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14 votes

How to compute Laplace Transform in Python?

Given the approach started in the OP's Github code I have this suggestion: Observe that the unilateral Laplace Transform given as: $$X(s) = \int_0^\infty x(t)e^{-st}dt$$ Is just the Fourier Transform ...
Dan Boschen's user avatar
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11 votes

Is the Laplace transform a special case of Fourier transform? (Not the other way around)

The Fourier Transform is the Laplace Transform with the complex variable s restricted to be the imaginary axis on the s plane. For this reason the Fourier Transform only exists when the imaginary axis ...
Dan Boschen's user avatar
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10 votes
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Why does the separable filter reduce the cost of computing the operator?

Assume you have a $N\times M$ sized image. If you know take what is classically used, a square filter kernel, of let's say size $L\times L$, you'd need to convolve that with the picture – which gives ...
Marcus Müller's user avatar
7 votes
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Why the unilateral Laplace transform?

The widespread use of the unilateral Laplace transform reflects the fact that in practice we often deal with causal systems and signals that have a defined starting time (usually chosen as $t_0=0$). ...
Matt L.'s user avatar
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7 votes

How to calculate critical damping of a system with two springs and a damper (or two springs and two dampers)?

Re-writing the strain-stress equation $$ 0 = E_1Ɛ + \frac{η(E_1+E_2)}{E_2}\dot{Ɛ} - \frac{η}{E_2}\dot{σ} - σ $$ for displacement/restoring force variables $$ 0 = E_1x + \frac{η(E_1+E_2)}{E_2}\dot{x} + ...
V.V.T's user avatar
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6 votes
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How can a system be unstable if $L(j\omega)$ is never exactly $-1$?

You cannot make conclusions about the stability of a system by only considering its transfer function evaluated on the imaginary axis $s=j\omega$. Replacing $s$ by $j\omega$ in the transfer function ...
Matt L.'s user avatar
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6 votes
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Laplace Transform of Cosine, Poles and Mapping to Frequency Domain

You're comparing the transforms of two different functions. You consider the Fourier transform of the function $x_1(t)=\cos(\omega_0 t)$, but you took the Laplace transform of the function $x_2(t)=\...
Matt L.'s user avatar
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6 votes

Why Fourier transform is not sufficient and we have to use Laplace transform?

The Laplace Transform is more representative of real systems that have a starting point, which is why the integral starts at 0, and also why the unit step function is generally talked about alongside ...
soultrane's user avatar
  • 241
6 votes

What are the advantages of Laplace Transform vs Fourier Transform in signal theory?

Laplace transforms can capture the transient behaviors of systems. Fourier transforms only capture the steady state behavior. Of course, Laplace transforms also require you to think in complex ...
Cort Ammon's user avatar
6 votes
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Help with my first (simple) Z-transform

First of all, it's important to understand that there is no single best way to transform a continuous-time system to a discrete-time system. The method you're using is called backward Euler method, ...
Matt L.'s user avatar
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6 votes

Confusions regarding differences between Fourier transform & Laplace transform?

Concerning your first question, both, the Laplace and the Fourier transform, are frequency domain representations of a function or signal. In the Fourier transform we deal with a real-valued frequency ...
Matt L.'s user avatar
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5 votes
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Inverse Laplace transform Using Inversion Formula

In engineering practice, the complex inversion integral is hardly ever used. As an engineer, you will almost exclusively need to invert rational functions, and this can be done by partial fraction ...
Matt L.'s user avatar
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5 votes

Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

The short answer is yes, if you have the Laplace or Z-transform of a function you do not need the Fourier transform. This is because the CFT is a special case of the Laplace transform and the DTFT ...
Matt's user avatar
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5 votes

Why do these 2 methods give different solutions?

The problem is that you took the derivative of the function $$\hat{x}_u(t)=2e^{-3t}-e^{-4t}\tag{1}$$ whereas using the Laplace transform you implicitly assumed that $x_u(t)$ equals zero for $t<0$:...
Matt L.'s user avatar
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5 votes

A question about the meaning of pole in time domain

Let $H(s)$ be a transfer function of the form $$H(s) = \frac{1}{s-p}$$ where $p$, which is a pole of $H(s)$, can be written as a complex number $a+jb$. Taking the inverse Laplace transform of $H(s)$ ...
anpar's user avatar
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5 votes

Laplace Transform of $f(t+a), a>0$ where $f(t)$ is not periodic

Let $s = \sigma + j\omega$, the inverse Laplace transform of $f(t+a)$ is given by $$f(t+a) = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} F(s)e^{s(t+a)} \mathrm{d}s = \frac{1}{2\pi j} \int_{...
anpar's user avatar
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5 votes
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Can a Fourier Transform exist even if the j$\omega$ axis is not in the Region of Convergence in it's Laplace Transform

You're right that the Laplace transform is not more general than the Fourier transform. They are just different. There are several (theoretically) important functions for which the Laplace transform ...
Matt L.'s user avatar
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5 votes
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When to use Fourier, Laplace and Z transforms?

It's natural consequence of applying a transform to a convolution relation. The output $y(t)$ of an (continuous-time) LTI system is described by a convolution integral : $$y(t) = h(t)\ast x(t) = \int_{...
Fat32's user avatar
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5 votes
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Why not use the same "standard" exponentials for both continuous and discrete time

This has to do with the way the Laplace transform and the $\mathcal{Z}$-transform are defined: $$\mathcal{L}\big\{x(t)\big\}=\int_{-\infty}^{\infty}x(t)e^{-st}dt\tag{1}$$ $$\mathcal{Z}\big\{x[n]\big\}=...
Matt L.'s user avatar
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5 votes
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Product of Doublet and Arbitrary Function

Your first equation is correct. For derivatives of the Dirac delta impulse you get slightly more involved expressions. For $\delta'(t)$ the following holds: $$f(t)\delta'(t)=f(0)\delta'(t)-f'(0)\delta(...
Matt L.'s user avatar
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4 votes
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What are the advantages and disadvantages of Laplace transform over Z transform?

Both transforms are equivalent tools, but the Laplace transform is used for continuous-time signals, whereas the $\mathcal{Z}$-transform is used for discrete-time signals (i.e, sequences). You can ...
Matt L.'s user avatar
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4 votes
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How to modify an LTI differential equation to be acausal or anti-causal?

It's important to realize that generally the differential equation (DE) alone doesn't tell us anything about causality. You claim that the system given in your question is causal. However, an anti-...
Matt L.'s user avatar
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4 votes
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Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

The answer to your last question is definitely 'no'. The point hotpaw2 makes in his answer is very relevant: the FFT is an efficient implementation of the DFT, and there are no equivalently efficient ...
Matt L.'s user avatar
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4 votes
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What's the meaning of a complex zero/pole?

We usually talk of $j\omega$ when we're also interested in the Laplace transform of a signal / system, but want to just talk about the frequency response. The physical meaning of the imaginary part ...
Peter K.'s user avatar
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4 votes

Relation between Laplace and Fourier transforms

Just as an effort to increase the post's didatics to the future visitors of this question (just as me): I have noticed you commented you were taught the imaginary axis of the Laplace plane is the ...
ren1's user avatar
  • 41
4 votes
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How can I plot a 3D graph of a given Laplace Transform of a function?

[X,Y] = meshgrid(-10:.1:10); s=X+j*Y; Z= abs((s+3)./((s+3).^2+25)); mesh(X,Y,Z)
Dan Boschen's user avatar
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4 votes

Laplace of step and integration are same?

This is because the impulse response of an integrator is $h(t)=u(t)$. The output which is the convolution with the impulse respoponse is $$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$$ and with $...
msm's user avatar
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4 votes
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What is wrong with my residue partial expansion method? (Transfer Function into State-Space Modal/Diagonal Form)

The discrepancy between your derivation and matlab's computation results because of a convention mismatch you used during the partial fraction expansion: Given that the function to be expanded is $H(s)...
Fat32's user avatar
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4 votes
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Inverse Laplace transform of two-sided and one-sided Laplace transform

The inversion formula is the same for both types of transforms: $$f(t)=\frac{1}{2\pi j}\int_{\alpha-j\infty}^{\alpha+j\infty}F(s)e^{st}ds\tag{1}$$ The difference is in the choice of the constant $\...
Matt L.'s user avatar
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