15 votes

Is the Laplace transform redundant?

The Fourier and the Laplace transform obviously have many things in common. However, there are cases where only one of them can be used, or where it's more convenient to use one or the other. First ...
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What is the difference between $X(j\omega)$ and $X(\omega)$ notation?

Both notations are common and correct. As pointed out by Yuri Nenakhov, the advantage of the argument $j\omega$ is that it coincides with the complex (Laplace transform) variable $s$ when its real-...
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10 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 ...
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10 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 ...
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9 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 ...
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Relation between Laplace and Fourier transforms

The Laplace transform evaluated at $s=j\omega$ is equal to the Fourier transform if its region of convergence (ROC) contains the imaginary axis. This is also true for the bilateral (two-sided) Laplace ...
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8 votes

Intuitive interpretation of Laplace transform

Why is the fourier transform a special case of the laplace transform? The Laplace transform produces a 2D surface of complex values, while the Fourier transform produces a 1D line of complex values. ...
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7 votes

Confusion Regarding Bi Linear Transform

The bi linear transform is the transform from the Laplace Transform Domain to the Z Transform. The Laplace Transform Domain is a regular plane. This transform transforms vertical lines in the Laplace ...
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  • 41.2k
7 votes

Why is a negative exponent present in Fourier and Laplace transform?

Matt is correct that the sign is convention. I think that there is a reason for it beyond that though. If we look at complex frequencies in the complex plane, they look like a constant vectors that ...
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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 $t_0=0$). ...
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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} + ...
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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 ...
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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 ...
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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)=\...
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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 ...
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5 votes

What is the difference between $X(j\omega)$ and $X(\omega)$ notation?

$X(j \omega)$ (frequency response) is a Fourier transform of system's impulse response. It's actually a function of frequency ($\omega$) but usually is written as $X(j \omega)$ because replacing $j \...
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Confusion Regarding Bi Linear Transform

As already mentioned by other people, the bilinear transform is often used to map a continuous-time system described in the $s$-domain to a discrete-time system described in the $z$-domain. However, a ...
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5 votes

Intuitive interpretation of Laplace transform

The best intuitive description of Laplace transform I've ever seen: At first glance, it would appear that the strategy of the Laplace transform is the same as the Fourier transform: correlate the ...
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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 ...
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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 ...
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  • 59
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 $t<0$:...
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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 $H(s)$ ...
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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_{...
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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, ...
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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)\star x(t) = \...
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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\}=...
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5 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 ...
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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(...
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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 ...
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  • 41
4 votes

Finding Laplace Transform without ROC

Strictly speaking you can't because without specifying the ROC, the inverse Laplace transform is generally not unique. However, in many contexts there is the implicit assumption of causality of the ...
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