Questions tagged [finite-differences]

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1 vote
1 answer
96 views

Not getting the same step response from Laplace transform and it's respective difference equation

I am trying to simulate a plant on a microcontroller. The transfer function of the plant is $$ G_{p} \left( s \right) = \frac{2}{\left( s + 3 \right) \left( s - 1 \right)} \tag{1} \label{1}$$ The step ...
1 vote
1 answer
453 views

How to calculate a 3D Fourier Transform?

I am currently employing a centered finite difference method to simulate the excitation of a circular membrane by a forcing term in Python, and the code is functioning as intended. To further enhance ...
0 votes
0 answers
75 views

Derivation of 9 point Laplacian filter

I'm reading a paper on how construct isotropic laplacian filter, and perhaps because it's an old paper, the notation in it really bothers me a lot. So can someone please explain it to me? For example, ...
1 vote
1 answer
996 views

Particular Solution to Difference Equation

I have a system given by $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$ I want to solve for $y[n]$ for $x[n]=(\frac{1}{2})^nu[n]$. The complementary solution evaluates to $[k_1(\frac{1}{2})...
3 votes
1 answer
397 views

The Gradient Operator of a Vectorized Image in Matrix Form

I have this optimization problem: $$ \arg \min_{ X \left( i, j \right) } \sum_{i, j} \left\| X \left( i, j \right) - 255 \right\|_{2}^{2} + \lambda \sum_{i, j} \left\| \nabla X \left( i, j \right) - \...
0 votes
1 answer
32 views

Undo finite difference with arbitrary timesteps

I'm wondering if there is a way to undo a finite difference filter with arbitrary timesteps. In the simplest case of a two-sample finite difference of a time-series $x[n]$, \begin{equation} y[n] = x[n]...
1 vote
1 answer
236 views

FM demodulation using complex differentiation

I'm trying to implement the complex differentiation discriminator which is taken from Software-Defined Radio Using MATLAB, Simulink, and the RTL-SDR. The final result is $$s(t) = \frac{s_q'(t) s_i(t) -...
0 votes
0 answers
21 views

Square pulse test of Upwind Finite Differences

I`m analyzing the numerical methods for the 1D convection equation for stability, consistency, and accuracy. I want to see How does this square pulse move in domain and time? Here is my code % ...
2 votes
1 answer
964 views

Backward finite difference differentiation filter frequency response

As title says what would be frequency response of backward finite difference differential filter, or what would be error of this differential filter, analyzed upon frequency of a signal?
5 votes
1 answer
794 views

Compute the Second Order Derivative of Digital Image with Finite Differences

I was looking for how to compute second order derivative of an image and came across the question kernels to Compute Second Order Derivative of Digital Image. In the top voted answer, it gives an ...
0 votes
1 answer
73 views

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 ...
0 votes
0 answers
27 views

Numerical computation of the derivative of a curve valued in $\operatorname{Diff}(\mathbb{R}^2)$

Assume we have a (discretized) sufficiently smooth curve $\alpha : [0, 1] \to \operatorname{Diff}(\mathbb{R}^2)$. Further assume that this curve is generated via a large deformation diffeomorphic ...
3 votes
2 answers
462 views

Multiplying the imaginary part of DFT with a linear ramp to get a derivative

I am trying to understand the statement in a relatively old publication from 1970s, when Fourier transforms found applications in chemical analysis. The author quotes the derivative theorem citing ...
1 vote
1 answer
460 views

Different type of padding in image?

I want to implement a gradient operator on an image in Matlab. Should I have to pad the image before its implementation? How do I decide which padding to use e.g. whether to use zero-padding or ...
1 vote
1 answer
129 views

Numerical differentiation on circular data

I have multiple time series of position data (x, y, z) and orientation data expressed as Euler/Tait-Bryan angles (yaw, pitch, roll) obtained from a head tracking device. I need to obtain velocity ...
1 vote
1 answer
93 views

Calculate the Derivative of Gradient Field of an Image

I meet a confusing thing in image processing recently.... Assume the image $x \in \mathbb{R}^n$, with its derivative (difference) matrix: $D^+ = \begin{bmatrix} D_h \\ Dv \end{bmatrix} \in \mathbb{R}...
1 vote
0 answers
93 views

Is non-causal, non-LTI solution to difference equation correct?

Find a solution to the difference equation $y[n]-\frac{5}{6}y[n-1]+\frac{1}{6}y[n-2]=\frac{1}{3}x[n-1]$ that is neither casual nor LTI, where $y[0]=y[1]=1$ and $x[n]=\delta[n]$ The homogenous ...
5 votes
3 answers
789 views

Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $?

A continuous state space model is defined as follows. $$ \dot{\boldsymbol{x}}(t) = A \boldsymbol{x}(t)+ B \boldsymbol{u}(t) \\ \boldsymbol{y}(t)= C \boldsymbol{x}(t)+ D \boldsymbol{u}(t) $$ If we ...
0 votes
1 answer
365 views

Time invariance and linearity of recursive system

I am confused about the definition of linearity and time invariance of recursive system Given $$y[n] = y[n-1] - y[n-2] - x[n]$$ To test time invariance, we shift the input $x[n]$ for shifting and ...
4 votes
4 answers
645 views

Causal Noise Free 1st Order Derivative in Discrete Domain

I need to have causal noise free first order derivation (Derivative). Now I am using a simple finite differences formula: $$ \frac{ x \left( n \right) - x \left( n - 1 \right) }{ {T}_{s} } $$ The ...
2 votes
1 answer
73 views

Understanding the resulting image matrix when differentiating image

Let $A$ be a image matrix and let $P(i,j)$ be the gray level of pixel $i,j$. Let $0$ be black and $255$ be white Assume I want to differentiate this image with respect to the columns $(x)$ as in I ...
0 votes
1 answer
268 views

Given a Horizontal Derivative Kernel What Would Be the Output Image?

Hi, im solving this problem and everything went well until i went and solved the position p(5,3) and i got the answer of 0. However, the solution guide lists the solution as 150 which is pointed at ...
3 votes
1 answer
168 views

Linear Difference Equation and Method of Least Squares

I'm reading the book "Fault-Diagnosis Systems" by Isermann in the par. 9.2.1a. The author explains how to estimate the parameter of a linear difference equation using Least Squares. We start with a ...
2 votes
2 answers
418 views

Frequency response of numerical differentiation by polynomial interpolation / finite difference

One can use polynomial interpolation (or finite difference) to do numerical differentiation. However, there seems to be a surprising lack of interest in obtaining frequency response of this numerical ...
0 votes
1 answer
668 views

Using Parks-Mcclellan (or others) for FIR differentiator?

I have never seen the use of Parks-Mcclellan for differentiator - most uses tend to be lowpass filters. Is this because Parks-Mcclellan implementation of FIR differentiator has numerical problems and ...
1 vote
0 answers
78 views

How far can numerical differentiation be accurate for sinusoids?

Suppose you have samples of uniform interval of a sinusoidal signal that is some sum of sinusoids. Forget about aliasing, as the signal is bandlimited. If we have infinite number of samples at ...