44

Convolution is correlation with the filter rotated 180 degrees. This makes no difference, if the filter is symmetric, like a Gaussian, or a Laplacian. But it makes a whole lot of difference, when the filter is not symmetric, like a derivative. The reason we need convolution is that it is associative, while correlation, in general, is not. To see why this ...


10

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 ...


8

$\underline{\text{Prologue :}}$ 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? Multiply them and compare? Since any complex number involves two entities (one for magnitude $\lvert z \rvert$ and other for argument $\theta$), any comparison ...


7

The (linear or aperiodic) convolution of two vectors $\mathbf x = (x[0], x[1], \ldots, x[N-1])$ and $\mathbf y = (y[0], y[1], \ldots, y[N-1])$ is a vector $$\mathbf z = {\mathbf x}\star {\mathbf y} = (z[0], z[1], \ldots, z[2N-1]).$$ On the other hand, their cross-correlation is a vector $$\mathbf w = {\mathbf x}\otimes {\mathbf y} = (w[-(N-1)], w[-(N-2)], \...


7

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 speech related applications. Cross-correlation is used all the time in hearing research as a model for what the left and ear and the right ear use to figure ...


7

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 $K$ sequences of length $N$ and energy $\sum_{n=0}^{N-1}|x[n]]|^2 = N$. In a seminal paper published in 1974, Welch proved that $$\max\big(\theta_a, \theta_c\big)\...


7

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 $$\operatorname{E}[X Y] = \operatorname{E}[X]\operatorname{E}[Y].$$ Consider your $X(t_1)$ and $Y(t_2)$ as $X$ and $Y$ in this answer. If both $\operatorname{E}[X] \ne ...


6

There a few commercial algorithms that do exactly that (Dolby Prologic, DTS Neo 6, Lexicon Logic 7, Bose Videostage, etc.). If your project has the funds, you can simply try to license one of those. The inner workings of these algorithms are rather complicated and typically a mixture of time domain and frequency domain feature extraction, some steering ...


6

[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, for a tool at the heart of JPEG and MP3. Natural images are not very stationary, but locally, their covariance is often modeled by a first- or second-order ...


5

I am trying to answer your question about incoherence here rather than update my previous answer on another question of yours. Compressive sensing requires low coherent pairs. So the lower $\mu(\Phi,\Psi)$, the better. Actually is $\Phi$ is spike basis (identity matrix) with $\phi_k(t) = \delta(t-k)$, and $\Psi$ is Fourier basis with $\psi_j(t) = 1/\sqrt n ...


5

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 autocorrelation function of the process is $$R_X(t, \hat{t}) = E[X(t)X(\hat{t})], -\infty < t, \hat{t} < \infty$$ and is a two-dimensional function (meaning it has ...


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

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 negative sign in front of it, i.e., $x_1 \approx -x_2$. Whether or not this makes sense depends a lot on the actual application. In the application you describe, ...


4

I understand your confusion because the equation is barely understandable from the information given in the book. It becomes more clear from the original paper by Classen and Meyr [1] from which it has been taken. They propose a two stage frequency offset estimation that consists of an acquistion stage and a tracking stage. The equation you've cited ...


4

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 statistics. Which implies one can, using only linear functions, estimate one from the other to some level.


4

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 the expression you're asking about is the cross-correlation of $x(t)$ and $y(t)$ evaluated at lag $\tau=0$: $$R_{xy}(0)=\lim_{T\rightarrow\infty}\frac{1}{2T}\...


4

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 $(1)$ and $(2)$ we see that correlation can be written as the following convolution: $$r_{x_1,x_2}[n]=x_1[n]\star x_2[-n]\tag{3}$$ The $\mathcal{Z}$-transform of $...


4

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 you need. Iterating through all pairs is not a big ask really - you can still use numpy to perform the cross correlation, you'll just need to have two loops (nested) to determine which signals to ...


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

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 $ \boldsymbol{\mu} $ is not zero. Since if there were more than 1, the matrix $ \boldsymbol{R} $ wasn't diagonal. Update Let's define $ \hat{\boldsymbol{x}} = \...


4

HINT: What is the physical meaning of the autocorrelation evaluated at lag $\tau=0$?


4

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 Equivalent to Convolution / Correlation with Periodic / Circular Boundary Conditions. It means that if you need different boundary conditions (Like padding with ...


4

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 transform of $x\star y$, which is $X(f)Y^*(f)$ (or $X^*(f)Y(f)$ for left-handed folks), must also have value $0$ for all $f$. But, finite-duration signals have ...


4

Convolution: $$ y(t) = h(t) \circledast x(t) = \int\limits_{-\infty}^{\infty} h(u) \, x(t-u) \ \mathrm{d}u $$ Cross Correlation: $$ R_{xy}(t) = \int\limits_{-\infty}^{\infty} y(u) \, x(t+u) \ \mathrm{d}u $$ The difference between the two is effectively the sign on $u$ in $x(t-u)$ in the integral. That correlation is like convolution but with one of the ...


4

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 be to do a windowed fft and pick the direction where the «crest factor» of the fft magnitude was largest (best explained by a single sinoid). Edit: It appears ...


4

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, the symbol "1" matching symbol "1" should carry just as much weight as symbol "6" matching symbol "6", but in the ...


3

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 directions on the scope screen. These directions are typically chosen so that the signal channels make an angle of 45 degrees to the horizontal. 1) The ...


3

First: you should really read some basic theory of autocorrelation. For example. I also don't know why we subtract the mean First, to subtract the mean is the usual and right thing to do – it's so standard that often it is straightly assumed that the signal has zero mean. Recall that the covariance of two variables is $E[ (x-\mu_x)(y-\mu_y)]$ – of which ...


3

A relatively simple application where correlation is used is in determining timing of a communications signal. A known synchronization signal will be sent periodically so that the receiver can use it as a point of reference. How does the receiver use it as a point of reference? It correlates the incoming signal stream against the known sequence, and when ...


3

I cant add comment due to low reputation. I think you misunderstood @lennon310's meaning. I reviewed his answer in the link, he treated Phi as a row selection matrix. @lennon310, please consider change your word 'rectangular identity'. I know what you mean, but that is not called identity matrix. Phi (in his context) is something like 0 1 0 0 0 0 0 0 0 ...


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