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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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35 views

why this frequency response is added to the power of e?

From where did this ${\angle}H{(\Omega)}$ came into power of $e$ ?
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5answers
560 views

Why does the Kalman filter remove only Gaussian noise?

What and where in the derivation of the Kalman filter is the assumption of Gaussian noise? Why and how does this assumption help?
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1answer
532 views

causality of the system $y[n] = x(2n)$

Can somebody please tell me why the system $y[n] = x(2n)$ is non-causal ? I know that causal systems depend on the past and present values of input and this system satisfies the condition. So why is ...
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2answers
542 views

Is this system time invariant?

I've been working at this problem for a while now, and can't seem to come to a solid conclusion - is this system time invariant? $y(t) = \int_{-\infty}^{t} e^{-9(t-\tau)} x(\tau)d\tau $ My reasoning ...
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1answer
2k views

How do I calculate the step response of a discrete-time system?

I am given the impulse response $$h[n]=2\left (\frac{1}{2} \right )^n\sigma [n]-\delta [n]$$ where $\sigma$ is the unit step function and $\delta$ is the unit impulse function. I know that $$a[n]=\...
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1answer
932 views

Impulse response from difference equation without partial fractions

I have a system with input $x[n]$ and output $y[n]$, described by the difference equation $$y[n] - 0.3y[n-1] + 0.1 y[n-2] = 3x[n] - x[n -1]$$ and I am trying to find the frequency and impulse ...
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1answer
300 views

Question on convolution

I am stuck at question number 2.8 This is how I have gone about solving it: I have calculated $y(t)$ by convolving $x(t)$ with $h(t)$ using the fact that $x(t)$ convolved with an impulse at $t=t_0$ ...
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1answer
266 views

Solving a second order difference equation

I need to solve the following 2nd order difference equation $$3y[n]-y[n-1]+y[n-2]={\bigg(\frac{1}{2}\bigg)}^{n} u[n]$$ where $u[n]$ is Heaviside's step function. I started by trying to find the ...
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2answers
74 views

whether the system is linear or not for the given problem

Given the system: $$y(t)=x(t+1)+x(t−1)$$ is the system linear? For a system to be a linear first it should satisfy zero input and zero output. How can we calculate output at 0 input if the system ...
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1answer
97 views

Overshoot computations not matching

Take the following transfer function of a 3rd order system: $$H(s)=\dfrac{2.302~s+0.3548}{s^3+0.739~s^2+3.223~s+0.3548}$$ with poles: ...
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1answer
51 views

Can linear transforms of vector space be seen as LTI systems? [duplicate]

Linear transforms of vector spaces has the linearity (i.e., homogeneity and superimposability), which is shared by LTI system. There are also shared concepts such as eigenvectors/eigenvalue... I am ...
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1answer
459 views

Time Variant or Time-Invariant system?

$y(t)=\mathcal{T} \{ x(t) \} = \int_{-\infty}^{x(t)} x(\tau) d\tau$ Is the above system time invariant ?
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1answer
346 views

Is this system invertible or not?

Prove that the following system is invertible. $$y(t) = \mathcal{T}\{x(t)\} = \int_{-\infty}^{3t} x(\tau) \,\mathrm d \tau$$ Answer: yes, the system is invertible. I need some hint here, not the ...
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1answer
145 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
424 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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2answers
401 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
291 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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1answer
50 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
141 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
322 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
102 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
2k 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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1answer
43 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
51 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
210 views

Estimate a transfer function from ARX models - Is ARIMAX better?

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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2answers
418 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
316 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
76 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
109 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
974 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
37 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
491 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
92 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
65 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
142 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
320 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
162 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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0answers
157 views

How to determine if a system is LTI from the frequency response function?

I was told that the FRF should be a linear function of $\omega$ for the system to be linear. So for a harmonic oscillator, with an FRF given by $$H(i\omega) = \frac{1}{\sqrt{\left(\omega_0^2 - \omega^...
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3answers
454 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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2answers
293 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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0answers
59 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
117 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
274 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
185 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
170 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
51 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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0answers
24 views

Is there a difference between system frequency response and system transfer function? [duplicate]

Is the system 'frequency response' the name given to the DTFT of the impulse response? And the system 'transfer function' the z-transform of the impulse response? Or can these terms be used ...
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1answer
63 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
185 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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2answers
665 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{...