Questions tagged [differential-equation]

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How to prove this system is not causal?

I set $x_1(t)=0$ and use the final rest property to get $y_1(t)=0$, and $y_2(t)=0$ for $t>1$. However, I cannot find general $y_2(t)$ for $x_2(t)$.
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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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Finding inverse $Z$ Transformation of a LTI system and switching of input output signal regarding to difference equation

I have 2 stuffs that are bothering me: I have the equation: $y[n]=8x[n]-2x[n-2]-x[n-4]$ When $x[n]$ input and $y[n]$ output. Now, the system is swapped, $x[n]$ output and $y[n]$ input The difference ...
Ben Shaines's user avatar
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1 answer
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Not getting the same step response from Laplace transform and it's respective difference equation

I am trying to simulate a plant on a microcontroller. The transfer function of the plant is $$ G_{p} \left( s \right) = \frac{2}{\left( s + 3 \right) \left( s - 1 \right)} \tag{1} \label{1}$$ The step ...
Diptopal's user avatar
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Block Diagram 2nd Butterworth Low-pass

I would like to represent a second-order butterworth low-pass filter in a block diagram, how can I get started? As per the attached image
Natan Vaz's user avatar
1 vote
1 answer
133 views

Impulse response of an LTI system

I want to find the impulse response of the linear time invariant system given by $y''(t)+4y'(t)+3y(t)=x'(t)+2x(t)$ using time domain analysis. As solved in the textbook "Continuous and Discrete ...
kobaltishblue_312's user avatar
1 vote
1 answer
78 views

Show that decomposition does not hold for non-linear system

The solution to an inhomogeneous differential equation can be split up into homogeneous solution and a particular solution (forced response). Another way to split up the solution to an inhomogeneous ...
Carl's user avatar
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Fit a harmonic oscilator via step response

Suppose we are studying a dynamic system believed to be well modeled via $$ x'' + \frac bm x' + \frac cm x = \frac Fm$$ $x''$ - acceleration $b$ - damping factor $c$ - spring rate $F$ - external force....
Vorac's user avatar
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1 answer
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Solving for the Kernel of a system using impulse balancing with diracs delta's derivative

I wanted to practice Impulse balancing and tried using it to solve this IAR system in one go as opposed to using its LTI attributes. The exercise wants me to find the impulse response of the system: $$...
Piratemetaldrinkingcrew's user avatar
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When $x(t)$ and the output $y(t)$ are related by linear differential equations, why is the system unstable when $M>N$?

This is two linear differential systems equating each other. \begin{equation} \frac{d^Ny(t)}{dt^N}+a_1\frac{d^{N-1}y(t)}{dt^{N-1}}+\cdots+a_{N-1}\frac{dy(t)}{dt}+a_Ny(t)=\\ b_{N-M}\frac{d^Mx(t)}{dt^M}+...
JellyTree's user avatar
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Why Is PDE Based Image Processing Not as Active as It Used to Be?

I can see from the literature that PDE-based image processing (especially inpainting) was a "hot topic" at one point. Why does it not seem to be an active area of research anymore, yet it ...
KaRJ XEN's user avatar
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How to solve this degree $2N$ polynomial equation of filter's cut-off frequency?

In a paper I am interested in, the following equation is given to determine the cut-off frequency $\omega$ of a filter: $$t^{2N}+\left[-2\left(1-\frac{1}{2Q^2}\right)\right]t^N +\left[-\left(\frac{|H(\...
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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-...
Riesz98's user avatar
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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 ...
Joako's user avatar
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Why is this minimum cost function returning an insane cost of registration value?

I have a sample program that uses the minimum cost function to align image B to image A. The method takes two arrays of points and does the summation of all the points in the images for both Matrix A ...
Dong Li's user avatar
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1 answer
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Difference equation from a transfer function of a low pass filter

I need to get the difference equation from this transfer function: $H(z) = g \frac{1+a_1}{1+a_1z^-1}$ My math skills are too many years old, but I remember I need to get the Y(output) on one side and ...
pizzafilms's user avatar
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0 answers
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Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $f(t)$ to have a bounded maximum slew rate?

Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $f(t)$ to have a bounded maximum slew rate? I want to know about which conditions must fulfill a real-valued time-...
Joako's user avatar
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2 answers
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Why does Simulink generate this code for a PID controller?

For the Simulink PID Controller model The Simulink generated code (rewrite for better understanding) is: ...
Jackoo's user avatar
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1 answer
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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 ...
Harsh's user avatar
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Square pulse test of Upwind Finite Differences

I`m analyzing the numerical methods for the 1D convection equation for stability, consistency, and accuracy. I want to see How does this square pulse move in domain and time? Here is my code % ...
Abdelrahman Mabrouk's user avatar
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1 answer
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RC high pass filter differential equation

I have the question to derive: $$RC \frac{dy(t)}{dt} + y(t) = RC \frac{dx(t)}{dt}$$ which describes a high pass filter or $CR$ circuit with $x(t)$ the input and $y(t)$ the output. I am really ...
Joesphtimeheist's user avatar
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Causal and Non-memoryless LTI sytems described by LCCDE

I was wondering about the nature of stable systems (in the BIBO sense) that are causal with memory for which we wish to represent them by LCCDE (if they may exist). How frequent do LCCDE exist such ...
SPARSE's user avatar
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2 answers
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From transfer function to differential equation

I have the below detailed solution (boxed in blue) that I don't understand completely: I can reconstitute the differential equation from: $$ (1+Ts) X(s) = K_v U(s) $$ $$ x(t) + T\dot x(t) = K_v u(t) $...
Dovendyr's user avatar
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1 answer
990 views

Particular Solution to Difference Equation

I have a system given by $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$ I want to solve for $y[n]$ for $x[n]=(\frac{1}{2})^nu[n]$. The complementary solution evaluates to $[k_1(\frac{1}{2})...
Orpheus's user avatar
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0 answers
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Augment State Space Model with Derivative of State

Problem How do you augment a state space model with the derivative of a state? I know how to augment a state space model with the integral of a state by doing the following. Given a linear system $$ \...
eball's user avatar
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Parameter tracking using Augmented state vector approach and unscented Kalman filter

I'm trying to reproduce and extend figure 9 results in Nonlinear dynamical system identification from uncertain and indirect measurements"HU Voss, J Timmer, J Kurths - International Journal of ...
ccc's user avatar
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1 answer
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Stability of a system in time-domain

Consider a system for which the input $x(t)$ and output $y(t)$ are related by the differential equation $$\frac{d^2y(t)}{dt^2} + \frac{3}{2}\frac{dy(t)}{dt} - y(t) = x(t) \tag{1}$$Determine $h(t)$ for ...
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1 answer
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How does the intuitive notion of causality fit in with control systems?

Edit: By causality, in this question, I do not mean the traditional mathematical definition in the theory o signals and systems; I mean causality as in an intuitive 'what's moving/pushing what notion'....
Mohamad Hussein Naim's user avatar
1 vote
1 answer
694 views

Confusion in initial condition of differential equation using Laplace transform transform

I'm confused in solving linear constant coefficients differential equations (LCCDEs) by Laplace transform if initial conditions are given at time just before $t=0$ just after $t=0$ exactly at $t=0$ ...
user215805's user avatar
3 votes
1 answer
2k views

Impulse response if initial conditions are given

In most books, impulse response of LTI systems are calculated by assuming initial rest condition, but how to calculate response of an impulse input if there are some initial conditions present ? My ...
user215805's user avatar
1 vote
1 answer
460 views

Finding the system output by convolution

pretty new to this concept, so do bear with me. A linear dynamic system is described by the following differential equation: Transfer function H(s) is calculated to be = I've already found the ...
Jisbon's user avatar
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Output Noise Power for ODE

My system is described by the following ODE: $$ \frac{\mathrm{d}y(t)}{\mathrm{d}t} = a-y(t)x(t) $$ Where $a$ is a constant and x(t) is a poisson process so that: $$ E[x(t)x(s)] = qI\delta(t-s) $$ ...
Knyq's user avatar
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1 answer
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Is a LTI system with dead-time equivalent with a LTI system with no dead-time but delayed input signal?

I wonder if the system parameters would remain the same after all. According to the definitions, the gain is the change of the output induced by the change of the input if we assume that the gain is ...
Sam Farjamirad's user avatar
1 vote
0 answers
93 views

Is non-causal, non-LTI solution to difference equation correct?

Find a solution to the difference equation $y[n]-\frac{5}{6}y[n-1]+\frac{1}{6}y[n-2]=\frac{1}{3}x[n-1]$ that is neither casual nor LTI, where $y[0]=y[1]=1$ and $x[n]=\delta[n]$ The homogenous ...
Shukant Pal's user avatar
3 votes
1 answer
98 views

Proof for the solution of homogenous difference equation

Suppose the output $y_h[n]$ of a linear, time-invariant system is described by by the following equation for input $x[n]=0$, $$\sum_{k=0}^{N}a_k y_h[n-k] = 0$$ My book states the $y_h[n]$ is in fact ...
Shukant Pal's user avatar
0 votes
1 answer
115 views

Difference equation with variable coefficients in MATLAB

Given the following difference equation $$y[n]-\left(\dfrac{n}{n+1}\right)y[n-1] = n\cdot x[n]$$ How can we use MATLAB to solve it? I know if the coefficients are constant we can simply use ...
Morcus's user avatar
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1 vote
1 answer
2k views

Determine the system function H(s) of a system and find out the differential equation

I have created the following system for practice purposes. From this system I want to determine the system function H(s). In the picture I have worked with auxiliary (dummy) variables, which should ...
P_Gate's user avatar
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4 votes
2 answers
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LCCDE in simple words?

What is LCCDE?I only know its abbreviation/full form :linear constant-coefficient difference equation I know that in s domain we have differential equations and in z domain we have difference ...
DSP_CS's user avatar
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4 votes
1 answer
171 views

Damped spring mass system - parameter estimation

I've tried to calculate the parameters of a damped spring mass system of the form $m~ y''(t)+d~y'(t)+c~y(t)=F(t)$ but I have some problems determining the mass m of the system. The damped ...
Phobos's user avatar
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3 votes
1 answer
497 views

Transform a Continuous Time Differential Equation of a Low Pass Filter to Discrete Time

Assuming I have a linear differential equation of first order low pass filter: $$RC*y(t)'+y(t) = x(t) $$ And Assuming i know now that the input x(t) is being sampled at a rate of $T_s$ how can now ...
user3921's user avatar
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