?

I would like to find the impulse response, $h[n]$, of an LTI system given the input $$x[n] = [1,-3,2]$$ and the output $$y[n] = [1,-1,-4,4]$$ I know that $y[t]=x[t]*h[t]$, but I am having hard ...
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47 views

Unsure if my unit step function is correct in relation to a diagram [closed]

Here is my function: $$3(t)(u(t)-1)) + (u(t-1)+3) - (u(t-3)+3) - 2(t)(u(t-3)+3) + u(t-4) -2(t)(u(t-6))$$ Is my function correct? If not, could you please tell me where I am wrong. Thank you in ...
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1answer
52 views

Is my answer incomplete? Checking the stability of a system

Yesterday, during my exam, I had the following exercise: Given $$H(s) = \frac{1}{s^2+2s+4}$$ check if it's stable. which was supposed to be the hardest (since it was the last one). From my ...
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1answer
276 views

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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505 views

BIBO stability of $y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$

How can I prove that the LTI system with (output $y(t)$, input $x(t)$) $$y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$$ is BIBO (bounded-input/bounded output) stable?
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1answer
593 views

LTI system $y(t)=x(t-T)$ with or without memory

How can I verify if the LTI system ($T$ is a real number, $y(t)$ output, $x(t)$ input): $$y(t)=x(t-T)$$ is with or without memory? I know that, if $y(t)$ depends only on $x(t)$, the LTI system will ...
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0answers
88 views

Why prediction by AR model causes time lag

Why does prediction by AR model cause a time lag? Please tell me why theoretically.
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1answer
112 views

Analyzing a particular discrete-time LTI system for input signal $x[n]=(1/3)^n$ for *all* $n$

I'm considering the following problem from some course notes. Suppose the following is known about a discrete-time LTI system: Given the input $x[n]=(1/3)^n$ for all $n$, the system ...
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3answers
1k views

Why cosine is not an eigen signal?

According to this website: If the output of a system has the same type as its input signal, then the input signal is referred to as the eigen function of the system. but in this question it is ...
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1answer
46 views

Time Delay Margin

I have a question regarding the time delay margin. I know the definition of time delay margin now I want to know that for the stability of the system is it good to have large time delay margin or ...
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1answer
775 views

What is $H_2$ and $H_{\infty}$ control? [closed]

I can create a Linear Quadratic Gaussian Integral(LQGI) controller very easy by using GNU Octave. LQGI is in the area of Optimal Control Theory. But there is something called Robust Control Theory. ...
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0answers
107 views

Check if I has right: Is this the Extended Kalman Filter

I have learn the Kalman-Buncy filter for the LQG controllers. I know that this is a signal processing forum and not robotics not math forum. But Extended Kalman Filters are daily discussed here. ...
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1answer
66 views

Extended Kalman Filter in mechanics, electronics and hydraulics?

Extended Kalman Filter is most used in GPS and navigation systems. But how much do I gain to switch to Extended Kalman Filter to the linjear Kalman Filter if I do LQGI controller for the industry? ...
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1answer
183 views

What is the spectrogram of an impulse response used for?

I saw that in acoustics an estimated impulse response of a room can be transformed using STFT/wavelets to a spectrogram as an analysis tool. However, I did not understand how the spectrogram is ...
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6answers
355 views

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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1answer
188 views

Can I assume system is LTI when given by DTFT of impulse response

I'm having hard time to grasp it probably because i don't fully understand it. I understand that when a system is given by $h(t)$ (in general $h(t-\tau)$) i can assume that it is a LTI system. So i ...
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3answers
507 views

Help in understanding the formula of Signal-to-Noise-Ratio (SNR) - Part 1

Question 1: Consider an Autoregressive model : \begin{align} y[n] &= y[n-1] + x[n]\\ z[n] &= y[n] + w[n] \end{align} where $y$ is the output observation, $x$ is a random input and $w$ is ...
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383 views

LTI system with Laplace transform

Given the input $$x(t)=u(t)$$ and the corresponding output signal measured as $$y(t)= 2 e^{-3t} u(t)$$ determine the impulse response $h(t)$. This what have done so far: $$ h(t)= \mathscr{L}^{-1} \...
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62 views

Linear Systems, Sparse Solutions, and $4 \times 4$ Sudoku Algorithm [closed]

I am unable to understand the paper Linear Systems, Sparse Solutions, and Sudoku. I have to form a $4 \times 4$ Sudoku using the algorithm in this paper. Can somebody please provide me the algorithm ...
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3answers
119 views

Linear difference equation [closed]

Give the equation $$y[n]-y[n-1] +\frac14y[n-2]= x[n]$$ With initial conditions $$y[0]=0\\ y[1]= 0$$ Find the step response. I have been able to get the solution $$y[n] =C_1(\frac12)^n +C_2(\frac12)^...
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1answer
330 views

Steady State of a Difference Equation using MATLAB

I'm given this difference equation, and I'm being asked to plot its response to an input $x[n] = 4\sin^3\left(0.25\pi n+\frac \pi 3\right)u(n)$: $$ y(n) = 0.9051y(n-1)-0.598y(n-2)+0.29y(n-3)-0.1958y(...
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1answer
229 views

Step response of a differentiating system

Consider I can find the step response of a system with the following method: We can easily find the step input of a system from its transfer function. Given a system with input $x(t)$, output $y(t)$ ...
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1answer
225 views

Under what conditions do the phase margin and Nyquist criteria give the same results?

When designing feedback systems, I often evaluate stability by thinking about phase margin: the closed loop system $$T(s) = \frac{L(s)}{1+L(s)}$$ is stable if $L(s)$ has positive phase margin, i.e., $...
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1answer
52 views

Notation of an LTI system consisting of LTI filters

I would like to find a reference for two notations of an LTI system consisting of LTI filters. In z-domain, the LTI system is given by $$ \mathbf{y}(z) = \mathbf{C}(z) \mathbf{s}(z) + \mathbf{D}(z) \...
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1answer
64 views

Is applying a linear operation to the input and output of a continuous linear operator OK?

The scientist and engineers guide to digital signal processing, Steven W. Smith, p. 134-135 states: [...] imagine a linear system receiving an input signal, $x[n]$. and generating an output signal, ...
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1answer
247 views

Is ideal LPF a LTI system? And is it time invariant?

Ideal filter time-domain impulse response is $$h(t) = 2B \operatorname{sinc}(2Bt)$$ It is not causal nor BIBO stable. Is it time invariant? And how can i prove it?
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811 views

Why does reversing the order of these two transfer functions give me different outputs?

Consider these two systems: \begin{align} &u\ {\longrightarrow}\boxed{s}{\longrightarrow}{\boxed{\frac 1s}}{\longrightarrow}\ y\\ &u\ {\longrightarrow}\boxed{\frac 1s}{\longrightarrow}{\boxed{...
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1answer
1k views

Determining if the given system is LTI

Problem Given the compound system below, with the input $x(t)=\operatorname{sinc(t)}$, the output of A is $y(t)=\operatorname{sinc(2t)}$ and the output of B is $z(t)=\operatorname{sinc(t)}$, ...
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2answers
549 views

Is a filter/control transfer function with positive phase “causal”?

In control we often use transfer functions with positive phase, i.e., a "lead filter" has transfer function $$G_c(s) = \frac{\alpha \tau s+1}{\tau s+1}$$ (with $\alpha>1$). Since the zero occurs ...
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1answer
158 views

Create distortion from basic linear (and non-linear if neccessary) DSP elements

I'm studying mechatronics and I'm intrested in DSP basics. My lecturer said that there are four basic linear DSP elements: Adder (and other mathematical operations) Amplification (shown on diagrams ...
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1answer
330 views

In the context of transfer functions, what is the relationship between the terms “proper”, “causal”, and “realizable”?

I am thinking about these terms in the context of linear control. A transfer function is proper if the degree of the numerator is not greater than the degree of the denominator. I've read often that ...
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2answers
485 views

Control design: under what conditions can closed-loop poles be placed arbitrarily?

Say we have a single-input linear system $\dot{\mathbf{x}} = A\mathbf{x}+Bu$. With full-state feedback ($u=-G\mathbf{x}$), it is straightforward to arbitrarily place the $n$ closed-loop poles (i.e., ...
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99 views

Why is it necessary to average signals with filters?

As someone who is new in working with filters, why do signals need to be averaged? What are the advantages, by performing filters on signals? As I know, there are different types of filters for ...
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2answers
168 views

Why do we assume zero mean noise in sensor data?

I am reading a paper on measuring respiratory patterns from video data. In defining the model, the authors formulate the problem mathematically as: $x_i(t)=h_i(t) \ast g(t) + n_i(t) $ Where $n_i(t)...
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0answers
64 views

identifying overshoots in given state space system with step inputs

Suppose I have the following state space system: $$ \dot{x}(t) = Ax(t) + Bu, \quad y(t) = Lx(t), \quad x(0) = x_0 $$ where $A$, $B$ and $L$ are real matrices, $u$ is a constant real vector (so that ...
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1answer
672 views

Differential equations and LTI systems [duplicate]

I've seen in many textbooks on Signals and Systems that an LTI (Linear Time-Invatiant) system can be described as a constant-coefficient linear differential equation, such as $$\sum_{k=1}^N a_k \frac{...
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1answer
977 views

Stability of a logarithmic function and system

A system such that $$y[n] = \log_{10}(|x[n]|) $$ was given as a stable one. I am not able to understand it though. When $x[n] = 0$ (which is bounded), the output $y[n]$ tends to get unbounded. So ...
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1answer
279 views

Linearity and shift-invariance of 2-D system on lattice

I know how to check a 1-D system for these conditions, but am confused about translating this to a 2-D system over a lattice. The system $H$ is define as: $$w[\mathbf x] = H\left\{u[\mathbf x]\right\}...
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1answer
1k views

Using the linear operator to check for time invariance of a differential equation?

I have a differential equation $$\frac{d^2y(t)}{dt^2}+y(t) = \frac{dx(t)}{dt} + x^2(t)$$ and I need to see if this system with input $x(t)$ and output $y(t)$ is time invariant and linear. I tried to ...
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3answers
829 views

Deconvolution and Polynomial division

I found this comment inside MATLAB's deconv.m function Deconvolution and polynomial division are the same operations as a digital filter's impulse response $B(z)/A(z)$. What does this statement ...
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0answers
53 views

Variance and Co-variance of a Linear Forecast

Consider a linear forecasting problem where all shocks $\{\epsilon_i\}_1^n$ are independently distributed with $\epsilon_i\sim N(0,\sigma_i^2)$ for all $i$. Suppose you want to forecast $\theta = \...
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373 views

BIBO stable LTI system frequency response for this input signal?

For an LTI system with bounded input and bounded output, I have the input $$x(t) = 5 + \cos(12t+\pi/4)$$ and output $$y(t) = 6\sin(12t)$$ It is said that the magnitude of the frequency response $|H(...