# 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, ...
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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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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 $G$...
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### 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/~...
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### 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 ...
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### 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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### 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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### 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 ...
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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 $$\delta[n] \... • 211 0 votes 0 answers 245 views ### What is the reasoning behind the deviration of propogation of uncertainty? When considering the uncertainty of a signal which is determined by multiple inputs propagation of uncertainty states that for measurement$$ y = f(x), x=\{x_1, x_2,..., x_N\}  uncertainty in $y$ is ...
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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 ...