Questions tagged [noise]

This tag is for questions about noise. In signal processing, noise is a general term for unwanted (and, in general, unknown) modifications that a signal may suffer during capture, storage, transmission, processing, or conversion.

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84 views

What is the point of using “white noise” term in maths?

I've been studying on my master thesis about "Stochastic Differential Equations". Since I'm new to this topic, I couldn't understand the relation between "white noise" and mathematics. I searched ...
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7 views

Relationship between noisy observations

Assume $\mathbf X_1^n$ is a vector of size $n$ whose elements are either $+1$ or $-1$. Then, we define $$\mathbf Y^n=\mathbf X_1^n+\mathbf N^n$$ where $\mathbf N^n$ is Gaussian additive noise with ...
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1answer
49 views

Do i.i.d. stochastic processes exist in continuous time?

Does there exist a stochastic process $X_t$, $t \in [0,\infty)$, such that $X_t$ is distributed according to some distribution $f(x)$ that possesses finite variance and such that $X_t$ and $X_s$ are ...
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27 views

White noise RMS vs. its bandwidth

From numerical simulation and regression analysis I discovered that the root-mean-square amplitude of white noise with bandwidth $\Delta\!f$ is proportional to $\sqrt{\!\Delta\!f}$. How can this be ...
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35 views

Would sampling the decimal digits of $\pi$ generate a white noise signal?

Discrete r.v. $X = \pi(d)$ (defined in another q of mine). Discrete r.v. $Y = X - 4.5$. q1: Would it be incorrect to deduce $Y\sim U(-4.5,4.5)$ from $X\sim U(0,9)$? q2: If you answered no to q1, ...
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30 views

Measuring information preserved by a slowly varying function

Let $\phi$ be a continuous and injective mapping $\phi: \mathbb{R} \to \mathbb{R}$. Intuitively, this is an information preserving map, but the information is not `stably' preserved. This is because $\...
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36 views

How is the noise gain function defined for higher order discrete piecewise white noise in a Newtonian system?

Background I have been trying to understand Kalman filters and implement them in a project I have. I have been following Roger Labbe's online book (https://nbviewer.jupyter.org/github/rlabbe/Kalman-...
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1answer
16 views

Upper and Lower shapes in Perlin Noise Generated Terrain

I am trying to learn about Perlin Noise and procedural generation. I am reading through an online tutorial about generating landscapes with noise, but I don't understand part of the author's ...
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1answer
28 views

Unit used in continuous time process noise matrix in kalman filters, when STD is from discrete time data

I'm trying to make a process noise matrix in continuous time. But i can't seem to find a clear definition of what "unit" the matrix should contain in continuous time. From our control book we have $...
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66 views

Cross-correlation of a deterministic signal and white Gaussian noise

I'm trying to describe the cross-correlation of a finite length input signal x[n] with the same signal corrupted by white Gaussian noise. If the signal would be infinitely long, the noise would be ...
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15 views

gaussian white noise implies gaussian arma process

An ARMA(p,q) process is a (weakly) stationary process $x_t=\sum_{i=1}^p\phi_ix_{t-i}+z_t+\sum_{j=1}^q\theta_j z_{t-j}$ where $z_t$ is white noise. Lets assume that $z_t$ is Gaussian white noise. ...
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23 views

Notation for adding Gaussian Noise to some function

This is a question of mathematical notation rather than a technical one. If I have some transient signal e.g. $f(t) = A_{0} \sin(\omega_{0} t + \phi)$. I add Gaussian-white noise to $f(t)$, what is ...
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27 views

Determine stationarity of time series containing sin of white noise [closed]

Could someone help me determine the stationarity of the the following time series Y? $ Z_t $ represents white noise with variance $ \sigma^2 $. $ Y_t = \sin(Z_t) + Z^2_t - Z_{t-1}$ I have tried ...
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44 views

Autocorrelation in a Brownian model

I have the following brownian model: $$ \dot{x}=v_0cos(\theta(t))+\sqrt{2D_t}\xi_x(t) \\ \dot{y}=v_0sin(\theta(t))+\sqrt{2D_t}\xi_y(t) \\ \dot{\theta}=\sqrt{2D_r}\xi_\theta(t)\\ $$ with $v_0$ ...
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23 views

How to Modify Measurement-Noise in Kalman Filter from 2D Const-Velocity to 2D Const-Acceleration

After extending a Kalman Filter from 2D Linear Velocity (code) to 2D Constant Acceleration, I realized the State-Predictions have the Y-Position pinned to roughly zero. As you can see, the yellow-...
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42 views

normalizing by using gaussian distribution for negative and positive numbers and feed in Min,Max normalization

I'm dealing in Python with a dataset which has 6 million float numbers belongs to 3 main parameters A, B , C and I map them in 24x20 matrices for each cycle and I plot them 480-values by 480-values. ...
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24 views

Simulating a random process with AWGN

I am trying to simulate the following random process: $$ \frac{dy}{dt} = f(y) + AN(t) $$ where $N(t)$ is additive Gaussian white noise, which, if I remember right, is defined by $\int\limits_0^t N(t)...
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1answer
95 views

How to code oscillator driven by Gaussian white noise? Edit: How to convert ODE to a system of SDE's?

I have written some python code which was designed to try to solve the following differential equation: $$\ddot{x}+\omega_0^2x=\eta(t),$$ where $\eta(t)$ is the gaussian white noise, with mean 0 and ...
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23 views

$E(X(t)X(t))=\sigma^2\delta(\tau)$ or $E(X(t)X(t))=\sigma^2$ [duplicate]

Let's say we have a white noise process $x(t)$ such that: $E(X(t)X(t+\tau))=N\delta(\tau)$ $E(X(t))=0$ In particular, with $\tau=0$, $E(X(t)X(t))=E(X^2(t))$ is infinite. Now, I want $X(t)$ at each ...
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322 views

Will the energy of a randomly driven harmonic oscillator grow to infinity or oscillate about a finite value?

The equation of motion for an undamped harmonic oscillator, with driver $f=f(t)$ is given by: $$\ddot{x}+x=f.$$ Let the initial conditions be given by: $$x(0)=\dot{x}(0)=0.$$ If $f=\cos(t)$ ...
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1answer
86 views

What is a white noise ? What is the derivate of the Brownian motion? [duplicate]

Could someone explain me what is a whit noise ? In my course it's written that it's the derivate of a Brownian motion, but how can it be the derivative of something that doesn't exist ?
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12 views

Inversion of Gradient Noise Function

Brief As Possible Given any equation $f(x)$ which adheres to a predefined pattern, there is always a way (sometimes not directly calculable) to get $f^{-1}(x)$ (which may be a set) such that $f^{-1}(...
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85 views

Error propagation for complex numbers from real/imaginary to modulus/phase

Suppose I have a complex number $Z = a+bi$ which I want to add relative Gaussian noise to. I want to add noise to the real and imaginary component independently. I write this as: $$Z_n = [a(1+\...
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25 views

Possible cause of noise in cosmic radio signals [closed]

In any region of space there is a background of cosmic radiation - such as that from the big bang. This radiation may be detected by an aerial and then amplified - resulting in a detected signal. If ...
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33 views

Scaling properties of 1/f Noise don't make sense

I understand that the Hurst exponent of $\frac{1}{f}$ (pink) noise is $0$ which means the statistics of $\frac{x(kt)}{k^0}=x(kt)$ are the same as $x(t) $ assuming $x$ is pink noise. However, I don't ...
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109 views

How to exclude noise from a polynomial

Consider a polynomial $P\in{\mathbb F}[X]$ where ${\mathbb F}$ is a finite field and $P$ is of degree $f$. Given the set of points $(1,y_1),\ldots,(n,y_n)$ where $n=3f+1$ and $f$ is the upper bound ...
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33 views

Holder Continuity of 1/f (Pink) noise

For $\omega=\omega(t)$ which is 1/f noise, does anyone know the supremum of all its possible Holder exponents i.e. supremum of all $\alpha$ such that $$|\omega(t)-\omega(s)|\leq C|t-s|^\alpha $$ for ...
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54 views

Will it rain the same on a static object vs a moving one?

The following is a silly little problem a friend and me got to talk about, but it got stuck in my mind and cannot shake it. A few days ago we caught in a heavy rain (lot of them these days) and, while ...
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1answer
127 views

Integral of white noise is a continuous space Markov process?

I just read the chapter 4.1 in the Gardiner. He defines that the so called "white noise" $\xi$ has the properties: $\left<\xi\right> = 0$, $\xi(t)$ is independent of $\xi(t')$ for $t\neq t'$. ...
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1answer
88 views

binary vector generation (by hamming distance) limit?

I have the following problem : I generate n-bits long binaries (0|1), where every bit is 0 OR 1 with probability 50%. So, say I generate 'm' such binaries. The question is what is the probability ...
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45 views

Probability of the distance between two noisy measurements being less than a threshold.

Imagine we have two signals: $ x_1[t]=\hat{x}_1[t]+N(0, \Sigma_1)$ $ x_2[t]=\hat{x}_2[t]+N(0, \Sigma_2)$ Where $x_i,~\hat{x}_i $ are the measured, true values, respectively, and $ N(0, \Sigma_i)$ ...
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1answer
38 views

Dealing with noisy points

The following picture represents a graph with price over time. I am a mathematical student, but also a trader. I want to create a function which could localize the good entry and exit points for sale ...
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31 views

Interpretation of autocorrelation function

In my study material I came across this: Interpretation of the autocorrelation function of a WSS process: The autocorrelation function RX ( τ ) measures the correlation between two random variables ...
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1answer
20 views

Interpolation with almost constant slope E

I have a curve $(\sigma_t,\varepsilon_t)$ described parametrically: Data here for (implicit) $t_i=(0,1,...n)$ I have two data series $\sigma_{t_i}=\sigma_1, \sigma_2, ... \sigma_n$ $\varepsilon_{...
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1answer
83 views

Rotational diffusion in Brownian motion

Suppose you have a particle which diffuses as: $$\dot\theta(t)=\sqrt{2D_r}\xi_\theta$$ where $\xi_\theta$ is a random gaussian noise with: $$<\xi_\theta(t)>=0 \\ <\xi_\theta(t)\xi_\theta(...
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1answer
43 views

Is wavelet noise reduction just removing the higher frequency coefficients?

I read some tutorials in noise reduction using wavelets, and they seem to be too simple. With Fourier transforms, there is a distinction between types of noise, and some attempts to estimate the ...
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0answers
106 views

Riemann integral of discrete white noise

I have a time series of values from a Gaussian white noise which are evenly spaced ($\Delta t$). I'd like to approximate $$\int_{0}^{T}\mathsf{A}(t)\mathbf{f}(t) \,\mathrm{d}t$$ where $\mathbf{f}$ is ...
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1answer
40 views

Decision rule that minmize the probability of error

Given We consider a real-valued, discrete-time communication system with a channel gain $h$ and additive white Laplacian noise of unit scale with two possible signals $s \in (-\mu,+\mu)$ that are ...
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23 views

How to utilize the right-hand side in inverse problems

Consider the inverse problem $A \, x = b$ with right-hand side $b$, using SVD: $\qquad A = \sum s_i \, U_i \otimes V_i \ $ — singular values $s_i, \ U_i$ and $V_i$ orthonormal bases $\qquad ...
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1answer
78 views

Relationship between noise term ($\epsilon$) and MLE solution for Linear Regression Models.

In Linear Regression models, given observed variables $x_1, x_2, x_3, ..., x_k$, unobserved (or predicted) variable $y$, and model parameters $\beta_0, \beta_1, \beta_2, \beta_3, ..., \beta_k$, it can ...
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25 views

remove uniform distributed noise with 0 mean and j variance from my data

I have the following problem. I have some points coordinates (x, y), that i affected with random noise. We can suppose that my points are: P1 , P2, P3,..., PN And after the noise addiction: P1 + x1,...
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51 views

How do derive the statistical representation of white noise in the frequency domain.

Setup: A Gaussian white noise process $x(t)$ is defined such that The autocovariance $R(\tau) = \left< x(t) x(t-\tau) \right> = \sigma^2\delta(\tau)$ (uncorrelated in time) The ensemble ...
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1answer
70 views

Information transmitting rate of a noisy discrete channel with Omission/Adding errors

So for a discrete channel, we have a source which sends signal $i$ at probability $p(i)$ that $\sum p(i) = 1$, and the signal is transmitted through a noisy channel. In Shannon's original paper, the ...
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51 views

Calculating average speed from co-ordinates vs time

I'm completely certain that I'm re-inventing the wheel, but struggling to think of search terms to make progress, so please bear with me. I'm gathering a person's location track in an app, and want ...
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1answer
60 views

How do you prove that a code is capacity-approaching?

Specifically, for low-density parity check (LDPC) codes, how do you apply Shannon's Noisy Coding Theorem to prove that not only do codes with zero-approaching maximum word error probability exist, but ...
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1answer
99 views

White Noise Sequence

Would this process be a white noise sequence? Consider the process $\{tY_t\}_{t = 1, . . . , 100}$, where $Y_t$ are independently, normally distributed with mean 0 and variance 1. The process ...
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1answer
945 views

What is a symmetric channel?

In one of the assignments I was asked to explain whether the given channels are symmetric. For instance, $Q_1$ and $Q_2$, the matrix describes a conditional distribution, e.g., $p(decode_1|encode_1)=p(...
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2answers
939 views

Deterministic noise and overfit relation when target function and hypothesis are changed

This question is in reference to Exercise 4.3 in the 'learning from data' book. Here is the question where H is the hypothesis set and f is the target function. Deterministic noise depends on H, as ...
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1answer
48 views

Interpreting equation from article with Ornstein-Uhlenbeck process

I was just reading an article in which the authors simulate an equation with added noise based on an Ornstein-Uhlenbeck process. This is the equation: They describe the term $ n(t) $ in equation A1: ...
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186 views

Correlation matrix and eigenvalues

I have a correlation matrix which has the first 6 eigenvalues substantially greater than the last 11. Also, the last eleven are nearly 1's. Does that mean that the last values symbolize the noise. I ...