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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, ...
MBaz's user avatar
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12 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 ...
Dan Boschen's user avatar
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7 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 ...
AlexTP's user avatar
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7 votes
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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 ...
Matt L.'s user avatar
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5 votes
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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]. ...
Maximilian Matthé's user avatar
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 ...
LJSilver's user avatar
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5 votes
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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 \...
Peter K.'s user avatar
  • 25.8k
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 ...
Chandler's user avatar
  • 204
5 votes
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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 ...
Tendero's user avatar
  • 5,020
5 votes
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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 ...
Dan Boschen's user avatar
  • 51.4k
5 votes
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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 ...
Juancho's user avatar
  • 5,016
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 ...
Dilip Sarwate's user avatar
4 votes
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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)...
Peter K.'s user avatar
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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 ...
Peter K.'s user avatar
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4 votes
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What will be the distribution of a given signal

The way to proceed is to use the definition of Marginal distribution, and also Law of the unconscious statistician, $$p_\mathbf{R}(\mathbf{r}) = \mathbb{E}_\delta\big\{p_{\mathbf{R}\mid \delta}(\...
AlexTP's user avatar
  • 6,595
4 votes
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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 ...
Royi's user avatar
  • 19.6k
4 votes

Quantization error standard deviation

The author is modeling quantization noise as being white (i.e., each sample is independent of previous or following samples) with each sample being a zero-mean, uniformly distributed random number ...
TimWescott's user avatar
  • 12.7k
3 votes

Interpretation of Histogram in Statistical Image Processing

Histograms of images can differ, widely. However, when features are inspected, one often uses derivative filters at different scales, or morphological decompositions, or independent component ...
Laurent Duval's user avatar
3 votes

Why is Gaussian noise called so?

It is named after extremely influential and famous german-speaking mathematician Carl Friedrich Gauss who lived during the 18th and 19th centuries and influenced among other things early statistics in ...
mathreadler's user avatar
3 votes

Histogram Counts of a 2d Matrix

Are you looking for a specific function to perform this task? Otherwise this could probably be done with a simple for loop. Determine your unique rows using the function bins = unique(A, 'rows'); ...
goldrik's user avatar
  • 460
3 votes

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

One must be careful when asking questions about the relationships between the elements of a complex random vector. The short answer to your question is that you cannot say much for either cases ...
Learn_and_Share's user avatar
3 votes
Accepted

Why doesn't law of large numbers apply to this stationary time-series?

I believe that you are thinking that each value of $X_t$ is determined by a different realisation of $Y$, which in this example is not true. Suppose that $Y$ is the value that comes out from a dice ...
Tendero's user avatar
  • 5,020
3 votes

Example of non-equivalence of the two PSD definitions

Work in progress: wait till I am done before reading (and throwing brickbats!) This question is difficult to answer without getting into a lot of details about basic signal analysis and Fourier ...
Dilip Sarwate's user avatar
3 votes
Accepted

Complex gaussian random variable

Answer and hint: Just add the mean to expression of zero mean random variable. Along with this scale the expression for unit variance gaussian with the standard deviation of the required variance. You ...
Dsp guy sam's user avatar
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3 votes
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How to Calculate Local Spatial Variance of an Image?

Non-Uniformity Correction often refers to the notion that, all things being equal (with respect to the object observed), the actual pixel value may be affected by bias (offset) and noise (deviation ...
Laurent Duval's user avatar
3 votes

Why is a random process strictly stationary when its joint Probability density function is time invariant?

Why is a random process strictly stationary when its joint probability density function is time invariant? This query, taken from the title of the question cannot be answered because there is no Why. ...
Dilip Sarwate's user avatar
2 votes

Magnitude-squared Coherence calculation inconsistence

To expand on John's answer. Averaging over 8 windows may not enough to produce a reliable estimate for some applications. According to the paper 1987 Coherence and Time Delay Estimation, G. Clifford ...
VMMF's user avatar
  • 1,110
2 votes

Features of a non-periodic signal for comparison

It is just an idea but how about a mean of the terms in a sliding window. ...
Amal's user avatar
  • 353
2 votes
Accepted

Detection of sine signals with random amplitudes

Since both the null hypothesis $\mathcal H_0$ and the alternative hypothesis $\mathcal H_1$ are signal + noise, this is not only the detection of a random signal in WGN, but also a discrimination ...
Gilles's user avatar
  • 3,396
2 votes
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What does the frequency axis of a Power Spectral Density mean?

The fact that the frequency variable of a power spectral density (PSD) equals the one of a Fourier transform of a "normal" time-domain signal can be seen more easily by considering the ...
Matt L.'s user avatar
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