10 votes
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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 ...
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  • 513
9 votes

Why do we need to conjugate complex signals in autocorrelation and cross correlation

$\underline{\text{Prologue :}}$ Let me ask you another question. How will you compare two complex numbers $U$ ($a+jb$) and $V$ ($c+jd$)? By comparing magnitude? Subtract them and take real part? ...
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  • 420
7 votes
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Auto Correlation vs Cross Correlation vs Convolution and their applications

I can tell you of at least three applications related to audio. Auto-correlation can be used over a changing block (a collection of) many audio samples to find the pitch. Very useful for musical and ...
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  • 1,093
7 votes
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perfect sequences

Let $\theta_a$ and $\theta_c$ respectively denote the maximum magnitudes of the off-peak or out-of-phase periodic autocorrelation functions and the periodic crosscorrelation functions of a set of $K$ ...
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7 votes
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Correlation of independent random processes

No. Quoting Wikipedia's article Independence (probability theory): If $X$ and $Y$ are independent random variables, then the expectation operator $\operatorname{E}$ has the property $$\...
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 ...
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6 votes
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What is the Fastest Way Ever to Do 2D Correlation?

In your case, since you have multiple images while you have a given set of kernels the DFT based Correlation would be the best fit. Pay attention that the DFT Based Convolution / Correlation Is ...
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6 votes
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How does DCT decorrelate images?

[EDIT] In 1991, Nasir Ahmed wrote: "How I Came Up with the Discrete Cosine Transform". Interesting to read, on how he was inspired by Chebyshev polynomials, and on how he didn't get funding,...
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5 votes
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explanation of correlation of stationary stochastic processes

You are mixing up two different notions. Your random process is a collection of random variables $\{X(t)\colon -\infty < t < \infty\}$, one random variable for each time instant. The ...
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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 \...
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5 votes
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Cross correlate a 2D array

What you have (conceptually) is not a 2D array but a collection of 1D arrays. correlate2D is designed to perform a 2D correlation calculation, so that's not what ...
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  • 1,329
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 ...
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  • 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 $ \...
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5 votes
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Maximum cross-correlation coefficient value for time delay estimation

As your plot shows, the second form allows for the correlation peak to be negative. Now, what does a strong negative cross correlation mean? It means the signals are very similar, except one has a ...
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  • 2,288
4 votes
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PSD and $\lim_{T\rightarrow \infty} \frac 1 {2T} \int_{-T}^T x(t)\bar y(t)\,dt$

For power signals $x(t)$ and $y(t)$, the function $$R_{xy}(\tau)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x(t)\bar{y}(t+\tau)dt\tag{1}$$ is the cross-correlation of $x(t)$ and $y(t)$. So ...
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4 votes

Why correlation property of $\mathcal Z$-transform contains a time reversal operation

Note that (discrete-time) convolution is defined as $$x_1[n]\star x_2[n]=\sum_kx_1[k]x_2[n-k]\tag{1}$$ and correlation is defined as $$r_{x_1,x_2}[n]=\sum_kx_1[k]x_2[k-n]\tag{2}$$ Comparing $(1)$ ...
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4 votes
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Time Alignment of 2 Sensors Sampling the Same Signal with Different Hardware Delays

If the above is a good representation you should just try to infer when there is energy in the signal to align them. As it seems they start with nothing (Zero value). Then all needed is just to find ...
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4 votes
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Cross Correlation, how can any signals except the trivial cases be uncorrelated?

Tl;DR version: You are not missing anything; finite-duration signals cannot be uncorrelated signals. If the crosscorrelation function $x\star y$ of $x$ and $y$ is zero everywhere, then the Fourier ...
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4 votes

Signals Cross Correlation from Their Power Spectral Densities?

No, For instance 2 instances of AWGN with the same STD have the same PSD while being totally uncorrelated (Moreover, Statistically Independent). Also, pay attention that Correlation is not ...
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  • 468
4 votes
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convolution vs correlation?

Convolution: $$ y(t) = h(t) \circledast x(t) = \int\limits_{-\infty}^{\infty} h(u) \, x(t-u) \ \mathrm{d}u $$ Cross Correlation: $$ R_{xy}(t) = \int\limits_{-\infty}^{\infty} y(u) \, x(t+u) \ \...
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4 votes

Applying Circular Cross Correlation in MATLAB

For 1D signals MATLAB has the function cconv(). In order to apply correlation and not convolution just flip the signals: ...
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  • 415
4 votes

Identify which of the three signals is closest to a sinusoidal curve

Is it sufficient to identify the «most sinuoidal» of those 3, or would you also want linear projections of those (consistent with a IMU sensor tilted vs the plane of motion)? A simple solution might ...
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  • 2,274
4 votes
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Estimate the time delay of two signals

The issue is we are looking for likeness but the values are scaled beyond that metric. In the OP’s construct all symbols used should have equal weight toward the correlation determination; for example,...
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4 votes

How to interpret values of the autocorrelation sequence?

The autocorrelation signals you have shown are biased autocorrelation. The issue is that the higher lags have fewer data points to be used to estimate the correlation at those lags. Another issue is ...
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4 votes

How to interpret values of the autocorrelation sequence?

I think of auto correlation as «self similarity». In practice: line the signal up with a copy of itself. Step the copy one sample at a time. Multiply corresponding pair of samples from the two and ...
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  • 2,274
3 votes

How to measure the time dependent correlation of two signals

Your signals look like they have about the same ordinal scale, and little lag. Hence, you can perform local correlations: take a segment of both signals centered at a time location, you can normalize ...
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3 votes

PSD and $\lim_{T\rightarrow \infty} \frac 1 {2T} \int_{-T}^T x(t)\bar y(t)\,dt$

Usually, for power signals, we define the inner product to be \begin{align} \left<x\,,\,y\right> &= \lim_\limits{T\rightarrow \infty} \frac 1 {2T} \int\limits_{-T}^T x(t)\bar y(t)\,dt \end{...
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3 votes

Correlation meter

Let me answer in slightly different order: 3) It seems you're really asking for a "Goniometer", which is a 2 channel oscilloscope with the left and right signal components plotted in two orthogonal ...
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3 votes

Is a wavelet-based correlation measure worth any additional computational overhead?

This is very late, but maybe it's worth it anyway... The time-scale plane is not the same as the time-frequency plane, although it might be useful as well. Signals at different places in the time ...
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3 votes
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Echo cancelling using autocorrelation function

Haven't you already got it there? $$ \hat{k} = \arg \max_{l > \frac{k}{10}} r_{xx}(l) $$ and $$ \hat{\mu} = \sqrt{\frac{r_{xx}(0) }{ r_{ss}(0) } - 1} $$ The R code below outputs the figure and: ...
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