Questions tagged [laplace-transform]

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18
votes
1answer
1k views

Is the Laplace transform redundant?

The Laplace transform is a generalization of the Fourier transform since the Fourier transform is the Laplace transform for $s = j\omega$ (i.e. $s$ is a pure imaginary number = zero real part of $s$). ...
16
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3answers
25k views

How poles are related to frequency response

I have recently fallen into fallacy, considering pole s=1 as there is infinite response at frequency 1. Yet, response was only 1. Now, can you derive the frequency response, given the poles? ...
9
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3answers
3k views

Intuitive interpretation of Laplace transform

So I am getting to grasps with Fourier transforms. Intuitively now I definately understand what it does and will soon follow some classes on the mathematics (so the actual subject). But then I go on ...
8
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3answers
2k views

What's the meaning of a complex zero/pole?

I have been studying signal processing and control for a while now, and I use Laplace and Fourier transforms almost everyday. Also another tools such as Nyquist or Bode plots. However, I had never ...
7
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2answers
1k views

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

I am trying to understand Fourier Transform and Laplace Transform. What is the difference between $X(j\omega)$ and $X(\omega)$ notation? what is the meaning of $j\omega$ ? Is it represent frequency? ...
7
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3answers
4k views

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

could anyone explain why there is a need of negative exponent in fourier and laplace transform.I looked through the web but I couldn't get anything.Does anything happen if a positive exponent is ...
7
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1answer
3k views

Why does the separable filter reduce the cost of computing the operator?

A separable filter in image processing can be written as product of two more simple filters. Typically a 2-dimensional convolution operation is separated into 2 onedimensional filters. This reduces ...
4
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3answers
4k views

Is $y[k] = y[k-1] + x[k]$ an integrator?

It looks exactly like an integrator to me. Since $$y[k] = y[k-1]+x[k] = y[k-2] + x[k-1] +x[k] = \sum{x}$$ Applying the Z-transform gives \begin{align} Y(z) &= Y(z)\cdot z^{-1} + X(z)\\ \...
4
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1answer
235 views

Why the unilateral Laplace transform?

Why is the Laplace transform commonly taught as the unilateral Laplace transform? I mean, for the Fourier transform, we commonly have the bilateral transform... if the signal is 0 for $t<0$, then ...
4
votes
2answers
2k views

blur detection using opencv

I'm writing a script to detect blur images using OpenCV by applying Laplacian filter and calculate the std but there is a problem the std for images that contain motion blur is very close to those ...
4
votes
2answers
757 views

Response of a system to a step function (heaviside)

I'd like to compute the response to a step function of a electrical/thermal system. Generally I can "easily" compute the transfer function $H$: $$H(\omega) = \frac{V_{out}(\omega)}{V_{in}(\omega)}$$ ...
3
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2answers
1k views

Fourier transform of exponent: Delta pulse or hyperbola?

Why do some tables say that Laplace (or Fourier?) inverse of exponential is a time-shifted delta pulse \begin{align} \delta (t) &\overset{\mathcal F}{\Longleftrightarrow} 1\\ \delta (t-t_0) &...
3
votes
1answer
192 views

Find state space model from transfer function

Let's suppose we have: G(s) = (s+1)/(s^2-2s+1) how can we find the state space representation of the transfer function: x_dot = x2 x2_dot = 2*x2-x1+u where u is an arbitrary input. I am very new ...
3
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1answer
620 views

Basic difference between Fourier transform and laplace transform? [duplicate]

I have read few links about difference between Fourier transform and Laplace transform but still not satisfied Please correct me if i am wrong Simply put, the main difference between Fourier ...
3
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4answers
3k views

On the meaning of s-plane and it's link to a transfer function

Considering Fourier analysis and let's say I'm walking on the blue frequency axis in the below 3D plot from zero towards infinity: So each time I encounter a non zero blue bar, I check the frequency ...
3
votes
1answer
2k views

Can the inverse system of a stable system be unstable?

Can the inverse system of a stable system be unstable? For the class of LTI systems, the criteria for stability of a system with impulse response $h(t)$ and systems function $H(s)$ are: $h(t)$ be ...
3
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1answer
2k views

Find transfer function from root locus and step response diagram?

I am given the response of a step of magnitude of 3 and the root locus and I have to find the transfer function of the system. The function I find gives me the step response(magnitude of 3 again) of ...
2
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3answers
351 views

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

Fourier transform has different types like continuous Fourier transform (CFT), Discrete time Fourier transform (DTFT) and Discrete Fourier transform ( DFT). CFT can be used in case of continuous ...
2
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3answers
2k views

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

Is there an easy way to explain the motivation behind the use of Laplace transform instead of Fourier transform? Isn't that any periodic function can be represented by sines and cosines? - Why to ...
2
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3answers
341 views

Transfer function of a frequency shifting system

There is a system which shifts frequencies of input by -Fc such that: Y(S) = X(S).H(S) But X(S) has value zero from 0 to Fc. I am confused on how the product of X(S) and H(S) becomes a positive ...
2
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2answers
1k views

How to modify an LTI differential equation to be acausal or anti-causal?

I'm trying to wrap my head around causality in LTI-systems. Considering continuous time only, I'm happy with the fact that the system is causal iff the impulse response function $h(t)=0$ for $t<0$. ...
2
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2answers
324 views

Help with my first (simple) Z-transform

I need to transform this Laplace function to the z-domain: From the answer I received: $s=(1-z^{−1})/T$ Then substitution into my Laplace function would give: $t(z) = 2R/(m*(1-z^{−1})/T + 2R)$ Is ...
2
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1answer
460 views

How can a system be unstable if $L(j\omega)$ is never exactly $-1$?

Say we have a linear system with unity feedback, with loop transfer function $L(j\omega)$. The closed-loop transfer function from reference to output is $T(j\omega) = \frac{Y(j\omega)}{R(\omega)}=\...
2
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1answer
5k views

Inverse Laplace transform Using Inversion Formula

Use the complex inversion formula to calculate the inverse Laplace transform $f(t)$ of the following Laplace transform: $$F_L (s) = \frac{1}{(s+2)(s^2 +4)}.$$ When the region of convergence is: ...
2
votes
1answer
51 views

Why there is Difference between shapes of ROC of z domain and s domain?

ROC(region of convergence) of Z domain is shown by a circular region while ROC in S domain is shown by a rectangular(approximately looking like rectangle) region What is the reason of this difference ...
2
votes
1answer
144 views

What is wrong with my residue partial expansion method? (Transfer Function into State-Space Modal/Diagonal Form)

I'm using reference from here and here. This is Laplace transfer function of DC Control Speed System: $$\frac{\omega(s)}{V(s)}=\frac{K}{(Js+b)(Ls+R)+K^2}$$ Where, $\omega$ is the motor angular ...
2
votes
1answer
938 views

Relating transfer functions with step responses

Relate the transfer function to its' corresponding step response. First, I tried setting up the poles and zeros of the transfer functions. This helped a bit since I know that $G_A (s)$, $G_B(s)$ and $...
2
votes
1answer
100 views

Finding Laplace Transform without ROC

While studying Laplace Transform i found that region of convergence (ROC) is important because for some problems we have same Laplace Transform but different ROC helps us to take correct inverse ...
2
votes
1answer
472 views

Meaning and unit of frequency in Laplace (Fourier) transform

Imagine transfer function obtained by Laplace transform, for example: $G(s) = \dfrac{1}{s+1}$ Now, I would like to do some frequency analysis, so I replace the $s$ with $\omega i$ (let's consider ...
2
votes
1answer
323 views

Servo motor analysis

I'm studying a mathematical behaviour of a servo motor and I need some help to understand it. The output signal is $\beta(t)$, representing the angle rotated by the axis at instant t, in relation to ...
2
votes
1answer
220 views

Impulse response of a continuous system sampled with zero-order hold

I've a continuous system $$F(s) = \frac{K}{Ts+1}.$$ I sample it with zero-order hold with sampling period $T_s$. The discrete system transfer function is $$ \begin{aligned} G(z) &= % \frac{z-1}{z}...
2
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1answer
345 views

Control systems and convolution

I think i am not understanding the concept of convolution well. Lets say we are given a system impulse response in the S-domain, and we have implemented a controller $G_c(s)$ that will adjust the ...
2
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1answer
269 views

Confusion in proof of Inverse Laplace Transform

For the proof of inverse Laplace transform, we change the integral from $\omega$ to $s$. I want to know the reason why we need to change the integral?
2
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2answers
73 views

Why we take Laplace Transform of functions which converged using Fourier Transform

There are several functions for which we know that Fourier Transform will exist but still we calculate its Laplace Transform. Can I know the reason why we need to take Laplace transform for which we ...
2
votes
2answers
49 views

How do I convert this simple Laplace equation to Z-domain?

A basic model of coupled strings (eg. piano) is provided here as: The principle is that it has two identical string simulations each formed by a delay line and LPF. The outputs of these are summed at ...
2
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0answers
102 views

How to transform a Fractional Order Laplace Transfer Function into a digital filter?

I'm working with loudspeaker impedance analysis. Electrical behavior of loudspeakers can be modeled with RLC networks. But real loudspeakers have components, that exhibit some non-linear and frequency ...
2
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0answers
214 views

Laplace Transform of $-e^{-at}u(-t)$

I have found a problem in applying Laplace Transform to $-e^{-at}u(-t)$ I am doing these steps: $$ = - \int_{-\infty}^{+\infty} e^{-at}u(-t) e^{-st}dt$$ $$ = - \int_{-\infty}^{0} e^{-at} e^{-st}dt$$...
1
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2answers
137 views

Confusion Regarding Bi Linear Transform

I was reading my book where the z-transform of a signal is derived to be ${1-e^{-2bT}z^{-1}}$ . Then it goes on to say that by applying the bilinear transform we can get $$\frac{2(1+bT+(bT-1)z^{-1})}...
1
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1answer
63 views

Laplace of step and integration are same?

Why do we have Laplace transform of a step function and integrator is same. \begin{align} \mathcal L\left[u(t)\right] &= \frac 1s\\ \mathcal L \left[ \int dt\right] &= \frac 1s \end{align}...
1
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1answer
87 views

How do bode plots work with unstable systems work?

If I had a system with right-half s-plane poles, how would a frequency response work? Since a purely imaginary value for s, would cause the Laplace transform to diverge for such a system, what meaning ...
1
vote
2answers
131 views

From Fourier transform to Laplace Transform

It's well known that you can estimate the Fourier Transform $X(f)$ of a signal $x(t)$ via its Laplace Transform $X(s)$, just by setting $s = j2\pi f$ to the latter, as long as the region of ...
1
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1answer
198 views

Can a Fourier Transform exist even if the j$\omega$ axis is not in the Region of Convergence in it's Laplace Transform

A couple of confusions have been occurred. The Signal I'm considering is f(t) = sin(t)*u(t) Fourier Transform of it can be derived. $-i \pi (\delta (\omega -1)-\delta (\omega +1))$ According to my ...
1
vote
3answers
364 views

Why use transfer functions than differential equations?

I have some simple questions regarding DSP and Control Systems, but found no simple answers. I just need simple and easy to understand answers, not complex answers with huge maths. Why we use ...
1
vote
1answer
630 views

Bilinear transformation confusion

Wikipedia says in bilinear transformation from \$s\$ domain to $z$ domain relation is $$\boxed{s \longleftarrow \frac{2}{T}\frac{z-1}{z+1}}$$ But here this relation is given like this $$\boxed{w=\...
1
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1answer
126 views

Confusion in basics of Laplace Transform

I have few confusions while starting Laplace Transform. So far I have studied, Fourier series and Fourier Transform. The basic difference which I found from different books is Fourier Transform is ...
1
vote
1answer
79 views

Laplace transform of a finite duration signal

Consider the following signal: $$ x(t) = e^{-2t}[u(t) - u(t-5)] $$ This signal exists only from 0 to 5 time units. Elsewhere, it is zero. Now, let's find the laplace transform of this signal using ...
1
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1answer
36 views

Why does subbing $s = j\omega$ into the Laplace transform of a cosine wave yield a purely imaginary result?

The Laplace transform of a cosine starting at $t=0$ is given by $$F(s) = \frac{s}{s^2 + \omega_0^2}$$ If I sub in $s = j\omega$, I get the Fourier transform of a cosine starting at $t=0$: $$F(j\...
1
vote
1answer
189 views

Output of an LTI system given its transfer function and input

Given the transfer function $$T(s) = \frac{100}{1 + \frac{s}{10^{6}}}$$ and the input $$v_i(t) = 0.1 \sin(100t)$$ find the output, $v_o(t)$. My approach was to use $v_o(t) = \mathcal{L^{-1}}\left\{T(...
1
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2answers
4k views

Why do poles in the left half of the S plane make a system stable?

A point on the S-plane (where $s=\sigma+j\omega$) represents a signal with a given frequency (given by the imaginary component) and which either decays, increases or stays stable (depending on the ...
1
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1answer
77 views

Claim: Given sampling time T, the hold operator is approximated at low frequency by a time delay of T/2

Can someone verify this statement? The hold operator is assumed to be a zero order hold Then the laplace transform of this hold operator has a well known form $(1-e^{-sT})/s$ Let w approach 0, we ...