Questions tagged [derivative]

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9
votes
2answers
837 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&...
7
votes
6answers
12k 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 ...
5
votes
3answers
5k 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 $x$, ...
5
votes
2answers
2k views

Ways to compute the n-the derivative of a discrete signal

This is a pretty general question about how to compute derivatives of a digital signal $x[n]$. I would like to know what are the different approaches (from naive to complex) and how are they compared ...
5
votes
1answer
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 ...
5
votes
1answer
172 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 ...
4
votes
5answers
9k views

Derivative of noisy signal

My input signal is phase vector. I want to differentiate it to get frequency vector. My input signal is somewhat noisy. Here is the input signal. This is the derivative of the input signal as ...
4
votes
1answer
310 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 ...
4
votes
2answers
4k 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 ...
4
votes
2answers
116 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 ...
3
votes
2answers
655 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 ...
3
votes
4answers
384 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 ...
3
votes
2answers
5k 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 ...
3
votes
1answer
137 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 ...
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 ...
3
votes
2answers
51 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 ...
3
votes
1answer
205 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 ...
3
votes
1answer
61 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 $f'(t)\theta(t)+f(0)\delta(t)$, ...
3
votes
1answer
1k 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 ...
2
votes
1answer
79 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-\...
2
votes
1answer
42 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
39 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 ...
2
votes
2answers
48 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 ...
2
votes
1answer
800 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 ...
2
votes
1answer
55 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 ...
2
votes
0answers
106 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 $...
2
votes
0answers
118 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 ...
1
vote
2answers
1k 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 ...
1
vote
1answer
1k 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 ...
1
vote
2answers
748 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 ...
1
vote
1answer
101 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?
1
vote
1answer
53 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
175 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 ...
1
vote
1answer
360 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 ...
1
vote
2answers
86 views

How do I estimate the derivative of the current position, when I have only values from past to present?

If I have a discrete real-time signal $x[n]$, with its latest value $x[i]$ and all its past values $x[i-t]$, how can I estimate the derivative at $x[i]$?
1
vote
1answer
30 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 ...
1
vote
1answer
63 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 ...
1
vote
1answer
868 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
98 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
vote
0answers
39 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 $$ \...
0
votes
3answers
324 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 ...
0
votes
2answers
216 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
1answer
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 ...
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 ...
0
votes
1answer
60 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) $...
0
votes
1answer
311 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 ...
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
1answer
54 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 ...
0
votes
1answer
68 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
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 ...