Questions tagged [derivation]

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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 ...
2
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0answers
140 views

Derivation of ZOH Discretization

I'm trying to understand the derivation of the zero order hold discretization method, and I have a couple of questions about some of the steps. I think I understand the first part, this is just the ...
1
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4answers
184 views

Causal noise free 1st order derivation in discrete domain

I need to have causal noise-free first order derivation. Now I am using a simple (x(n)-x(n-1))/Ts. It is super-sensitive to noises. Do we have a noise-free or noise-reduced first order differentiation ...
0
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1answer
689 views

What will the impulse response of a matched filter look like if the input is complex?

If $h(t)$ is the impulse response of a filter matched to a signal $s(t)$, I read that $h(t) = ks(t_o - t)$. But what if the signal is complex? I went through the derivation of the matched filter and ...
5
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1answer
143 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 ...
0
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2answers
434 views

my Butterworth lowpass formulas do not agree with Fisher webpage

I want to implement Butterworth low-pass filter. Thanks to this question, I have found out that the filter coefficients can be generated using Tony Fisher web-site or using his code. But the problem ...
1
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1answer
2k views

Butterworth filter approximation: derivation and output poles

I am having trouble understanding the exact derivation of the butterworth filter and how it results in the output of the poles. I have researched multiple lecture series and textbooks and this is my ...
2
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1answer
167 views

Derivation of range migration algorithm

Problem: In Walter G.Carrara's book on synthetic aperture radar, the equation is presented: $\Phi(K_X, K_R) = -K_XX_t - R_B\sqrt(K_R^2 - K_X^2) +K_RR_S $ (10.30) And this is said to come from ...
3
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1answer
193 views

Doubts on LMS derivation

I have been trying to follow the Least Mean Square(LMS) algorithm derivation given by Wikipedia here and have the following questions. Here I expected $y(n)$ is to be computed by convolving $x(n)$ ...
1
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1answer
448 views

Unable to understand the derivation of the update equation for LMS

I am trying to follow the derivation of the Least Mean Square https://en.wikipedia.org/wiki/Least_mean_squares_filter#Proof but I cannot get the update rule. I am stuck in the following steps and ...
0
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1answer
479 views

Derivation of downsampling in the frequency domain

I'm having some trouble with this derivation $$s_d = \begin{cases}1&\text{for}\quad m\quad\text{multiples of}\quad D\\ 0 &\text{otherwise} \end{cases}$$ then $s_d$ is somehow rewritten to: $...
0
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1answer
131 views

Beam Former linear array antenna

How can I design a beam former (with weigth vector $w$) that protects a signal coming from an angle of arrival $\theta_0$ and completely suppresses signals coming from $\theta_1,\theta_2,...,\theta_k$?...
1
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1answer
281 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
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1answer
82 views

Unable to understand how the paper simplifies the covariance matrix - Kalman filter

The paper Convergence Analysis of the Unscented Kalman Filter for Filtering Noisy Chaotic Signals presents the convergence analysis of Unscented Kalman Filter download http://www.eie.polyu.edu.hk/~...
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 ...
0
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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 ...
3
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2answers
264 views

Deriviation of the “Twiddle Sum” property

I can't seem to understand how to derive the "twiddle sum" property: $$\sum_{n=0}^{N-1}W_{N}^{kn}=N \ \delta[k\bmod N] $$ where $$ W_{N} \triangleq e^{\frac{j 2 \pi }{N}} $$ and $$ \delta[n] \...
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0answers
215 views

What is the reasoning behind the deviration of propogation of uncertainty?

When considering the uncertainty of a signal which is determined by multiple inputs propagation of uncertainty states that for measurement $$ y = f(x), x=\{x_1, x_2,..., x_N\} $$ uncertainty in $y$ is ...
4
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2answers
594 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 ...