Questions tagged [derivative]

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Proving Fourier transform pair with derivatives using duality

I want to use duality to prove the Fourier transform pair $t^nx(t) \overset{\mathscr{F}}{\longleftrightarrow} j^n\frac{d^nX(\omega)}{d\omega^n}$ but I am struggling. I learned that if $x(t) \overset{\...
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5 votes
1 answer
55 views

How to perform Spatial derivative calculation?

I'm trying to solve the following question about "Harris Corner Detection". Consider the following image: As the first step of the "Harris Corner Detection", we should compute the ...
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What is the unit of velocity calculated using Savitzky–Golay filter?

I have data from an eye tracker collected while subjects were doing a saccadic task (i.e. fast eyes movements). Specifically, I have the position (in degree) on the x and y axes, collected at 120Hz. ...
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2 votes
1 answer
58 views

How to find point of inflexion of a digital signal?

Let's say I would like to find a point of inflexion of a digital signal which has been gathered via sampling of a continuous time domain signal with sampling period $100\,\mu s$. The digital signal ...
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2 answers
693 views

Are scipy second-order Gaussian derivatives correct?

For an edge detection algorithm, I need to compute second-order derivatives of an image, and I do this with use of Gaussian derivatives. I assumed that the ...
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0 answers
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How to measure smoothness of a signal

Lets say I have two signals with a different number of data points - a step function and a smooth spline as you can see below. What is mathematical operation I can use to quantify the "smoothness&...
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1 vote
1 answer
111 views

What phase rotation occurs when you take the derivative of an audio signal?

If you take the derivative of an audio signal, it provides a 6 dB/oct upward sloping filter (increasing high frequencies / cutting low frequencies) all the way across the spectrum. What is the result ...
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2 votes
1 answer
51 views

Relation between Gaussian derivatives and Gausian-windowed Cosine function

I am doing some research on UWB radars which transmit a pulse given by: $$ g(t) = p(t) cos(2\pi f_{c}t) = A e^{(-\frac{t^{2}}{2\tau ^{2}})} cos(2\pi f_{c}t) $$ In some other papers, the transmitted ...
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3 votes
1 answer
97 views

Bounds of higher order derivative of bandlimited signals

Here is a bandlimited function f(t) with bandwidth Ω: The function f(t) is bounded in [-A,A]. Then the bound of the derivative of f(t) is bounded as: |f'(t)|≤2πΩA. So, what is the bound of its n-th ...
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1 answer
90 views

Expression for Frequency-shifted Gaussian pulse

I am doing some research on UWB radars which transmit frequency-shifted Gaussian pulses. These pulses are given by: $$ g(t) = p(t)cos(2\pi f_{c}t) = V_{TX}exp(-\frac{t^{2}}{2\tau^2})cos(2\pi f_{c}t) $...
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2 votes
1 answer
84 views

How to handle a logarithmic term in Kalman filter?

I am trying to implement a Kalman filter for an echo pulse detection application as similar to this paper. (an open source version is here (pg 16)) The measurement variable is $h(x,t)=A_0 (\dfrac{t-\...
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1 vote
1 answer
367 views

kernel to calculate second order partial derivative of digital image

I'm working on image stacks, and I need to calculate second order partial derivatives of it. I already know how to calculate derivative on x and y axis, using Finite Cameron Taylor - Difference ...
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0 answers
51 views

Gradient of transfer function (z-transform) with respect to coefficients/parameters?

my problem is perhaps very simple but I just can not find the answer, even though this is used (though not explained) in several books: What is $\frac{\partial G (z, \theta)}{\partial \theta}$ when $G$...
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1 vote
0 answers
56 views

Augment State Space Model with Derivative of State

Problem How do you augment a state space model with the derivative of a state? I know how to augment a state space model with the integral of a state by doing the following. Given a linear system $$ \...
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5 votes
1 answer
341 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 ...
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3 votes
2 answers
54 views

How to detect start and finish of temperature control in temperature time series

I have a huge dataset containing temperature data inside a building. I want to extract the time that the building starts and stops controlling the temperature (approximately around the vertical black ...
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2 votes
2 answers
52 views

What is a speaker behaviour for out of range frequencies?

I was watching this video. The experiment is to create a square signal and record what is outputted by a speaker. We see two diracs signal (+ and -) that looks like the derivative of the signal. Can ...
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3 votes
1 answer
352 views

Laplacian of Gaussian operator

This might be a silly question. I was reading about the Laplacian of Gaussian (LoG) operator and got confused about the alternative equivelant ways we can make use of it. Let's assume we have a 2D ...
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14 votes
6 answers
32k views

What is the first derivative of Dirac delta function?

Could you please help me in a simple way, what is the first derivative of a Dirac delta function? I found this answer: The informal answer is a positive Delta function immediately followed by a ...
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0 votes
1 answer
628 views

Implementing a finite impulse respone (FIR) filter for computing the derivative of a discrete signal

I am trying to reimplement an algorithm on my own. In the description of the implementation, it's written that they compute the derivate of a series of value using a [-1/2, +1/2] finite impulse ...
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1 vote
1 answer
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Relation between original points and 1st/2nd derivative points

I have points $\{p_0, p_1, ... p_n\}$, I create a discrete derivative consisting of $\{d_0, d_1, ..., d_{n-1}\}$ like this: $d_k=p_{k+1}-p_k$. I'd like to choose a point in the original signal by ...
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1 vote
1 answer
116 views

Validity of differentiation property of Fourier transform

we know the differentiation property of Fourier transform says that, if $$x(t)\longleftrightarrow X(j\omega)$$ then $$\dfrac{d}{dt}x(t)\longleftrightarrow j\omega X(j\omega)$$ We know that we can use ...
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0 votes
1 answer
80 views

Why we need Laplacian for edge detection?

I am studying some trivial computer vision processing techniques and I came across edge detection algorithms. IMO sharp changes in the gradient are enough indications to detect the edges in an image ...
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  • 101
1 vote
1 answer
231 views

Frequency response of FM modulation/demodulation chain with phase derivative demodulation

Frequency modulating a carrier by white noise and then demodulating the complex signal using discrete derivative of phase it appears that the discriminator is acting as a low-pass filter. How do I ...
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4 votes
1 answer
491 views

Intuition behind image derivative using Fourier Transform for edges detection

This equation can be shown mathematically: $\frac{\partial f}{\partial x}=\frac{2\pi i}{N} \mathcal F^{-1}\left(u\cdot \mathcal F(f(x,y)\right)$ I am struggling to understand the intuition behind it ...
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0 votes
0 answers
62 views

Second (numerical) derivative as estimation of oscillation

I have a discrete signal (vector of numbers) coming from a measurement. This signal has been filtered so that the noise has been removed. Now I am looking for an analytical representation of the ...
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0 votes
1 answer
85 views

Numerical higher order derivatives and time axis

I have a rather elementary question. Suppose we wish to study even-derivatives of an instrumental signal say second fourth and sixth derivatives and plot it as a function of time. With each successive ...
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-1 votes
1 answer
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What is the form of the spectral derivative in the all-positive-frequency notation in DFT?

The Discrete Fourier Transform (DFT) of a function $u:[0,2\pi] \to \mathbb R$ sampled over $N$ equidistant points $\theta_j = 2\pi j/N,\, j = 0, \dots, N-1,$ is defined by $$ \tilde U_k = \frac1N \...
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0 votes
3 answers
521 views

Frequency response of numerical derivative

Analytical derivative of a function is equivalent to convolution of that function with $s$ in Laplace domain. Numerical derivatives are limited in bandwidth due to finite sampling rate, so they are ...
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2 votes
1 answer
56 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 ...
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2 votes
1 answer
1k views

Why is a first/second derivative useful in spectroscopy?

I'm currently working on Raman spectroscopy and while reading some literature I came across the first and second derivative of a spectrum. It's not clear to me why they are useful to look at since ...
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0 votes
1 answer
106 views

Time derivative of signal - effect on noise distribution

I have an angular velocity measurement that has a certain amount of ripple that yields an histogram shaped like a gaussian curve. I quantify that ripple using a standard deviation. I am interested in ...
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1 vote
1 answer
2k views

Bilinear Transform (Tustin's Method) applied to the Derivative

I hope that I have not misunderstood something terribly wrong, but the continuous derivative $D=d/dt$ can be considered a transfer function in Laplace space $D(s) = s$, right? So when I try to ...
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2 votes
0 answers
128 views

Correct way of derivating in frequency domain with FFT

I believe I am very close to the answer and only need a small nudge to get to the answer. What I want: I want to take a signal, use FFT to transform it to the frequency domain (FD), multiply it by $...
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0 votes
0 answers
39 views

derivative filter for computing speed from robot rangefinder

I have a laser rangefinder on my robot and I need to compute the speed of the detected object. I have seen many different filters for that, but all of them apply only to offline data, or introduce ...
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5 votes
1 answer
4k views

What exactly is Savitzky-Golay differentiation filter?

I could understand Savitzky-Golay filter as being smoothing filter, but then there also seems to be Savitzky-Golay differentiation filter, though for some reason, details do not seem to be clear. So ...
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5 votes
1 answer
197 views

Bounds of the difference of a bounded band-limited function

For a continuous signal (function), we have Bernstein inequality : $$ |{df(t)}/dt| \le 2AB\pi $$ where $A=\sup|f(t)|$ and $B$ is the bandwidth of $f(t)$. The question is: is there a relationship ...
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-1 votes
1 answer
259 views

Digital Derivative

I need to calculate the derivative of a digital signal (sinusoidal). In one of the papers they have mentioned Gilbert transformation is used to calculate the derivative. But I have searched in the net ...
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6 votes
2 answers
144 views

Estimating a Signal Given a Noisy Measurement of the Signal and Its Derivative (Denoising)

I have a signal and its derivative simultaneously measured, both including additive noise. The measurement is completed before the analysis, so it can be looked ahead. Now I want to reconstruct a less ...
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12 votes
2 answers
1k views

Bounds of the derivative of a bounded band-limited function

Let $f(t)$ be a function with properties: $$\begin{array}{ll} t\in\mathbf{R}&t\text{ is in reals}\\ f(t)\in\mathbf{R}\text{ for all } t&f(t)\text{ is in reals}\\ |f(t)|<A\text{ for all }t&...
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6 votes
3 answers
6k views

Derivative with respect to complex conjugate

I have a real function $C$ of a complex vector $x$. While taking the gradient of the function $C$ for minimising the same, why do we take the derivatives with respect to the complex conjugate of $x$, ...
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  • 133
1 vote
2 answers
941 views

should I apply low-pass filter when calculating central derivative?

Let us say that we have a discrete signal $I_n$, $n=0, 1, 2, ...$. According to Nyquist theorem the maximum frequency for such discretization is $f_{max} = 0.5$. Now imagine that I want to calculate ...
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1 vote
2 answers
230 views

MLE parameter estimation -- confusion regarding some terms in the pdf of complex normal r.v (Part 2)

This question is based on the application of the pdf which was an earlier question of mine asked here Confusion regarding pdf of circularly symmetric complex gaussian rv If $v \sim CN(0,2\sigma^2_v)$ ...
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0 votes
2 answers
414 views

Ramp function as derivative in frequency domain?

It is said that to get Laplacian of Gaussian in frequency domain, we may multiply the Fourier transform of Gaussian with two differentiating ramp function (1 ramp gives 1 order of derivative). The ...
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1 vote
2 answers
2k views

Why taking derivative amplifies noise

In image processing we may use derivatives to help us detect the edges. While at mean time, this operation would also make the data noisier. But why do we have this effect? My intuition is that if we ...
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3 votes
2 answers
6k views

Derivative filter in Python

In Alaa Kharbouch, Ali Shoeb, John Guttag, Sydney S. Cash, An algorithm for seizure onset detection using intracranial EEG, Epilepsy & Behavior, Volume 22, Supplement 1, 2011 (section 2.1, 3rd ...
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2 votes
0 answers
120 views

Helmholtz decomposition implementation

I need to perform the Helmholtz decomposition of a 2D flow. An old and obvious problem is the numerical differentiation (largely amplifying the noise). I do understand the procedure I am just ...
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0 votes
1 answer
2k views

First derivative analog filter

I'm reading about fault detection via signal processing in time domain. One possibility is to check that first derivative of the signal is in some predefined bounds. The text says that to obtain the ...
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3 votes
1 answer
71 views

How to differentiate the product signal $f(t)\theta(t)$, where $\theta(t)$ is Heaviside's unit step function?

What is the derivative (in the engineer's sense) of the causal function $f(t)\theta(t)$, where $\theta$ is the Heaviside unit step function? I've seen the formula $f'(t)\theta(t)+f(0)\delta(t)$, ...
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0 votes
1 answer
43 views

Image Geometry Terms

I getting confused over some basics of image geometry terms due to different definitions, implementations and different outputs in the internet, here is an example. The terms are: Image derivative ...
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