Questions tagged [laplace-transform]
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242 questions
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Is it possible to approximate non-causal time delay?
I have been struggling with a problem regarding a digital control theory. Namely it's a feedforward compensation for a system with a transport delay $T_d$. The problem is that in such a case the ...
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How to exploit feedforward compensation for a system with transport delay?
I have a combination of a feedback control loop and feedforward compensation
where $\tilde{E}$ represents a fluctuation of the system input (namely a input voltage of a dc-dc converter) which is ...
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33
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Confusion about a symbol in Chen's Linear System Theory and Deisgn
In pg. 123 of Chen's Linear System Theory and Deisgn 3rd edition, Theorem (5.2) states:
What is the expression after $\sin$? is this a typo?
Later he writes, again using this symbol:
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What is relationship between the Laplace transform of the ideally-sampled signal and that of the original continuous signal?
Suppose a continuous signal $x(t)$, the Laplace transform of $x(t)$ is $X(s)$. Suppose the ideally-sampled signal of $x(t)$ is $\hat{x}(t)=\sum\limits_{n=-\infty}\limits^{\infty}x(nT)\delta(t-nT)$, ...
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How is causality in Laplace transform related to Fourier transform?
Taking the Laplace transform of a system given by a differential equation yields its transfer function $H(s)$. The region of convergence of the causal impulse response of the system lies right of the ...
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181
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Relation between the damping ratio and the phase margin in 2nd order systems
I can't remember how to derive the relation between the damping ratio $\zeta$ and the phase margin. I cant remember where to start from to end up with a numerical relationship between them for a ...
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45
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Partial derivative of transfer function in Laplace domain
For some LTI transfer function $g(t,\rho)$ with a constant parameter $\rho$,
with the transformed equivalent:
$$\begin{align}
g(t,\rho)&\leftrightarrows G(s,\rho)\\
\end{align}$$
Is it equivalent ...
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140
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Don't we need both negative and positive discrete complex exponentials to make a real discrete time signal?
For a continuous time periodic signal , the Fourier spectrum has both negative and positive complex exponentials in equal numbers ,but I have seen for some discrete time periodic signals it is not the ...
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why we don't get equal number of negative and positive complex exponentials for the DTFS of a discrete time periodic signal?
When we compute the Discrete time Fourier series of a discrete time periodic signal , why don't we get the same number of negative complex exponentials and positive complex exponentials ?
Even though ...
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48
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$y(0)$ terms in the Laplace transform
When taking the Laplace transform (in my case, for building a transfer function) of a signal $y(n)$ the substitution below is often made directly:
$$\mathscr{L} \big\{ y^{(n)}(t) \big\} = s^n \mathscr{...
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Two meanings for "innovation" in Wiener filter are the same?
This is related question to A question about Wiener filter based on Linear Estimation by Kailath, based on the textbook Linear Estimation by Kailath. In that link I talk about how I first learned what ...
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A question about Wiener filter based on Linear Estimation by Kailath
In my linear estimation class based on the textbook Linear Estimation by Kailath, we went through the process of finding LLSE of $\hat{x}(t+\lambda)$ for fixed $\lambda$ given $\{y(\tau)|-\infty<\...
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Impulse response of a causal LTI system without using Laplace transform
I have this differential equation that models a causal LTI system:
$$
\ddot{v}(t) - \dot{v}(t) - 2v(t) = \ddot{u}(t) + 2\dot{u}(t) + u(t)
$$
I was asked to find the impulse response both by using ...
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How to compute the 2D coordinate in the s-domain?
I am not sure if my question is a right question to ask or not since I am still learning about Laplace Transform and S-Domain.
If we have 2D coordinates (x,y) in the spatial domain (i.e., Euclidean ...
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Laplace Transform and Inverse laplace Transform for 2D images python code available?
I am wondering if there is any implementation of Laplace Transform and Inverse Laplace Transform available for 2D data (i.e., images). For example, a batch of N ...
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Are complex exponentials real thing?
Is there any physical significance of complex exponentials.
I mean can we produce them like how we can produce sinusoidal signals using a signal generator?
OR are they just pure mathematical ...
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Why do singularities on the imaginary axis affect the Fourier transform differently than the Laplace transform?
(Please note that I'm aware there are already several questions asking about the difference between the two transforms. However, none of them that I could find touch on this specific issue of the ...
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Laplace transform of this simple parallel RLC circuit? (For audio speaker simulation ...)
SPEAKER AS RLC CIRCUIT
I read this article here which demonstrates a simulation of a speaker as a simple RLC circuit where the RLC components are in parallel:
MY GOAL
I am interested in creating a ...
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78
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Bode plot phase shift equation when poles and zeros are not at the origin
Let $$H(s)=\frac{s^{n}}{s^{m}}$$
For $n \ne m$ the phase shift between output and input will be
$\frac{\pi}{2}(n-m)$.
For situations where the poles and zeros are not at the origin, I could find the ...
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Mixer and control systems
A mixer in the time domain usually multiplies 2 signals of the time domain , however what does it do in the Laplace domain?Is there a equivalent block diagram of a mixer in the Laplace domain?The ...
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79
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Simulate Op-Amp low-pass transfer function in Python
I am struggling to simulate the frequency response of a simple op-amp low-pass transfer function in Python. The results I get are not accurate. The transfer function is $H(s)=\frac{1}{1 + s\tau}$, ...
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138
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I have a transfer function in s domain convert to time domain
I have a transfer function of $\frac{1}{s^2}$ in $s$ domain that represents a $\frac{\text{output}(s)}{\text{input}(s)}$.
Say I put a constant DC input as input (s) or just input in time domain that ...
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78
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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$
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115
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Correlation gives contradictory results
I am trying to find the correlation between the signals $u(t)$ and $\sin(t)[u(t)-u(t-2)]$
The correlation function $C(t) = \int^{\infty}_{-\infty} u(\tau+t)\sin(t)(u(t)-u(t-2))d\tau$
This is my ...
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112
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Derivation of 9 point Laplacian filter
I'm reading a paper on how construct isotropic laplacian filter, and perhaps because it's an old paper, the notation in it really bothers me a lot. So can someone please explain it to me? For example, ...
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Is there a Fourier Transform generalization that lets you analyze arbitrary complex frequencies?
Suppose you have a function that can be described as $$f(s) = \sum_{n=0}^{\infty} a_n e^{f_n s}$$ where each $f_n$ is a complex number. I am looking for a transform $T$ to act on $f$ which produces a ...
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3
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145
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How to separate Transient and Steady-State Expression from Periodic Summation Response?
Background
My question comes from here, it's a response of 1st order LPF RC circuit from an arbitrary periodic input.
How to determine the transient response of a circuit to causal periodic inputs?
...
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Laplace domain transfer function from system sampled at discrete times
I'm trying to understand an analysis of a sampled continuous time system in the Laplace domain. The source analysis is here (PDF page 6, slide marked 11); I'll explain further below. Suppose I have a ...
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Finding the region of stability of a system
Suppose we have a closed loop system controlled by some microcontroller $K$
First we take the open loop gain which is $\frac{K}{s(s+6)}$.It has 1 pole at the origin and at $s=6$ and 0 zeros.
So we ...
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639
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Calculating transfer function of a linear time varying system?
If we excite a LTI system with the Dirac delta $\delta(t)$, the system outputs the impulse response $h(t)$. For a LTI system, it doesn't matter when we excite the system with the Dirac delta, we will ...
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Max input of a system given it's transfer function and an assumed step change (beginner)
I have an exercise that gives me the following transfer function
$$
\frac{0.5}{s+0.5}
$$
and an assumed step change in the target of 20
I am asked to calculate the maximum input for the assumed step ...
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63
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LLTV Systems breakdown(2)
In this question I proved that for a linear linearly time varying system $S$ such that if $t_{2} = t_{1}+t_{0}$ then $h(t,t_{2}) = h(t,t_{1})+h(t,t_{0})\rightarrow$ $h(t,t_{0}) = g(t_{0})h(t) $ where $...
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What exactly are the assumptions behind Tustin's formula? Application on state space models
I was parsing the forum when I saw this post surging out of the depths of this forum like an old Kraken. The problem is quite simple.
You have a continuous time state space model :
$$
\begin{split}
\...
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Linear linearly time varying systems Laplace transform
Suppose that for a system $S$ if we have $t_{2} = t_{1}+t_{0}\rightarrow h(t,t_{2}) =h(t,t_{1})+h(t,t_{0}) $ .Then if we take the double Laplace transform to$t,t_{2}$ we will get:
$$L_{t_{2}}(L_{t}(h(...
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Transfer function $h(t)$ of a positive feedback system
I want to find the transfer function h(t) of the below positive feedback system. I came out till this.
How can i get the inverse laplace of this function? say β = 1 and γ = 1
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Inverting transformation $\displaystyle m(t)=\sum_{i=1}^d x(i)^t$
Suppose there's a vector of $d$ of positive numbers $x(1),\ldots,x(d)$ which I need to obtain from a vector of $d$ derived quantities $m(t_1),\ldots,m(t_d)$ where $\{t_i\}$ is some conveniently chosen ...
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Discrete version of this transform?
I have the following transform for $t>0, a_i>0$
$$f(t)=\sum_{i=0}^d a_i \exp(-t a_i)$$
And I need to invert it for a set of target values $b$:
Find $(t_0,t_1,\ldots,t_d)$ such that $f(t_0),f(t_1)...
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How we determine type of filter with pole(s), zero(s)? [duplicate]
Let's say we have this Laplace transform:
$$H_{1}(s)=\frac{1}{(s+1)(s+3)}\;, \; \Re{e} (s)>-1 $$
So, we know that there is a poles at $s=-1$ and $s=-3$.
With these informations, we found that to be ...
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Transfer function of LTI causal system
I have $$y(n)-3y(n-1)+2y(n-2)=4x(n)-2x(n-1)$$ that is the equation for a causal, discrete time LTI system. Using the Laplace Transform I rewrote it as: $$Y(s)-3e^{-s}Y(s)+2e^{-2s}Y(s)=4X(s)-2e^{-s}X(s)...
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Finding transfer functions from a system of multiple inputs
If I have a system: $$sX(s) = AX(s)+BY(s)+CZ(s)$$ How would I find the transfer functions $\frac{X(s)}{Y(s)}$ and $\frac{X(s)}{Z(s)}$?
Do I simply disregard one of the inputs? I am quite confused and ...
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How does the state estimate selection work?
I have been solving following problem. I have two open loop state estimators used for estimation of the unmeasurable states of a given linear dynamic system. The first estimator provides estimate $\...
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371
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First Order State Space Question
I am trying to understand the state space equation for a simple first-order LTI system. Suppose I have a system with impulse response
$$h(t) = \frac{1}{\tau}e^{\frac{-t}{\tau}} \ \theta(t)$$
In this ...
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Does the definition of stable system contradict itself?
A system is said to be stable when any of its poles are <0.
However I don't get why that is the case. Negative poles mean negative angular frequency, and negative angular frequency is equal to ...
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What is the intuitive interpretation of the transfer function of this system?
If I have the system that could be observed in the next Image:
I want to know the transfer function, where the external force $f$ is the entry and $x_1$ is the output. The direction and positive ...
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987
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Transfer function and Laplace domain
If we give a input $x(t)=u(t)$ to a system $\mathcal{S}$ we get an output $y(t) = e^{-t} u(t)$.
After we Laplace-transform both the input and the output we get the transfer function
$$H(s) = 1-\frac{1}...
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Name of property of Laplace transform
\begin{align}
L[e^{-at}u(t)] &= \frac{1}{s+a}\\
L[\cos(\omega_{o}t)u(t)] &= \frac{s}{s^{2}+\omega^{2}_{o}}\\
L[e^{-at}\cos(\omega_{o}t)u(t)] &= \frac{s+a}{(s+a)^{2}+\omega_{o}^2}
\end{...
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2
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Sampling with impulse train
There are many times when the textbooks used by all university students like me make an introduction to the notion of sampling of a continuous-time signal with an impulse train as shown below.
Why do ...
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219
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Prove that the filter is stable, causal and minimum phase
I have a system which has the following transfer function
$$H(s)=\frac{\beta + s}{s^{2} + 2\alpha s + \beta^{2}}$$
where $\beta = \sqrt{\omega^{2} + \alpha^{2}}$ and $\alpha>0$.
This system, as ...
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Study the stability of $H(s)=\frac{\beta + s}{s^{2} + 2\alpha s + \beta^{2}}$
I have a system which has the following transfer function
$$H(s)=\frac{\beta + s}{s^{2} + 2\alpha s + \beta^{2}}$$
where $\beta = \sqrt{\omega^{2} + \alpha^{2}}$ and $\alpha>0$. The degree of ...
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