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# Tag Info

### 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 ...
• 53.1k
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 ...
• 6,640
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 ...
• 90.8k
Accepted

• 25.9k

### 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 ...
• 204
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 ...
• 5,030
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 ...
• 53.1k
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 ...
• 5,006

### 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 ...
• 20.5k
Accepted

• 6,640
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 ...
• 20.1k

### 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 ...
• 12.9k

### 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 ...
• 32.1k

### 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'); ...
• 460

### 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 ...
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 ...
• 5,030

### 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 ...
• 20.5k
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 ...
• 2,620

### Averaging data from 2 sensors

Here are two papers that deal with generic sensor fusion under unknown statistics: Multiple sensor fusion under unknown distributions, Journal of the Franklin Institute, 1999 On fusers that perform ...
• 32.1k
Accepted

### 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 ...
• 32.1k

### 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. ...
• 20.5k
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 ...
• 3,416

### 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 ...
• 1,142
Accepted

### 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 ...
• 90.8k
Accepted

### Independence of Noise at Each DFT Output

Let $m=k-k'$. Then the sum in your final equation becomes $$\sum_{n=0}^{N-1}e^{-2\pi imn/N}=\frac{1-e^{-2\pi im}}{1-e^{-2\pi im/N}},\quad m\neq 0\tag{1}$$ where I've used the formula for the ...
• 90.8k
A cascade of $L$ binary symmetric channels (BSCs) with cross-over probability $p$ is the same as a single BSC with cross-over probability p_L = \left.\left.\frac 12 \right(1 - (1-2p)^L\right).\tag{...