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Questions tagged [derivative]

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1answer
221 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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0answers
63 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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0answers
18 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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0answers
20 views

What is the derivative of Radon Transformation?

We have the Radon transformation, in the most common form: $g(\phi,r)=\int\int f(x,y)\delta(x\sin\phi-y\cos\phi-r)dxdy$ Question 1: So what is the closed form of the derivative of Radon transform w....
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0answers
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Constructing a lowpass digital differentiator such that maximum error is $n$ dB for passband

I want to construct a lowpass digital differentiator with uniform-interval samples such that error from ideal differentiator is $n$ dB in magnitude at maximum for the passband up to frequency $\...
3
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1answer
396 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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1answer
126 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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1answer
84 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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2answers
57 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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2answers
324 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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3answers
543 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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2answers
249 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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2answers
96 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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2answers
132 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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2answers
223 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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2answers
2k views

Derivative filter in Python

In a research paper (section 2.1, 3rd paragraph) about EEG, the authors note that the spectral amplitude profile of a signal is inversely proportional to frequency. To correct for this trend, they ...
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0answers
89 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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1answer
920 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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1answer
48 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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1answer
41 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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1answer
120 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 ...
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1answer
195 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 ...
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1answer
427 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 ...
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1answer
394 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?
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1answer
73 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?
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2answers
489 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 ...
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1answer
744 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 ...
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1answer
20 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 =...
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2answers
159 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
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2answers
1k 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 ...
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0answers
321 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 $y[n]$ ...
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2answers
2k 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
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1answer
810 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 ...
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2answers
2k 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 ...
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2answers
1k 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 ...
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1answer
70 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, ...
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1answer
93 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. ...
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2answers
140 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 ...
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5answers
5k 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 ...
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2answers
64 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 $x[i]$?
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2answers
290 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 ...