Questions tagged [laplace-transform]

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Reference for Wiener-Kolmogorov whitening procedure

In an old paper, accessible only for a fee, I found: (where the LHS of Eq (5) lack the symbol of derivative). What is the Wiener-Kolmogorov whitening procedure? I would like to study this procedure, ...
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How does the phase/gain margin method work?

We have the closed loop transfer function: $$T(s)=\frac{L(s)}{1+L(s)}$$ So as far as I understand we check along the $j\omega$-axis on the Bode plot whether $L(s)=-1$, cause that's when $T(s)$ has ...
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Time invariance in transfer functions

I read this in a book: here ${\cal L}[x(t-a)u(t-a)]$ is the laplace transform of a time shifted function $x(t)$ shifted by $a$ seconds and we know that transfer functions have the formula $H(s)=Y(s)/...
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Finding and displaying Laplace or Z transform ROC(region of convergence) using MATLAB

Is there any way, we can use MATLAB for finding and displaying Laplace or Z transform Region of convergence?
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Confusion regarding Laplace transform calculation in MATLAB

I am trying to learn about Laplace transform and especially about ROC and i found out on this weblink. I have also attached a snapshot of this link and highlighted where it is being said that although ...
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making a laplace s-domain plot from numerical data to decompose signal into decaying sinusoids

I would like to make a 3D laplace s-domain plot from experimental data I have. The examples I have seen for this are when the function is already known and an analytical solution can be obtained (...
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Laplace transform of derivative

Here is a short proof that Laplace Transform of $x'(t)$ is laplace transform of $x(t)$ multipled by s: On the other hand, the proof that I know uses integration by parts: One condition is that $x(t)...
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Bandwidth of a given function

Suppose the signal given is $x(t)=(cos(5t)+e^{-2t})u(t)$ and the goal is to find the bandwidth of this signal. We know the bandwidth, $B$, is just $f_{max}-f_{min}$. Our first step is taking the ...
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Product of Doublet and Arbitrary Function

We know that the product of the delta function and another function samples the latter function. That is, $$ \delta(t-\tau)f(t)=\delta(t-\tau)f(\tau) $$ Does the doublet function retain this same ...
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Step response of a given input and output (Laplace or Fourier)

I am trying to calculate the step response of the following given: Should I use Laplace transform or Fourier transform?
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Filter filters out more than needed

I am currently coding a school assignment. I have a 1.5s recording of someone's speech that has 4 rogue cosines mixed in. With sampling rate 16000Hz, I divided the recording into frames of 1024 ...
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The Laplace transform - Steven W. Smith Book - impulse response cancellation method

While studying the Laplace transform using Steven W. Smith Book I found something uncomprehending. In the 32th chapter - The Laplace Transform, page 590, last paragraph describes the cancelling ...
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What will the output of a system which has no Fourier transform?

Let's assume a system $h(t)= e^{j2t}$. This system has no region of convergence. What will be the output if I provide any input to this system?
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Confusions regarding differences between Fourier transform & Laplace transform?

Although this topic has already been addressed in multiple popular questions of SE but i have few confusions in this regard Number 1) Link of question https://electronics.stackexchange.com/questions/...
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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 ...
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How are pole-zero plots, magnitude response plots, and phase response plots related?

Given that the Laplace transform of a continuous-time signal $h(t)$ is $H(s)$, what can a plot of the poles and zeros of $H(s)$ on the $s$-plane tell me about the magnitude response and phase response ...
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Connection from Fourier to Laplace Transform

I have a basic understanding of Laplace and Fourier but having trouble making a connection. Every time I attempt to look at reasons these are connected I'm told about the s-plane and regions of ...
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Optimal FWHM for Difference of Gaussians edge detection

When we apply a Gaussian blur to a a dataset, we set the FWHM to optimally detect specific frequencies. For example, the human cortex is around 2.5mm thick, so a blur with 2.5mm FWHM is nicely tuned ...
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Can we tell if a system is linear and time-invariant from its frequency response?

Given a system with a known frequency response in the S-domain. Is there a way to find whether the system is linear and time invariant? My current understanding is that we need to take the inverse ...
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How to get state-space equations form from a block diagram?

This is the block diagram that I'd like to transform into a state-space representation, where u1 and u2 are inputs and y1 and y2 are the outputs of the system I tried to place state variables on the ...
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Mixed - Discrete and Continuous system Laplace domain stability - Effect of Sampler and DAC

I have a system whose the plant transfer function is continuous and the compensation is discrete. I have an ADC which allows to measure the output of the system and a DAC which allows to control the ...
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Find $A$ and $G$ value to satisfy the requirement

Given a disturbance reduction system Create a system that will reduce $U(s)$ noise to $100$ times its value. Find the A and G gain value to satisfy the requirement My attempt: I've analyzed the ...
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Laplace transform plot isn't right

I'm trying to plot the Laplace transform of a function. Here's my MatLab script ...
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1 answer
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Why RHP zero phase is not 180° to 90°

The asymptotic phase behavior of an RHP zero is from 0 degrees to -90°, the mirror of an LHP zero. Graphically, I'm confused about why this is the case and the phase is not from +180° to +90°. See the ...
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A system having impulse response $ h(t)=u(t) $ stable or not?

I know that for a system to be BIBO stable its impulse response must be absolutely integrable and the impulse response $ h(t)= u(t)$ integrates to approach infinity (i guess) I proceeded as$$ \int_{-\...
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Unilateral Laplace Transform's Differentiation Property

I've read in numerous places that the unilateral laplace transform is extermely useful in solving differential equations with initial conditions based on the differentiation property of the unilateral ...
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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) $...
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What is z equal to in z-transform?

In some places, it is said that z is equal to: $$z = e^s \quad where \quad s = \sigma + j \Omega $$ But in some places, it is said that z is equal to: $$z = e^{sT_s} \quad $$ where Ts is a sampling ...
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Pole Magnitude and Damping Ratio relationship

I know that the damping ratio of a system is defined by the angle of the pole, calculated with respect to the left hand side $x$-axis. Could one infer though, that if the magnitude of the poles is ...
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2 votes
1 answer
247 views

How can we prove the correctness of the integration property of the Laplace transform?

I was going through an Electrical Engineering textbook for understanding the Laplace transform and came across the following proof for one of the properties of the Unilateral Laplace transform. ...
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how to find the inverse response of a system

For the system described by the differential equation below find its inverse zero-state unit step response $$\dfrac{d^2y(t)}{dt^2}-2\dfrac{dy(t)}{dt}-8y(t)=\dfrac{d^2x(t)}{dt^2}-2\dfrac{dx(t)}{dt}-3x(...
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Part 2: Root Locus, Transfer Functions and Unit Step Response?

I'm continuing my question referenced here: Part 1 Question / Problem Description Say I have a new Root Locus shown below Consider the generic feedback loop, and the transfer function $G(s)$ shown by ...
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Root Locus, Transfer Functions and Unit Step Response?

Consider the generic feedback loop, and the transfer function $G(s)$ shown by the following root locus plot. Where $\mathbf{x}$ denotes the open-loop poles and $\square$ denotes the closed loop poles....
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transfer function of a sampler in the s domain

I would like to modelize my whole system into the S-domain. This is a mixed system, there a numerical part (corrector, ADC, DAC) and an analogic part (plant transfer function, sensors, etc...). I know ...
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How to calculate critical damping of a system with two springs and a damper (or two springs and two dampers)?

Background For a simple system where you have a mass attached to a spring and damper in parallel: We can calculate the critical damping from the equation of motion: $mx_{tt} + cx_t + kx = 0$ $ms^2 + ...
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How to solve this Laplace integral for an averaging function in an iterative numerical (finite difference) model?

I am trying to implement a mathematical model for vibrational damping described in this article. They provide an equation for damping force ($F$) as a function of: a spring constant ($k$) a damping ...
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Why not use the same "standard" exponentials for both continuous and discrete time

In continuous time the standard exponential signal is usually defined as $$ e^{st}, \quad\text{with}\quad s = \sigma+j \omega $$ In discrete time the standard exponential signal is usually defined as ...
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If the impulse response of a causal LTIC system is $h(t) = \delta(t) + \sin(t)u(t)$, is it marginally stable or unstable (BIBO)?

If you take the $H(s) = \mathcal{L}[h(t)]$ the poles are on the imaginary axis so the system should be marginally stable, but is it?
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2 votes
1 answer
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Confusion for two-sided signal

Given a continuous LTI system with transfer function $$H(s)= -\frac{2s}{(s+6)(s+2)}$$ Plot the location of the pole(s) and zero(s) Find all possible regions of convergence From the problem above find ...
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Why can you use the one-sided laplace transform to solve differential equation describing a causal LTI-system?

In an example, an equation describing a causal LTI-system is $$ (D^2 + 5D + 6) y(t) = (D+1) x(t) $$ where $y(t) = y_{zs}(t) + y_{zi}(t)$ and the initial conditions are $y(0^-) = 2, \dot{y}(0^-) = 1$. $...
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2 votes
2 answers
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Laplace Transform: zeros and corresponding impulse response $h(t)$

Poles and the impulse response If our impulse response is in the form : $$h(t) = e^{-\sigma_0 t}\cos(\omega_0 t) \, u(t)$$ (where $u(t)$ is the unit step function) And its Laplace transform is : $$H(s)...
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RoC and Stability of a Rectangular Signal

If we have a system with an impulse defined as: $$h(t)=u(t)-u(t-2)$$ Then the Laplace Transform of h(t) would be the transfer function: $$H(s)=\frac{1}{s}-\frac{e^{-2s}}{s}, \quad Re(s)>0$$ We also ...
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3 votes
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Laplace transform : integral vs poles & zeros

If Laplace transform is expressed as : $$\int_{-\infty}^{+\infty} h(t)e^{-st}dt $$ with : $$s = \sigma + j\omega$$ and $h(t)$ an impulse response expressed as : $$h(t) = Ae^{-\sigma_0t}\cos(\omega_0t+\...
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Discrete time Final Value Theorem applied to feedback system

I wish to calculate the Final Value of systems in which a high pass filter of the output feeds back into the input. A simple example would be: where is a 1st order high pass filter with transfer ...
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Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)

Assume we have the following system (coming from control systems theory, hence in s-domain) $ Y(s) = H_A (s) \cdot A(s) - H_B (s) \cdot B(s) $ I now wish to consider $a(t)$ and $b(t)$ as white noise ...
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Inverting a sampled system

I'm doing some self-study for an upcoming exam and came across the following question: My first idea to solve it was using the bilinear transform to get some approximation of $H(Z)$ (or just using ...
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Why is Fourier space not adequate for (theoretical or digital) filters?

As far as I have seen, almost all theoretical filter design occurs in Laplace or Z-space. Also, there is a pervasive connection to real life analog filters in the design. If one is just thinking in a ...
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Real impulse response

It will be great if someone can explain me what exactly means "real impulse response". Further more , what is the effect of such a response on ROC (Laplace plane) and in particular if its ...
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Fourier Transform of an Exponential Sine Sweep

The Exponential Sine Sweep (ESS), according to Farina [1], can be described by the following formula: $$x(t)=\sin\left(\frac{2\pi f_1 T}{R}\left(e^{\frac{t R}{T}} -1\right) \right)$$ where, $t$ - ...
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1 vote
1 answer
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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 $t=0$ ...
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