14

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, keeping track of each result. Then, calculate the histogram of the result; you'll find that you got each of 1, 2, 3, 4, 5 and 6 approximately the same number of ...


11

A signal having a mean-value or DC component of zero is commonly referred to as mean-free or as having no DC component. It does not mean that it cannot be averaged, just that the average comes out as zero. Might be a little inexact but it is very common.


9

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 off with a relatively simple example. Take a look at the following figure (pulled from dspguide... this is actually a great online book for knowing the basics of ...


9

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) \cos(\Omega t) \\ Q(t) &= - \xi(t) \sin(\Omega t) \, . \end{align} We express the effect of the filter as a convolution with the time response function $h$, ...


6

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 entropy function: A $L-$concatenated BSC, which is also a BSC characterized by $p_L$, can be visualized as in the figure below The complemented probability $p_L$ ...


5

For any single chunk (window) of data the coherence will, as you observed, be 1. In order to properly estimate coherence you must average the spectra and cross-spectra for multiple windows, and THEN calculate coherence. The auto-spectra XX and YY can be averaged the conventional way. For the cross-spectrum XY you must average the real and imaginary ...


5

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: minexponent = 0; maxexponent = 2; targetsum = 10; precision = 32; loop precision times { exponent = (minexponent + maxexponent)*0.5; if (sum(s.^exponent) < ...


5

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 orthogonal if $\langle x,y\rangle=0$, by definition. If your Hilbert Space is $L_2(\mathbb{R})$ (the space of real square integrable functions) then the scalar ...


5

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: $$(f \star g)[n]\ \stackrel{\mathrm{def}}{=} \sum_{m=-\infty}^{\infty} f^*[m]\ g[m+n].$$ will not exist (or be infinite) if $f$ and $g$ have non-zero mean. So, in ...


5

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 $$ C = E[nn^H]. $$ The nomenclature of variance is more suitable for scalar random variables, in my opinion (or it refers to the diagonal of the covariance matrix, i.e. the ...


5

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 relevant here but that you might find interesting) is ergodicity. Most of us find it difficult to understand the difference between both concepts, but then you ...


5

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 actively characterized. This can be easily done for the case the OP has presented of taking a reading from a single rotating axis. The optimum combining for the ...


4

As mentioned above, both signals 3 and 5 appear to be quite correlated and have a similar period. We can think of two signals being correlated if we can shift one of the sources left or right and increase or decrease its amplitude so that it fits on top of the other source. Note we are not changing the frequency of the source, we are just performing a phase ...


4

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 $x(t)$ exists, and it can be computed as the Fourier transform of the autocorrelation function of $x(t)$. Having said that, if for some reason one assumes that the ...


4

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 $P_w$ is the noise poiwer. If the noise $w$ has a mean of zero, then $P_w=\sigma^2_w=1$. From (1) (with $P_w=1)$ you get the desired value of $P_s$ for a given ...


4

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 the input signal, which is approximately flat, then the power spectrum of the output signal is $$S_Y(\omega)=|H(\omega)|^2S_X(\omega)$$ where $H(\omega)$ is ...


4

Given data $ { \left\{ {x}_{i} \right\} }_{i = 1}^{N} $ the Empirical STD of the data is well defined: $$ STD = \sqrt{ \frac{1}{N - 1} \sum_{i = 1}^{N} { \left( {x}_{i} - \bar{x} \right) }^{2} } $$ Where $ \bar{x} $ is the empirical mean of the data given by: $$ \bar{x} = \frac{1}{N} \sum_{i = 1}^{N} {x}_{i} $$ Now, if there's a model on the data (Such ...


4

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 the CRLB (Cramer Rao Lower Bound) for Delay Estimation (Easy to derive for white noise). If you calculate the CRLB for Time Delay Estimation you'll see it ...


4

OK, let's have a look at one of the problematic terms: $$ \frac{\delta}{\delta x} \bigg[ \bigg(\frac{\tilde{d}[n]}{x}-\mu_A\bigg)^2 \bigg ] = - \frac{2 \tilde{d}[n] \bigg (\tilde{d}[n] - \mu_A x\bigg) }{x^3} $$ which can be verified by Wolfram Alpha. The full derivative of the summation term is then just this summed over $n$.


4

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 useful) If you plot all the RGB colors of a standard RGB image of the natural world you will notice something (see below), the values tend to fall on a ...


4

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 does) of jointly Gaussian random variables is a Gaussian random variable. You can find an in-depth treatment of filtering random processes in this MIT ...


3

It comes from the Nyquist rate. The Nyquist sample rate for a signal with bandwidth $B$ is $2B$. In other words, that is the lowest sample rate that will contain all of the information in the signal without distorting it (i.e. without losing information). The question implicitly assumes that the sample rate is $2B$, and thus that in time $T$ you will get $...


3

A kernel density estimation based on your data: Depend on your programming language, you may find the corresponding fitting functions with different distributions. EDIT I tried to use the allfitdist function with statistic toolbox in Matlab in your data fitting. x = [24.988 25.224 25.212 25.066 25.310 24.963 24.826 24.944 25.490 25.176 24.740 24.988 .....


3

There is no best pdf for a periodic signal. There is also no way to find the 'exact' pdf of a measured signal. What you have to do is to measure the pdf from the data. Use a histogram to approximate the pdf of your data. Define a number of intervals within the amplitude range of your signal and simply count the numbers of data samples per interval. This ...


3

After many weeks I give the answer to my own question. There is a limit in which we can solve this problem in a reasonably simple way. Suppose we sum enough points in our DFT that the central limit theorem guarantees that the distribution of the sum's real and imaginary parts are Gaussian distributed. Then we only need to compute the variance. If we ...


3

An individual inner product does produce a scalar, but often when a cross correlation is calculated multiple individual cross correlations (i.e. dot products) are calculated at different time offsets. These individual scalar results form a vector that is indexed by the relative time offset.


3

A complex random variable with dimension being 2 $Z=X+jY$ can be defined as a Random vector with components $X$ and $Y$ which take values along 2 different dimensions. The Probability Density Function of $Z$ is defined using the PDFs of $X$ and $Y$. The Expectation is defined as $\mathrm{E}(Z)=\mathrm{E}(X)+j\mathrm{E}(Y)$ and the variance is defined the ...


3

If $h(t)$ is continuous at $a$, then $h(t)\delta(t-a) = h(a)\delta(t-a)$. Since the $\operatorname{sinc}$ function is continuous everywhere, $$\begin{align} Y(t) &= \sum_{n=-\infty}^{\infty} Z_n\delta(t-n\tau)h(t)\\ &= \sum_{n=-\infty}^{\infty} Z_n\delta(t-n\tau)h(n\tau)\\ &= \sum_{n=-\infty}^{\infty} Z_n\delta(t-n\tau)\operatorname{sinc}(n)\\ &...


3

Similar to the preferred answer above (Jason S.), and also derived from the formula taken from Knut (Vol.2, p 232), one can also derive a formula to replace a value, i.e. remove and add a value in one step. According to my tests, replace delivers better precision than the two-step remove/add version. The code below is in Java, mean and s get updated ("...


3

Correlation between 2 signals means you can say something about one of them by observing the other. If you mean the standard correlation, $ E[xy] $, it means you knowledge second moment statistics.


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