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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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598 views

determine the impulse response of given second order difference equation

$y[n]-3y[n-1]-4y[n-2]=x[n]$ i solved it as follows : 1.first calculated natural-response(ZIR i.e,zero input response) by using characteristic equation it came out as $y_n[n]=[c_1(-1)^{n}+c_2(4)^...
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69 views

How to derivate in the frequency domain

I have two Time Domain functions, $f_1(t)$ and $f_2(t)$. I have both Fourier Transforms, $F_1(\omega)$ and $F_2(\omega)$. Functions $f_1$ and $f_2$ are not independent and, in fact, $f_1$ is also a ...
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184 views

Is $y[n]=x[n] * x[n^2]$ invertible?

Is the following system invertible or not? $$y[n]=x[n] * x[n^2]$$ where $*$ stands for the aperiodic convolution operator. I have not been able to find a mathematically sufficient argument for it...
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347 views

Inverse system of a system with integral

I am trying to find the inverse system of the following (I tried finding the mathematical inverse function but since it is not the same I am not so sure) . Can someone help me find it? $$ y(t)=\int_{...
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571 views

Fourier transform of function division in time domain

$F_1(\omega)$ is the Fourier Transform of $f_1(t)$. $F_2(\omega)$ is the Fourier TRansform of $f_2(t)$. Can I obtain the Fourier Transform ($F_3(\omega)$) of $$ f_3(t) = \frac{f_1(t)}{f_2(t)} $$ ...
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2answers
133 views

Is system $h(t) = u(t)-u(t-1)$ LTI or not?

$$y(t) = \int_{0}^{\infty} x(\tau)h(t-\tau) \ \mathrm{d}\tau$$ where $h(t) = u(t)-u(t-1)$. I have some problem with understanding whether system LTI or LTV just looking the impulse response of the ...
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84 views

The Spectrum of the Impulse Response of Linear Time Invariant (LTI) System

Consider an LTI system with impulse response $$h[n] = \frac{1}{5^n} u[n].$$ (a) Determine and sketch the magnitude and phase response $|H(\omega)|$ and $\arg(H(\omega))$ respectively. (b) ...
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363 views

For a system to be causal, number of finite zeros <= number of finite poles. Why?

I read in this pdf that for a system to be causal, the number of finite zeros must be no greater than number of finite poles. Why? I know that for a system to be causal, $h[n]=0$ for all $n<0$. ...
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1answer
139 views

Is the sum of discrete sinusoids an eigenfunction of an LTI system?

Does $$x[n]=e^{j\omega n}+e^{2j\omega n}$$ represent an eigenfunction of an LTI system?
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40 views

Confusion regarding model order and lags

I have similar questions as the one asked in these posts: https://stackoverflow.com/questions/47083890/fir-filter-length-is-the-intercept-included-as-a-coefficient-matlab/47085339?noredirect=1#...
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454 views

Is $y[n] = n x[n]$ an LTI system?

How can I test if $y[n] = n x[n]$ is an LTI system? And any other system for that matter? For example, how come $y[n] = \left( \frac{1}{2} \right)^n u[n]$, where $u[n]$ is the unit step, is an LTI ...
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1answer
527 views

Covariance matrix, Q, for a Kalman filter given the stochastic differential equation for the state of the system?

Given that I have a stochastic differential equation describing the motion of my system like so: $$ \ddot{x}(t) + \Omega_0^2x(t) - C\dfrac{dW(t)}{dt} = 0$$ Where $\Omega_0$ and $C$ are constants. I ...
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2answers
66 views

Is there any connection between frequency response of a system and spectrum of a signal?

For instance, an AR process can be generated through a white noise $w[n]$ passing through an all-poles system (IIR), $$y[n] = \sum_{k = 1}^{p}a_ky[n-k] + w[n]$$ and we can analyze the IIR system, ...
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5k views

How to determine if the system is linear or nonlinear

Given a signal $$y(n)=x(n)\cos(wn)$$ How to determine if the given signal is linear or non linear? I am getting so confused reading the theory. If someone could help step by step to determine this....
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64 views

Create an input so that poles show up as outputs?

Let's say we have a rational, causal, stable LTI system with transfer function $$H(z) = \frac{A(z)}{B(z)}$$ If $H(z)$ has $N$ poles, we can in theory have only 1 of those poles, $p_i$, show up at the ...
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388 views

Hilbert transform properties

Here Its says Hilbert transform is a non-causal,linear,and time-invariant system How can I prove it mathematically? wikipedia says the input output relation like this $$\boxed{y(t)=\frac{1}{\pi}\...
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2answers
107 views

why exponential term neglected in equation?

where does that exponential term gone, is this because it is a constant term or it has to do something with stablity?
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1answer
33 views

Question on LSI system

Why second option (1) is not correct ?
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1answer
36 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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686 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
648 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
653 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
1k 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
372 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
306 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
77 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
103 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
53 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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523 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
415 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
181 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
553 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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420 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
350 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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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
154 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
395 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
112 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
46 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
236 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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466 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
433 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
84 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
110 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
38 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 ...