Questions tagged [derivation]

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Fourier Transform and Music Analysis

I am a senior in high-school and am currently trying to conduct an exploration on Fourier Analysis, specifically using the Discrete Fourier Transform to analyze a chord played on my piano. Basically, ...
Ralph Khouri's user avatar
2 votes
1 answer
169 views

Frequency domain derivation of Hilbert transform of $\cos(\omega t)$

I'm reading "Understanding Digital Signal Processing, 3rd Edition" by Richard Lyons. Chapter 9 derives Hilbert transform impulse response by defining it in frequency domain first and then ...
Aleksander Alekseev's user avatar
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1 answer
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Time Frequency Analysis Equation Derivation

I have been reading Leon Cohen's book "Time Frequency Analysis" as part of a project for university. On page twelve or equation (1.57) during his derivation of a representation of the ...
RickarySanchez's user avatar
2 votes
1 answer
85 views

Derivation of a non-ideal low-pass rectangular windowed FIR filter

I have been trying to understand certain aspects of FIR filter design which have frankly annoyed me for some time such as exactly why the critical frequency $\omega_c$ in a low-pass FIR filter is ...
RickarySanchez's user avatar
4 votes
0 answers
50 views

Deriving the probability distribution function of the cross-spectral phase

I came across a research paper (\ref{Ref 1}) where the probability distribution of the phase of the cross-spectrum was derived. Unfortunately, I could not understand how the authors arrived at the ...
rill's user avatar
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6 votes
1 answer
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Alternative derivation for a Thiran filter

Following this reasoning (link on ee.se) I'm trying to derive the transfer function for a Thiran filter, but I get stuck. I know of the original paper, I just thought I'd go this way. This is what I'm ...
a concerned citizen's user avatar
1 vote
1 answer
63 views

Cyclic spectrum equality to spectral correlation density

As long as I know, the cyclic auto-correlation is defined as: $$R_x^\alpha\left(\tau\right)=\lim_{\Delta t\rightarrow\infty}\frac{1}{\Delta t}\int_{-\Delta t/2}^{\Delta t/2}x\left(t-\frac{\tau}{2}\...
Gideon Genadi Kogan's user avatar
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1 answer
68 views

Derivation of Inverse Fourier transform from forward Fourier transform

Consider the Fourier pairs: $$\psi(x,t) \stackrel{\mathrm{FT}}{\longleftrightarrow} \Psi(k,t)$$ $$\text{If } \quad \quad\Psi(k,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \psi(x,t) e^{-ikx} \, dx \...
Suresh's user avatar
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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 $G$...
smallStackBigFlow's user avatar
0 votes
2 answers
118 views

Looking for a Mathematical Derivation for the Energy Formula in Continuous-time Domain

I have just started my signal and system course and I would like to know how we derive the corresponding formula for the energy of a continuous-time signal $x(t)$ over an interval $[t_{1},t_{2}]$ : $$...
B E I R U T's user avatar
4 votes
1 answer
392 views

I Q sampling and baseband version of analytic signal

Is it correct to say that if we have a radio signal $s(t)$ centered around the angular frequency of $\omega$ as $\omega\pm\omega_B/2$ (where $\omega_B$ is the bandwidth of the signal) and the ...
axk's user avatar
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2 votes
1 answer
72 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 ...
caesar's user avatar
  • 123
2 votes
0 answers
809 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 ...
tttapa's user avatar
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1 answer
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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 ...
Aditya P's user avatar
  • 171
5 votes
1 answer
255 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 ...
dt128's user avatar
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2 votes
1 answer
234 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 ...
matthew's user avatar
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3 votes
1 answer
396 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 $x(n)$ ...
Sajil C K's user avatar
  • 149
1 vote
1 answer
1k 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 ...
Srishti M's user avatar
  • 616
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2 answers
975 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: $...
Niwol's user avatar
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1 answer
202 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 $\theta_1,\theta_2,...,\theta_k$?...
Rhea's user avatar
  • 1
1 vote
1 answer
484 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 ...
arash javan's user avatar
0 votes
1 answer
116 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/~...
SKM's user avatar
  • 611
2 votes
1 answer
3k 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 ...
ConfusedCheese's user avatar
3 votes
2 answers
3k 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 ...
Clément F's user avatar
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2 answers
3k 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 ...
matteopuc's user avatar
  • 101
0 votes
2 answers
653 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 ...
John Smith's user avatar
3 votes
2 answers
333 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] \...
Mike's user avatar
  • 211
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0 answers
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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 ...
nivag's user avatar
  • 715
6 votes
2 answers
4k 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 ...
JustGoscha's user avatar