Questions tagged [derivative]
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83
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Numeric Derivative Frequency Response
It seems quite common to use np.diff() for removing DC components of signals when plotting spectrograms.
I noticed some unexpected behaviour during analyis which ...
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45
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How to calculate the derivative of a sequence (discrete function) with respect to another discrete function?
In Learning Recursive Filters for Low-Level Vision via a Hybrid Neural Network, the hidden state of the recurrent neural network (RNN), after some simplification, presented as
$$
h[k] = (1-p)\cdot x[k]...
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The Second Derivative of an Image Using the Sobel Filter
For computing the Harris corner detector, I want to calculate $I_{xy}$ and $I_{xy}$. Obtaining $I_x$ and $I_y$ are clear for me. But what about the second order derivatives?
This and this and this one ...
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How to detect decrease of a digital signal?
I have below given digital signal (sampled with period $T = 1\,\mathrm{ms}$)
for which I need to evaluate its decrese. Namely I am interested in regions where the signal decreases faster than $-1000\,\...
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Could the conjugate derivatives of two independent random signals be uncorrelated?
Suppose there are two independent signals, $s(t-\tau)$ and $n(t)$, and they are doubtly uncorrelated so that $\mathbb{E}${$s(t-\tau)\times n(t)$}=0. I wonder if the equation
$\mathbb{E}${$\frac{\...
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How to compensate phase delay introduced by the digital integrator?
Let's say I have a digital integrator with transfer function in following form
$$
\frac{Y(z)}{U(z)} = \frac{T}{2}\cdot\frac{z + 1}{z - 1}
$$
I have been looking for a mechanism how to compensate the ...
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How is a discretized 1D signal be considered a 'vector'?
This question is perhaps related to the semantics or jargon of signal processing. I have checked plenty of advanced books/monographs on multivariable calculus and signal processing but could find an ...
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Is there an intuition or analytic explanation for the multi-taper form of power spectrogram derivatives using Slepian windows?
I am using $x_k(t,f)$, the FFT of the 1-d signal x weighted by the k'th Slepian function in a window around time t. k goes from 1 to K.
This allows defining the multi-taper spectrogram:
$$S(t,f) = \...
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Selective Discrete Time Derivative Filter
I have hit a problem which could be probably solved via discrete time derivative filter able to calculate the derivative of two specific harmonics of the input signal (namely frequencies: $f_1 = 100\,\...
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Fourier Derivative of Discrete Values in Python
I am attempting to find the time derivative of physical system measurements in the Fourier domain. The system can be modeled as an ODE and has a periodic solution.
$$\frac{d}{d t} y{\left(t \right)} = ...
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Instantaneous frequency vs time for a piecewise signal
I sample a signal that consists of a train of pulses which are amplitude and frequency modulated. I would like to generate a plot that is instantaneous frequency vs time. I compute the Hilbert ...
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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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546
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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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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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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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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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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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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)
$$
In some other papers, the transmitted ...
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Bounds of higher order derivative of bandlimited signals
Here is a bandlimited function f(t) with bandwidth Ω:
The function f(t) is bounded in [-A,A].
Then the bound of the derivative of f(t) is bounded as:
|f'(t)|≤2πΩA.
So, what is the bound of its n-th ...
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Expression for Frequency-shifted Gaussian pulse
I am doing some research on UWB radars which transmit frequency-shifted Gaussian pulses. These pulses are given by:
$$
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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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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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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126
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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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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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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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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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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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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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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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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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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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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)$$
We know that we can use ...
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Why we need Laplacian for edge detection?
I am studying some trivial computer vision processing techniques and I came across edge detection algorithms. IMO sharp changes in the gradient are enough indications to detect the edges in an image ...
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Frequency response of FM modulation/demodulation chain with phase derivative demodulation
Frequency modulating a carrier by white noise and then demodulating the complex signal using discrete derivative of phase it appears that the discriminator is acting as a low-pass filter.
How do I ...
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Intuition behind image derivative using Fourier Transform for edges detection
This equation can be shown mathematically:
$\frac{\partial f}{\partial x}=\frac{2\pi i}{N} \mathcal F^{-1}\left(u\cdot \mathcal F(f(x,y)\right)$
I am struggling to understand the intuition behind it ...
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Second (numerical) derivative as estimation of oscillation
I have a discrete signal (vector of numbers) coming from a measurement. This signal has been filtered so that the noise has been removed. Now I am looking for an analytical representation of the ...
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Numerical higher order derivatives and time axis
I have a rather elementary question. Suppose we wish to study even-derivatives of an instrumental signal say second fourth and sixth derivatives and plot it as a function of time. With each successive ...
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What is the form of the spectral derivative in the all-positive-frequency notation in DFT?
The Discrete Fourier Transform (DFT) of a function $u:[0,2\pi] \to \mathbb R$ sampled over $N$ equidistant points $\theta_j = 2\pi j/N,\, j = 0, \dots, N-1,$ is defined by
$$
\tilde U_k = \frac1N \...
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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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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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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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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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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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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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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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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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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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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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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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