Signal Processing Stack Exchange
Signal Processing Stack Exchange

Questions tagged [derivation]

23 questions
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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 to one another? Is it possible with FIR/IIR…
JustGoscha
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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 for a discrete function $x[n]$ like this? $$ |x[n]…
dt128
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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 corresponding analytic signal is $s_a(t)=s(t)+j…
axk
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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 $x(n)$ with $h(n)$, but in the equation given by…
Sajil C K
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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 use numpy? Thanks in advance!
Clément F
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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] \triangleq \begin{cases} 1 & \text{if } n=0 \\ …
Mike
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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 want $P(i,j)=P(i,j+1)-P(i,j)$ in the new image. I…
caesar
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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 solution to a system of linear equations using the…
tttapa
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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 substituting the value for $X_a$ using the Principle…
matthew
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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 understanding so far: An idealised low-pass…
ConfusedCheese
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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}\right)x^*\left(t+\frac{\tau}{2}\right)e^{-2\pi…
Gideon Genadi Kogan
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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 shall appreciate help where I am going wrong. The…
Srishti M
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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 \frac{f(x+1) - f(x-1)}{2} \longrightarrow 0.5[1\…
arash javan
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Derivation of Inverse Fourier transform from forward Fourier transform

Consider the Fourier pairs: $$\psi(x,t) \stackrel{\mathrm{FT}}{\longleftrightarrow} \Psi(k,t)$$ $$\text{If } \quad \quad\Psi(k,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \psi(x,t) e^{-ikx} \, dx \quad \quad \dots(i)$$ $$\text{then, can we…
Suresh
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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 $G$ is some transfer function, and $\theta$ the…
haavbj
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