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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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In EKF should Kalman Gain converge to a specific value?

I have implemented an EKF using the standard predict and update equations in order to perform state estimation of a vehicle with multiple sensors. The model has process covariance matrix $Q$ and each ...
useeeeer132's user avatar
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Why is $y(n)=Ax(n)+B$ not relaxed if $B\neq 0$?

I'm wondering why $y(n)=Ax(n)+B$ not relaxed if $B\neq 0$? The definition of a relaxed system as I've understood is that $y(n)$ does not depend on any $y$-values. So what is the definition of a ...
per persson's user avatar
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Inverse of a causal and stable system

Consider a discrete-time causal and stable LTI system $S_1$​. The inverse system $S_2$​ is defined as the system that takes the output of $S_1$​ as its input and provides the input of $S_1$​ as its ...
Nyquist-er's user avatar
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How does a time delay affect the difference equation of a LTI-system?

I'm right now working on my digital signal processing homework and among the exercises is a tough LTI-system in a canonical-like form, however right before the output is a time delay. Suppose the ...
Luxdragon's user avatar
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DT System with input initially at rest

This is the very basics of Signals but my lecturer has provided barely any notes and I cannot find answers online (not just to this, but most of signals for some reason) Dealing with DT system: y[n]=y[...
Lucy's user avatar
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Finding the impulse response of an c.t LTI system through the step response

Say we have a continuous LTI system of which we know the analytical expression of its step response. Let's call it $y_{\text{step}}(t)$. Having $y_{\text{step}}(t)$ (and therefore also $Y_{\text{step}}...
Nyquist-er's user avatar
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Impulse response of a causal LTI system without using Laplace transform

I have this differential equation that models a causal LTI system: $$ \ddot{v}(t) - \dot{v}(t) - 2v(t) = \ddot{u}(t) + 2\dot{u}(t) + u(t) $$ I was asked to find the impulse response both by using ...
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Excitation signals for system identification - Applications

I have a practical question regarding the various applications of excitation signals in identifying Linear Time-Invariant (LTI) systems. Specifically, I am curious about the usage of the following ...
Muhammad's user avatar
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Cross-correlation of two processes generated from the same signal through different LTI systems

A problem in Statistical and Adaptive Signal Processing (problem 10.15) presents two WSS signals both generated from zero-mean white Gaussian noise with $\sigma_w = 1$. They are described by $v_1(n) = ...
Matthew Faigan's user avatar
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is this system linear? causal?

$$y(t) = \int_{t_0}^t u(\tau)\, d\tau + y(t_0)$$ I have trouble determining whether this system is causal or not and linear or not. I think this system is causal because it integrates input signals ...
karma123's user avatar
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Does the impulse response of every stable system have finite energy?

I need to show that the impulse response of every stable system has finite energy. I have trouble solving this problem. Since impulse input signal is unbounded, I can't apply BIBO stability of system ...
karma123's user avatar
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If the input of the system depends on the future outputs then is the system non-causal?

While I'm aware of the fact that causality implies that the present output is only dependent on present and past inputs, something that is bugging me is what if the input is dependent on future ...
Ahan Shetty's user avatar
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truncating a system and plotting it on bode

Let system G(s) be: $$ G(s)=\sum_{i=0}^{10}\frac{(-1)^i}{(2i+1)^2}\frac{\omega_i}{s^2+2\zeta_i \omega_i s+\omega_i ^2}$$ $$\omega_i=\frac{(2i+1)\pi}{T}\, T=1\, \zeta_i=0.2$$ Its impulse response is an ...
Rice's user avatar
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Is heat equation considered as LTI system?

The heat equation $$ \frac{\partial u}{\partial t} = \alpha \nabla^2 u + f(x,t) \tag 1 $$ Here, $u$ represents the temperature field, $\alpha$ is the thermal diffusivity of the material, $\nabla^2$ is ...
AOJ keygen's user avatar
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Is $y(t) = y(t-4)+x(t-4)$ time invariant or not?

I want to check the time invariability of this recursively defined function $$y(t) = y(t-4)+x(t-4)$$ We can check time invariability of functions expressed in terms of x(t), but I couldn't find ...
Rohit Joshi's user avatar
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Doubt on LTI systems (Zero input-Zero Output)

So, I have a problem where the input and output of a system are given : \begin{align} x(t) &= \sin(10t) \\ y(t) &=5\cos(10t+\frac{\pi}{6}) \end{align} So, I need to determine whether this ...
Naveen's user avatar
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How to find impulse response for the given system?

How can I find the impulse response for the following system in time domain? I actually would like to find my mistake in my attempt. Below is what I have tried according to the answer given for this ...
Lars Smith's user avatar
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Correlation between two signals

I am having an input signal x[n]=[1 2 3 4 5 6] and another signal y[n]=[2 3 4]. In order to check whether y[n] is present in the signal x[n],I did cross correlation and the result obtained was [ 0 0 0 ...
Aami's user avatar
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How to design a system that stretches input by factor of 2?

In general, How do we design a system that takes input x(t) and outputs x(t/2)? I know that if system outputs x(t/2) then the frequency domain of output signal has the form X(2f). But what should be ...
user133933's user avatar
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Linearity of a system to biomedical applications

Suppose $x(t)$ is temperature and $y(t)$ is sweat. If the following equation describes the system, answer if is it linear or not. $y(t) = [H(x)](t)$ . I would say that it isn't linear because we dont ...
Panos's user avatar
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Interpreting eigenvalues of non-normalized covariance matrix of time-series measurements

Cross-posted from physics stackexchange Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). ...
user3716267's user avatar
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Linear System: Symmetric Under Time Reversal?

In class, my professor mentioned that "Linear systems must be symmetric under time reversal" in an off-handed way and did not clarify further. I assume this is true, but I'm not sure how one ...
Camellia99's user avatar
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What math should I study to really understand signal processing?

I am reading an elementary book on signal processing - "Signals and Systems". It never struck until recently, the math involving signal processing seemingly has a lot more depth than the ...
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Output of time-continuous linear system with a phase shifter as impulse response

I stumbled upon a false/true statement which goes: A time-continuous linear system, whose impulse response $c(t) = \frac{1}{\pi t}$ has a pole at the origin, always produces an output signal $y(t)$ ...
Yevtee's user avatar
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How to find time-varying impulse responses?

Given is a system that can be described as $y(t) = x(t)\cdot \sigma(t)$ with $\sigma(t) = \left\{\begin{array}{ll} 1, & t \geq 0 \\ 0, & t<0\end{array}\right. .$ The output of a ...
Kaiser F's user avatar
1 vote
1 answer
149 views

Continuous-time convolution of signals with negative amplitudes

While preparing for a mid-term exam, I encountered negative amplitudes for the first time while convolving two signals. I've already solved the problem, but my result and results from others conflict ...
kesetovic's user avatar
1 vote
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Why lag compensator is preferred over PI for sinusoidal reference?

In this post, LJSilver mentioned that a PI compensator is not appropriate for a constantly changing reference, such as a sinusoidal waveform. In this scenario, a lag compensator is considered the ...
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how does steady-state error decrease as the pole of the compensator moves closer to the origin?

The steady-state error improves when the pole moves closer to the origin, as seen in lag or integral compensators with step input. Is there an intuitive explanation for this phenomenon?
internet's user avatar
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Function Transfer of Block Diagram Discrete System

I need to find out the transfer function in Z from this diagram, how can I extract this information from this diagram Obs: Doing the math, my H_z gave, H_z = 1 + 0.5z^-1 + 2z^-2 + 2z^-3 + 0.5z^-4 + z^-...
Pedrimbus's user avatar
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1 answer
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Is a PI-controller considered an LTI system?

Is a PI-controller considered an LTI system? Intuitively it seems that the integral part would break the time-invariant requirement requirement, because the output depends on how wound up the ...
K0ICHI's user avatar
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1 answer
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Are complex exponentials the only eigenfunction for arbitrary LTI systems?

After reading a few posts, like this. I know that arbitrary LTI systems always have complex exponential eigenfunctions. And that for specific LTI systems you can also have other types of ...
roobee's user avatar
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System Identification Using Sinusoidal Inputs

I have a system I would like to model using experimental data. I input several sinusoidal signal and measured the outputs. I can vary the frequency and the amplitude of the input signal: Input 1: $A \...
takfuruya's user avatar
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What is the adjoint of a linear operator and why is it useful?

The concept of linear operators and their adjoints arises frequently in some corners of signal processing, but is not particularly well documented, at least from a signal processing perspective (you ...
Gillespie's user avatar
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random signals through LTI systems, why are these two signals joint wide sense stationary?

I’m trying to solve this problem but I don’t understand an assumption the solution makes: The question: let $\hat{W}$ be the best linear approximation of $W_t$ out of $Y_t$, find $\text{CoV}(W_4, \...
Piratemetaldrinkingcrew's user avatar
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Equipment to test my theory on time varying systems

How can I make a linearly time varying system to test my theory? Can I ask for a PID microcontroller available from my university such that $K$ is gradually increased from 0 to $c$
Volpina's user avatar
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2 votes
3 answers
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Sine as input to an LTI system

Everywhere in theory (books , online) there is this statement "For sinusoidal inputs, any LTI has a sinusoidal output with the gain of $|H(s)|$, the same frequency, and a phase shift equal to $\...
tonythestark's user avatar
7 votes
2 answers
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If the convolution of two signals is a unit impulse, what does this tell us?

I have two discrete-time LTI systems whose transfer functions satisfy $h_1[n] * h_2[n]= \delta[n]$. We also know that system 1 is causal and stable. Does $h_1[n] * h_2[n]= \delta[n]$ tell us anything ...
tonythestark's user avatar
1 vote
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386 views

Why does convolution give the output of a passing a signal through a filter?

I have a rudimentary understanding of Convolution, the Convolution Theorem and why the output z(t) of an LTI system can be found using the convolution of input signal x(t) and the impulse response h(t)...
SNIreaPER's user avatar
1 vote
0 answers
306 views

How to recover the LTI system step response by the known output and input signals?

Having the input signal as a step-like pulse and the output as its distorted version after passing through the system: is it possible to somehow recover the step response of the system? In the Figure ...
Curious's user avatar
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2 votes
1 answer
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Why do they say that complex exponentials are eigenfunctions of LTI systems, when there are still transient responses?

Let $$\dot{x} = Ax+Bu$$ $$y = Cx + Du$$ be a linear ODE with $x(0)=0$. Here, I am assume $A$ is invertible. As you can see, the relation $$H:u(.) \mapsto y(.),$$ where $(u(.),y(.))$ is a solution to ...
Spencer Kraisler's user avatar
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How to find system output by its step response?

Inspired by this post, I tried to reproduce the procedure described in the answer in Python considering rectangular pulse: ...
Curious's user avatar
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Magnitude spectrum of LTI system output signal

First year student so please excuse my lack of knowledge. As i understand i need to use convolution which is: $$ y^{out}(t) = u^{in}(t) * h(t) $$ Or maybe my thoughts are wrong so please correct me. ...
Illia Zakharov's user avatar
2 votes
1 answer
136 views

State space transformation

I have some governing equations of the form: $$\begin{align} \ddot \theta(t) &= \frac{MgL + mgl}{J} \theta(t) + \frac B J \dot x(t) - \frac \alpha J V + \frac {mg}{J} d - \frac{c_1}{J} \dot \theta(...
skylinev2's user avatar
3 votes
1 answer
343 views

How can I show that an LTI system can be expressed as a difference equation?

I'm in the process of re-learning DSP (not a subject I've visited since University) and in quite a few resources I see this general form of a DT-LTI difference equation: $$y[n] + a_1y[n-1] + a_2y[n-2] ...
andowt's user avatar
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1 vote
1 answer
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Effect of BIBO-Instability on the frequency response of a ideal LPF

I recently came across this post stating that the continuous ideal LPF is BIBO-unstable since the impulse response is not absolutely integrable, and this post stating some examples. I have been trying ...
Finn Heijink's user avatar
2 votes
1 answer
88 views

LTI system tradeoff between gain, bandwidth, and delay

For first-order LTI systems, the gain-bandwidth constant is often discussed. I've seen the claim that in general, gain and bandwidth don't directly trade off with each other as much as delay. For an ...
Halleff's user avatar
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2 answers
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Routh's stability condition

Assume we have a LTI system which has poles in the half left plane of the s domain. Before I learnt Routh's stability condition I had imagined that this was enough to decide whether a LTI system was ...
Volpina's user avatar
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Verifying Linearity and Shift Invariance Under Summation

I am having some trouble working through verifying linearity and shift invariance when the transformation is under a summation. The given transformation is as follows: $$ y(m,n)=\sum_{i=-1}^{i=1}\sum_{...
pflykyle's user avatar
1 vote
2 answers
461 views

Calculating transfer function of a linear time varying system?

If we excite a LTI system with the Dirac delta $\delta(t)$, the system outputs the impulse response $h(t)$. For a LTI system, it doesn't matter when we excite the system with the Dirac delta, we will ...
Volpina's user avatar
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0 votes
1 answer
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Max input of a system given it's transfer function and an assumed step change (beginner)

I have an exercise that gives me the following transfer function $$ \frac{0.5}{s+0.5} $$ and an assumed step change in the target of 20 I am asked to calculate the maximum input for the assumed step ...
Tom Callan's user avatar

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