14 votes

Why is Gaussian noise called so?

Noise is random, but like most random phenomena, it follows a certain pattern. Different patterns are given different names. Consider rolling a die. This is clearly random. Roll the die 1000 times, ...
user avatar
  • 13.8k
10 votes
Accepted

What Does It Mean Exactly When Two Parts of a Signal Are Correlated?

Yeah, it can mess you up pretty badly if you don't get the fundamentals right off the get-go. This is how I interpret correlation, and it has worked for me for what I do for a living. Let's start ...
user avatar
  • 503
10 votes
Accepted

What is the connection between analog signal to noise ratio and signal to noise ratio in the IQ plane in a quadrature demodulation system?

Because each step in the processing chain is linear we consider a case with only noise and no coherent signal. Denote the noise $\xi(t)$. The $I$ and $Q$ signals are \begin{align}\ I(t) &= \xi(t) \...
user avatar
  • 1,016
10 votes

Is Allan variance still relevant?

I currently work in the design of atomic clocks and precision frequency sources and pleased to report that the Allan Variance is still quite relevant and useful. In fact it's utility extends to ...
user avatar
  • 37.1k
8 votes
Accepted

Concept About Estimated Standard Deviation

Given data $ { \left\{ {x}_{i} \right\} }_{i = 1}^{N} $ the Empirical STD of the data is well defined: $$ STD = \sqrt{ \frac{1}{N - 1} \sum_{i = 1}^{N} { \left( {x}_{i} - \bar{x} \right) }^{2} } $$ ...
user avatar
  • 40.7k
8 votes
Accepted

Adding Variance \ Weights Information When Solving a Basis Pursuit Denoising Problem (BPDN)

Your formulation: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| \boldsymbol{x} \right\|}_{1} $$ Has 2 elements: The ...
user avatar
  • 40.7k
7 votes
Accepted

Subsample Time Delay Estimation

Even though the signals are sampled you can get accuracy which is well above the accuracy offered by the samples as long as you sample using Nyquist. Actually, Using the Matched Filter you can achieve ...
user avatar
  • 40.7k
6 votes

What Does It Mean Exactly When Two Parts of a Signal Are Correlated?

Correlation between 2 signals means you can say something about one of them by observing the other. If you mean the standard correlation, $ E \left[ x y \right] $, it means you knowledge second moment ...
user avatar
  • 40.7k
6 votes
Accepted

Capacity of cascade binary symmetric channels

A binary symmetric channel (BSC) can be characterized by its complemented probability $p$. Its well-known capacity is $$C = 1 - H(p) = 1 - (-p\log(p) - (1-p)\log(1-p))$$ where $H(p)$ is binary ...
user avatar
  • 5,760
5 votes
Accepted

How is cross-correlation related with orthogonality?

I suppose you mean the cross-correlation at lag zero. Well take an Hilbert space $H$ (i.e. a metric space in which you can define a scalar product $\langle\cdot ,\cdot\rangle$). Then $x,y\in H$ are ...
user avatar
  • 738
5 votes
Accepted

Cross-correlation or cross-covariance of non-zero mean signals

What are reasons to choose for cross-correlation or cross-covariance when comparing signals with non-zero mean? Well, part of the issue is that cross-correlation as defined in your equation: $$(f \...
user avatar
  • 22.5k
5 votes
Accepted

Variance of correlated noise

Even though your calculation yields the correct result (in this case), the steps are not completely correct. First of all, the covariance matrix of a random variable $n$ is given by $$ C = E[nn^H]. ...
user avatar
5 votes

How to Apply Statistical Algorithms of Signal Processing to Regulate Variation of a Curve?

You can regulate its second derivative which is the curvature. Something like: $$ \hat{x} = \arg \min_{x} \frac{1}{2} \left\| x - y \right\|_{2}^{2} + \frac{\lambda}{2} \left\| D D x \right\|_{2}^{2}...
user avatar
  • 40.7k
5 votes

what does it mean to have a decorrelated colour space?

edit: to be clear this answer describes why Lab can be described as a decorrelated color space. This does not imply that decorrelation is the main benefit of using Lab (see many answers on why Lab is ...
user avatar
  • 194
5 votes

Difference between $\mathbb{E}[\mathbf{x} \mathbf{x}^{\rm{H}}]$ and $\mathbb{E}[(\mathbf{x}-\boldsymbol{\mu}) (\mathbf{x}-\boldsymbol{\mu})^{\rm{H}}]$

For the first case, as you wrote, it means the elements are not correlated. Since this is a Gaussian Random Vector it means the elements are independent. It means that at most only one element of $ \...
user avatar
  • 40.7k
5 votes
Accepted

The Distribution of Filtered Gaussian White Noise

If you filter a Gaussian random process with an LTI system, the output will also be Gaussian. You can make intuitive sense of this by considering that a linear combination (which is what filtering ...
user avatar
  • 80.2k
5 votes
Accepted

Redistributing Color in a RGB Image According to a Gaussian Distribution

After you equalize the histogram you can think of your data as a stream of variables $ {X}_{i} $ where $ X \sim U \left[ 0, 1 \right] $. Now all you need is to transform samples of Uniform Random ...
user avatar
  • 40.7k
5 votes

Isolate High Variance vs Low Variance Sections of Signals

What you did is the reasonable solution. From here you can do 2 things to mitigate your issues: Computation Efficiency You can use online calculation of the Mean and the STD. Remember when you move ...
user avatar
  • 40.7k
5 votes
Accepted

Random Signals - statistical properties are time dependant?

The answer to the question (a counterexample) Properties of random processes will in general be time-dependent. They are not only when talking about stationary processes. Another related concept (not ...
user avatar
  • 4,872
5 votes
Accepted

Averaging data from 2 sensors

If sensor A has a defect, the clear answer is to only use sensor B. A preferred solution to minimize noise would be to do a weighted average based on the quality of each sensor, when that can be ...
user avatar
  • 37.1k
5 votes
Accepted

Quantization error standard deviation

In the case of uniform quantization, and under some light hypothesis for the signal, the error can be modeled as an additive IID signal, independent of the signal, and with uniform distribution ...
user avatar
  • 4,701
5 votes

Autocorrelation and the dot product of complex signals

The autocorrelation value at zero lag is a positive real number regardless of whether the sequence is real-valued or complex-valued, except in the trivial case of the sequence being the all-zeroes ...
user avatar
4 votes
Accepted

adjust mean of signal using exponential

You need to do it numerically, as a related toy problem $2^a + 3^a = 4$ appears to have no symbolic solution. Bisection method (binary search) is probably the easiest solver to implement: ...
user avatar
4 votes
Accepted

Filtering random Signal

The output signal will still be normally distributed, but its power spectrum, i.e. its frequency content, will obviously be different from the input signal. If $S_X(\omega)$ is the power spectrum of ...
user avatar
  • 80.2k
4 votes
Accepted

Fixed SNR with unitary noise variance

If you define the SNR as the ratio of the signal power and the noise power in dB, you have $$SNR_{dB}=10\log \left(\frac{P_s}{P_w}\right)\tag{1}$$ where $P_s$ is the power of the desired signal and $...
user avatar
  • 80.2k
4 votes
Accepted

What is a covariance matrix?

It's the key point of array signal processing, I suppose. Say $x$ is the input vector of $[N,1]$ dimension collected from $N$ array sensors. $x(k)$ is its realization at the $k$ moment of time. By its ...
user avatar
  • 1,023
4 votes
Accepted

Minimum Mean Square Estimator - Equivalent Expressions to Minimize

Since $ M \in \mathbb{S}^{N}_{++} $ (In other convention $ M \succ 0 $) by Cholesky Decomposition there is a Triangular Matrix $ R \in \mathbb{R}^{N \times N} $ such that $ M = {R}^{T} R $. Using ...
user avatar
  • 40.7k
4 votes
Accepted

Covariance between real and imaginary parts of Fourier transform of a stationary time series

Note that in general the Fourier transform of a stationary process $x(t)$ does not exist. The Wiener-Khinchin theorem only states that under certain conditions the power spectral density of $x(t)$ ...
user avatar
  • 80.2k
4 votes
Accepted

Maximum likelihood estimator for multiplicative Gaussian noise

OK, let's have a look at one of the problematic terms: $$ \frac{\delta}{\delta x} \bigg[ \bigg(\frac{\tilde{d}[n]}{x}-\mu_A\bigg)^2 \bigg ] = - \frac{2 \tilde{d}[n] \bigg (\tilde{d}[n] - \mu_A x\bigg)...
user avatar
  • 22.5k
4 votes

Interpretation of Histogram in Statistical Image Processing

Does it assume that each pixel in images obey the same probability distribution for the histograms of images? Images of different scenes will definitely not obey the same probability distribution of ...
user avatar
  • 22.5k

Only top scored, non community-wiki answers of a minimum length are eligible