Questions tagged [derivative]
The derivative tag has no usage guidance.
53
questions
1
vote
1answer
49 views
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 ...
1
vote
1answer
26 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 ...
0
votes
1answer
39 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 ...
1
vote
1answer
69 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 ...
4
votes
1answer
146 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 ...
0
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0answers
46 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 ...
0
votes
1answer
43 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 ...
0
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0answers
20 views
Which frequency bins give the best interpolation for the derivative of a function?
A function $u:[0,2\pi]\to\mathbb R$ sampled over $N$ equidistant points $\theta_j=(2\pi/N)j,\, j = 0, \dots, N-1,$ can be interpolated by a set of functions $\{u_{k_0}\}$ enumerated by integers $k_0\...
0
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1answer
39 views
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 \...
0
votes
3answers
166 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 ...
2
votes
1answer
28 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 ...
2
votes
1answer
285 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 ...
0
votes
1answer
49 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 ...
1
vote
1answer
709 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 ...
2
votes
0answers
93 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 $...
0
votes
0answers
32 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 ...
4
votes
1answer
2k 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 ...
5
votes
1answer
149 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
$$
where $A=\sup|f(t)|$ and $B$ is the bandwidth of $f(t)$. The question is: is there a relationship ...
-1
votes
1answer
141 views
Digital Derivative
I need to calculate the derivative of a digital signal (sinusoidal). In one of the papers they have mentioned Gilbert transformation is used to calculate the derivative. But I have searched in the net ...
3
votes
2answers
90 views
Estimating a Signal Given a Noisy Measurement of the Signal and Its Derivative (Denoising)
I have a signal and its derivative simultaneously measured, both including additive noise. The measurement is completed before the analysis, so it can be looked ahead. Now I want to reconstruct a less ...
8
votes
2answers
602 views
Bounds of the derivative of a bounded band-limited function
Let $f(t)$ be a function with properties:
$$\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&...
4
votes
3answers
3k 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 $x$, ...
1
vote
2answers
485 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 ...
0
votes
2answers
164 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 $v \sim CN(0,2\sigma^2_v)$ ...
0
votes
2answers
229 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 ...
1
vote
2answers
690 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 ...
0
votes
2answers
3k 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 ...
2
votes
0answers
111 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 ...
0
votes
1answer
1k 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 ...
3
votes
1answer
53 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 $f'(t)\theta(t)+f(0)\delta(t)$, ...
0
votes
1answer
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 ...
0
votes
1answer
127 views
Contour sharpening: Optimal direction for derivation
Talking about sharpening a contour in an image. What's the optimal direction for derivation? What's the maximum value of derivative?
I think that the optimal direction for derivation is the direction ...
1
vote
1answer
320 views
Different approaches for partial image derivation
I know there are different ways for partial derivation of an image, among others: Sobel kernel, LoG, Prewitt and so on.
But the simplest one is the central difference:
$$
\frac{d}{dx} f(x) \approx ...
0
votes
1answer
830 views
Image derivative in the Fourier space
It can be shown that the image derivative in the $x$ direction is given by:
$$
\frac{\partial f}{\partial x}=\frac{2\pi i}{N} \mathcal F^{-1}\left(u\cdot \mathcal F(f(x,y)\right)
$$
where $N$ is the ...
1
vote
1answer
632 views
How do derivative masks work for finding edges in image?
As I know, masks like follow are derivative mask,
but I don't get it why they can detect edges and which one of them can find vertical edges and which one can find harizontal edges?
1
vote
1answer
86 views
Why level of noise can be magnified twice through each numerical differentiation?
I was reading a paper and saw this is mentioned there, but I cannot figure out how this can analytically be proven?
3
votes
2answers
550 views
Differentiation of sine in Fourier domain
The derivative of $\sin(\omega_o t)$ is $\cos(\omega_o t)$.
The Fourier transform of $\sin(\omega_o t)$ is $\frac{\pi}{j}[\delta(\omega-\omega_o) - \delta(\omega+\omega_o)]$.
Differentiation in the ...
0
votes
1answer
1k views
discrete-time (slew) rate limiting algorithm
Consider uniformly-spaced samples of smooth, bandlimited signal observed in noise and subject to some artifacts (small jumps). Physical restrictions impose a constraint on the maximum magnitude of ...
0
votes
1answer
23 views
Derivative of equation containing Gaussian filtered image
Let $w$ be our image. For example, consider the following with the vectorized $w$:
$$ E(w) = \frac 12 \|Aw+b\|_2^2 $$
I know the optimal condition of the above equation:
$$ \nabla_wE = A^TA\hat w + b =...
0
votes
2answers
221 views
What are the implications of resampling the derivative of a signal with a higher frequency than the original signal?
Suppose I have a digital signal measured with sampling time, $T_s=1$ sec. If I take it's derivative, it will, naturally have $T_s=1$ sec. But what are the implications if I re-sample this derivative ...
3
votes
2answers
2k views
Is there a difference between filtering a signal before or after differentiating it?
I have a time series and I want to apply:
a differentiation
a Butterworth filter
Does the order theoretically (mathematically) make any difference? Does it make any difference in real life when I ...
2
votes
4answers
232 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 ...
0
votes
0answers
372 views
Smoothing the discrete acceleration
In order to know if my signal is increasing or decreasing, I'm using the discrete derivative $y[n] = x[n] - x[n-1]$
or a smoothed version of it (for example Exponential Weight Moving Average of $y[n]$ ...
0
votes
2answers
3k views
Compute the time derivative of a noisy digital signal?
The issue is that my signal is very noisy. I need extract its time derivative as accurate as possible. P.S. I do not have any prior knowledge on the signal (black box).
On forums some suggested ...
3
votes
1answer
950 views
Wrong estimation of derivatives with an extended Kalman filter
I am trying to implement an extended Kalman filter (EKF) in MATLAB for the estimation of joint trajectories (angular position, angular velocity and angular acceleration) from noisy motion capture ...
4
votes
2answers
3k views
derivative filter transfer function
In many of the papers it is said that the derivative filter transfer function is given by: $$H(z) = \dfrac{1}{8T}\left(-z^{-2} - 2z^{-1} + 2z + z^{2}\right)$$ But no one gave the detailed information ...
0
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2answers
2k views
Digital signal derivative
I'm new to signal processing and I need your help: I have an array of 128 elements (call it Window) filled with 128 samples taken from a sensor. I was wondering how ...
0
votes
1answer
74 views
Derivative Notation as a factor
In the figure below, from the book "Principles of Linear Systems and Signals - Lathi", the author uses the notation D to replace the Leibniz's notation $d/dt$, and after that he rewrites the equation, ...
1
vote
1answer
94 views
Derivative of signal with missing samples
I have software that tracks an object moving (in the x-dimension only) across a video shot from a stationary camera. I need to find the velocity and acceleration of the object as functions of time. ...
-1
votes
2answers
173 views
How to find derivative of 2-D elliptical Gaussian function with different standard deviations along $x$ and $y$ directions?
I am trying to find the 2-D derivative of an elongated Gaussian density.
The Gaussian has standard deviations $\sigma_x$ and $\sigma_y$. How can I get the scale-normalized 2-D Gaussian derivative in ...
