Questions tagged [laplace-transform]

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24 votes
1 answer
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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 $s$). ...
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20 votes
3 answers
33k 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? ...
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13 votes
4 answers
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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 ...
  • 249
11 votes
4 answers
6k 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 ...
  • 442
10 votes
2 answers
2k views

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

What is the difference between $X(j\omega)$ and $X(\omega)$ notation? What is the meaning of $j\omega$? Does it represent frequency, and if yes, what is the meaning of an imaginary frequency?
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9 votes
1 answer
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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 ...
  • 245
9 votes
5 answers
5k 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 ...
  • 4,890
7 votes
2 answers
30k views

Relation between Laplace and Fourier transforms

I know that $$X_L(s=j\omega)=X_F(\omega)$$ if $x(t)$ is one sided and absolutely integrable and hence the imaginary axis of the Laplace transform is the Fourier transform. But Fourier transform ...
  • 225
7 votes
1 answer
251 views

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 + ...
  • 427
6 votes
1 answer
997 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 ...
5 votes
2 answers
173 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})}...
  • 253
5 votes
2 answers
2k views

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

Always had a thought about why Laplace transform reveals the transient properties of the system? My doubt is based on the following fact, Fourier transform is given as  \begin{equation} \mathscr{F}\...
  • 420
5 votes
1 answer
10k views

How to compute Laplace Transform in Python?

I am trying to do practicals for signal processing where I need to Laplace Transform a function. Used 'fft' of numpy before. Nothing of Laplace is found in the documentation. Do we have any other ...
  • 153
5 votes
1 answer
122 views

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 ...
  • 261
5 votes
1 answer
214 views

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 ...
5 votes
1 answer
333 views

What are the properties of continuous-time improper systems?

I am trying to better understand the properties of improper systems $H(s) = \frac{b(s)}{a(s)}$, for which the order of the numerator $b(s)$ is greater than the order of the denominator $a(s)$ (in the ...
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4 votes
2 answers
409 views

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 ...
4 votes
2 answers
72 views

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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4 votes
1 answer
416 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 ...
4 votes
1 answer
92 views

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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4 votes
2 answers
84 views

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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4 votes
2 answers
3k 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
2 answers
1k 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 $H$: $$H(\omega) = \frac{V_{out}(\omega)}{V_{in}(\omega)}$$ ...
4 votes
1 answer
224 views

s-Domin or z-Domain - What to Use for Mixed systems

I recently had to deal with a power electronic system where I had to implement a dynamic model of a power converter in order to design a suitable controller for that converter. The control will be ...
4 votes
1 answer
583 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 ...
4 votes
1 answer
3k 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 ...
3 votes
2 answers
2k 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) &...
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3 votes
1 answer
786 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)}=\...
  • 352
3 votes
2 answers
251 views

Response of an ideal integrator to a cosine wave

It sounds like a very elementary question on system theory but I got quite confused about it, so hopefully you guys can enlighten me. I'm considering an ideal analog integrator, i.e., a system with ...
  • 2,303
3 votes
1 answer
73 views

What did I do wrong with this simple filter build?

I tried to put everything I have learned from people here together to code my first filter from scratch. Unfortunately, it didn't go well and I'm not getting the expected output. The math/code became ...
  • 427
3 votes
4 answers
5k 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 ...
  • 131
3 votes
1 answer
276 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 ...
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3 votes
1 answer
44 views

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{...
3 votes
2 answers
163 views

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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3 votes
1 answer
104 views

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 ...
  • 123
3 votes
1 answer
166 views

ROC of the function in the problem 9.14 of Oppenheim's Signals and Systems textbook

I have solved the problem 9.14 in Oppenheim's Signals and Systems textbook, but my solution and the one in Slader is different. Problem is given above. And Slader solution is here. I have also ...
3 votes
1 answer
1k views

Inverse Laplace transform of two-sided and one-sided Laplace transform

As I read in Wikipedia, there are two types of Laplace transforms One-sided Laplace transform: $F(s) = \int_{0}^\infty e^{-st} f(t) dt$ Two-sided Laplace transform: $F(s) = \int_{-\infty}^\infty e^{-...
3 votes
1 answer
937 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 ...
  • 131
3 votes
1 answer
3k 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 votes
1 answer
900 views

Impulse response if initial conditions are given

In most books, impulse response of LTI systems are calculated by assuming initial rest condition, but how to calculate response of an impulse input if there are some initial conditions present ? My ...
3 votes
2 answers
409 views

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

A basic model of coupled strings (eg. piano) is provided in DSP Related - JULIUS O. SMITH III - PHYSICAL AUDIO SIGNAL PROCESSING - FOR VIRTUAL MUSICAL INSTRUMENTS AND AUDIO EFFECTS- Two Coupled ...
  • 427
3 votes
0 answers
176 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 votes
3 answers
625 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 ...
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2 votes
3 answers
4k 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 ...
  • 297
2 votes
3 answers
982 views

Transfer function of a frequency shifting system

There is a system which shifts frequencies of input by $-F_c$ such that: $$Y(S) = X(S).H(S)$$ But $X(S)$ has value zero from $0$ to $F_c$. I am confused on how the product of $X(S)$ and $H(S)$ ...
2 votes
2 answers
2k 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 $t<0$. ...
  • 123
2 votes
1 answer
763 views

When to use Fourier, Laplace and Z transforms?

If we have an LTI system, with an input signal $x(t)$, impulse response $h(t)$ and output $y(t)$, I was under the assumption that if the input and impulse response were continuous in time, then you ...
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2 votes
2 answers
419 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 ...
  • 427
2 votes
1 answer
6k 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: ...
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2 votes
2 answers
598 views

Why Are There Two Different Common $ 3 \times 3 $ Kernels for the Laplacian?

I find both of these 3x3 Laplacian kernels to be commonly used: 0 -1 0 -1 4 -1 0 -1 0 and: -1 -1 -1 -1 8 -1 -1 -1 -1 ...